A Unified Electro-Gravity (UEG) Thery of Nature

A Uniﬁed Electro-Gravity (UEG) Theory of Nature

Nirod K. Das New York University

Table of Contents

Introduction...... (Page 1)

Part-I: Unified Electro-Gravity (UEG) Theory in the Small Scale: Particle Physics and Quantum Mechanics ...... (Page 3) 1. A New Unified Electro-Gravity Theory for the Electron...... (Page 4) 2. A Generalized Unified Electro-Gravity (UEG) Model Applicable to All El- ementary Particles ...... (Page 14) 3. Unified Electro-Gravity (UEG) Theory and Quantum Electrodynamics ...... (Page 31)

Part-II: Extension of the Unified Electro-Gravity (UEG) Theory to Physics in the Large Scale: Stellar, Galactic and Cosmology Models ...... (Page 43) 4. Unified Electro-Gravity (UEG) Theory Applied to Stellar Gravitation, and the Mass-Luminosity Relation (MLR)...... (Page 44) 5. Unified Electro-Gravity (UEG) Theory Applied to Spiral Galaxies...... (Page 59) 6. A Unified Electro-Gravity (UEG) Model as a Substitute for Super-Massive Black Holes (SMBH) at Galactic Centers...... (Page 69) 7. The Unified Electro-Gravity (UEG) Theory Applied to Cosmology...... (Page 74)

Part-III: Derivation of the Maxwell’s Equations of Electromag- netic Theory from First Principles, which Supersede Newtonian Mechanics ...... (Page 87) 8. A New Approach to Teaching Maxwell’s Equations, as Derived from Simple Relativistic Transformation Principles: A Tutorial ...... (Page 88) 9. Deriving the Newton’s Laws from the Maxwell’s Equations - Basic Concepts of Charge and Space-Time Supersede All Mechanical Principles ...... (Page 121)

Introduction A collection of papers, describing a new theory unifying electrical, gravitational as well as mechanical concepts are presented. The papers demonstrate the generality and validity of of the unified theory, as applied to a diverse set of fundamental problems, covering elementary particles in the small scale, as well as stellar, galactic and cosmological models in the large scale. The papers are listed in three major sub-groups: The Part-I covers three papers applied to problems in the smallest scale of nature. The first paper introduces the new unified electro-gravity (UEG) theory to the most basic elementary particle - the electron. The UEG theory allows the electron to be modeled first as a static charge, without any spinning effects, which is self-consistently balanced as a stable particle. The second paper extends the static UEG theory, by including higher- order UEG effects, to model other elementary particles, which may be made of a single spherical charge layer located at different specific radii, or may consist of multiple charge layers to form composite charged or neutral particles. These particles may include all known particles - fermions or bosons - of the standard model of particle physics, such as a proton, neutron, neutrino and possibly even a Higgs, W or Z boson. The static UEG theory of the first paper is extended in the third paper, to model a dynamic electron structure that includes spinning, where the spinning motion is self-supported by new UEG effects due to the moving charge. Although the third paper explicitly models the electron, the basic concepts can be extended and generalized as well to other elementary particles. This electrodynamic UEG model is consistent with the basic quantum mechanics, and can consistently explain many quantum electrodynamic phenomena without having to simply accept them as some “special quantum effects”, as they are currently understood. These include discovery of the origin of the fine-structure constant and the Plank’s constant, explaining charge quantization, wave-particle duality and the photon concept. The Part II of the papers extends the UEG theory to problems in the large scales of nature, presented in four papers. The fist paper models stellar gravitation, as per the new UEG theory, in order to re-evaluate the stellar mass-luminosity relation (MLR). This is accomplished by modeling UEG effects of stellar radiation on gravitation between binary stars, which determines the orbital dynamics of the binary stars, as well as on self- gravitation of a star, which determines the star’s light output through nuclear reaction. Measured luminosity and orbital parameters of binary stars, supported by the new theory, lead to a revised MLR, where the actual mass of a star is found to be less - significantly less for highly luminous stars - than the currently believed mass. It is discovered that the actual mass of Sun, the lone star in our solar system, is about half of the gravitational mass which is currently estimated based on Newtonian gravitation. The other half of the Sun’s gravitational mass is the result of the new UEG effect due to the Sun’s radiation. The theory is similarly extended in the following two papers, in order to model gravitation in a spiral galaxy by including the new UEG effect due to the galaxy’s light distribution. The new UEG effect would explain galactic rotation without need for any hypothetical “dark matter” (second paper) as well as explain gravitation at the galactic center without requiring any “super-massive black hole (SMBH)” (third paper). The fourth paper extends the UEG theory in the largest possible scale of nature, to model cosmic expansion without need for any hypothetical “dark matter” or “dark energy”. This is possible by including the new UEG effects due to the cosmic back-ground (CMB) radiation, as well as due to future radiation from any new stars made from collapsing of the remaining hydrogen in the universe. As per the new UEG theory, the electric charge and its force fields are the origin of all material mass, as well as of all forms of forces and energy. Accordingly, one may expect

that the fundamental electro-magnetic fields of an electrical charge, which is governed by the Gauss’ Law and certain basic principle of charge invariance, must be consistent with and supersede all mechanical principles. This is because the mass, force and energy, upon which all mechanical principles are empirically founded, trace their physical origin to the basic electric fields. The above expectation is examined in two steps, as presented in the two papers of Part-III. The first paper was written as a tutorial on a “derivation” of the Maxwell’s equations, established from first principles, based on the discovery of a new concept of charge invariance, which is relativistically consistent across different inertial frames. The new concept of charge invariance allows a direct derivation of the Maxwell’s equations from first principles, based only on the basic definition of a charge using Gauss’ Law, and simple relativistic space-time transformation. This is accomplished without any apriori knowledge of the mechanical theory of the Newton’s Laws, or on the associated principles of relativistic transformation of mass, energy and momentum. Instead, the mechanical principles of the Newton’s Laws can now be “derived” from the Maxwell’s equations, as presented in the second paper, by deriving and relating the forces between specific charges under different conditions of motion, as they are observed in two inertial frames. The resulting transformation relations for the forces, so derived directly from the Maxwell’s equations, would dictate the basic relations of the Newton’s Laws, that must be satisfied in order to be consistent with the electrical principles of the Maxwell’s equations. The combined results of the two papers essentially unify the basic electrical principles with the mechanical principles of the Newton’s Laws, making the Newton’s Laws theoretically redundant. The unification of electrical or electromagnetic concepts with the Newton’s Laws in Part-III, together with the unification of electricity and gravitation in Parts I and II, would establish a new unified paradigm to model any physical phenomena of nature. This would make the weak, strong forces and quantum theory of elementary particles, and the hypothetical dark matter and dark energy of astrophysics and cosmology, theoretically redundant. This is the closest we have come to a complete unified theory of everything (TOE), which has been the ultimate aspiration of physical sciences. The basic UEG theory is remarkably validated in all the collected papers, but it is still not fully rigorous. A fully rigorous UEG theory would require future development of a complete, dynamic unified electro-gravito-magnetic (UEGM) theory, including all possible higher-order UEG static as well as dynamic effects. Nirod K. Das New York, May 2018

2 Part-I: Uniﬁed Electro-Gravity (UEG) Theory in the Small Scale: Particle Physics and Quantum Mechanics

1-1

A New Uniﬁed Electro-Gravity Theory for the Electron

Nirod K. Das Department of Electrical and Computer Engineering, Tandon School of Engineering, New York University, 5 Metrotech Center, Brooklyn, NY 11201 (Dated: May 9, 2018, Revised July 15, 2019) A rigorous model for an electron is presented by generalizing the Coulomb’s Law or Gauss’s Law of electrostatics, using a unified theory of electricity and gravity. The permittivity of the free-space is allowed to be variable, dependent on the energy density associated with the electric field at a given location, employing generalized concepts of gravity and mass/energy density. The electric field becomes a non-linear function of the source charge, where concept of the energy density needs to be properly defined. Stable solutions are derived for a spherically symmetric, surface-charge distribution of an elementary charge. This is implemented by assuming that the gravitational field and its equivalent permittivity function is proportional to the energy density, as a simple first- order approximation, with the constant of proportionality referred to as the Unifield Electro-Gravity (UEG) constant. The stable solution with the lowest mass/energy is assumed to represent a “static” electron without any spin. Further, assuming that the mass/energy of a static electron is half of the total mass/energy of an electron including its spin contribution, the required UEG constant is estimated. More fundamentally, the lowest stable mass of a static elementary charged particle, its associated classical radius, and the UEG constant are related to each other by a dimensionless constant, independent of any specific value of the charge or mass of the particle. This dimensionless constant is numerologically suspected to be closely related to the the fine structure constant. This finding may carry greater fundamental significance, with scope of the UEG theory covering other elementary particles in the standard model of particle physics.

I. INTRODUCTION ized concept of gravity produced due to energy density associated with the electric field, that would be consistent with the Newton’s Law of Gravity [18, 19]. The The electron is the most fundamental charged parti- permittivity of the “free-space” around a charge, which cle of nature [1], carrying the smallest mass among all is conventionally assumed to be a fixed constant in the known charged particles, and is classified as a lepton in Coulomb’s Law or Gauss’ Law, is now modeled as a func- the standard model of particle physics [2, 3]. It plays tional distribution, dependent on the distribution of the a fundamental role in our everyday nature as a basic electric field or its associated energy density. The per- building block of all chemical elements, which consist of mittivity function needs to be consistent with the New- one or more electrons orbiting in different spatial forms ton’s Law of gravity, where a gravitational field is recog- around an oppositely charged, massive central nucleus nized to be directly proportional to the gradient of the [4, 5]. Different physical parameters of the electron - its inverse-permittivity function. Accordingly, such an “uni- charge, mass, as well as the spin angular momentum and fied electo-gravitational (UEG)” field may be modeled as the magnetic moment [6–8]- have been measured in great a non-linear field, where the permittivity distribution is a precision. The electron’s characteristics in an electro- general function of the source charge, or equivalently the magnetic field have also been successfully modeled using electric field is a non-linear function of the source charge. quantum mechanical wave functions [9–11] and quantum Under this non-linear condition, the definition of energy electro-dynamics [12]. However, any internal structure of density and its expression in terms of the source charge the electron, and the origin of its mass, remain mysteri- or the electric field, used in conventional electromagnetic ous. It is sometimes considered to be a “point particle” theory, may have to be properly modified. with no particular internal structure [13]. However, the With a proper definition of the energy density associ- electromagnetic energy, or its equivalent mass, for the ated with the non-linear UEG field, and a suitable re- point-particle would be infinite [14], which is unphysi- lationship between the gravitational field and the en- cal and inconsistent with the finite measured mass of the ergy density, the permittivity function surrounding a electron [6]. Further, the question of how the electronic spherically symmetric surface-charge distribution may be charge could withstand the repulsive force due to its own solved, either analytically or numerically. Consequently, electric field [14], which is infinite for the point-structure the total energy, or its equivalent mass as per special with a zero radius (or even a finite value if the electron relativity, may be derived as a function of the charge ra- had a non-zero radius), can not be properly answered. dius. It is discovered that stable solutions, where the first In this paper we model an electron using a new Uni- derivative of the total energy with respect to the charge fied Electro-Gravity (UEG) theory. The theory attempts radius is zero, and the second derivative positive, are to unify the concept of the electric field surrounding a possible for certain discrete values of the charge radius. source charge, as defined by the Coulomb’s Law or Gauss’ The derivation assumes a simple proportional relation- Law of electrostatics [15–17], together with a general- ship between the energy density and the UEG field, with

Copyright # TXu 2-128-436, c 2018 by Nirod K. Das. All Rights Reserved. 4 N. Das, A UEG Theory of Nature, 2018 1-2 the constant of proportionality referred to as the UEG constant. It may be reasonable to assume that the stable solution having the smallest possible mass/energy is associated with the mass/energy of an ideal “static electron” that does not spin around itself. Further, the mass of the static electron may be ideally assumed to be half of the total mass of an electron that includes its spin contribution. Accordingly, by reverse deduction, the UEG constant can be calculated, and is recognized as a new fundamental constant of nature. This is a significant fundamental development. The new UEG constant is defined as the gravitational acceleration per unit energy density, carrying a dimension of (m/s2)/(J/m3). More significantly, a dimensionless constant relating the UEG constant, the stable static mass, and its associated classical radius, is identified which would apply to any basic charge particle, independent of the specific charge or mass of the particle. The value of this dimensionless constant is numerologi- FIG. 1. cally recognized to be closely related to the fine structure constant [20]. This general finding may suggest a 2 much broader scope of application of the UEG theory to W = mc , where c is the speed of light in an isolated other known elementary particles in the standard model free-space, is inversely proportional to the permittivity , of particle physics [2, 3, 21, 22], which might be associ- or directly proportional to = 1/. This is in consistency ated with different effective values of the UEG constant, q2 with the energy W = 8πr of a spherical surface charge resulting in different mass and classical radii of the par- q q of radius rq, placed in a medium with permittivity . ticles, while they carry the same value of the elementary charge as the electron. Considering the broad reach of the fine structure constant in quantum mechanics and 1 1 (¯r) = 0r(¯r), = 0r, = = r = 0r, electro-dynamics [20, 23, 24], the recognition that the 0 = 1 , = 1 ; fine-structure constant may have its fundamental origin 0 0 r r in the UEG theory may carry profound theoretical and m(¯r) α (¯r), m(¯r) = m0r(¯r), fundamental implications. m = m( → , → 1), = 1 = = m ; 0 0 r r r 0 m0 F −∇W (¯r) −∇[m(¯r)c2] Eg = = = m0 m0 m0 II. GRAVITY AS GRADIENT OF FREE-SPACE −∇[m (¯r)c2] PERMITTIVITY = 0 r = −c2∇ (¯r). (1) m0 r

A massive body in a gravitational field Eg experiences a force F in a certain direction in space. In the theory of A. Gravitational Field and Permittivity Function general relativity this force is seen as a result of curvature in a Region with Energy/Mass Distribution of the surrounding “free-space” [25]. The force may be alternatively modeled by considering the permittivity of the surrounding “free-space” to be a non-uniform func- Consider the gravitational field produced by a body of mass of m , as per the Newton’s Law of Gravitation, ex- tion (¯r) of the location r¯ (unlike a constant value = 0 0 normally used), and assuming that the mass of a given erting a force on an external mass δm0. The permittivity body at a particular location is a function of the local function around the mass m0 may be expressed using the permittivity (see Fig.1). As the mass is displaced from model (1) developed above. one location over an incremental distance along a given Gm δm direction, its mass or equivalent energy is also incremen- F = − 0 0 r,ˆ tally changed due to the incremental change in the per- r2 mittivity associated with the displacement. This change F Gm0 2 2 ∂r(r) Eg = δm = − 2 rˆ = −c ∇r(¯r) = −c ∂r r,ˆ in energy per unit displacement in the given direction 0 r ∂ (r) Gm Gm would be equal to the force component in the particular r = 0 , = 1 − 0 , (r → ∞) = 1 . (2) ∂r 2 2 r 2 r direction. Accordingly, the gravitational field is modeled c r c r in terms of gradient of the permittivity function of the The above result would be applicable for all distances “free-space” medium. r > 0 for an ideal point-body, and would apply only out- We assume that the mass m or the equivalent energy side the body for a body of non-zero radius.

5 N. Das, A UEG Theory of Nature, 2018 1- 3

the charge, in the presence of a permittivity distribution (r) = 1/(r) may be expressed using the Coulomb’s Law.

E = q r,ˆ D = q r,ˆ E = D = (r)D. (4) 4πr2(r) 4πr2 (r)

Let us calculate an incremental energy dW required in moving an incremental charge dq from inﬁnity to a radius r = rq, using the above electric ﬁeld. This is equivalent to having dW = V (q)dq using a potential concept, where V (q) is the potential (function of q) at the radius r = rq. Integrating the dW over the total charge q would give the total energy W .

∞ dW = dq R E(q) · dr rq = V (q; r = r )dq = RRR dW dτ, q τ;r>rq τ FIG. 2. q q W = R dW = RRR ( R dW )dτ. τ;r>rq τ (5) q=0 q=0 The permittivity function for a body with distributed mass/energy, such as an electric charge, may be simi- The incremental charge dq may be expressed in terms of larly developed (see Fig.2), by relating the divergence of an incremental change in the electric flux density dD us- the gravitational field Eg in (2) to the mass-density mτ0. ing Gauss Law. The incremental energy dW can then be The mass-density mτ0 of a distributed body at a partic- expressed as an integral over the external volume τ; r > rq ular location is defined as the mass per a unit elemental using the divergence theorem. volume dτ = 1 at the given location. The equivalent 2 energy-density Wτ0 = c mτ0. dW = V (q; r = r )dq = RR V (q)dD · ds q S,r=rq+δ RRR 2 = τ;r>r ∇ · (−V (q)dD)dτ ∇ · Eg = −4πGmτ0 = ∇ · (−c ∇r), q 4πGm 4πGW = RRR (−∇V (q) · dD)dτ ∇ · ∇ = τ0 = τ0 . (3) τ;r>rq r c2 c4 + RRR −V (q)(∇ · dD)dτ τ;r>rq = RRR E(q) · dDdτ τ;r>rq III. MODELING ENERGY DENSITY IN A RRR = τ;r>r (q)D(q) · dDdτ NON-LINEAR MEDIUM AROUND AN q ELECTRIC CHARGE = RRR (q) 1 ∂ (D(q) · D(q))dqdτ τ;r>rq 2 ∂q ∂|D|2 = RRR 1 (q) dqdτ = RRR dW dτ; In the unified electro-gravity (UEG) model, the per- τ;r>rq 2 ∂q τ;r>rq τ mittivity distribution of the free-space is dependent upon ∇ · dD = 0 in τ. (6) the energy density distribution, which is dependent upon the source charge. This is unlike a linear dielectric We have now established an expression for an incremen- medium where the permittivity function is independent tal energy density dWτ , which may be integrated over of the field strength or the source charge. Having the the total charge q to obtain the required expression of permittivity distribution to be a function of the source the energy density Wτ . The general expression may be charge, is equivalent to having the electric field distri- verified to be the conventional energy density for a linear bution to be a non-linear function of the source charge. medium, when the permittivity is a constant indepen- The energy density in such a non-linear medium needs dent of the charge q. The total energy W can then be to be properly modeled, starting from the fundamentals. calculated as the volume integral of the energy density This would result in a general expression for the energy Wτ . density for a non-linear medium, which may be verified with a standard expression of the energy density for a 1 ∂|D|2 linear medium, as a special case when the permittivity is dWτ = (q) dq, a constant independent of the charge. 2 ∂q q The electric field E and the electric flux density D pro- RRR RRR R 2 W = τ Wτ dτ = τ ( dWτ )dτ = m0c . (7) duced due to a charge q, at a distance r from the center of q=0

6 N. Das, A UEG Theory of Nature, 2018 1- 4

In equivalency to a conventional definition of the en- electromagnetic theory [26, 27], in order to re-establish ergy density for a linear medium, it may be useful to proper relationship between different energy and power define a new variable 0 for a non-linear medium. The associated with an electromagnetic field. conventional expression of the energy density for a linear Theoretically, there are many possible expressions for medium, with the inverse-permittivity for the linear the vector function U. A simple, physically meaningful medium simply substituted by the new equivalent vari- proposition is to express the function U (11), referred able 0, would be valid as well for the non-linear medium. to as the UEG function, proportional to the original energy density Wτ , and directed toward the center of q q mass/gravity of the particle. R R 1 ∂|D|2 1 0 2 1 0 2 Consider the external free-space region of a “neutral” Wτ = dWτ = 2 (q) ∂q dq = 2 |D| = 2 0r|D| , q=0 q=0 material body, that appears to be charge-less to an ex- q q ternal observer, with the electromagnetic field and its ∂|D|2 2 0 = 1 R (q) dq = 1 R (q) ∂q dq |D|2 ∂q q2 ∂q associated energy density in the external region equal to q=0 q=0 zero. With the above choice of the UEG function U (11), q q 2 R 20 R 0 no new, special treatment would be required to model = 2 (q)qdq = 2 r(q)qdq = 0r. (8) q q=0 q q=0 the gravitational field in the external region, because the original as well as the revised energy densities of (9), Wτ 0 and Wτ respectively, would be zero in this region. Fur- IV. A UNIFIED ELECTRO-GRAVITY MODEL ther, with the choice of the UEG function (11), the total FOR AN ELEMENTARY CHARGE, WITH A energy W , or its equivalent mass m = W/c2 of the neu- NEW DEFINITION OF THE ENERGY DENSITY tral body, as seen by an external observer, would remain the same whether the W is calculated by integrating the For a given total energy W , the energy density Wτ original or the revised energy density in the internal re- we derived may not be unique. An alternate expression gion, as per the deduction in (10). Accordingly, New- 0 of the energy density Wτ may be defined by adding a tonian gravitational field in the external region of such distribution f to the original energy density Wτ , such neutral material bodies would remain unaffected by the 0 that the Wτ would result in the same total energy W new UEG theory, which would be consistent with obser- when integrated over the total volume τ as that due to vation. the original energy density Wτ . Accordingly, a fixed to- The selected UEG function U (11) could be non-zero tal energy W is redistributed into the different energy in the internal region of a neutral body discussed above, 0 densities Wτ and Wτ inside the volume τ. This can be due to non-zero electromagnetic fields associated with accomplished by having the additional distribution f ex- any charged sub-structure internal to the body. This 0 pressed as divergence of a suitable vector distribution U, would lead to having the revised energy density Wτ in which is identically zero everywhere outside the volume (9) to be different from the original energy density Wτ τ. in the internal region. Accordingly, it would require a revised treatment for modeling the gravitational field, in the internal charged region of such a neutral material RRR 2 W = τ Wτ dτ = m0c , body, or for that matter in any general region in the q presence of a non-zero electromagnetic field. 4W 1 0 2 1 0 Wτ = 4τ = 2 |D| = 2 4 ∫ qr(q)dq, The new alternate expression for the energy density Wτ 16π r 0 0 of (9), using the new UEG function U of (11), may now 0 W = Wτ + f, τ be substituted for the original energy density Wτ0 = Wτ f = ∇ · U = ∇ · (Uuˆ); U = 0 outside of τ, (9) in the UEG modeling of the gravitational field in (3).

0 2 0 4πGWτ ∇ · Eg= −c ∇ · ∇ = −4πGmτ = − RRR 0 r c2 W = τ Wτ dτ 4πG 4πG RRR RRR = − (Wτ + ∇ · U) = − Wτ − ∇ · (γWτ rˆ), = τ (Wτ + f)dτ = τ (Wτ + ∇ · U)dτ c2 c2 = RRR W dτ + RRR ∇ · Udτ 4πG 4πG τ τ τ 2 U = 2 ζWτ rˆ = γWτ rˆ . (12) RRR RR RRR c c = τ Wτ dτ + S U · ds = τ Wτ dτ, (10) It may be observed from the above expression of the gravitational field Eg, that the new UEG function U, which was introduced for an alternate definition of the 0 U(Wτ ): U(Wτ = 0) = 0, uˆ =r ˆ; energy density Wτ in (11), would be equivalent to having Wτ = 0 outside of τ, U = ζW τ r.ˆ (11) an additional gravitational field equal to −γWτ rˆ, referred to as the UEG field. The parameter γ in (12) is a new 0 An alternate expression of the energy density Wτ , as in scalar constant, referred to as the UEG constant, which (9), would require revision of the Poynting theorem of the is related to the constant ζ used in (11).

7 N. Das, A UEG Theory of Nature, 2018 1- 5

A. Series Solution for r, with a Strong UEG Force Assumption 2 ∇ · Eg = −c ∇ · ∇r ' −∇ · (γWτ rˆ) q ¯ γrˆ R We will solve for the inverse-relative permittivity func- = −∇ · ( 2 4 qr(q, r)dq) 16π r 0 0 tion (r), by expanding it as power-series of r−i with un- r 2 0 r 2 known coefficients b , and then solve for the coefficients in ¯ γq r(r)ˆr Gq i = −∇ · ( 2 4 ), r >> r0 = 4 , order to satisfy the above UEG relation (12). In the limit 32π r 0 8π0c q of large distance r, the r(r) needs to satisfy the Newto- ¯ γrˆ R Eg ' −γWτ rˆ = − 2 4 qr(q, r)dq nian gravitational field (2) due to the particle mass m0, 16π r 0 0 approaching unity at infinite distance r → ∞. The limit- 2 0 r γq r(r)ˆr Gq2 ing conditions would fix the first two coefficients b0 and = − 2 4 , r >> r0 = 4 . (14) 32π r 0 8π0c b1. The expression (8) for the energy density Wτ in a non- linear medium is used in the above derivation. Assuming that the charge distribution and the UEG solution are spherically symmetric, the differential operators in the above expression can be expressed in terms of deriva- ∞ tives with respect to the radius. Substituting the series P −i Gm0 r(r, q) = bir , b0 = 1, b1 = − 2 . (13) expression of (13) in (14) we get, i=0 c

q ∂ 1 γ ∂r r ' 2 (γWτ ) = 2 4 2 ∫ qr(r, q)dq c 16π r c 0 0 3 q 3rµ 3 γq2 2 = 2 4 ∫ qr(r, q)dq, rµ = 2 2 ∝ q , This assumes that the surrounding medium at infinite 2q r 0 24π c 0 2 distance from the particle is a free-space with = 0, γq 1/3 −16 1/3 rµ = ( 2 2 ) = 5.14 × 10 γ , r = 1 = 1/r = r, and the m = m0 is the mass of the 24π c 0 3 particle when measured in the free-space medium. If the ∂2 (r,q) r r ' − µ (r, q) . (15) surrounding medium is different from the free-space, with ∂(r−3)∂(q2) 4q2 r = r0, 1/r = r 6= 1, then the above solution (13) needs Gm to be scaled with b = and b = 0 r . It may be 0 r 1 c2 ∞ 3r3 ∞ q shown from the following iterative solution for the r(r), P −(i+1) µ P −(i+4) bi(−i)r ' 2 (∫ qbi(q)dq)r that each term in the series expression of (13), and there- i=0 2q i=0 0 fore the entire expression of (13), would be multiplied by 3 ∞ q 3rµ P −(i+1) the (r → ∞) of the surrounding medium, in order to = 2 (∫ qbi−3(q)dq)r . (16) r 2q i=3 0 obtain the r(r) for the particle in the given surrounding medium. Further, the mass function m = m(r = rq) for The above relation provides an iterative solution for the particle measured in the given surrounding medium, the coefficients bi. as derived in section IV B using the above scaled r(r), may be shown to be equal to m = m (r → ∞), as 0 r 3 q expected in section II. For simplicity, in the following 3rµ bi ' − 2 (∫ qbi−3(q)dq), derivations we will assume the surrounding medium to 2q (i) 0 be free-space, the results from which may be properly b0 = 1, bi = 0; i 6= 3k = 0, 3, 6, 9 ··· . (17) scaled as needed for any other surrounding medium. The series may be re-sequenced with ak = b3k, because We may assume that the new UEG field −γWτ rˆ is all coefficients bi for i other than i = 3k = 0, 3, 6, 9 ··· are much stronger than the conventional Newtonian gravi- zero. tational field of the charge particle, contributed due to the original energy density Wτ . This is because the con- 3 q ventional Newtonian gravitational field of an elementary rµ ak = b3k ' − 2 (∫ qb3k−3(q)dq) charge is known to be very week, having a negligible (es- q (2k) 0 3 q sentially no) effect on the permittivity function. It may rµ be shown, that this assumption would be valid given the = − 2 (∫ qak−1(q)dq) . (18) q (2k) 0 radius r of the charge particle is much larger than the radius r0 of a black-hole produced by an elementary charge From the above iterative relation it may be recognized 2 2 2k q, with a mass equal to the classical mass q /(8π0r0c ) that ak would be proportional to q . This condition of the charge with the radius r0. may be used to simplify the iterative relation for ak and

8 N. Das, A UEG Theory of Nature, 2018 1- 6 then solve for all the coefficients ak starting with the Notice in the Fig.3 that the function r of (22) (and 0 known coefficient a0 = 1. the corresponding effective function r of (23)), derived using the rigorous definition of the energy density (8) 3 for a non-linear medium, exhibits an oscillatory behavior 2k rµ ak(q) ∝ q , a0 = 1, a1 = − 2×2 , changing its sign from positive to negative values and vice versa. This is in contrast with the result for = 1/r = r6 r3 r a = µ ; a = −a µ . 0 = 1/0 from (24) (using a simplistic (incorrect) UEG 2 2×2×4×4 k k−1 (2k)(2k) (19) r r model), which monotonically approaches zero with no The series expression for the inverse-relative- oscillatory behavior. The rigorously derived, oscillatory 0 0 permittivity function r(r) may be re-formatted as behavior of the r = 1/r and r = 1/r functions is a a power series of t2k, where t is a normalized variable key development, which would lead also to an oscillatory 1.5 t = (rµ/r) , with corresponding normalized coefficients behavior of the total energy/mass of the charge particle 0 ak. as a function of radius, to be established in the following section. This would allow the charge particle to main- ∞ ∞ tain a stable structure at discrete values of radius, where P −i P −3k r(r) = bir ' akr the total energy/mass of the particle would be locally i=0 k=0 minimum. ∞ ∞ P 0 rµ 3k P 0 2k From Fig.3, it may be noted that at discrete locations = ak( r ) = akt , 0 k=0 k=0 where r of (22) is zero, the corresponding r of (23) is 0 3k rµ 1.5 0 0 1 non-zero, and vice versa. Accordingly, the energy den- akrµ = ak, t = ( r ) , ak = −ak−1 , (20) (2k)(2k) sity Wτ of (8) would be non-zero, at the discrete locations where the field E of (4) is zero, and vice versa. This is unlike a conventional field in a “free-space” medium hav- 0 (−1)k ak = 2k 2 , k! = (k)(k − 1)(k − 2) ··· (1), (21) ing a fixed relative permittivity r = 1, in which case a 2 (k!) non-zero or zero electric field is respectively associated with a non-zero or zero energy density. The above non- 2 4 6 conventional behavior is a result of the non-conventional (r) ' 1 − t + t − t + ··· r 22[1!]2 24[2!]2 26[3!]2 nature of the “free-space” medium, as per the UEG theory, which is no longer a fixed but is a “flexible” or vari- = J0(t). (22) able medium with a non-linear behavior. The electric The above series is recognized as as the zeroth-order field in such a flexible medium would be a non-linear Bessel function J0(t) [28]. The corresponding effective function of the source charge, and the equivalent permit- 0 0 function r = 1/r may be deduced from (22) using the tivity is a function of the source charge and location. The definition (8), and similarly recognized in relation to the energy density in such a non-linear medium needs to be first-order Bessel function J1(t) [28]. properly re-defined as in (8), resulting in the effective in- 0 verse relative-permittivity r of (23), which leads to the q non-conventional disconnect between the energy density 0 2 R r(r) = 2 qr(q, r)dq and the electric field, discussed above. q 0 Further, the relativity permittivity r from (22) in t2 t4 t6 Fig.3 is allowed to be negative, which may be theoret- ' 1 − 2 + 2 − 2 + ··· 22[1!] ×2 24[2!] ×3 26[3!] ×4 ically associated with a negative speed of light. The ef- 0 = (2/t)J1(t), (23) fective permittivity r from (23) in Fig.3 is also allowed to be negative, which as per its definition in (8) would allow The inverse-relative permittivity function r = 1/r of 0 the energy density to be negative. These possibilities of (22), as well as the corresponding effective function r = 0 negative light speed and negative energy density are re- 1/r of (23) are plotted in Fig.3 as a function of the nor- markable new developments, not encountered in conven- 2/3 malized radius rµ/r = t . tional physical problems, which may carry far-reaching The function r = 1/r that would have resulted if physical and philosophical implications. a conventional energy density for a linear medium (see (14,8)) were used (incorrectly) in the above derivation of section IV A, where the effective function 0 = 1/0 r r B. Particle Energy and Mass, as a Function of the from (8) that defines the energy density would be equal Charge Radius to the function r = 1/r, is expressed in (24), and is also plotted in Fig.3 for reference. Once the inverse-relative permittivity function r(r) is solved, the energy density can be expressed in terms of 2 4 6 0 t t t the (r) using (8), which can then be integrated over the r(r) = r(r) ' 1 − 2 + 4 − 6 + ··· r 2 1! 2 2! 2 3! total volume outside the charge radius (there is no field 2 = e−t /2. (24) inside the charge radius) to obtain the total energy or

9 N. Das, A UEG Theory of Nature, 2018 1- 7

FIG. 3.

the equivalent mass m (=m0 in (13)) of the particle. mass (26) normalized with respect to mµ is also plotted in Figs.4,5 for reference, showing no stable radius. Also plotted in Figs.4,5 for reference is the normalized W = RRR W dτ = m c2 τ τ 0 mass (m/mµ) = (rµ/r), based on a simple Coulomb’s q RRR 1 field, which asymptotically approaches the normalized = τ [ 2 4 ∫ qr(q, r)dq]dτ, 16π r 0 0 masses of (25) and (26) for r → ∞, as should be ex- ∞ q pected. Clearly, the Coulomb mass does not allow any W 1 1 m = m0 = 2 = 2 ∫ 2 ∫ qr(q, r)dqdr stable radius. c 4πc 0 r r 0 (2k+ 2 ) ∞ k 3 r P (−1) t µ 1.5 2 ∞ = mµ 2k 2 , t = ( r ) , W q R r(r) k=0 2 (k!) (k+1)(3k+1) m = m0 = 2 = 2 2 dr c 8πc 0 r r 2 q −30 −1/3 2 mµ = = 2.49 × 10 γ , ∞ (2k+ ) 8πc2 rµ (−1)kt 3 0 = mµ P . (26) 22kk!(3k+1) γq2 1/3 −16 1/3 k=0 rµ = ( ) = 5.14 × 10 γ . (25) 24π2c2 0 The smallest possible stable mass deduced from the The charge radius in (25) is maintained as a general oscillatory mass of (25) (Figs.4,5) is expected to be the variable (=r). The general mass function m(r) in (25) mass of an electron (or a positron) without any spin. This 2 0 would also represent the equivalent energy (=c m(r)) is referred to as the static UEG mass me of an electron. 0 contained in the field external to a sphere of radius r, We will assume that the static UEG mass me of an elec- produced due to the charge placed at any radius less than tron is about half of the total electron mass me, that in- r. cludes additional mass/energy due to the electron’s spin. 0 Fig.4 and Fig.5 (with different mass scales/resolutions) This factor of about 2 between the me and me is sug- plot the normalized mass m/mµ of (25) as a function of gested by recognizing that the electron’s spin g-factor, as the normalized radius rµ/r, showing the oscillatory be- defined below in (27), is approximately equal to 2. The 0 havior of the mass function, as we anticipated earlier. bare static UEG mass me of an electron spins effectively Any of the minimum points of the mass function would at the same speed and at the same radial distance as correspond to a possible stable particle with the partic- the electron’s charge q. This would result in having the ular charge radius, as we also anticipated. The mass ratio of the spin magnetic moment M and the spin angu- 0 m = m0 that would have resulted, if the inverse-relative lar momentum p equal to q/(2me). This is equivalent to 0 0 permittivity function of (24) were used in the derivation having a total electron mass me = gme ' 2me spinning of (25,8), based on a simplistic (incorrect) UEG model at about half of a given speed or about half of a given assuming a linear medium, is expressed in (26). This radius (or at about half of a given speed-radius product),

10 N. Das, A UEG Theory of Nature, 2018 1- 8

FIG. 4. in order to produce the same given angular momentum p. This factor of about 2 is represented by the electron’s m0 spin g-factor. e = me = 1.5425, mµ 2mµ me −30 −1/3 mµ = 3.085 = 2.49 × 10 γ , M q gq p = 0 = 2m , 1/3 −30 2me e γ = 3.085 × 2.49 × 10 /me, 0 me me γ = 5.997 × 102(m/s2)/(J/m3). (29) g ' 2, me = g ' 2 . (27)

The same conclusion may also be suggested by observ- As per the UEG theory of the electron, the constant γ ing that the orbital magnetic moment of an atomic elec- is declared to be a new natural constant, which is equal 2 tron with an orbital angular momentum is approxi- to a new gravitational acceleration in m/s toward the ~ 3 mately equal to the magnetic moment of a spinning elec- center of gravity, produced due to one J/m of energy density. tron with spin angular momentum ~/2. The approximately same magnetic moments in the two cases means the velocity-radius product of the orbital and the spinning electrons are about the same. With about the same C. General Relationship Between the UEG speed-radius product, having the spin angular moment Constant γ, the Particle Mass and Classical Radius. (= ~/2) half of the orbital angular moment (= ~) suggests 0 that the bare UEG static mass me of the spinning elec- The above estimate of the value of the UEG constant −31 0 tron is about half of the total mass me = 9.109 × 10 kg requires the actual UEG static mass me of the electron. of the orbiting electron. However, a general relationship between the smallest sta- 0 ble UEG static mass me of an elementary particle, the 0 corresponding classical radius re, and the UEG constant 0 ~ γ required to produce the mass m0 , can be derived based me × (vr)spin = 2 , me × (vr)orbital = ~, e on the expressions for the reference mass mµ (25) and (vr) = g × (vr) ' (vr) , spin 2 orbital orbital reference radius rµ (15) used in the above analysis. 0 me me me = g ' 2 . (28) 2 0 mµ 3 3q4 3r0 π With the assumption of me = me/2 for the minimum ( ) = = e , m0 4 2 0 3 γm0 stable mass in Figs.4,5, the value of the normalization e 64πc 0γme e constant mµ can be calculated, from which the value of γm0 m0 2 e = 3π( e )3, m0 = q . (30) 0 2 mµ e 0 2 the UEG constant γ is estimated. re 8π0rec

11 N. Das, A UEG Theory of Nature, 2018 1- 9

FIG. 5.

The value of the ratio m/mµ = 1.5425 from the Figs.4,5, related to the small diﬀerence between the actual value 0 for the smallest possible stable mass m = me. Using of the g-factor and its ideal value of 2 suggested in (27). 0 0 this value, the γ, me and re may be related in term of a This may point to possible physical origin of the g-factor dimensionless constant. associated with the spin, governed by a more rigorous version of the new UEG theory.

γm0 m0 e = 3π( e )3 = 34.590 . (31) 0 2 mµ re Leaving aside any small computational inaccuracy, or any small difference due to lack of generality or rigor of If we simply assume the total mass me of the elementary the basic UEG model, the close relations of the above 0 particle with spin to be twice the UEG mass me, and the dimensionless constant (31 or 32) to the fine-structure 0 classical radius re associated with me half of that (= re) constant is intriguing. First, the very existence of a 0 with me, the γ, me and re may be related using a new dimensionless constant based on the UEG theory, and dimensionless constant, which would be eight times the its intriguing close numerological relationship with the above constant. known fine-structure constant α, may strongly suggest certain fundamental basis and significance of the new 0 UEG theory. The close numerological relationship may γme me 3 2 = 24π( mµ ) = 8 × 34.590 = 276.720 . (32) also strongly suggest an explicit close relationship be- re tween the UEG constant γ associated with the dimen- Notice that the above constant is close to twice the sionless constant (31 or 32) from the UEG theory, and inverse-fine structure constant 1/α = 137.036, and the the particle’s quantum-theoretical spin angular momen- earlier constant in (31) is one fourth of the 1/α, with less tum ~/2 (consequently, the Plank’s constant ~) associated than one percent of difference. It may be possible that with the fine-structure constant α. However, any model- the normalized stable mass in Figs.4,5 is not accurate. ing of a physically spinning particle is beyond the scope This may reflect possible inaccuracy in computation due of the present UEG theory, which is valid only for a static to poor convergence of the power series in (25), when the charge. A more advanced modeling, extending the static normalized parameter t is sufficiently greater than unity UEG theory to model an electrodynamic problem of a (t is close to 4 at the smallest stable mass of Figs.4,5). physically spinning charge, would be needed in order to More significantly, the small difference may also be due to study any direct physical relationship between the UEG lack of generality or rigor of the basic UEG static theory theory and the quantum spin theory (and quantum the- for the particle, presented in this paper with assumption ory in general), and consequently between the associated of a simple UEG function in (11), and without including dimensionless constant (31 or 32) and the fine-structure the particle’s spin. The small difference may perhaps be constant α, respectively.

12 N. Das, A UEG Theory of Nature, 2018 1-10

V. SUMMARY AND FUTURE SCOPE. The basic UEG theory models only a static elementary charge without spin. Further, the energy density associated with the electric field around a charge, which is revised in this paper in terms of a new UEG function, A new unified electro-gravity (UEG) theory is pre- is still not a uniquely-defined concept. The simple UEG sented to model an elementary charge particle, based theory used in this work may need to to be extended to on a non-linear permittivity function of the empty space model the electrodynamic problem of a spinning electron around the charge, which is dependent on distribution of [11]. The theory may be further refined and extended the energy density. A new fundamental physical constant using higher-order UEG functions to model other ele- γ, referred to as the UEG constant, is introduced in order mentary charge particles [3, 21, 22], such as a proton, in to redefine the energy density around the charge, leading the standard model of particle physics [2, 29]. The basic to a new gravitational field. The value of the constant γ theory for a charged particle could also be extended for is estimated to be about 600 (m/s2)/(J/m3), by recogniz- neutral particles composed of concentric layers of oppo- ing that the lightest possible elementary charge particle site charges, and similarly for other possible composite is an electron (or a positron). A fundamental dimension- charged or neutral particles consisting of many layers of less constant exists, relating the mass of an elementary charge particles in definite concentric patterns. Accord- charge particle, its classical radius, and the UEG con- ingly, the fundamental basis of the new UEG theory may stant γ required to produce the particle as the lightest open research avenues, providing an alternate paradigm possible stable particle based on the UEG theory. This to the existing standard model of the particle physics. dimensionless constant is shown to be closely related to This could succeed in achieving the long-pending unifi- the fine-structure constant α used in quantum electrody- cation of the electromagnetism and gravity into one com- namics [20, 23], with less than one percent of difference. plete theory, which would allow modeling of all charged This would strongly suggest a deeper fundamental basis and neutral particles of the standard model without need of the UEG theory, with fundamental relationship with for any other additional force, possibly making the weak the quantum-mechanical concepts, that could possibly be and strong forces currently used in the standard model extended to model any other elementary particles. redundant.

[1] J. J. Thomson, Philosophical Magazine Series 5 44, 293 [16] C. A. de Coulomb, Histoire de l’Acadimie Royale des (1897). Sciences , 569 (1785). [2] N. Cottingham and D. Greenwood, An Introduction to [17] C. A. de Coulomb, Histoire de l’Acadimie Royale des the Standard Model of Particle Physics (2Ed) (Cam- Sciences , 578 (1785). bridge University Press, 2007). [18] S. I. Newton, Principia: Mathematical Principles of Nat- [3] Wikipedia, “Leptons, Table of Leptons,” http://en. ural Philosophy. I. B. Cohen, A. Whitman and J. Bu- wikipedia.org/wiki/Lepton, Retrieved (2013). denz, English Translators from 1726 Original (University [4] E. Rotherford, Philosophical Magazine, Series 6 21, 669 of California Press, 1999). (1911). [19] S. I. Newton and S. Hawking, Principia (Running Press, [5] N. Bohr, “Nobel Lecture: The Structure of the 2005). Atom,” Nobel Foundation: (Retrieved August 2017) [20] A. Sommerfeld, Atomic Structure and Spectral Lines. http://www.nobelprize.org/nobel_prizes/physics/ (Translated by H. L. Brose) (Methuen, 1923). laureates/1922/bohr-lecture.html (1922). [21] Wikipedia, “List of Baryons,” http://en.wikipedia. [6] P. J. Mohr, B. N. Taylor, and D. B. Newell, Review of org/wiki/List_of_baryons, Retrieved (2013). Modern Physics 88, 1 (2016). [22] Wikipedia, “List of Mesons,” http://en.wikipedia. [7] R. A. Millikan, Physical Review (Series I) 32, 349 (1911). org/wiki/List_of_mesons, Retrieved (2013). [8] G. Gabrielse and D. Hanneke, CERN Courier 46, 35 [23] R. P. Feynman, QED: The Strange Theory of Light and (2006). Matter (p. 129) (Princeton University Press, 1985). [9] E. Schrodinger, Annalen der Physik 384, 361 (1926). [24] M. H. McGregor, The Power of Alpha (p. 69) (World [10] R. de L. Kronig and W. G. Penney, Proceedings of the Scientiﬁc, 2007). Royal Society A 130, 499 (1931). [25] A. Einstein, Annalen der Physik 354, 769 (1916). [11] P. A. M. Dirac, Proceedings of the Royal Society A: [26] R. F. Harrington, Time Harmonic Electromagnetic Fields Mathematical, Physical and Engineering Sciences 117, (McGraw-Hill Co., 1984). 610 (1928). [27] J. D. Jackson, Classical Electrodynamics, 2 Ed. (John [12] R. P. Feynman, Physical Review 80, 440 (1950). Wiley and Sons, New York, 1975). [13] E. J. Eichten, M. E. Peskin, and M. Peskin, Physical [28] M. Abramowitz and I. Stegun. (Editors), Handbook Review Letters 50, 811 (1983). of Mathematical Functions: Bessel Functions of [14] R. P. Feynman, R. B. Leighton, and M. Sands, Lectures Integer Order (Cambridge University Press, 1964). on Physics, Vol.II, Ch.28 (Addision Wesley, 1964). [29] M. D. Schwartz, Quantun Field Theory and the Standard [15] S. Ramo, J. R. Whinnery, and T. V. Duzer, Fields and Model (Cambridge University Press, 2013). Waves in Communication Electronics, 3rd Edition (John Wiley and Sons, 1993).

13 2-1

A Generalized Uniﬁed Electro-Gravity (UEG) Model Applicable to All Elementary Particles

Nirod K. Das Department of Electrical and Computer Engineering, Tandon School of Engineering, New York University, 5 Metrotech Center, Brooklyn, NY 11201 (Dated: May 9, 2018; Revised December 20, 2018) The Unified Electro-Gravity (UEG) theory, originally developed to model an electron, is generalized to model a variety of composite charged as well as neutral particles, which may constitute all known elementary particles of particle physics. A direct extension of the UEG theory for the electron is possible by modifying the functional dependence between the electro-gravitational field and the energy density, which would lead to a general class of basic charged particles carrying different levels of mass/energy, with the electron mass at the lowest level. The basic theory may also be extended to model simple composite neutral particles, consisting of two layers of surface charges of equal magnitudes but opposite signs. The model may be similarly generalized to synthesize more complex structures of composite charged or neutral particles, consisting of increasing levels of charged layers. Depending upon its specific basic or composite structure, a particle could be highly stable like an electron or a proton, or relatively unstable in different degrees, which may be identified with other known particles of the standard model of particle physics. The generalized UEG model may provide a new unified paradigm for particle physics, as a substitute for the standard model currently used, making the weak and strong forces of the standard model redundant.

I. INTRODUCTION to a basic UEG model of [1] with a fixed UEG constant, by associating different discrete values of the UEG con- The Unified Electro-Gravity (UEG) theory was suc- stant to the different levels of stable mass. Accordingly, cessfully established in [1] to model an ideal “static the derivations and results in the basic UEG model of the electron,” without spin. In this theory, the electro- electron [1] maybe directly applicable to model the higher gravitational field, referred to as the UEG field, is as- levels of stable particles, by simply substituting the UEG sumed to be proportional to the energy density, with constant with a specific different value for a different level the constant of proportionality referred to as the UEG of stable mass. Consequently, a fundamental dimension- constant. Stable solutions in this model include dis- less constant, which relates the UEG constant, the stable crete levels of mass/energy, where the lowest possible mass and the associated classical radius, as established mass/energy state is recognized as a static electron, and in [1] in relation to the fine structure constant, would in differences between the energy levels are found to be principle remain valid for all levels of the stable solutions. small as compared to the energy of the electron. From This would be a significant development, which would in- the solutions, it is clear that the basic form of the theory dicate that the UEG theory, to which the dimensionless [1] can neither model a proton, which is another stable fine-structure constant may trace its fundamental origin, charged particle in common occurrence, with significantly can be much general in its scope of application to a broad larger mass than electron, nor a neutron, which is also class of - possibly all - charged particles, independent of in common occurrence but carries a zero total charge, any specific configuration or mass of the particle. nor many other known charged or neutral particles in The derivations of the UEG theory of [1] may also the standard model of particle physics. In order that the be extended, with reasonable additional effort, to model UEG theory can be established as a truly unified the- a general class of composite neutral particles, consist- ory, it needs to generalized for application to all known ing of a general internal charge structure which is en- charged or neutral particles. closed by an external layer of charge of equal magnitude The UEG theory may be extended by having the UEG but opposite in sign, as compared to the total internal field to be dependent on higher powers of the energy den- charge. A special class of such neutral particles (the sity, expressed in terms of a general function of the en- zeroth kind) may be identified as neutrinos, when two ergy density, referred to as a UEG function. With a oppositely charged layers are very closely spaced, result- suitable form of the UEG function, stable solutions for a ing in significantly lower mass of the neutral particles charged particle with higher levels of mass/energy would than those associated with their internal charge structure be possible, where a stable solution in the next higher without the external charge layer. A composite neutral level following the electron maybe identified as a pro- particle, made of a negative charge layer that encloses ton. The general UEG function may be properly approx- a positively-charged layer of a proton, having the total imated, with discrete values of the UEG function for the mass close to that of the proton, maybe identified as a different levels of the stable solutions. A general UEG neutron. The mass/energy of any general composite neu- model with such a discretized approximation of the gen- tral particle may be related to that of the internal charge eral UEG function maybe treated analytically equivalent structure, and values of different critical energies, using

Copyright # TXu 2-128-436, c 2018 by Nirod K. Das. All Rights Reserved. 14 N. Das, A UEG Theory of Nature, 2018 2-2 simple formulations, based on the results of [1]. For a II. A GENERAL UEG THEORY, WITH HIGHER special configuration of the neutral particles (the first ORDER FUNCTIONAL DEPENDENCE OF THE kind), the above formulations may need to be numeri- UEG ACCELERATION ON ENERGY DENSITY cally solved. In this case, the factor (meson factor) relating the mass of a synthesized neutral particle and that In the basic UEG theory of [1], the UEG acceleration associated with its internal charge structure maybe nu- 4πG 4πG Eg = − 2 U = − 2 ζWτ rˆ was expressed in the sim- merically calculated, that can be tabulated or plotted in c c plest form, proportional to the energy density Wτ , with a general normalized form for convenient use to model the proportionality constant γ = 4πG ζ. The basic UEG any neutral particle of the special kind. c2 theory of [1] may be extended using a general functional form of the function U =rU ˆ (Wτ ), dependent on the en- The above modelings may be extended, formulated ergy density Wτ . This may be treated as equivalent to in the most general form, to model increasingly com- substituting the UEG constant parameter γ used in [1] plex structures of composite charged or neutral particles. with a general UEG function γ(Wτ ). For analytical sim- They can be synthesized using multiple charge layers, ar- plicity, the UEG function γ(Wτ ) maybe treated with a ranged in multiple levels and sub-levels (shells), and be “stair-case” approximation, having different discrete val- associated with different orders of stability depending on ues of γ for different ranges of the flux density D, or the specific structure. Such a large class of general par- for the corresponding ranges of the radial distance r, as ticles maybe identified with all known particles in the shown in Fig.1. standard model of particle physics. Depending on the specific charge structure and associated stability, a particle may be identified as a baryon, meson, lepton, or a U =rU ˆ (Wτ ) =rζ ˆ (Wτ )Wτ , basic boson, representing all observed particles covered 4πG 4πG ∇ · Eg = − (Wτ + ∇ · U) ' − ∇ · U by the standard model of particle physics [2–4]. Such a c2 c2 4πG generalized UEG model would provide a new paradigm = − ∇ · (ˆrζ(Wτ )Wτ ) = −∇ · (γ(Wτ )Wτ rˆ) . (1) that may completely replace the standard model, making c2 the basic weak and strong forces of the standard model [5, 6] redundant. In other words, the electromagnetic and gravitational forces, which were successfully unified 0 2 0 4 0 6 4πG 1+α1Wτ +α2Wτ +α3Wτ +··· through the basic UEG theory of [1], could be effectively γ(Wτ ) = 2 ζ(Wτ ) = γ1 2 4 6 c 1+α1Wτ +α2Wτ +α3Wτ +··· unified as well with the hypothetical weak and strong Wτ 2 Wτ 2 Wτ 2 (1+( 0 ) )(1+( 0 ) )(1+( 0 ) )··· forces of particle physics, through the generalization es- Wτ10 Wτ20 Wτ30 = γ1 tablished in the present work. That would be a remark- (1+( Wτ )2)(1+( Wτ )2)(1+( Wτ )2)··· able development in modern physics. Wτ10 Wτ20 Wτ30 D 4 D 4 D 4 (1+( 0 ) )(1+( 0 ) )(1+( 0 ) )··· ∼ D10 D20 D30 = γ1 All the models presented in the paper are explicitly (1+( D )4)(1+( D )4)(1+( D )4)··· valid for ideal static particles, that do not include spin. D10 D20 D30 0 0 0 The Plank’s constant, which is twice the spin angular r10 8 r20 8 r30 8 (1+( r ) )(1+( r ) )(1+( r ) )··· momentum of a fermionic (baryon and lepton) particle, = γ1 r r r , (2) (1+( 10 )8)(1+( 20 )8)(1+( 30 )8)··· should be indirectly related to the UEG constant through r r r their shared relationship with the fine-structure constant, discussed earlier. This may suggest that spin dynamics, 0 described by the Plank’s constant, could be closely re- r10 8 γ ' γ1, r > r10; γ ' γ2 = γ1( r ) , r10 > r > r20; lated to the UEG theory. Accordingly, the UEG theory 10 0 0 could conceivably be extended to dynamic modeling of r10 8 r20 8 γ ' γ3 = γ1( r ) ( r ) , r20 > r > r30; (3) a spinning particle at any general level, where the cen- 10 20 0 0 r0 tral acceleration of the spinning particle would be sup- r10 8 r20 8 (i−1)0 8 γ ' γi = γ1( ) ( ) ··· ( ) , r > r > ri0. ported by the UEG effects of the particle’s own electric r10 r20 r(i−1)0 (i−1)0 and magnetic fields. Such an extended dynamic UEG model maybe separately explored [7], beyond the scope The basic mass (energy) function m(r)(W (r) = of the present paper. However, for useful mass estima- m(r)c2), which is the total mass (energy) of an elementary tions in this paper, we may simply assume that the to- spherical charge layer placed at radius r, and the inverse tal mass/energy of an elementary spinning charge at any relative-permittivity function r(r) which is the inverse given level is about twice that of a static charge at the of the relative permittivity seen at the charge layer at level without the spin [1, 7]. Accordingly, for all calcu- radius r, maybe derived (see the sketches in Figs.2, 3) lations in this paper, the mass in a given level maybe using the basic UEG theory of [1], based on the stair- assumed to be twice or equal to the UEG static mass [1] case approximation of (3) for the γ. The radii where the in the particular level, depending on if the spin contribu- mass function is stable would represent stable elemen- tion in the level is included or not, respectively. tary charge particles. The mass and radii of such stable

15 N. Das, A UEG Theory of Nature, 2018 2-3

FIG. 1.

FIG. 2.

charged particles may be indexed as mij and rij, respec- at the boundary of the level (i−1) at r = r(i−1)0, resulting tively, in reference to a particular level i of the general in the effective truncated mass mt(r) to be valid only for j 1 2 UEG theory, and a particular shell (= , ) in the given r < r(i−1)0, with an initial condition mt(r = r(i−1)0) = 0, level (see Figs.4, 5). and mt(r < r(i−1)0) equal to m(r) − m(r = r(i−1)0). Therefore, to be particular, any mass mij = m(r = rij) Note that the mass profile m(r) in a given level i refers listed in Table V actually refers to the respective trun- to an ideal derivation in the absence of all levels lower cated value mt(r = rij), which would be exactly and ap- than i, with a fixed UEG constant γ for the given level proximately equal to its ideal non-truncated value, re- that is valid for all r to infinite distance. However, in the spectively for levels 1 and 2, but be somewhat lower than presence of a neighboring lower level (i−1), the mass m(r) the ideal value for levels 3 and 4. associated with the level i needs to be properly truncated

16 N. Das, A UEG Theory of Nature, 2018 2-4

FIG. 3.

In addition to the stable elementary charge particles, various kinds of neutral particles that carry zero total charge, as well as various composite charge particles combining the neutral and the stable elementary particles, can be synthesized based on certain basic principles of the UEG theory [1]. A basic neutral particle may consist of two elementary charges of opposite signs, placed at two different radii. Alternatively, a neutral particle may consist of an internal composite charge (±q) structure surrounded by an external elementary charge (∓q) layer, with zero total charge. This way a large number of particles could be synthesized from the UEG theory, consisting of all particles (leptons, baryons and mesons) as well as different force carriers (bosons), and possibly even other particles that have not yet been discovered or are not practically realized because suitable decay paths might not be realized in particle-collision experiments. Such particle synthesis using the UEG theory would provide a complete, alternate model to the existing standard FIG. 4. model of the particle physics [5, 6]. The stability of a neutral or a composite charge particles is determined by the stability of the individual parts of its total structure. Accordingly, such particles may be stable or “quasi-stable” depending on if all or most parts are definitively stable, while any remaining parts are only quasi-stable. The two different kinds of stability of the parts, referred to here, are analogous to having a massive ergy valley”, may decay into other stable or quasi-stable particle on earth placed inside a bowl, or on top of an in- particles of lower mass/energy. This will determine pos- verted bowl, where the first kind is definitively stable and sible decay paths and associated transient times for the the second kind is conditionally stable or quasi-stable. A different particles. quasi-stable state would represent a transient state that would decay into stable particles, or other quasi-stable particles that are relatively more stable, having lower total mass/energy. Even a definitively stable particles, with In the following we will separately discuss the individ- availability of enough energy to overcome its local “en- ual types of neutral or composite charge particles.

17 N. Das, A UEG Theory of Nature, 2018 2-5

msn(r1, r2) = [m(r1) − m(r2)]/r(r2), r1 < r2. (4)

The electric fields due to the two layers of charges of equal magnitude but opposite signs would cancel with each other, resulting in zero total field and its associated energy density, in the external region (r > r2). Ac- cordingly, the equivalent mass m(r2) associated with the energy in the external region, produced due the inner charge layer (+q) placed at r = r1 (without presence of the outer charge (−q) layer at r = r2), is first sub- tracted in (4) from the total mass/energy m(r1). Further, the inverse relative-permittivity r in the external region (r > r2) of the composite neutral body is assumed to be unity, which needs to be continuous with that in the region between the two charge layers across the external boundary at r = r2. Therefore, the original r(r) func- FIG. 5. tion due to the the inner charge (+q) (without presence of the external charge (−q)) needs to be scaled by multiplying it with the factor 1/r(r = r2), in order to obtain the new inverse relative-permittivity function of the com- III. NEUTRINO: A NEUTRAL PARTICLE OF posite neutral particle that would be valid in the region THE ZEROTH KIND, WITH OPPOSITELY between its two charge layers r1 < r < r2. Consequently, CHARGED LAYERS CLOSE TO EACH OTHER. the original energy content (m(r1) − m(r2)) between the two charge layers, as discussed above, also needs to be A. Neutrino for the First Level, Based on the multiplied by the same factor (1/r(r = r2)) in order to Basic UEG Model find the actual new mass of the composite neutral particle. This is because the mass/energy scales in proportion to the inverse relative permittivity [1]. We will first consider the basic UEG model, which is Now, based on (4), the msn would be zero as r1 → r2, applicable to the level one. Similar results can then be except when r(r2) is zero. extended to the general UEG model as applicable to any higher level.

The inverse-permittivity function r(r) (see Fig.2) of r1 → r2, msn → 0, r(r2) 6= 0; the basic UEG theory [1] (or a general UEG theory msn 6= 0, r(r2) → r(rn0) = 0. (5) for any higher level) oscillates around r = 0. Given r(r = rn0) = 0, two layers of opposite charges at radii An approximate model for the m(r) and r(r) near r = close to but slightly (infinitesimally) larger than rn0, rn0 may be used, in order to simplify the model for the r2 > r1 > rn0, would produce a stable synthesized neutral resulting mass msn of the synthesized neutral particle, particle having a relatively small, non-zero mass. Such a and its derivative with respect to r, from which specific particle is recognized as an electron neutrino [8–10]. The conclusions may be conveniently established. Note that oppositely charge layers in the above configuration, when the derivatives of m(r) and r(r) with radius r, at r = rn0, they are closely spaced around any other radii where are of opposite signs (see Figs.2, 3 and [1]), represented r 6= 0 would result in a theoretically stable body but by the variables ±α and ∓β; α, β > 0, respectively. with a zero mass/energy. It may be noted, if the two radii r1 and r2, r2 > r1, are close to each other but both are smaller than rn0, it can be shown to result in a neg- r = rn0 + δr, r2 = rn0 + δr2, r1 = rn0 + δr1, ative, unstable energy. This case is not considered in the m(r) ' m(r ) ± α(δr) = m ± α(δr), α > 0, following detailed analyses because the resulting nega- n0 0 tive, unstable energy would not represent any physical r(r) = r(rn0) ∓ β(δr) = ∓β(δr), r(rn0) = 0, β > 0, particle. msn(δr1, δr2) = [m(r1) − m(r2)]/r(r2) α δr1 Based on the UEG theory [1], given the mass m(r) and = [±α(δr1) ∓ α(δr2)]/(∓β(δr2)) = [1 − ]; β δr2 the inverse relative-permittivity (r) profiles of an ele- r ∂msn ∂msn > 0, δr1 > 0; < 0; δr2 > 0. (6) mentary charge particle, the mass msn(r1, r2) of a com- ∂(δr2) ∂(δr1) posite neutral body, synthesized with two elementary charges (±q) of opposite signs, placed at radii r2 and As mentioned earlier, we assume r2 ≥ r1 as required or r1, r2 > r1, can be expressed as: assumed in the above mass formula.

18 N. Das, A UEG Theory of Nature, 2018 2-6

c(r → ∞) is larger than the standard light speed c0, approaching infinity for r2 → rn0. Accordingly, the speed r2 ≥ r1; r2 = rn0 + δr2, limit in the medium between the charges r1 ≤ r ≤ r2 is r1 = rn0 + δr1, δr2 ≥ δr1. (7) no longer governed by the standard light speed c0, but by the new light speed c(r → ∞) which could approach When r2 = r1 = rsn, the resulting msn of (6) can be infinity for r2 → rn0. This would allow the charges to shown to be zero, as anticipated earlier, for all rsn other spin at speeds greater than c0, even approaching infinite than rsn = rn0, or for all δrn 6= 0. speed for r2 → rn0. This is a remarkable new understanding, which would allow the neutral particle to have a significant, non-zero spin angular momentum (as ex- δr2 = δr2n, δr1 = δr1n; r1 = rsn1, r2 = rsn2; pected from a fermion), even though the total mass is rsn1 ≤ rsn2, δr1n ≤ δr2n; expected to be relatively small or negligible. Such a particle with a small mass, which could even approach zero, rsn1 = rsn2 = rsn; δr2n = δr1n = δrn, rsn = rn0 + δrn; but with a non-zero spin angular momentum (= ~/2), msn(δr1 = δr1n = δrn, δr2 = δr2n = δrn) may clearly be identified as an electron neutrino [8–10], α δr1n = β [1 − δr ] = 0, δrn 6= 0. (8) which is a spin-half particle grouped under leptons in the 2n standard model of particle physics [4, 6]. Following the above case with r1 = r2 = rsn, only when The ratio (α/β) in (10) maybe estimated based on the rsn > rn0, δrn > 0, it is a stable point as can be shown m(r) and r(r) profiles for the level 1 ([1], Figs.2, 3), to from the derivative of the approximate mass expression be of the order of 5000 eV or so. As per (10), this places (6) . Note that when r1 = r2, the stability condition is only an upper limit, predicting the mass of an electron different from a standard stability condition (having the neutron to be actually any value between zero and about first derivative of the energy/mass function with respect 5000 eV, likely much smaller than the 5000eV limit, as to the radius equal to zero and the second derivative neg- per measured estimates [8]. The upper limit could also ative for both the radius variables r1 and r2) used else- be significantly reduced by a more rigorous UEG model. where when r1 6= r2. In this case with r1 = r2 = rsn, we A reference (data-fit) value of 50 eV (= 0.5Mev (electron need a positive energy derivative with respect to r2 for mass) x 0.0001 (neutron factor)) for this upper limit is r2 larger than the stable point, and a negative derivative adopted in Table III, such that an extension of the UEG with r1, for r1 less than the stable point (so called, a “V ” theory of the electron neutron to predict similar upper type of stability). limits for the masses of the muon- and tauon- neutrinos, as presented in the following, would also be consistent with respective measured estimates (see Table III) [8]. rsn = rn0 + δrn > rn0; δr2n = δr1n = δrn > 0, ∂msn > 0, δr1 = δrn > 0; ∂(δr2) B. Neutrino at Higher Levels, Based on a General ∂msn UEG Model < 0, δr2 = δrn > 0. (9) ∂(δr1) Consider the limiting case, when the above stable The above neutrino analysis using the basic UEG the- point rsn approaches rn0 from the larger side, which ory for the level 1, may be similarly extended to a general is equivalent to having δrn positively approaching zero model applicable to any level. The resulting neutrino in (δrn → 0+). More specifically, we have δr2n ≥ δr1n ≥ 0, the second and third levels may be identified as the muon and they both approach the same value δrn = 0, but the and tau neutrinos, respectively [8, 11, 12]. For such a δr1n is closer to zero than the δr2n. The limiting stable general model, the basic mass function m(r) and inverse- mass in this case is not necessarily zero, having a range permittivity function r(r) in the above analysis maybe of possible positive values between zero and (α/β). substituted by the respective functions mi(r) and ri(r) for a particular level i. Accordingly, the final neutrino mass msni for the level i can be obtained from the mass α δr1n msn(δr1n, δr2n) = [1 − ], δr1n ≤ δr2n; msn1 for the level 1 by simply multiplying msn1 by a β δr2n α normalization factor mi2/m12. β ≥ msn((δr1n → 0) ≤ δr2n, δr2n → 0) ≥ 0. (10) It may also be noted, that the original inverse relative- msni = msn1mi2/m12 = (msn1/m12) × mi2. (11) permittivity r(r), which is unity at r → ∞, corresponds to a standard light speed c(r → ∞) = c0. In contrast, the (msn1/m12) is a useful factor, referred as the neutrino scaled inverse relative-permittivity r(r)/r(r = r2), valid factor, which describes the neutrino mass msni at a given for r1 ≤ r ≤ r2 between the charges at radii r = r1, r2 (dis- level as a fraction of the mass mi2 of an elementary charge cussed earlier), is equal to 1/r(r = r2) at r → ∞, which at the respective level. is greater than unity in magnitude, approaching infinity It may also be noted that in the general higher-order for r2 → rn0. The corresponding light speed in this case UEG model, for notational and formulational conve-

19 N. Das, A UEG Theory of Nature, 2018 2-7 nience a synthesized neutral mass msni at any given level i is a standard theoretical mass which assumes that the surrounding external medium has a reference relative permittivity equal to that (=(r(i−1)0r(i−2)0 ··· (r00 = 1))) seen by an elementary charge particle in the level i. This is a hypothetical situation. The actual mass in a practical case, when the external medium is the free space with r00 = 1, would be equal to the standard mass msni multiplied by the above reference relative permittivity.

msni(actual) = msni/ (r(i−1)0r(i−2)0 ··· r10(r00 = 1)). (12)

FIG. 6. Such a relationship between an msni(actual) and its theoretical value msni would apply as well for any other kind of synthesized neutral particle, covered in the following sections IV and V. 0 0 ri0 = ri0j0, ri = rsni, i ≥ i, j = 1, 2,

msni = Min[(mi0j0 + m(i0−1)0 + m(i0−2)0

IV. NEUTRAL PARTICLE OF THE FIRST + ··· + mi0 − mi(ri))/ri(ri)]ri=rsni KIND, WITH AN EXTERNAL CHARGE IN A = (m + m + m “MESON SHELL”, AND AN INTERNAL i0j0 (i0−1)0 (i0−2)0 0 OPPOSITELY CHARGED BODY PLACED AT + ··· + mi0 − mi(rsni))/ri(rsni), i > i, THE SAME OR A DIFFERENT LEVEL msni = Min[(mi0j0 − mi(ri))/ri(ri)]ri=rsni = (m − m (r ))/ (r ), i0 = i. (14) A neutral particle may be synthesized with opposite i0j0 i sni ri sni charges placed at the same or different levels i and i0; i0 ≥ i. The charge at the inner level i0 may be placed at a The expression of msn(ri0, ri) in (13) is similar in prin- 0 ciple to that of msn(r , r ) in (4), sharing the same basic shell j = 1, 2 at a radius r 0 0, and the opposite charge of 1 2 i j concepts of the UEG theory [1]. With reference to (13), the outer level i is placed at a special shell n referred to and Figs.2, 3 [1], for a given r the m (r ) increases, and as the “neutral shell” or “meson shell”, which is different i0 i i therefore the numerator of m reduces, whereas the fac- from a regular shell j = 1, 2. The name of this special sn tor 1/ (r ) first remains relatively unchanged but then shell is in reference to synthesis of mesons, which often ri i rapidly increases, as the radius r is reduced from r → ∞ uses this cell to produce relatively lower-mass particles i i closer to the central core of the level i (r > r ). As we (compared to baryons). Unlike the regular shells j = 1, 2 i i2 expected, a minimum (stable) value of the m = m that are defined with fixed radii r pre-determined as sn sni ij can be clearly established by balancing the two oppos- per the basic UEG model, the meson shell radius r for sni ing trends indicated above, at a suitable location with the synthesized neutral particle at the level i is a variable r = r outside of the core region, referred to as the determined by the mass m of the internal charge particle i sni i0 “meson shell”. and masses m of all levels i ≤ k < i0. The mass m is k0 sni Note that the standard theoretical value of m in (14) stable at the radius r , determined by having the first sni sni needs to be properly scaled using (12), in order to obtain derivative of the mass with respect to the radius to be its actual value realized when the external medium is the zero and the second derivative positive. free space with relative permittivity r00 = 1. General Neutral Particle of the First Kind, with a Regular or a Composite Charge at an Inner Level:

msn(ri0, ri) = (mi0(ri0) + m(i0−1)0 + m(i0−2)0 This is a general treatment for the primary kind of neu- 0 tral particle discussed above. In this case, the inner level + ··· + mi0 − mi(ri))/ri(ri), i > i, 0 charge in the above model maybe substituted by a gen- msn(r 0, ri) = (m 0(r 0) − mi(ri))/ (ri), i = i, i i i ri eral charge with eﬀective mass mci seen at the level i 2 2 ∂msn = ∂msn = 0, ∂ msn > 0, ∂ msn > 0, (13) (see Fig.6), which may be a regular charge, or a gen- ∂ri ∂r 0 ∂r2 ∂r2 i i i0 eral composite charge, in the same or diﬀerent level. If

20 N. Das, A UEG Theory of Nature, 2018 2-8 the inner regular or composite charge is stable or quasi- i0 is associated with a shell j0 = 1, 2, as in the first kind stable without the external opposite charge, it would also of neutral particle (meson, Fig.6) discussed above. How- be stable/quasi-stable with the external opposite charge. ever, unlike the first kind of neutral particle, in this case This should be evident from the formula for the synthe- the charge layer in the external level is placed in a con- sized neutral mass msni. ventional shell j = 1, 2, not in the “meson shell”. For a special case, if i0 and i maybe the same level, then j0 < j, 0 2 which means that the j is the internal shell whereas j is ∂msn ∂ msn 0 msn(ri) = (mci − mi(ri))/ri(ri), = 0, > 0, the external shell of the common level i = i. ∂ri ∂r2 i The charged body in the internal level i0 has two pos- ri = rsni, msni = Min[(mci − mi(ri))/ri(ri)]ri=rsni sibilities. In the first group, it is a layer of a standard = (mci − mi(rsni))/r(rsni), (15) elementary charge of mass mi0j0, located at radius ri0j0, 0 msni(actual) = msni/(r(i−1)0r(i−2)0 ··· r10(r00 = 1)), with i > i.

m (r , r ) = [m (r ) + m + m 0 sn i0 i i0 i0 (i0−1)0 (i0−2)0 mci = mci0j0 + m(i0−1)0 + m(i0−2)0 ··· + mi0, i > i; + ··· + m − m (r )]/ (r ), i0 > i , 0 0 i0 i i ri i mci = mci0j0, i = i; j = 1, 2, 2 2 ∂msn = ∂msn = 0, ∂ msn > 0, ∂ msn > 0, m ≥ m = m /2; m ' m , m >> m . (16) ∂r 0 ∂ri ∂r2 ∂r2 ci i2 ei sni ci ci ei i i0 i 0 0 r 0 = r 0 0, r = r , i > i; j, j = 1, 2, Normalized values for (msni/mci) versus (2mci/mei) = i i j i ij (m /m ) ≥ 1 are plotted in Fig.7, that maybe applica- ci i2 msni = (mi0j0 + m(i0−1)0 + m(i0−2)0 ble for general use at all levels i. These plots are de- + ··· + mi0 − mij)/rij, (17) rived using the normalized functions mi(ri/ri2)/mi2 = m(r/re)/me and ri(ri/ri2) = r(r/re), which were origi- msni(actual) = msni/(r(i−1)0r(i−2)0 ··· r10(r00 = 1)). nally derived from the UEG analysis [1] for the first level i = 1, but are assumed to be approximately valid as well The expression of msn(ri0, ri) in (17) is similar in prin- for all levels. This is due to primary similarity of the ciple to that of msn(ri0, ri) in (13), and of msn(r1, r2) in basic UEG model in all levels. In principle, however, the (4), sharing the same basic concepts of the UEG theory chart in Fig.7 should be separately established with dif- [1]. ferent best-fit data for each different level. This would General Neutral Particle of the Second Kind, accommodate secondary differences in the UEG function with the External Charge Placed in a Shell j = 1, 2: γ(Wτ ), and in the associated mass mi(ri/ri2) and inverse- permittivity ri(ri/ri2) profiles, in the different levels, as This is the second group of neutral particles of the second well as differences in any inter-level interactions. kind, following the first group discussed above. This sec- However, we will ignore the secondary deviations be- ond group is essentially a general treatment of the first tween the levels, and instead propose to use the same group of particles, by replacing the inner charge layer by chart of Fig.7 for all levels. This would be accomplished a composite charge particle of mass mci0j0. If the com- by simply substituting the ideal non-truncated mass me posite charge is stable/quasi-stable without the external 2 in Fig.7 with the effective truncated mass mi2 = Wi2/c opposite charge, the total neutral charge including the ex- from Table V (see section II, Fig.3), without having to internal opposite charge would also be stable/quasi-stable. troduce additional truncation parameters for each level. Any resulting deficiency in using the common chart of Fig.7, due to the above non-ideal substitution, appears msn(ri) = [mci − mi(ri)]/ri(ri), to be approximately compensated by all the different sec- 0 mci = mci0j0, i = i; ondary effects in the given level. This would allow uni- m = m + m + m form use of Fig.7 for all levels, maintaining the same ci ci0j0 (i0−1)0 (i0−2)0 0 0 required trend across the levels, resulting in a simplified + ··· + mi0, i > i; j = 1, 2, mass estimation of any synthesized neutral body of the 2 ∂msn = 0, ∂ msn > 0; r = r , j = 1, 2; first kind. ∂r 2 i ij i ∂ri msni = (mci − mij)/rij, (18) V. NEUTRAL PARTICLE OF THE SECOND msni(actual) = msni/(r(i−1)0r(i−2)0 ··· r10(r00 = 1)). KIND, WITH THE EXTERNAL CHARGE PLACED IN A SHELL j = 1, 2 VI. COMPOSITE CHARGED PARTICLES A second kind of a neutral charge may be synthesized with oppositely charged bodies placed in different levels A composite charged particle consists of an elementary 0 0 i and i ; i ≥ i. The charged body in the internal level charge layer at a radius r = rij, at a particular level i

21 N. Das, A UEG Theory of Nature, 2018 2-9

FIG. 7. and shell j, together with a synthesized neutral particle placed internal to this charged layer at level i0. The mass 0 or energy of the total particle is quasi-stable at the radius r = rij, mcij = mij + msni0rij, i = i; r, deﬁned by the derivative of the mass with the r to mcij = mij + (20) be zero, whereas the second derivative may be positive 0 m 0 /( 0 0 ··· ), i > i. or negative for the part of the mass contributed by the sni rij r(i −1)0 r(i −2)0 ri0 external charge layer or the internal synthesized neutral It may be noted that, for notational and formulational body, respectively. Due to the quasi-stable nature of the convenience, the mass mij or mcij, respectively of an ele- particle, the particle is associated with a transient state mentary or a composite charged particle at a given level that would naturally decay to other particles of lower i, refers to only a theoretical number which is the con- mass. The mass mcij of such a composite particle may tribution of mass at the given level i and internal to the be expressed in terms of the mass mij of an elementary level. The actual mass, if there is no other charge layer charge particle at the particular level and shell (ij), with external to the level i, would be equal to this reference radius rij, and the mass msni0 of the synthesized neutral theoretical mass plus sum of masses m associated with 0 0 k0 particle at the level i (see Fig.8). The level i is normally all levels k < i. greater than the level i, but it maybe at the same level as i if the synthesized neutral particle is of the second kind 0 (see section V), and the shell j of the outermost charge mcij(actual) = mcij, i = 1; layer of the synthesized neutral particle is internal to the mcij(actual) = mcij + shell j (j0 < j). m(i−1)0 + m(i−2)0 + ··· + m10, i > 1. (21)

VII. CALCULATIONS FOR KNOWN PARTICLES USING THE GENERALIZED UEG 0 THEORY mci(r) = mi(r) + msni0ri(r), i = i; mci(r) = mi(r) + We will apply the generalized UEG theory that we 0 msni0ri(r)/(r(i0−1)0r(i0−2)0 ··· ri0), i > i; have developed to known particles (leptons, baryons and ∂m ∂m ∂2m mesons), and compare the resulting mass/energy esti- ci = 0, i = 0, i > 0, ∂r ∂r ∂r2 mates from the UEG theory to best available measured values [2–4]. The generalized UEG models in sections III- ∂ ∂2 ri = 0, ri < 0, (19) ∂r ∂r2 VI are based on a stair-case approximation of the UEG

22 N. Das, A UEG Theory of Nature, 2018 2-10 constant γ in section II for different regions of energy any such general particle, are proposed and successfully density (see Fig.1), and also assume that the total mass applied to model and predict the masses of a large class including spin is twice that of the static UEG mass. This of basic and composite particles, including some “force ignores any higher-order effects in a rigorous UEG theory, carriers”. that would require a variable γ having a smooth func- The purpose of the proposed particle configurations tional dependence on the energy density, as well as in a and the resulting mass estimates based on the new UEG rigorous spin model based on a dynamic UEGM (Unified theory, for such a large class of known particles, is not Electro-Gravito-Magnetic) theory that would be valid for to focus on any definite study of the individual parti- simple as well as composite particles. The rigorous mod- cles. In fact, it should be reasonable to expect that els can be significantly more complicated to compute, and the actual configurations and mass estimates may devi- may at this point be pre-mature to establish accurately. ate somewhat or significantly from those proposed here, However, the effects of the rigorous models, presum- almost certainly for a handful of the large number of ably small over a simplified (first order) general UEG particles studied. The real purpose is to provide a new model, may be accounted for by introducing reasonable theoretical paradigm for particle physics, that is convinc- corrections to various key parameters obtained from the ingly shown here to have the capacity to model a large simple UEG theory. The corrected parameters are listed class of, possibly all, known basic and composite parti- in Table V, and are estimated by fitting the simple UEG cles. The clear success of this exercise is a remarkable theory for selected particles to match their measured scientific development. It establishes that the new UEG mass/energy, as shown in Appendix. The correspond- theory, which obviously unifies the electromagnetic and ing parameters obtained from a simple UEG theory are gravitational theories in explicit terms, could also unify also listed alongside the corrected values in Table V for the entire standard model of particle physics [5, 6] un- comparison. The simple and the corrected parameters der its general scope, thus making the strong and weak are seen to maintain certain relative trends from shell forces, as well as all classification schemes of elementary to shell in a given particle level, which may imply that particles (leptons, quarks and force carriers), of the stan- the same general foundation is shared by the simple and dard model physically redundant. Accordingly, a rigor- the rigorous models, except with reasonable numerical ous version of the new UEG theory may provide a definite adjustments in the parameter values due to higher-order physical basis for a grand-unified theory (GUT) [23, 24] effects. and a theory of everything (ToE) [25], which have been These adjusted parameters from Table V are then used the grand aspiration of modern physics in recent decades. in a simple UEG theory, by employing the synthesis rules of sections II-VI, to estimate the mass of all leptons (Ta- ble I), and all principal baryons (Table II) and mesons Appendix: Estimation of Parameters of the Unified (Table III), and the resulting mass estimates are com- Electro-Gravity (UEG) Theory, Using Available pared to available measured data [2–4]. Possible UEG Energy Data of Known Particles configurations that may emulate other basic particles (Higgs Boson [13–15], W and Z bosons [16, 17], Top [18– 20] and Bottom [21, 22] quarks) that have been experi- Refer to the UEG synthesis rules for different particles mentally observed are also listed in Table IV. Note that (see sections II-VI). Tables I-IV show the charge struc- the Top and Bottom quarks are modeled in the Table IV tures for all basic and composite particles, synthesized as equivalent neutral, boson-like particles, which might using the UEG theory. The mass/energy formulas be detected in pairs that transitionally represent the associated in the synthesis of the different particles are respective quark-antiquark combinations. Close agree- not explicitly shown, but should be self-evident in the ments between the masses estimated using the UEG the- following calculations. For reference, see Table-I for an ory and available measured data in Tables I-IV for such example calculation of such synthesis. a large class of basic and composite particles clearly suggests the power of the new UEG theory as a potential (1) Electron (e), Proton (p) and Neutron (n), masses substitute for the Standard Model of particle physics. determine the energies for levels 1 and 2 :

me = W12 ≈ W11 ≈ W10 = 0.5MeV VIII. CONCLUSION mp = W22 = 938.3MeV mn = W21 ≈ W20 = 939.6MeV The basic UEG theory, ﬁrst developed to model an electron [1] and then separately validated through quan- (2) Use mµ, mπ±, mη data to calculate εr12 and εr11 : tum mechanics [7], is proposed to be generalized in this paper to model all basic and composite particles [2–4] εr11 = mµ/mη = 105.7/547.8 = 0.193 covered by the standard model of particle physics. A ε /ε = m /m = 139.6/105.7 = 1.32 general structural conﬁguration for a particle, and the r12 r11 π± µ associated theoretical and calculation rules to synthesize εr12 = 0.193 × 1.32 = 0.255

23 N. Das, A UEG Theory of Nature, 2018 2-11

(3) Use mΛ+, mΞ0c+, εr11 to calculate εr10 : (9) Λ0b, Σb+, W21, and W31 energies determine the energies of level i = 4 : εr11/εr10 > mΞ0c+/mΛ+ = 2576/2286 = 1.127 W42 = mΛ0b − (W20/2) − (W30/2) εr10 < εr11/1.127 = 0.193/1.127 = 0.171 = 5620 − (939.6/2) − (1816/2) = 4242.1 0 mc2 = mΛ+ = 2286, m e2 = mp/2 W41 ≈ W40 = m − (W20/2) − (W30/2) 0 Σb+ αc2 = mc2/m e2 = mΛ+/(mp/2) = 5807 − (939.6/2) − (1816/2) = 4429.1 = 2286/(938.3/2) = 4.87

αm2 = 0.95 (from chart) (10) Use Ξb, W20,W42 energies and results from (2), (3), m = m × α = 2286 × 0.95 = 2171.7 sn2 c2 m2 (6) and (8) to get εr30 : εr11/εr10 = mΞ0c+/msn2 = 2576/2171.7 = 1.186 [mΞb− × (εr10/εr11) − (W20/2)] × (εr20/εr21) εr10 = εr11/1.186 = 0.193/1.186 = 0.162 = [5790 × (0.162/0.193) − (939.8/2)] × (0.11/0.117)

(4) Λ+ and W20 = mn energies determine W31 : = 4390.2 × 0.937 = 4113.6 = Wsn42/εr30

W31 = mΛ+ − W20/2 = 2286 − 939.6/2 = 1816.2 = (W42/2) × 0.189/εr30 = 4242 × 0.0945/εr30 0 εr30 = 4242 × 0.0945/4113.6 = 0.097 (5) Ξ c+, Ξc+ and W31 energies determine W32. This 0 assumes that the meson factors αm2 for both Ξ c+, Ξc+ The above result assumes αm2 ≈ 1. This can now be are approximately the same, because the respective me- veriﬁed to be correct, because in this case we have son mass coeﬃcients αc2 are expected to be very close: αc2 = (4390.2 + (939.6/2))/(938.3/2) = 10.4 >> 1.

m 0 /mΞc+ = W31/W32 = 2576/2467 = 1.044 Ξ c+ (11) Use ηc, W21,W22,W31,W42 energies and results W32 = W31/1.044 = 1816.2/1.044 = 1739.7 from (2), (3), (6), (8) and (10) to estimate εr31, which would best-fit with the (αm ∼ αc) meson-factor chart (6) Muon mass determines the meson factor αm2 = 0 (Fig.7): msn2/mc2 for αc2 = mc2/m e2 = mc2/(W22/2) = 1, mc2 = W22/2, in level i = 2. The result would be valid for all Increasing ε would increase meson coefficient αc and levels i : r31 meson factor αm, and accordingly the particle mass. We αm2 = msn2/mc2 = Wsn22/(W22/2) assume that εr31 ≥ εr30 = 0.097. Try first the lowest value for ε = ε = 0.097: = 105.7 × (εr10/εr11)/(938.3/2) r31 r30 = 105.7 × 0.162/0.193/(938.3/2) = 0.189 mηc = (W42/2 × αm4 × (εr31/εr30) + W31 + W21/2) = Wsni2/(Wi2/2) = αmi2, for all i. × αm2 × (εr11/εr10) = (4242/2 × 0.189 × (0.097/0.097) + 1816 + 939.6/2) × 0.95 × (0.193/0.162) = 3041MeV; (7) Use Ξ− energy, and W31 from result (4), εr20 from result (8b) below, and the factor αm31 (see the end note), αc2 = ((4242/2 × 0.189 × (0.097/0.097) + 1816 to get εr22 : + 939.6/2)/(W22/2) = 5.73, αm2 = 0.95 (from chart). mΞ− − me2 = 1321 − 938.3 = 382.7 = Wsn31 × (εr22/εr20) The above calculated mass is reasonably close to the = αm31 × (W31/2) × (εr22/εr20) available data for the particle mass mηc=2980MeV, = 0.269 × 1816.2/2 × (ε /0.110) within about a few percent accuracy. Note that the cal- r22 culation is already larger than the available mass data. εr22 = 382.7 × 0.11 × 2/0.269/1816.2 = 0.172 Any increase of εr31 would increase the particle mass and (8a) Use Λ0,W32 energies and the ratio αm3 = increase the deviation from the mass data. Therefore, Wsn32/W32 = 0.189/2 = 0.0945 from result (6) to get the the best estimate for εr31 is equal to εr30 = 0.097. ratio ε21/ε20 : (12) Use B±,W21,W22,W32,W41,W42 energies and re- m − mn = 1115 − 939.6 = 175.4 = W × (ε /ε ) Λ0 sn32 r21 r20 sults from (2), (3) and (10) to estimate εr32, which would = 0.0945 × W32 × (εr21/εr20) = 0.0945 × 1740 × (εr21/εr20) best fit with the (αm ∼ αc) meson-factor chart (Fig.7): ε /ε = 1740 × 0.0945/175.4 = 0.937 r20 r21 mB± = (W41/2 × αm4 × (εr32/εr30) + W32 + W21/2) (8b) Use Tauon energy, and results from (2), (3) and (8a) × αm2 × (εr21/εr10) = (4429/2 × 0.269 × (εr32/0.097) to calculate ε and ε : r21 r20 + 1740 + 939.6/2) × αm2 × (0.255/0.162) = 5279MeV; mτ × (εr10/εr11) = 1776 × 0.162/0.193 = 1490.74 αc4 = (4429/2)/(W42/2) = (4429/2)/(4242/2) = 1.044, = Wsn32/εr20 = 0.0945 × 1740/εr20 αm4 = 0.269 (from chart) εr20 = 0.0945 × 1740/1490.74 = 0.110 αc2 = (4429/2 × 0.269 × (εr32/0.097) + 1740 + 939.6/2) εr21 = εr20/(εr20/εr21) = 0.110/0.937 = 0.117 /(W22/2) = (6141.24 × εr32 + 2209.8)/(938.3/2)

24 N. Das, A UEG Theory of Nature, 2018 2-12

A bit of trial iterations would be needed in the above calculations to get solution for εr32=0.206, αc2 = 7.407, αm2 = 0.965 (from the chart, Fig.7).

0 Note: αci = mci/m ei = mci/(Wi2/2) ≈ 1.044 for mci = Wi1/2, for i = 3 and 4. That is, W31/W32 = 1740/1816 ≈ W41/W42 = 4429/4242 = 1.044. Therefore, the corresponding αmi = msni/mci = Wsni1/(Wi2/2) would be same. For this value of αci = 1.044, αmi is estimated to be 0.269, which best ﬁt all particle data consistent with the UEG theory. Accordingly, we will use Wsni1/(Wi2/2) = FIG. 8. 0.269 = αmi1 for i = 3, 4, for all calculations.

[1] N. Das, “A New Unified Electro-Gravity (UEG) The- wiki/Higgs_boson, Retrieved (2017). ory of the Electron,” Paper #1, pp.4-13, in “A Unified [14] ATLAS Collaboration, Physics Letters B 716, 1 (2012). Electro-Gravity (UEG) Theory of Nature,” (2018). [15] CMS Collaboration, Physics Letters B 716, 30 (2012). [2] Wikipedia, “List of Baryons,” http://en.wikipedia. [16] Wikipedia, “W and Z Bosons,” http://en.wikipedia. org/wiki/List_of_baryons, Retrieved (2013). org/wiki/W_and_Z_bosons, Retrieved (2017). [3] Wikipedia, “List of Mesons,” http://en.wikipedia. [17] CERN Courier, “CERN Discoveries: Heavylight,” org/wiki/List_of_mesons, Retrieved (2013). (Retrieved August 2017) http://cern-discoveries. [4] Wikipedia, “Leptons, Table of Leptons,” http://en. web.cern.ch/cern-discoveries/Courier/Heavylight/ wikipedia.org/wiki/Lepton, Retrieved (2013). Heavylight.html (1983). [5] M. D. Schwartz, Quantun Field Theory and the Standard [18] Wikipedia, “Top Quark,” http://en.wikipedia.org/ Model (Cambridge University Press, 2013). wiki/Top_quark, Retrieved (2017). [6] N. Cottingham and D. Greenwood, An Introduction to [19] CDF Collaboration, Physical Review Letters 74, 2626 the Standard Model of Particle Physics (2Ed) (Cam- (1995). bridge University Press, 2007). [20] D0 Collaboration, Physical Review Letters 74, 2422 [7] N. Das, “Unified ElectroGravity (UEG) Theory and (1995). Quantum Electrodynamics,” Paper #3, pp.31-42, in [21] Wikipedia, “Bottom Quark,” http://en.wikipedia. “A Unified Electro-Gravity (UEG) Theory of Nature,” org/wiki/Bottom_quark, Retrieved (2017). (2018). [22] Fermilab, “Discoveries at Fermilab: Discovery of [8] Wikipedia, “Neutrino,” http://en.wikipedia.org/ the Bottom Quark,” (Retrieved August 2017) wiki/Neutrino, Retrieved (2017). http://www.fnal.gov/pub/inquiring/physics/ [9] F. Reines, “Nobel Lecture: The Neutrino - From discoveries/bottom_quark_pr.html (1977). Poltergeist to Particle,” Nobel Foundation: (Retrieved [23] H. Georgi and S. Glashow, Physical Review Letters 32, August 2017) http://www.nobelprize.org/nobel_ 438 (1974). prizes/physics/laureates/1995/reines-lecture. [24] J. Pati and A. Salam, Physical Review D 10, 275 (1974). html (1995). [25] J. Ellis, Nature 323, 595 (1986). [10] C. L. Cowan Jr., F. Reines, F. B. Harrison, and H. W. Kruse, Science 124, 103 (1956). [11] L. M. Lederman, “Nobel Lecture: The Neutrino: Obser- vations in Particle Physics from Two Neutrinos to the Standard Model,” Nobel Foundation: (Retrieved August 2017) http://www.nobelprize.org/nobel_prizes/ physics/laureates/1988/lederman-lecture.html (1988). [12] Fermilab, “Physicists Find First Direct Evidence of Tau Neutrino at Femilab,” (Retrieved August 2017) http://www.fnal.gov/pub/inquiring/physics/ neutrino/discovery/index.html (2000). [13] Wikipedia, “Higgs Boson,” http://en.wikipedia.org/

25 N. Das, A UEG Theory of Nature, 2018 2-13

Table I UEG Shell Model of Baryons

Name Energy Energy (Est) Level One Configuration Level Two Configuration Level Three Configuration Level Four Configuration (MeV) (MeV) Meson Shell Shell 2 Shell 1 Meson Shell Shell 2 Shell 1 Meson Shell Shell 2 Shell 1 Meson Shell Shell 2 Shell 1 p 938.3 938.3  n 939.6 939.6    0 1115 1114.5       2286 2285.8  0b 5620 5619.8     1189 1199.4     0 1192 1199.4       1197 1199.4    c(?)) 2454 c  2453 2353    c0 2454 2353     b  5807 5806.8  b0 5806.8   b  5815 5806.8   0 1314 1320.2       1321 1320.2    c  2467 2501    c0 2470 2501     c' 2576 2587    c '0 2578 2587     cc (?))  cc 3518 3495.2    b0 5796.5       b  5790 5796.5      c0 2695 2645.4     b  6165 6143      1232      1383     0 1384       1387    c  2517    c0 2518      0 1531       1535   

Lower block for selected J=3/2 baryons as examples. Top block for regular J=1/2 baryons. Compare J=3/2 baryons with corresponding J=1/2 baryons, in terms of their relative charge structure. They are different equivalent charge states of the same composite structure. Although the two charge states are equivalent electrically, but with spinning they lead to different (magnetically) dynamic states, having somewhat diffeerent energy/mass.

26 N. Das, A UEG Theory of Nature, 2018 2-14

Name Energy Energy (Est) Calculations (MeV) (MeV) p 938.3 938.3 n 939.6 939.6  0 1115 1114.5 1740/2*0.189*0.117/0.11+939.6=1114.5. Meson factor: 0.189, Level 3.   2286 2285.8 1816+939.6/2=2285.8 0b 5620 5619.8 4242+1816/2+939.8/2=5619.8   1189 1199.4 1816/2*0.269*0.117/0.11+939.6=1199.4. Meson factor: 0.269, Level 3.  0 1192 1199.4   1197 1199.4 c(?)) 2454 c  2453 2353 1740/2*0.189/0.11*0.255/0.162=2353. Meson factor: 0.189, Level 3. c0 2454 2353 b  5807 5806.8 4429+1816/2+939.6/2=5806.8 b0 5806.8 b  5815 5806.8  0 1314 1320.2 1816/2*0.269*0.172/0.11+938.3=1320.2. Meson factor: 0.269, Level 3.   1321 1320.2 c  2467 2501 (1740+939.6/2)*0.95*0.193/0.162=2501. Meson factor: 0.95 (alpha_c=4.71), Level 2. c0 2470 2501 c' 2576 2587 (1816+939.6/2)*0.95*0.193/0.162=2587. Meson factor: 0.95 (alpha_c=4.87), Level 2. c '0 2578 2587  cc (?)) cc 3518 3495.2 1816/2*0.269/0.11*0.255/0.162=3495.2. Meson factor: 0.269, Level 3. b0 5796.5 (4242/2*0.189/0.097*0.117/0.11+939.6/2)*(1.0)*0.193/0.162=5796.5. Meson factors: 1.0 (alpha_c=10.37), Level 2; 0.189, Level 4. b  5790 5796.5 c0 2695 2645.4 1816/2*0.269/0.11*0.193/0.162=2645.4. Meson factor: 0.269, Level 3. b  6165 6143 (4242+1816/2)*0.95*0.117/0.11+939.6=. Meson factor: 0.95 (alpha_c=5.92), Level 3.   1232   1383  0 1384   1387 c  2517 c0 2518  0 1531   1535 Refer to the UEG synthesis rules for different particles (sections II-VI). The mass/energy formula associated in the synthesis of a particular particle maybe evident from its calculation shown above. For example, the specific calculations for the particle (  0) are explained in the following:

Step 1: Neutral Particle of Kind 1, at level 3 (see section IV):

m31 W 31 / 2 1816 / 2 MeV (Table V),m 'e3W 32 / 2 1740 / 2 MeV (Table V), c m31/' me 3  m c/'m e3 1.044, m 0.269  mi1 (Table V, Fig.7), msn3   m m c  mi1 m 31 1816 / 2*0.269 MeV

Step 2: Composite Charge Particle, at level 2 (see section VI):

m22 W 22  938.3 MeV (Table V, assume full mass with spin for the level 2), r22 0.172,r20  0.11 (Table V), mc2  m22 [ msn 3 /r20 ]  r22  1816 / 2*0.269*0.172 / 0.11 MeV.

Step 3: Neutral Particle of Kind 1, at level 1 (see section IV):

mc m c2, m ' e 1 W 12 / 2  0.5 / 2 MeV (Table V), cm c/ m ' e1  1, m  1 (Fig.7), msn1   m m c m c m c2 1816 / 2*0.269*0.172 / 0.11 MeV=1320.2 MeV=mass of the particle 0 . (Notice that this last step is a trivial approximation. Such a trivial approximate step for synthesis of a neutral particle at the level 1 may not be explicitly shown in the above calculations table.)

27 N. Das, A UEG Theory of Nature, 2018 2-15

Table II UEG Shell Model of Mesons

Name Energy Energy (Est.) Level One Configuration Level Two Configuration Level Three Configuration Level Four Configuration (MeV) (MeV) Meson Shell Shell 2 Shell 1 Meson Shell Shell 2 Shell 1 Meson Shell Shell 2 Shell 1 Meson Shell Shell 2 Shell 1   139.6 139.6      139.6 139.6     0 135 139.6      547.8 547.3     957.8 937        c 2980 3041        b 9390 9227   K  493.7 495.4      K  493.7 495.4      K 0 497.6 495.4       D  1869 1883        D  1869 1883        D 0 1864 1883         Ds  1968 1942      Ds  1968 1942      B  5279 5278      B  5279 5278      B 0 5279 5278       0sB 5366 5335     Bc  6277 6349    Bc  6277 6349      775      775   0   775    

   0// are shown as examples of vector mesons, all others are pseudo-scalar mesons. Compare    0// mesons with corresponding scalar mesons   // 0 in terms of their relative charge structure. Vector mesons are equivalent composite charge states of the corresponding pseudo-scalar mesons. Although the two charge states look identical electrically, with difference in the spin state of level 2 they lead to somewhat different (magnetically) mass/energy.

Name Energy Energy (Est.) Calculations (MeV) (MeV)

  139.6 139.6 938.3/2*0.189*0.255/0.162=139.6MeV. Meson factor: 0.189, Level 2.   139.6 139.6  0 135 <139.6 Meson factors: <1, Level 1; 0.189, Level 2.  547.8 547.3 938.3/2*0.189/0.162=547.3MeV. Meson factor: 0.189, Level 2.   957.8 937 (1740/2*0.189*0.172/0.11+(938.3/2.0))*0.82*0.255/0.162=937. Meson factors: 0.837 (alpha_c=1.547), Level 2; 0.189, Level 3.  c 2980 3041 (4242/2*0.189*0.097/0.097+1816+939.6/2)*0.95*0.193/0.162=3041MeV. Meson facors: 0.95 (alpha_c=5.73), Level 2; 0.189, Level 4.  b 9390 9227 1740/2*0.189/0.11/0.162=9227MeV. Meson factor: 0.189, Level 3. K  493.7 495.4 (1740/2*0.189*0.117/0.11+939.6/2)*0.645*0.193/0.162=495.4. Meson factors: 0.645 (alpha_c=1.374), Level 2; 0.189, Level 3. K  493.7 495.4 K 0 497.6 495.4 D  1869 1883 ((4429/2*0.269*0.097/0.097+1816/2)*(0.775)*0.117/0.11+939.6)*(0.925)*0.193/0.162=1883MeV. D  1869 1883 Meson factors: 0.925 (alpha_c=3.64), Level 2; 0.775 (alpha_c=1.728), Level 3; 0.269, Level 4. D 0 1864 1883 Ds  1968 1942 ((4242/2*0.189*0.097/0.097+1816/2)*0.72)*0.117/0.11+939.6=1942MeV. Meson factors: 0.72 (alpha_c=1.504), Level 3; Ds  1968 1942 0.189, Level 4. B  5279 5278 (4429/2*0.269*0.206/0.097+1740+939.6/2)*(0.965)*0.255/0.162=5278MeV; Meson factors: 0.965 (alpha_c=7.407), Level 2; B  5279 5278 0.269, Level 4. B 0 5279 5278 0sB 5366 5335 4242/2*0.189/0.097*0.117/0.11+939.6=5335MeV. Meson factor: 0.189, Level 4. Bc  6277 6349 (4429+1816/2)*0.953*0.117/0.11+939.6=6349 Meson factor: 0.953 (alpha_c=6.13), Level 3. Bc  6277 6349   775   775  0 775 Refer to the UEG synthesis rules for different particles (sections II-VI). The mass/energy formula associated in the synthesis of a particular particle maybe evident from its calculation shown above. See Table-I for an example of such synthesis.

28 N. Das, A UEG Theory of Nature, 2018 2-16

Table III UEG Shell Model of Leptons

Name Energy Energy(Est.) Level One Configuration Level Two Configuration Level Three Configuration Level Four Configuration (MeV) (MeV) Meson Shell Shell 2 Shell 1 Meson Shell Shell 2 Shell 1 Meson Shell Shell 2 Shell 1 Meson Shell Shell 2 Shell 1 e  0.5 0.5 e  0.5 0.5 e /  <0.000005 <0.00005   105.7 105.6   105.7 105.6 /  <0.17 <0.57 0.0473   1776 1780   1776 1780 /  <15.5 <9.8

Note: For neutrinos, both charges are close to each other in the same shell, either shell #1 or #2, placed near one of the locations where permittivity is infinity

Name Energy Energy (Est.) Calculations (MeV) (MeV)

0.5 0.5 UEG parameter  1 for level 1 determines the electron/positron energy. 0.5 0.5 <0.000005 <0.00005 < 0.5*0.0001=0.00005MeV; Assume neutrino factor <0.0001. 105.7 105.6 938.3/2*0.189*0.193/0.162=105.6MeV. Meson Factor: 0.189, Level 2. 105.7 105.6 <0.17 <0.57 <938.3*0.0001/0.162=0.57MeV; Neutrino factor < 0.0001. 0.0473 0.5/2*0.189=0.0473MeV; Meson Factor: 0.189, Levcel 1. 1776 1780 1740/2*0.189/0.11*0.193/0.162=1780MeV. Meson factor: 0.189, Level 3. 1776 1780 <15.5 <9.8 <1740*0.0001/0.11/0.162=9.8MeV; Neutrino factor<0.0001. Refer to the UEG synthesis rules for different particles (sections II-VI). The mass/energy formula associated in the synthesis of a particular particle maybe evident from its calculation shown above. See Table-I for an example of such synthesis.

Table IV UEG Shell Model of Special Particles ( W, Z and H Bosons, Top (t) and Bottom (b) Quarks)

Name Energy Energy(Est.) Level One Configuration Level Two Configuration Level Three Configuration Level Four Configuration (GeV) (GeV) Meson Shell Shell 2 Shell 1 Meson Shell Shell 2 Shell 1 Meson Shell Shell 2 Shell 1 Meson Shell Shell 2 Shell 1 W  80.39 75.3      87.9    82.6      W  80.39 82.6(81.9)      Z 91.19 92.1     H 125.09 127.2     (t t  ) / 2 173.21 175.5     (b b  ) / 2 4.18 4.51    

Name Energy Energy (Est.) Calculations (GeV) (GeV) W  80.39 75.3 (4.429/0.097*0.2+1.740)/0.172/0.162*0.193=75.3GeV 87.9 (4.429/2*0.269)/(0.162*0.11*0.097)*0.255=87.9GeV. Meson factor: 0.269, Level 4. 82.6 (4.429/2*0.269)/0.097/0.11*(0.11/0.117)/0.162*0.255=82.6GeV. Similar to above. Level 2 (special 0th shell and shell 1 used). W  80.39 82.6(81.9) (75.3+87.9+82.6)/3=81.9GeV. W+, but level 1 charge negative. One state shown, average of three states (=81.9GeV) listed. Z 91.19 92.1 (1.740+(4.242/2*0.189)*0.2/0.097)/.172/.162=92.1GeV. Meson factor: 0.189, Level 4. H 125.09 127 (4.429/2.0*0.269/0.097*0.097+1.816+0.9396-0.9396)/0.117/0.162=127.2GeV. Meson factor: 0.269, Level 4. (t t  ) / 2 173.21 175.5 4.242/2.0*0.189/0.097/0.11/0.162*0.193/0.255=175.5GeV. Meson factor: 0.189, Level 4. (b b  ) / 2 4.18 4.37 (1.740/2.0*0.189*0.117/0.11+0.9396)/0.255=4.37GeV. Meson factor: 0.189, Level 3. Refer to the UEG synthesis rules for different particles (sections II-VI). The mass/energy formula associated in the synthesis of a particular particle maybe evident from its calculation shown above. See Table-I for an example of such synthesis.

29 N. Das, A UEG Theory of Nature, 2018 2-17

Table V UEG Parameters For Particle Modeling

Level One Parameters Level Two Parameters Level Three Parameters Level Four Parameters /1  /1  r12 /1  r11 /1  r10 /1  r22 /1  r21 r20 /1  r32 /1  r31 /1  r30 /1  r42 /1  r41 /1  r40 Simple UEG Theory 0.3 0.22 0.18 0.3 0.22 0.18 0.3 0.22 0.18 0.3 0.22 0.18 Data Fit 0.255 0.193 0.162 0.172 0.117 0.11 0.2 0.097 0.097 W (MeV) W 12 W 11 W 10 W 22 W 21 W 20 W 32 W 31 W 30 W 42 W 41 40 Simple UEG Theory 0.5 0.51 0.51 938.3 950.5 950.5 1740 1763 1763 4242 4297 4297 Data Fit 0.5 938.3 939.6 939.6 1740 1816 1816 4242 4429 4429

mi WWsnii22 /;2 mi2 0.0945(Data Fit ); mi2 0.192(UEGTheory ) WW/ ( / 2); 0.189(Data Fit ); 0.384(UEGTheory ) Meson Factors: mi sni22 i2 mi2 mi2 mi WWsnii11 /;1 mi1 0.1345(Data Fit ); mi1 0.226(UEGTheory ) mi WW sni11 / ( i1 / 2); mi1 0.269(Data Fit ); mi1 0.452(UEGTheory )

Notes: - Data-fit and UEG theoretical values for the meson factor for any general energy W, or its equivalent mass m, is provided separately in a graphical plot (see Fig.7). - Energy W, or its equivalent mass m, of a particular level and shell listed above is twice the associated UEG static (without spin) energy/mass. The listed energy/mass is the total energy/mass of the particular level and shell if there is a spinning charge layer at the particular shell and level.

30 3-1

Uniﬁed Electro-Gravity (UEG) Theory and Quantum Electrodynamics

Nirod K. Das Department of Electrical and Computer Engineering, Tandon School of Engineering, New York University, 5 Metrotech Center, Brooklyn, NY 11201 (Dated: May 9, 2018; Revised December 20, 2018) The Unified Electro-Gravity (UEG) theory, originally developed to model a stable static charge, is extended to a spinning charge using a “quasi-static” UEG model. The results from the new theory, evaluated in comparison with concepts and parameters from basic quantum mechanics (QM) and quantum electrodynamics (QED), show that the QM and the QED trace their fundamental origins to the UEG theory. The fine structure constant and the electron g-factor, which are key QED parameters, can be directly related to the proportionality constant (referred to as the UEG constant) used in the UEG theory. A QM wave function is shown to be equivalent to a space-time ripple in the permittivity function of the free space, produced by the UEG fields surrounding a spinning charge, and the basic QM relationships between energy and frequency naturally emerge from the UEG model. Further extension and generalization of the theory may also explain all other quantum mechanical concepts including particle-wave duality, frequency shift in electrodynamic scattering, and charge quantization, leading to full unification of the electromagnetics and gravity with the quantum mechanics.

I. INTRODUCTION able extension of the basic static UEG theory of [1]. The extended model, referred to as the “quasi-static” UEG model, will be guided by existing concepts from Newto- A new theory unifying the electromagnetic and grav- nian mechanics and gravity [2], relativistic mechanics [3], itational concepts, referred to as the Unifed Electro- electromagnetics [4] and general relativity [5], and build Gravity (UEG) theory, was proposed in [1] to model a upon the basic principles of the static UEG theory of [1]. stable, static electronic charge, referred to as a static The objective is to explain different quantum-mechanical UEG electron. In its most basic form, the UEG theory and quantum-electrodynamic concepts and parameters, introduces a gravitational field proportional to the en- such as the wave function [6], Planck’s constant [7] and ergy density surrounding the charge, with the constant angular momentum [8], fine structure constant [9, 10] of proportionality γ, referred to as the UEG constant, and g-factor [11], in terms of UEG concepts and param- which results in a strong gradient of the the permittivity eters such as the permittivity function, the UEG con- function (r) around the charge. With the success of the r stant(s) and different UEG forces [1]. Further exten- static UEG theory, an electron with a spin angular mo- sion of the principles of the quasi-static UEG model of mentum /2 may be conceived in terms of the static UEG ~ the spinning electron may physically explain energy and electron, that physically spins at a certain radial distance frequency shift in an electrodynamic scattering process, r to produce the given angular momentum. The central 0 charge quantization linked to quantization of the angular acceleration of the spinning electron would be supported momentum, and wave-particle duality based on a pilot- by suitable UEG forces produced by the surrounding elec- wave concept. In any event, development of the UEG tric and magnetic fields. The spinning electron would be theory would open a wider unified theoretical framework, self-supported by the radial forces due to the electron’s combining electromagnetic, gravitational, together with own UEG fields, in distinct contrast with orbiting of an the quantum mechanical and electrodynamic concepts, electron around the nucleus of an atom, which instead is that would be applicable to all elementary particles. This externally supported by the radial forces due the electric would provide a unified alternative to the standard model field of the central nucleus. The permittivity function of particle physics [12], without need for additional strong (r) of the static UEG electron, would transform into r and weak nuclear forces. a space-time-dependent permittivity function r(r, t) for the spinning electron, which would be equivalent to hav- The sequence of presentation in different sections ing a space-time ripple, representing a quantum mechan- maybe outlined as follows: ical wave function. The spinning radius, speed, associ- Section II presents the basic model of a spinning elec- ated wave frequency, angular momentum, energy/mass tron, in equivalence to an orbiting electron, and extracts may be modeled by extending the static UEG theory of basic relations between the mass of the static UEG elec- [1], by including additional dynamic UEG effects due to tron, the total mass of the spinning electron, the electron the magnetic field and field momentum-distribution of g-factor and the spin angular momentum. the spinning electron. This is followed in section III by identifying different A rigorous, dynamic version of the static UEG theory UEG forces in a spinning electron, and formulating the of [1] would be needed to fully model the spinning elec- total UEG acceleration that would support the spinning tron, which is premature at this point. In the absence motion, based on first-order estimates. This would allow of the rigorous dynamic UEG theory, we will use suit- relating the fine structure constant from quantum elec-

Copyright # TXu 2-128-436, c 2018 by Nirod K. Das. All Rights Reserved. 31 N. Das, A UEG Theory of Nature, 2018 3-2 trodynamics to the UEG constant in section IV, in an that could be sustained by the UEG forces due to its own approximate form, that can be verified with the UEG fields. The spinning of the static UEG electron maybe constant available from the static UEG model of elec- considered equivalent to orbiting of a complete electron tron in [1]. A much closer evaluation is explored in sub- structure (that already spins) around a central nucleus at section IV A, with deeper insights into the UEG spin a suitable orbital radius. Except, the complete-electron model, which may assist in future development of a rig- orbiting is sustained by the electric fields of an external orous UEG theory. body, the nucleus, whereas the spinning of the electron Section V models the UEGravito-Magnetic effect due is sustained by the UEG forces due to the fields pro- to field momentum associated with the spinning motion, duced by the electron itself. The spinning electron may which is shown to cancel with the basic UEG effect due be treated similar to an orbiting electron with orbital to the magnetic energy density of the spinning charge. quantum number equal to one, where the complete elec- This allowed the formulations presented in the sections tron mass me (already including the spin effects) for the II, III and IV using only the UEG acceleration due to the orbiting electron is substituted by the static UEG mass 0 electric energy density, in order to support the spinning me of the “bare” (without the spin) electron. We know motion. from quantum electrodynamics, that the magnetic mo- Section VI relates the quantum-mechanical wave fre- ment µJ due to an orbiting electron with orbital quantum quency to the spinning frequency in terms of the spin number equal to one, is about the same as that due to a velocity and the g-factor. This is based on relativistic spinning electron, different by a factor g = g/2 [13] close transformation between the spinning frame and a station- to unity. This means the velocity-radius products of the ary external frame. The value of the g-factor is shown orbiting and the spinning electrons are also close to each to be estimated from the UEG spin model in different other, different with the same above factor g. In addition, degrees of accuracy, as compared with its known mea- we also know from quantum electrodynamics, that the sured value. This is to reinforce validity of the spin UEG angular momentum J of the orbiting and spinning states model, and illustrate finer predictive power of the model. are ~ and ~/2, respectively, different exactly by a factor Fundamental significance of the spin UEG model, and of two. Based on the above information we already have, potential implications of the UEG theory in general, are it can be deduced that the total mass me (with spin) of discussed in section VII, outlining concepts for full unifi- the orbiting electron is g(= 2g) times, or about twice, the 0 cation of electromagnetics and gravity, together with the mass me of a static UEG electron (without spin, modeled physics of quantum mechanics and elementary particles. by a static UEG theory). Also can be deduced, that the angular momentum of the spinning electron is exactly 0 equal to the above mass me of the bare static UEG elec- II. ELECTRON SPIN MODELED AS ORBITING tron, multiplied by the velocity-radius product, (v0r0) of OF A STATIC UEG ELECTRON, AROUND ITS the spinning electron. OWN FIELDS

q Consider a static UEG electron, which is originally µJ = 2 × (vr), modeled as a stable charge body using the basic, static µ (spin) ' µ (orbital) = ~q = 1 µ (spin), J J 2me g J UEG theory of [1]. Then, consider the static UEG elec- (vr) ' (vr) = ~ = 1 (vr) , tron to spin at radius r0 at a speed v0, close to the spin orbital me g spin speed of light c, the central acceleration of which could g = g = 1.00115965218091, be sustained by suitable UEG force(s). Clearly, the ba- 2 sic UEG theory of [1] which rigorously models the static 1 ~ UEG charge without spin, may no longer be rigorously J = m × (vr),J(spin) = 2 J(orbital) = 2 , valid for the spinning charge. A dynamic UEG theory for 0 1 me(vr) = me(vr) = ~ a moving charge would be required, that would include spin 2 orbital 2 additional UEG forces due to energy density of the mag- 0 1 0 0 netic field, as well as UEGravito-magnetic effects due to me ' 2 me = gme, me = gme, the field momentum associated with the electromagnetic J(orbital) J(spin) = ~ = = me (vr) fields. The development of a such a complete dynamic 2 2 2 orbital 0 1 0 0 theory, referred to as a Unified Electro-Gravito-Magnetic = (gme) g (vr)spin = me(vr)spin = mev0r0. (1) (UEGM) theory, is premature at this point, beyond the scope of the present work. However, in absence of such As we deduced, the ratio between the spinning and a full dynamic theory, we will model the spin behavior the static electron masses is approximately equal to two. using a quasi-static UEG model, complemented by estab- One might casually expect the ratio to be equal to the lished results and insights from quantum electrodynam- relativistic boost factor, which is dependent on the spin ics. speed, as per special relativity. However, as mentioned A static UEG electron (modeled by a static UEG the- before, we anticipate the spinning speed v0 to be close to ory) may spin at a specific radial distance and velocity, the speed of light c, in which case the relativistic boost

32 N. Das, A UEG Theory of Nature, 2018 3-3 factor would be much larger than the above ratio close to stationary frame (unprimed frame) far from the spinning two. Accordingly, this might appear, at first, to be a con- center. We would estimate the electric field, electric en- tradiction to the casual expectation, but that may not be ergy density, and the associated UEG acceleration in the a valid observation. The mass transformation relation of external stationary frame. This UEG-acceleration due special relativity is applicable only to a complete, stable to the electric energy density of the spinning electron, massive particle in motion, but not for transformation of as seen in the external unprimed frame, would support mass of the static UEG electron as it spins. This is be- the central acceleration of the spinning body. Space-time cause, the static UEG electron is an ideal state that is not relations of special relativity may be used, for relativis- dynamically stable in motion, and therefore does not con- tic transformation between the external unprimed frame stitute a complete, valid particle by itself to which spe- and the spinning primed frame. cial relativity can be independently applied. The static UEG electron is only a part of the internal formation of 0 A length parameter, radius re of the charge, along the the complete electron structure, and the special relativ- orbital motion, as seen at a given instant in the orbit- ity can only be applied to the complete structure. Fur- ing primed frame, would be multiplied by the relativis- ther, the environment around a spinning electron, which tic boost factor (= N) as seen in the external unprimed is governed by the UEG theory involving non-linear de- frame. Accordingly, one may interpret that the electronic formation of the free-space structure around the electron, charge would “look stretched” along the orbit by the is significantly different from a simple free-space medium 0 boost factor (N). If N = 2πr0/re, the charge may appear assumed in the special relativity. Accordingly, the prin- wrapped around the spinning orbit, satisfying a suitable ciples of special relativity may not apply strictly in their periodic physical condition. The periodic condition may conventional forms, in this spinning environment, partic- be assumed to be a required “resonant” condition in a ularly for transformation of mass. dynamic UEG theory, yet to be rigorously developed, in The angular momentum relation of (1) may now be order to maintain a stable spin structure. Under this res- used to express the ratio of the spin radius r0 and an onant condition, the stretched charge may be viewed in 0 equivalent radius re of the static UEG electron, in terms the form of N number of virtual unit cells, that are peri- of the fine structure constant α [9, 14]. The ratio is ap- odically arranged with proper overlap (assumed overlap proximately equal to 1/α, assuming that the spin velocity factor 2) between neighboring units, over the entire cir- v0 is close to the speed of light c. cumference of the spin orbit (see Fig.1). Accordingly, the spinning charge may be viewed as a ring-charge, as well 2 2 as a ring-current distribution. In fact, because the spin- 0 q r0v0 0 q mer v = = ~ , me = , ning orbit is actually in random orientations, with the 0 0 8π r0 c2 2 8π r0 c2 0 e 0 e spin-axis pointing randomly in all possible directions, the r0 4π0~c c 1 c 1 0 = 2 ( v ) = α ( v ) ' α , v0 ' c. (2) charge may be viewed to be wrapped around the entire re q 0 0 surface of a sphere of radius equal to the spin radius r0. Accordingly, the structure may be approximately mod- III. UEG ACCELERATION COMPONENTS eled as a spinning surface-charge distribution, uniformly THAT SUPPORT THE SPINNING CENTRAL distributed over the sphere of radius r0, with axis of the MOTION) spinning randomly changing in time.

The fields and dynamics of a moving charged body As per the above model, we would derive the aver- with an expected speed close to the speed of light would age electric energy density at radial distance r0, as seen need special modeling and interpretation. We would by the external unprimed frame, by assuming a spher- show in the following section V that the UEG acceler- ically symmetric electric field. The average electric en- ation due to the energy density of the magnetic field of ergy density may first be expressed assuming that the the spinning charge would cancel with the UEGravito- uniform charge distribution on the sphere of radius r0 Magnetic acceleration produced by the momentum distri- is stationary with respect to the unprimed frame, hav- bution associated with the equivalent UEG mass/energy ing only radial electric field. When the charge spins, the distribution. Therefore, the central acceleration of the radial electric field at a given point, which is directed or- spinning charge would be sustained only by the remain- thogonal to the spinning velocity at the point, is expected ing UEG acceleration due to energy density of the elec- to be increased by the relativistic boost factor N, as per tric field. The central spinning motion of the electron, special relativity. Consequently, the average electric en- thus balanced by all the UEG and the UEG-Magnetic ef- ergy density that we derived first, assuming the station- fects, may all be viewed as gravitational in nature in the ary charge distribution, may now be multiplied by the fundamental sense. Accordingly, consistent with general square of the boost factor, in order to find the average relativity, a moving frame attached to the spinning orbit electric energy density of the spinning electron. This av- may be interpreted as an orbiting inertial frame (primed erage electric energy density would be used to calculate frame), moving along with the charge body, and the or- the UEG acceleration to support the central acceleration biting frame would be inertially equivalent to an external of the spinning charge.

33 N. Das, A UEG Theory of Nature, 2018 3-4

FIG. 1.

be considered an empirical “filling factor,” necessary for q 1 qN proper estimation of the effective UEG force seen at the E = 2 s = 2 , center of the spinning electron. 4π0r0 v2 4π0r0 1− 0 c2 r 2 v 2πr 2πr 2 q 2 2 q 2 2πr 2 N = 1 − 0 = a × ( 0 ) ' ( 0 ), a ' 1. W = 0 E = 0 ( ) (N) ' 0 ( ) ( 0 ) , 2 0 0 (3) τ 2 2 2 2 2 r0 c re re 4π0r0 4π0r0 e 2 2 4r0 q 2 2πr 2 4 v c2 Eg ' γWτ ( ) ' γ 0 ( ) ( 0 ) ( ) = 0 ' , 02 2 2 r0 π r0 r0 IV. PLANCK’S AND FINE STRUCTURE πr 4π0r0 e CONSTANTS RELATED TO THE UEG γm0 2 r e = γ q ' 0 , CONSTANT USING THE SPIN MODEL 0 2 0 3 2 4r0 re 8π0re c e r γm0 0 ' 4 e = 138.359 (UEG Theory), As discussed earlier, the central acceleration of the spin r0 0 2 e re is to be sustained only by the UEG force due to the r 0 = 1 ( c ) = 1 ( 1 ) average electric energy density at the radius r . The r0 α v0 α r 0 e 1− 1 UEG force due to magnetic energy density is assumed to N2 be balanced by the UEGravito magnetic force, as shown 4π c ' 1 = 0~ ' 137.036 (QED). (4) in section V. α q2 As per the charge model discussed above, the charge structure of Fig.1 maybe effectively interpreted in the The angular momentum associated with the above form of overlapping (overlapping factor 2) square grids spinning would be equal to ~/2, as expected from quan- 0 0 of 2re x 2re size each, wrapped in a ring configuration tum electrodynamics. The required ratio of the spin- r r0 around the sphere of radius r0. Further, because the axis radius 0 and the charge-radius e, as derived in (2) from of the ring structure of Fig.1 is randomly changing in quantum electrodynamics, is approximately equal to the time, the 2r0 x 2r0 sized grid structure would be effec- inverse of the fine structure constant α [14]. This ratio e e 0 tively overlapping in two dimensions, wrapped over the r0/re is shown above in (4) to approximately compare to 0 0 0 2 surface of the sphere. Therefore, the UEG force at the that r0/re = 4γme/(re ) independently estimated from center of the grid may be calculated by multiplying the the UEG theory, based on the spinning model of Fig.1, average energy density on the spherical surface by a fac- using the UEG constant γ derived in [1] from a static tor (4/π). The factor is equal to the ratio of the area of a UEG model of an electron. The small fractional differ- 0 0 0 re x re (excluding overlap region) square grid to that of ence between the above two results for the ratio r0/re is 0 an enclosed circle of radius re. This geometric factor may calculated to be of the order of the fine structure con-

34 N. Das, A UEG Theory of Nature, 2018 3-5 stant. Conversely, if one would estimate the UEG con- a lower γ, before introducing higher-order UEG terms. 0 2 0 stant γ = re /(4αme) from the above UEG theory and In other words, the actual value of γ in the dominant 0 spin model, using the ratio 1/α = r0/re (2) from quan- acceleration in a rigorous UEG theory, which is the the tum electrodynamics, and known values of the electron constant of proportionality between the UEG accelera- 0 mass me = me/g from (1) and corresponding electron ra- tion and the energy density, in the low energy-density 0 dius re from (3), the value of γ would be different from range, is expected to be lower than than the value of γ that in [1] derived from a static UEG electron model. in [1] obtained from a basic UEG theory without any The two different independent values of γ would intro- higher-order effects. This is under the condition that the duce a fractional ambiguity close to the fine structure rigorous and the basic UEG theories produce the same 0 constant, which we will address shortly in the following static electron mass me. The lower value of γ in the rig- section IV A. orous theory would be associated with a reduction of the 0 effective radius re, in proportion to the cube-root of γ [1]. Based on the above discussion, we may recognize that 0 2 re in principle there are three theoretical values for the con- γ(UEG) = 138.359 × 0 4me stant γ. (I) The value of γ obtained from a basic UEG 2 3 re g 2 -2 -3 theory [1], that produces a stable electron mass equal = 138.359 × 4m ' 6.017 × 10 (ms )/(Jm ), 0 e to me = me/g. (II) The actual value of the γ, which 0 2 re is the constant of proportionality between the UEG ac- γ(QED) = 137.036 × 0 4me celeration and the energy density, valid when the energy r2g3 density is sufficiently small. This would be the dominant = 137.036 × e ' 5.96 × 102(ms-2)/(Jm-3), 4me term, or the first order approximation, in a fully rigorous γ(UEG) − γ(QED) 0 2 0 UEG theory. And, (III) the value of the γ = αre /(4me) ' γ(QED) × 1.32α ∼ γ(QED) × α. (5) that is indirectly estimated from an ideal spin model of the electron, as derived in section IV, which expects the Leaving aside the small fractional ambiguity, discussed known fine structure constant from quantum electrody- above, the close results from the UEG theory and quan- namics to be related to the γ, and the known values of the 0 0 tum electrodynamics point to a definite fundamental con- electron mass me and the electron radius re. A higher- nection between the two theories, relating the UEG con- order UEG theory for a bare, static electron, combined stant γ to the fine structure constant α. This is a signif- with a rigorous, dynamic UEG model for a spinning elec- icant development, also providing a direct physical rela- tron would be needed to explain any differences or spe- tion between the UEG constant γ and the Plancks con- cific relationship between the three parameters, which is stant ~, via the fine structure constant α to which both beyond any scope of the present work. The parameters the γ and ~ are related to, founded on the dynamic mod- (I) and (III) are shown in (5) to be close to each other eling of the spin, sustained by the UEG forces. In other with a fractional difference of the order of the fine struc- words, the Planck’s constant ~, with its origin in quan- ture constant, and we may suspect similar closeness of tum mechanics, may no longer be considered a fully inde- the parameter (II) to the other two values. Additional pendent natural constant, but is rather unified together insight may assist in more accurate evaluation for the with the UEG constant γ. Accordingly, the UEG theory, parameter (II), which is the actual constant of propor- which already unifies the electric and gravitational prin- tionality between UEG acceleration and energy density, ciples, is now positioned to be fully unified with quantum in an environment with low energy density. mechanics as well. The effective radius used in an ideal spin model of sec- 0 tion II, Fig.1, is assumed to be the classical radius re of the bare electron, which determines the required rela- 0 A. Closer Relationship Between the UEG and tivistic boost factor to be ideally equal to N = 2πr0/re. Quantum Electrodynamics The actual value of the boost factor is slightly larger than this ideal value, represented by the factor a, the value of It may be noted that the UEG theory from which the which can be estimated as shown in (7), from the known constant γ was estimated in (4) is only a basic theory, values of the fine structure constant α and electron g- where the UEG acceleration is assumed to be propor- factor. This increase is equivalent to having an effective 0 tional to the energy density, with γ as the constant of the radius re to be smaller than the ideal value by the same proportionality. A rigorous UEG theory would include factor a. It may be reasonable to estimate that the ac- 0 higher-order acceleration proportional to higher powers tual increased value of the boost factor N = 2πr0a/re, 0 of the energy density, which is expected to reduce the and the corresponding reduced effective radius re/a, is 0 value of the stable mass me, as compared to that from associated with the average between the two values of the basic theory with a given γ without the higher-order γ (I) and (II), that is the average between the values terms [1]. Conversely, in order to have the same final sta- needed in a basic UEG model and in a rigorous UEG 0 0 ble mass me, one may need to start with a basic UEG the- model to produce the same stable mass me. Whereas, 0 ory producing a higher stable mass, or equivalently with the ideal value of the boost factor N = 2πr0/re, and the

35 N. Das, A UEG Theory of Nature, 2018 3-6

0 corresponding effective radius re, is associated with the constants. Conversely, when a rigorous dynamic UEG first value of γ (I), that is the value needed in a basic theory would be established and it validates the basic 0 UEG model to produce the stable mass me. As per the principles of the modeling (4), the fine structure and the earlier discussion on the effect of the higher-order UEG Planck’s constants could in principle be derived and pre- theory, the reduction in the effective radius would mean dicted exactly from the UEG theory. For now, we may that the average of the two values of γ (I) and (II) would evaluate the fine structure constant α, which is the in- be lower than the first value of γ (I), by a factor equal verse of ratio of the spin radius and the electron radius, to a3 ' 1.0045 ' 1 + 0.62α (see (7)). This would place the more accurately from the modeling of (4), guided by the estimate for the actual value of γ (II), fractionally about following insight. (1+1.24α) lower than the estimate of γ (I) from the basic We know from (7) that the boost factor N is slightly 0 UEG theory. This estimate for the actual value of the larger than the ratio (2πr0/re) obtained from the angular γ (II) is fairly close to the estimate (III) from quantum momentum in (2), by the factor a. This is equivalent to electrodynamics, which we know from (5) to be fraction- having increased overlap between the neighboring parti- ally about (1 + 1.32α) lower than the estimate (I). In cles in the ring model of section II, Fig.1. The average other words, the estimates of γ (II) and (III) would be energy density in (4) maybe properly redistributed over essentially equal to each other with a fractional difference the surface of the spin sphere, weighted in proportion of less than 0.1α, or within 0.1%. to the actual energy/mass distribution. Accordingly, the redistributed energy density at the particle center would γ(Basic UEG) be inversely proportional to the square of the overlap fac- γ(actual) ' (1+2a3) tor a, considering overlap of the spinning particle in two γ(Basic UEG) dimensions over the surface of the spin sphere. More is = = 5.963 × 102( -2)/( -3), (1+1.24α) ms Jm the overlap, which is on the outer edges of the particle, γ(Basic UEG) more energy density needs to be redistributed away from γ(QED) ' = 5.96 × 102(ms-2)/(Jm-3), (1+1.32α) the center, leaving less energy density at the particle cen- γ(actual) ter. This redistributed energy density, evaluated at the γ(actual) ' γ(QED), < 1.001 . (6) γ(QED) center of the particle, would be multiplied with the UEG constant γ, in order to obtain the UEG acceleration that The parameter a, used in the above discussion and de- supports the central acceleration. Accordingly, the UEG duction, is expressed as follows, using (2) for the ratio acceleration would be reduced by a factor a2. This may r /r0 , and (15) to relate the boost factor N to the fac- 0 e be introduced as an multiplying factor 1/a2, in addition tor g. The value of a may be calculated using known to the ideal redistribution factor 4/π that we already have measured values of the fine structure constant α and the in our model of (4). electron g-factor g = 2g .

2 0 2 0 q 2 2 0 q 2 2πr0 2 r0 4π0~c c 1 c 1 1 W τ = 2 E = 2 ( 2 ) (N) = 2 ( 2 ) ( 0 ) (a) , = ( ) = ( ) = ( ), 4π r 4π r re r0 q2 v0 α v0 α r 0 0 0 0 e 1− 1 2 0 2 2 2 N 4re 1 2 0 q 2 2πr0 4 v0 Eg = γWτ ( 2 )( a ) = γ 2 ( 2 ) ( 0 ) ( π ) = r , 2πr0 2πa 1 πr0 4π r re 0 N = a × ( ) = ( r ), e 0 0 r0 α 1 e 1− γm0 2 r v 2 N2 e = γ q = ( 0 )( 0 ) 0 2 0 3 2 4r0 c re 8π re c e 1 2π/α (2π/α)(g−1) 0 = r = q r0 1 r0 2 a 1 2 = ( )(1 − ) = ( )(1 − (1 − g) ). (8) N 1− 1−(g−1) 4r0 N2 4r0 N2 e e = (2π/α) × 0.0011596529 = 1 − 0.0015126724 . (7) The expression for the ratio of the spin and classical static electron radii, derived from spin angular momentum re- Based on the close estimated values (II) and (III) of γ, lation (2), may now be used for a closer relationship be- as discussed above, it may be suggested that the two val- tween the rigorous UEG constant γ to the ﬁne structure ues (II) and (III), namely, the value of γ from a rigorous constant α. 0 2 0 UEG model of electron, and the value γ = αre /(4me) derived from quantum electrodynamics, could be, after all, r 4π c 0 = 0~ ( c ) = 1 ( c ) = 1 ( 1 ) = 1 ( 1 ), r0 q2 v0 α v0 α r α q 2 equal to each other, or close to each other with relatively e 1− 1 1−(1−g) higher precision. This proposition may be supported by N2 4γm0 r q q further insights and more accurate modeling of the cen- e = ( 0 )(1 − (1 − g2)) = 1 1 − (1 − g2) = 1 1 − 1 , 0 2 r0 α α 2 tral acceleration of the spinning electron. re e N The above modeling in (4) of the spinning electron us- 4γm0 r 2 ( e )( 0 ) = ( 1 ) . (9) ing the UEG theory was established in an approximate 0 2 r0 α re e form, in the absence of a rigorous dynamic UEG the- 0 0 2 ory, as an initial estimate in order to illustrate funda- The dimensionless constant 4γme/re , where γ is the mental relations between the UEG and the ﬁne structure rigorous UEG constant, and the other dimensional con-

36 N. Das, A UEG Theory of Nature, 2018 3-7 stant 1/α, which is the inverse of fine structure constant, are now shown in (9) to be close to each other with fractional difference of approximately 1/(2N2). This fractional difference is of the order of the square of α/(2π), which would amount to having the above two dimension- 2 µ 2 γµµ 2 2 less constants essentially equal to each other, with their E¯gum = −rˆ < γ( H ) >= −rˆ S < (sin θ + 4cos θ) > ratio different from unity only in the sixth or higher dec- 2 32π2r6 π imal places. This is a significant development, which, in R (sin2θ+4cos2θ) sin θdθ γµµ2 addition to reinforcing unmistakable unified connection = −rˆ S 0 32π2r6 π between the UEG theory and quantum electrodynamics, R sin θdθ opens valuable insights for any future development of a 0 fully rigorous UEG theory. The effective γ for a rigor- γµµ2 2 2 S γµq ~ ous UEG theory is now very accurately estimated from a = −rˆ 2 6 = −rˆ 2 2 6 . (11) 16π r 64π mer basic UEG theory and available information from quantum electrodynamics. Additional details for a rigorous UEG theory could also be extracted from the g-factor, the measured value of which is available with very high precision. This would be possible through the parameter a in (7), which is the change of the effective radius of Unlike a static electron without any spin, which pro- 0 the particle as compared to its ideal classical value re, duces a UEG force field and is associated with a gravi- carrying information that would constrain any variation tational mass distribution (mass-density) as per Gauss’ of a general UEG function γ(Wτ ) in a rigorous UEG the- Law, a spinning electron would in addition be asso- ory. This is in addition to the effective γ deduced above, ciated with an effective UEG momentum distribution which would be the first-order constant coefficient of the (momentum-density) that may be expressed by multi- general UEG function, which is only an approximation plying the UEG mass density and the velocity derived of the general UEG function for low energy density Wτ . above. This momentum-density due to the moving UEG mass-density is expected to produce a gravito-magnetic field, in a very similar way as an electric current distribution due to a moving electric charge distribution produces V. THE UEG ACCELERATION DUE TO THE a magnetic field as per the Ampere’s Law of the electro- MAGNETIC FIELD, AND THE UEGM magnetic theory. Accordingly, the gravito-magnetic field (UEGRAVITO-MAGNETIC) ACCELERATION DUE TO THE FIELD MOMENTUM may also be derived from the UEG momentum density using an equivalent version of the Ampere’s Law.

We will find the expression of the velocity due to spin- The acceleration due to the gravito-magnetic field may ning at a given radius r. This may be derived from be expressed as the cross-product of gravito-magnetic the electromagnetic field momentum, using the Coulomb field and the velocity. Note that there are two ve- electric field due to the electron charge and the magnetic locity terms in the above derivation: (I) the velocity field produced due to spin magnetic moment µS. The ve- used in derivation of the gravito-magnetic field to be- locity may also be estimated from a quantum-mechanical gin with, and then (II) the velocity that multiplies with 0 model where the spinning of the static mass me is treated the gravito-magnetic field to find the gravito-magnetic similar to the orbital motion of the total electron mass acceleration. As a reasonable approach to estimate the me. The two velocity expressions from the electromag- average gravito-magnetic acceleration, we choose the two netic and the quantum models are similar except the sin θ velocity terms to be expressed differently as in (10) - the factors. former derived electromagnetically (¯v(EM)) and the later quantum-mechanically (¯v(QM)). Also note that we treat the gravito-magnetic acceleration E¯gm in (12) just like µ µ µ¯ = ~q z,ˆ H¯ = θˆ S sin θ +r ˆ S cos θ, an equivalent acceleration in an electromagnetic model- S 2me 4πr3 2πr3 ing, without any adjustment factor. This is unlike con- ¯ q E¯×H¯ ˆ2µS sin θ ˆ~ sin θ ventional gravito-magnetic modeling [15] where an addi- E =r ˆ 2 , v¯(EM) = ε 2 = φ qr = φ mer , 4πεr ( 2 E ) tional factor of 1/4 might be needed. This is because, ¯ 0 1 S = mer¯0 × v¯0 = 2 mer¯ × v¯ in conventional gravito-magnetic modeling [15] the mass, 1 ~ ˆ ~ which is the source of gravitation, relativistically varies =z ˆ mevφr sin θ =z ˆ , v¯(QM) = φ . (10) 2 2 mer sin θ with velocity. Whereas, the average mass density ρvu in the present modeling, associated with the azimuthal (φˆ- The energy density in the magnetic field would produce directed) is assumed to be independent of the velocity v¯, an UEG acceleration Egum, which may be expressed by just like the electric charge density, which is the source multiplying the average energy density in the magnetic of an electromagnetic field, would be in an equivalent field with the UEG constant γ. electromagnetic modeling.

37 N. Das, A UEG Theory of Nature, 2018 3-8

by the relativistic boost factor N between the rotating frame (primed frame) and a stationary frame (unprimed ε 2 γq2 E¯gue = −rγˆ ( E ) = −rˆ , frame) far from the spin center. Accordingly, the QM 2 32π2εr4 frequency ω may be shown to be slightly larger than the γq2 ρvu = −ε∇¯ · E¯gue = − , spin frequency ω , with a small diﬀerence of ω/N. The in- 16π2r5 0 tuitive relationships may also be established using space- ∇¯ × H¯ = J¯ = ρ v¯(EM), H¯ = θHˆ , gu gu vu gu guθ time transformation between the primed and unprimed ∂(rH ) 2 1 guθ γq ~ sin θ frames, and enforcing a periodic symmetry condition (β r ∂r = ρvuvφ(EM) = − 2 6 , 16π mer x (2πr0) =2π) around the circumference 2πr0 of the ro- 2 γq ~ sin θ tating frame. Hguθ = , 64π2mer5 This is a signiﬁcant development, which provides a di- E¯gm = −µ < v¯ × H¯gu >= −µv¯(QM) × H¯gu rect physical process that represents the QM wave, in the 2 form of ripples produced due to spinning in a a non-linear ~ γq ~ sin θ =rµv ˆ φ(QM)Hguθ =rµ ˆ ( )( ) mer sin θ 64π2mer5 free-space medium. This may be established by directly 2 2 relating the QM wave frequency ω to the physical spin- γµq ~ =r ˆ 2 2 6 . (12) 64π mer ning frequency ω0 of the charge.

It is shown that the gravito-magnetic acceleration (12) 0 0 0 0 t0+s0v /c2 s0+t0v ejωte−jβs = ejω t e−jβ s , t = 0 , s = 0 , due to UEG momentum density is negative of the UEG s 2 s 2 acceleration (11) due to the energy density in the mag- v0 v0 1− 2 1− 2 netic field. Therefore, the total UEG force is simply the c c 2 UEG force due to the energy density in the electric field, 0 ω−βv0 0 β−ωv0/c ω = s , β = s , independent of the magnetic field generated due to the v2 v2 1− 0 1− 0 spin. c2 c2 0 ω−v0/r0 ω−ω0 ω = ω = s = s ; v2 v2 E¯gm + E¯gum = 0, 1− 0 1− 0 c2 c2 ¯ ¯ ¯ ¯ Eg = Egue + Egm + Egum 1 β × (2πr0) = 2π, β = r . (14) ε 2 γq2 0 = E¯gue = −rγˆ ( E ) = −rˆ . (13) 2 32π2εr4 For a general interpretation of the above concept of the The theory developed in this section is an impor- quantum/UEG wave, first consider a “stationary” spin- tant recognition of the existence and significance of the ning charged body with a total mass m (including spin gravito-magnetic effect surrounding the electron, pro- and static UEG mass) and linear momentum p = 0, with duced as per the new UEG theory. The gravito-magnetic no linear motion of the center of spinning. The region effect constitutes a critical physical mechanism of the surrounding the charge will be associated with a space- complete internal structure of the electron. time dependent permittivity function r(r, t) expressed in the harmonic form (14), which would represent the quantum/UEG wave function of the stationary particle. The wave will be seen by a stationary observer to be oscil- VI. QUANTUM MECHANICAL WAVE IS A jωt RIPPLE IN THE “NON-LINEAR” FREE-SPACE lating as e with frequency ω, but having no spatial MEDIUM, WITH THE QUANTUM FREQUENCY dependence with wave number β = 1/r = 0, in the re- CLOSE TO THE SPIN FREQUENCY gion far from the center (r = ∞). We may assume that the wave amplitude in the far region is uniform in space, The quantum mechanical (QM) wave of frequency ω independent of the spatial variation of the UEG field. may be viewed as a ripple in the free space produced due This wave function would be consistent with quantum to the spinning of the electron, as a result of the strong mechanics, with the expected energy-wave frequency re- 2 UEG force. The non-linear permittivity function of the lationship w = mc = ~ω, and momentum-wave number free-space in the UEG (static) theory would transform relationship p = mv = ~β = 0. into the QM wave function of the “free-space” as a result Now, the above quantum-mechanical relationships for of the spinning. It was discussed in sections III, V, that the “stationary” spinning charge may be extended as well the strong UEG field around the spinning charge would when the charged body undergoes a linear motion of the produce an equivalent rotating inertial frame, dragged center of spinning, with velocity v in a given direction s. along with the moving charge due to the UEGravito- Applying space-time transformation of special relativity, jωt Magnetic (UEGM) effect. The frequency ω of the QM the above wave function e of the stationary charge in wave maybe intuitively “seen” as a difference-frequency the far region, dependent only on time, maybe shown jω(t−sv/c2)η jω0t −jβ0s ω − ω0 relative to the rotating frame spinning with the to transform into a wave e = e e , frequency ω0. The difference frequency ω − ω0, and the with both space and time variation, as seen by a sta- 0 2 actual QM frequency ω may be related with each other tionary observer. The new frequency ω = ηω = ηmc /~

38 N. Das, A UEG Theory of Nature, 2018 3-9

0 2 0 0 2 0 = m c /~, and the new wave number β = ω v/c = m v/~ der, consistent with the prediction from quantum elec- 0 0 = p /~, are related to the new mass m = ηm and mo- trodynamics (QED). This estimate for the g-factor, when mentum p0 = m0v, where η = (1 − v2/c2)−0.5 is the rela- rounded up, is accurate up to the 5th decimal point, as tivistic boost factor associated with the velocity v. The compared to the currently measured value. basic quantum mechanical energy/momentum and fre- 0 0 2 0 quency/wave number relationships, W = m c = ω ~ and p0 = m0v = β0 , are clearly established between the wave 2 q 2 q 0 ~ µJ = qf0πr0 = 2 w0r0 = 2 v0r0,J = mev0r0, parameters in the region far from the center of the mov- µ q q q r0 ing charge, and the mass m0 and linear momentum p0 of J = = 2(1 + 1 ) ' 2(1 + e ) J 2m0 2me N 2me 2πr0 the physical charged body moving at the center of wave. e = q 2(1 + α ) = q g, g = 2(1 + α ), Clearly, the above quantum-mechanical relationships 2me 2π 2me 2π for the quantum/UEG wave would not be valid in the g α 1 2 = (1 + 2π ) = 1.0011614097; α = 137.035999139 , region closer to the central charge. A full dynamic UEG g (measured) = 1.0011596521 . (16) theory may be needed to rigorously model the wave func- 2 tion in the central region, particularly in the immediate vicinity of the charge with strong energy density. C. Higher Order Corrections to the g-factor

A. Electron g-Factor Related to Relativistic Boost Higher order correction to the g-factor may also be es- factor, and to the Spin and Quantum Wave timated from the UEG/QM theory. This follows up on Frequencies the above result that the total electron mass is slightly larger than twice (factor of about 2(1 + α/(2π)) the UEG Based on the above quantum-mechanical interpreta- electrostatic mass, which is different from the ideal factor tion, the frequency ω in (14) may be related to the total of 2 assumed in a simple spinning model with an ideal 0 electron mass me. On the other hand, the spin frequency relativistic boost factor N = 2πr0/re = 2π/α Accordingly, 0 we need the electric and magnetic energies of the spinning ω0 is related to the static electron mass me through the electron to be each slightly larger than the static electric spin angular momentum ~/2. Accordingly, given that energy. This would be accomplished by having a slightly the two frequencies ω and ω0 are related to each other in (14) by the boost factor N, the total and the static larger relativistic boost factor than the ideal value of masses would also be related to each other by the boost 2π/α (boost factor increased to (2π/α)(1 + α/(2π))). Fol- factor. Consequently, the electron g-factor, which is the lowing the similar derivation for the g-factor presented ratio of the total and the static masses, would be directly earlier, this would lead to a smaller g-factor than the related to the boost factor. first order estimate above, the trend being consistent with the measured g-factor and the theoretical derivation from quantum electrodynamics. 1 ω−ω0 s = N, s = N(ω − ω0) = ω, v2 v2 1− 0 1− 0 c2 c2 α α g = 2(1 + 2π /(1 + 2π )) 1 2 ω(1 − ) = ω0, ~ω = mec , α α α α 2 N ' 2(1 + 2π (1 − 2π )) = 2(1 + 2π − ( 2π ) ), v2 g α α 2 0 0 0 ~ ' (1 + − ( ) ) = 1.0011600608 , J = mer0v0 = me ω = 2 , v0 = ω0r0, 2 2π 2π 0 g 2 (measured) = 1.0011596521 . (17) 0 2 mec2 1 2mev0 0 mec2 1 (1 − ) = , me = (1 − ) ~ N ~ 2 N The above estimate for the g-factor, when rounded up, 2v0 is accurate up to the sixth decimal point, as compared to = me (1 − 1 ) = me = me , 2(1− 1 ) N 2(1+ 1 ) g the currently measured value. This is one order improve- N2 N ment compared to the first order estimation deduced ear- 1 g N = 2 − 1 = 0.001159652 (measured). (15) lier. This above estimation is based on the assumption that the mass/energy of the spinning electron increases B. Estimating g-Factor from the Fine Structure proportional to the boost factor. This trend is consis- and UEG Constants, Based on the Spin Model tent with the special relativity, which is expected not to strictly apply in the dynamic UEGM model. Alternate The small difference between the quantum wave and improvement in accuracy of estimation of the g-factor is the spinning frequencies appears in the form of the g- possible by assuming that the difference between the fine factor of the electron. The value of the g-factor may structure constant and the UEG dimension-less constant 0 0 2 be estimated directly from the UEG constant, or equiv- 4γme/re is related to the higher-order correction term alently from the fine structure constant, to the first or- of the g-factor (see section IV A).

39 N. Das, A UEG Theory of Nature, 2018 3-10

to also govern the spin dynamics of the electron that de- r0 termines the spin angular momentum, and consequently g = 2(1 + 1 ) = 2(1 + e ) N 2πr0a is shown to be directly related (on physical basis) to the fine structure constant. Interestingly, the dimensionless α (1/α) 1/6 ' 2(1 + 2π × ( 2 ) ) constant from the UEG theory, which is now related to (4γm0 /r0 ) e e the fine structure constant, is a normalized-parameter α 137.0360 1/6 α = 2(1 + 2π × ( 138.3588 ) ) = 2(1 + 2π × (1 − 0.0016)), independent of any specific mass or charge of a parti- g ' (1 + α × (1 − 0.0016)) = 1.0011595514 , cle, and therefore is a mathematically-based number, re- 2 2π quired to maintain a stable static particle (based on the g (measured) = 1.0011596521 . (18) 2 UEG theory, before any spin is introduced) with a given This is improvement in the higher-order corrections of charge q and a given UEG constant γ. Considering that the g-factor, compared to the earlier estimation in (17), it is a mathematically-based number, independent of any with improvement showing in the seventh and eighth dec- specific particle mass or charge, the dimensionless UEG imal points. This estimation uses a simple averaging of constant or equivalently the fine structure constant is ex- the two UEG constants γ, one from the basic UEG the- pected to carry a general scope of application to any ele- ory of electron [1] and the other from QED using the fine mentary particle (electron/positron, proton/anti-proton, structure constant, as reasoned in section IV A, in order for example). By extension, the scope of the constant to deduce an effective γ. This effective γ determines an would cover composite charged as well as neutral par- 0 effective radius (=re/a) for estimation of the boost fac- ticles, consisting of of multiple charge layers. However, tor N, from which the g-factor is estimated as shown in in the present work the theory is specifically applied to (18). Accordingly, a more accurate prediction/estimation the spin dynamics of an electron, which is the simplest of the g-factor would be possible by deducing a more ac- particle. curate effective γ using a higher-order UEG model of [1]. An exact value of the g-factor can be predicted directly from an exact boost factor N, if it could be available, A. Quantization of Charge and Angular using the exact relationship g = 2(1+1/N) (see 15), (18)). Momentum as Complementary, Emergent Concepts q 2 In principle, the N = 1/ 1 − (v0/c) could be solved from a fully rigorous (both static and dynamic parts) UEG The discovery of the new UEG theory of such sig- model, as the required relativistic boost factor for an nificance, to which the fundamental origin of the fine 0 electron with a static UEG mass me, spinning at a radius structure constant of quantum electrodynamics could 0 r0 = ~/(2v0me) and speed v0, to acquire its total known be traced, is bound to open reexamination of many 0 0 dynamic mass me = gme = 2me(1 + 1/N) and an angular related physical phenomena, that remained mysterious momentum ~/2. Such a rigorous and dynamic Unified and unsolved to date. Consider an immediate conse- Electro-Gravito-Magnetic (UEGM) theory maybe at this quence of the discovery. Once the fine structure constant 2 point premature, and is beyond the scope of the present α = q /(4π0~c) is independently established as a funda- work. mental dimensionless constant that determines the stable mass and spin dynamics of an elementary particle, then the required constant α, for a given angular-momentum VII. DISCUSSION: FUNDAMENTAL parameter ~ and a reference value of c, would force the 2 √ IMPLICATIONS FROM THE UEG THEORY OF elementary quantity q /0 (or equivalently q/ 0) to be a QUANTUM ELECTRODYNAMICS fixed, quantized value. This would be the case, when any new charge is created in the form of a particle-antiparticle The fine structure constant α, first introduced by Som- pair. The available quantized angular momenta, in inte- merfeld [9] as a dimensionless number relating physical gral multiples of ~ from any transitional “photon packets” constants from quantum mechanics (~), electromagnetics (see later discussion on the photon concept), are expected (q and 0) and relativity (c), remained mysterious in its to dynamically force the two charges (positive and neg- origin [10, 16, 17], even though the constant has been ative) in the particle-antiparticle pair, to each acquire widely used in all quantum field theories [11, 12]. As a fixed value of magnitude q (given 0 and c as refer- per the current work, it is now clear that the fine struc- ence constants). This is a significant new understand- ture constant has its fundamental origin in a new Uni- ing of the elementary charge q as a dynamically emer- fied Electro-Gravity (UEG) theory, developed for model- gent, fixed quantity, no longer a pre-assigned parameter ing of elementary particles [1]. A dimensionless constant as currently understood. The new understanding could emerges in the UEG theory, relating a constant used in solve the current mystery of the natural quantization of the theory (the UEG constant γ) with an elementary all available charges, because they all would consist of particle’s stable mass and the particle’s classical radius, an integral number of the elementary charge (+q or −q), which appeared to be closely related (numerically) to the each having the same magnitude, which is dynamically fine structure constant [1]. In the present work, this di- fixed at the time of their production, enforced by the mensionless constant from the UEG theory of [1] is shown UEG theory and quantization of the available angular

40 N. Das, A UEG Theory of Nature, 2018 3-11 momenta. ter would also be guided by the surrounding quantum rip- Conversely, given the fixed magnitude q of an elemen- ples, that are constrained by suitable UEG principles, or tary charge already available, the required (dictated by equivalently governed by the quantum-mechanical princi- 2 the UEG theory) constant α = q /(4π0~c) would fix the ples [6, 8, 18]. These ripples in the free-space may repre- charge’s angular momentum ~/2 (given 0 and c as refer- sent the pilot wave proposed by de Broglie [19, 20], which ence constants), as well as its energy-frequency relation- could be used to physically explain the measured inter- 0 0 ship W = ~ω (section VI). Consequently, all “photon ference pattern of the electron when it passes through a packets” (light radiation), that are naturally produced screen with two closely-spaced slits. The central core of through a coupling process with the non-linear UEG the particle could be physically guided by the interference fields of the elementary charge (see later discussion on pattern created by its own surrounding pilot wave [21]. the photon concept), would be each associated with a This would result in having the physical locations, where quantized angular momentum (= ~) and energy (= ~ω), the central charged particle is actually detected by a suit- which are pre-fixed by the angular momentum ~/2 of the able measurement, to be probabilistically distributed by coupling charge. These available transitional “photons,” the same pattern as the pilot-wave’s interference. which are assumed to be general exchange media in the charge creation process discussed earlier, would, in turn, determine the magnitude of each new elementary charge C. Electrodynamic Scattering, Photoelectric ±q created. Accordingly, the Planck’s constant ~ and the Effect, and the Photon Concept elementary charge magnitude q would constitute a complementary pair of constants, that are naturally emer- Further, a non-linear UEG process similar to gent, balanced with each other through the dynamics of that responsible for generation of the UEG/quantum- the UEG theory and the classical electromagnetic theory. mechanical ripple or wave of a spinning electron, with The continual balancing process between the elemen- its energy (momentum) directly related in proportion to tary charge q and the Planck’s constant ~, as discussed the wave frequency (wave number), could also be respon- above, may be traced back to the beginning of the cur- sible for non-linear interaction of the UEG/quantum- rent universe. All naturally-existing elementary charges mechanical wave of an electron with a UEG/field wave were created in this beginning phase, possibly through a of an incident or outgoing light (photon). This would chained reaction process that was originally balanced by result in dynamic “mixing” between the UEG/quantum- the angular momentum of a transitional photon, which in mechanical/field waves of the electron and the photon. turn was synchronized with the annihilating elementary The process would be analogous to frequency up- or charges from a preceding collapsing universe. This may down-conversion in transistor electronic circuits [22], pro- presume a cyclic universe, where the same fixed mag- duced due to non-linear mixing of two time-dependent nitude of the elementary charge in the current universe electrical signals of different frequencies, having the con- would also be maintained through annihilation in a col- cept extended for both time- and space-dependent sig- lapsing phase in the future, into re-creation in a new nals. Based on a suitable non-linear mixing process, it bouncing universe, by repeating the above q − ~ synchro- is conceivable that any change of the light’s frequency nization process. (wave number) would be negative of that of the electron’s UEG/quantum-mechanical frequency (wave number). The change of the electron’s frequency (wave num- B. Wave-Particle Duality ber) would be in direct proportion to that of its energy (momentum), with the constant of proportionality equal As the new model of electron spin clearly establishes, to ~, as per the UEG theory of the electron. In addition, the charged center of the electron is surrounded by the the change of the electron’s energy (momentum) would “quantum ripples” which are actual ripples or variations be equal to the negative change of the light’s energy (mo- in the structure or characteristics (permittivity) of the mentum), as per the principle of energy (momentum) “free-space” itself. Accordingly, the electron would ex- conservation. Therefore, combining the above three con- hibit particle-like behavior governed by its central core, ditions, the change of the light’s frequency (wave num- and as well exhibit its wave-like behavior due to the sur- ber) would be in direct proportion to that of its own, rounding ripples. This would explain the wave-particle or equivalently negative of the electron’s, energy (mo- dual behavior of the electron, which has been experimen- mentum), with the constant of proportionality equal to tally observed, but is considered to be mysterious based ~. This mechanism could physically explain the Comp- on the current quantum-mechanical understanding. The ton scattering [23], without having to accept it as some ripples are produced by non-linear spin dynamics of the mysterious fundamental “quantum phenomenon”. central electron, based on the UEG theory which is fun- A similar non-linear, dynamic mixing process could damentally non-linear. The central particle and the sur- also explain the nature of quantized absorption/radiation rounding quantum ripples can not be de-linked from each of light (photon) energy, by/from a given material, by other, and are expected to complement each other in all associating the process with a known quantized energy physical processes. Any motion of the electron at the cen- transition of the material’s electrons (due to material’s

41 N. Das, A UEG Theory of Nature, 2018 3-12 atomic or molecular structure). As per the non-linear energy and the angular momentum (in a circularly- mixing process and the principle of energy conserva- polarized state), with their ratio equal to the radian fre- tion, the wave frequency and energy quanta of any ab- quency (= ω) as per the classical electromagnetic the- sorbed/radiated light (photon) could be explained to ory [26, 27], the spin-like angular momentum associated be equal to positive/negative changes in UEG-quantum- with each energy-quantum of light (= ~ω), as deduced mechanical wave frequency and energy of a transitioning above, would be equal to ~. All these combined prin- electron of the material, respectively. The changes in ciples of light would now provide a complete physical the electron’s radian frequency and energy are known to explanation, based on the UEG theory and classical elec- be proportional to each other, with the constant of pro- tromagnetic theory, for the nature of a “photon packet” portionality equal to ~, as per the UEG theory of the in the Einstein’s photoelectric effect [25], or the Comp- electron. Therefore, combining the above conditions, the ton/Raman type scatterings [23, 24], and similarly in the absorbed/radiated light’s radian frequency and the en- Planck’s black-body radiation [7]. Accordingly, Planck’s ergy quantum would also be directly related in propor- initial suspicion - that the quantum-mechanical “photon tion to each other, with the constant of proportionality packet” might not represent any “mysterious” fundamen- ~. Extending this principle, in case an incident light’s tal nature of the light itself, but could simply be a book- frequency exceeds the above threshold frequency of ab- keeping tool that happened to properly model the absorp- sorption, the process would be associated with a scat- tion/radiation of light [28, 29] - may after all be validated tered light of a lower frequency. In this process, using a by the new UEG theory. similar explanation as above, the energy quantum of elec- The underlying mechanism of a dynamic, non-linear tron transition can be shown to be proportional to the mixing processes, as discussed above, seem conceptually difference in the radian frequencies between the incident clear. However, its detailed understanding and modeling and scattered light, with the constant of proportionality may require development of a complete, dynamic Unified ~. This could physically explain Raman type scattering Electro-Gravito-Magnetic (UEGM) theory of an elemen- [24] as well as Einstein’s photoelectric effect [25], without tary charge, interacting or mixing in the presence of an invoking any “quantum mystery”. external electromagnetic radiation (light). Such a general theory is at this point premature, beyond the scope Further, using known relationship between the light’s of the present work.

[1] N. Das, “A New Unified Electro-Gravity (UEG) The- [15] B. Mashhoon, F. Gronwald, and H. I. M. Lichtenegger, ory of the Electron,” Paper #1, pp.4-13, in “A Unified Gyros, Clocks, Interferometers ...: Testing Relativistic Electro-Gravity (UEG) Theory of Nature,” (2018). Gravity in Space 562, 83 (2001). [2] S. I. Newton, Principia: Mathematical Principles of Nat- [16] M. H. McGregor, The Power of Alpha (p. 69) (World ural Philosophy. I. B. Cohen, A. Whitman and J. Bu- Scientific, 2007). denz, English Translators from 1726 Original (University [17] L. M. Lederman and D. Teresi, The God Particle: If of California Press, 1999). the Universe is the Answer, What is the Question (ch.2) [3] A. Einstein, Annalen der Physik 322, 891 (1905). (Dell Publishing, 1993). [4] J. C. Maxwell, A Treatise on Electricity and Magnetism, [18] W. Pauli, Journal of Physics 43, 601 (1927). Vol. I and II (Reprint from 1873) (Dover Publications, [19] L. de Broglie, Journal de Physique et le Radium 8, 225 2007). (1927). [5] A. Einstein, Annalen der Physik 354, 769 (1916). [20] D. Bohm, Physical Review 85, 166 (1952). [6] E. Schrodinger, Annalen der Physik 384, 361 (1926). [21] Y. Couder and E.Fort, Physical Review Letters 97 [7] M. Planck, Annalen der Physik 309, 553 (1901). (2006). [8] P. A. M. Dirac, Proceedings of the Royal Society A: [22] D. M. Pozar, Microwave Engineering, 2nd Edition (John Mathematical, Physical and Engineering Sciences 117, Wiley and Sons, 1998). 610 (1928). [23] A. H. Compton, Physical Review 21, 483 (1923). [9] A. Sommerfeld, Atomic Structure and Spectral Lines. [24] C. V. Raman and K. S. Krishnan, Nature 121, 501 (Translated by H. L. Brose) (Methuen, 1923). (1928). [10] R. P. Feynman, QED: The Strange Theory of Light and [25] A. Einstein, Annalen der Physik 17, 132 (1905). Matter (p. 129) (Princeton University Press, 1985). [26] Wikipedia, “Angular Momentum of Light,” [11] S. Brodsky, V. Franke, J. Hiller, G. McCartor, S. Paston, http://en.wikipedia.org/wiki/Spin_angular_ and E. P. and, Nuclear Physics B 46, 353 (2004). momentum-0f-light, Retrieved (2018). [12] N. Cottingham and D. Greenwood, An Introduction to [27] A. M. Stewart, arXiv:physics. class-ph, the Standard Model of Particle Physics (2Ed) (Cam- physics/0504082v3 (2005). bridge University Press, 2007). [28] T. S. Kuhn, Black-Body Theory and Quantum Disconti- [13] Wikipedia, “g-factor,” http://en.wikipedia.org/ nuity (Oxford: Clarendon Press, 1978). wiki/G_factor_(physics), Retrieved (2017). [29] H. Kragh, “Max Planck: The Reluctant Revolutionary [14] Wikipedia, “Fine-Structure Constant,” http: 1894-1912,” PhysicsWorld.com (2000). //en.wikipedia.org/wiki/Fine_structure_constant, Retrieved (2017).

Part-II: Extension of the Uniﬁed Electro-Gravity (UEG) Theory in the Large Scale: Stellar, Galactic and Cosmology Models

4-1

Uniﬁed Electro-Gravity (UEG) Theory Applied to Stellar Gravitation, and the Mass-Luminosity Relation (MLR)

Nirod K. Das Department of Electrical and Computer Engineering, Tandon School of Engineering, New York University, 5 Metrotech Center, Brooklyn, NY 11201 (Dated: May 9, 2018) The Unified Electro-Gravity (UEG) theory is applied to model gravitational effects of an individual star or a binary-star system, including that of the sun which is the only star of our solar system. The basic UEG theory was originally developed to model elementary particles, as a substitute for the standard model of particle physics. The UEG theory is extended in this paper (a) to model the gravitational force due to light radiation from an individual star, which determines its energy output due to nuclear fusion in the star, as well as (b) to model the gravitational force between two nearby stars, which determines the orbital dynamics in a binary-star system. The mass-luminosity relation (MLR) derived separately from each of the above two models are compared and studied together with the MLR currently available from measured orbital data for binary stars, as well as from an existing energy-source model for stellar nuclear fusion (Eddington’s model). The current MLR data uses conventional Newtonian gravity, where the gravitational force is produced only due to the gravitational mass of the star, which is assumed to be equal to the inertial mass as per the principle of equivalence. This Newtonian gravitational model is modified by including the new UEG effect due to the light radiation of a star, in order to establish the actual MLR which can be significantly different from the currently available MLR. The new UEG theory is applied to an individual isolated star (for modeling the force for stellar nuclear fusion), which is spherically symmetric about its own center, in a fundamentally different manner from its application to a binary-star system (for modeling orbital motion of a binary-star), which is not a spherically-symmetric structure.

I. INTRODUCTION a suitable effective energy density defined at the point of observation of the force, determined by the intensity distribution due to the star’s radiation in the vicinity A new Unified Electro-Gravity (UEG) theory was de- of the observation point. A rigorous UEG model that veloped in [1, 2], as a substitute for the standard model of would be applicable for any stellar or galactic structure, particle physics, which successfully modeled elementary with general distribution of radiation, appear premature particles. The UEG theory introduced a new definition at this point. In this paper, we postulate a suitable for the energy density in an electromagnetic field, which model for a binary-star structure, applied specifically effectively resulted in having a new gravitational force when its individual partner stars are approximately sim- proportional to the conventional energy density, directed ilar to each other and are positioned relatively close to toward the center of gravity of a particle. The center of each other (eclipsing binary), as useful special situations. gravity of the particle is located at its physical center due This would allow us to revisit the mass-luminosity rela- to spherical symmetry of the structure, because the par- tion (MLR) of [3, 4], deduced from measured observation ticle is assumed to be spherically symmetric with respect of mostly closely spaced, eclipsing binaries [5]. Consider- to itself and it is treated ideally in isolation from any ing the large range of star luminosities in [3], with upper surrounding object. This simplicity of spherical symme- 5 try may not be valid in general situations, for example limit as large as 10 solar luminosity, we expect the effec- in modeling gravitation due to light radiation from indi- tive gravitational mass of an individual star in a binary vidual stars in a binary-star system [3], which clearly is system to be in general appreciably different from its in- not spherically symmetric with respect to the total struc- ertial mass, due to additional UEG effect from the stellar ture, even though its two individual partner stars may be radiation. In other words, the mass term in the MLR of spherically symmetric with respect to their own physical [3], which should in principle be the gravitational mass centers. The basic UEG theory of [1] developed for par- of a star but is assumed to be equal to its inertial mass ticle physics, needs to be modified for a general radiating as per the principle of equivalence of general relativity structure with no simple symmetry, particularly to model [6], might not be really equal to the inertial mass of a gravitation in a binary-star system with approximately star. The actual inertial mass would be deduced using identical partner stars. the proposed model to provide a new stellar MLR ((actual inertial)mass-luminosity relation). This would be a Computation of an equivalent gravitational force due modern advancement in the physics of stellar gravitation, to stellar radiation from a given star, as per the new as it relates to the orbital dynamics of binary stars, based UEG theory, requires definition of (a) a suitable grav- on the new UEG theory. itational center toward which the force is directed at, determined by the gravitational parameters of the par- The MLR data from orbital measurement of binary ticular star as well as of its surrounding objects, and (b) stars were also believed to be confirmed in [4, 7],

Copyright # TXu 2-128-436, c 2018 by Nirod K. Das. All Rights Reserved. 44 N. Das, A UEG Theory of Nature, 2018 4-2 with sound theoretical results from a stellar energy- fusion as well as orbital dynamics of binary stars. source model based on balancing of gravitational pressure It may be noted, that the MLR of [3, 4, 7] for solar- with thermal pressure from nuclear fusion (Eddington’s mass stars is implicitly assumed to confirm with the mea- Model). Therefore, in order to definitively validate the sured luminosity of the sun and its effective gravitational UEG theory, the energy-source model also needs to be mass as observed from planetary motions in our solar modified based on the UEG theory, and the results for system. The sun is the only star in our solar system, both the models of [3, 7] must be shown to be consistent with orbiting planets having a fraction of the solar mass with each other. Accordingly, the new MLR ((actual in- [8]. Accordingly, the sun may be treated essentially as ertial) mass-luminosity relation) as derived from a stel- an isolated star, for modeling its gravitation using the lar energy-source model should be the same or sufficiently UEG theory. Under this condition, the effective gravita- similar to that deduced, as discussed earlier, from binary- tional mass mg of the sun would be equal to the sum star measurements, when the additional gravitation ef- of its inertial mass m and the UEG mass mu due to fects of the new UEG theory are included. In equivalent its total luminosity L0. That is, mg = m + mu. The terms, for a given actual mass (inertial) and its associ- UEG mass mu may be calculated using the known value ated light output of a star, the presence of the additional of solar luminosity L0 [9] and the UEG constant γ de- gravitational effects due to the new UEG theory in the duced in [1, 2] from modeling of elementary particles. two MLR models should lead to the same or comparable As mentioned, the total effective gravitational mass of “equivalent-mass” as the mass-term in the MLR data of the sun is also known [9], based on the observed orbital [3, 4, 7], which is derived using only conventional Newto- periods of planets. The inertial mass of the sun may nian gravity. The “equivalent-mass” parameter, which is then be estimated by subtracting the UEG mass from a function of the inertial mass and star light, is defined the total effective gravitational mass. Now, in order that here such that the expression of the star’s luminosity de- the UEG theory confirms with the two models of MLR rived using any particular model based on only Newto- [3, 4, 7], specifically for the mass parameters of the sun, nian gravity, and that using a rigorous derivation includ- we need to show that the mass (actual inertial) term in ing additional gravitation due to the star light as per the “new” MLR derived using the UEG theory is equal to the UEG theory, would appear in functionally identical the known inertial mass m of the sun, whereas the mass forms. Except, the mass term in the simple Newtonian (equivalent gravitational) term in the “existing” MLR case is replaced by the equivalent-mass in the rigorous [3, 4, 7] is equal to the known gravitational mass mg of derivation using the UEG theory. the sun, when the luminosity is equal to the solar lumi- The gravitational forces which determine nuclear fu- nosity. Conversely, the expressions of the MLRs (new and sion in the energy-source model of a star act differently existing) derived in functional forms, with the additional from those in a binary-star system which determine the functional constraint mg = m + mu required specifically orbital motion of the partner stars. In the former case, for the sun, could allow for an estimate solution of the the gravitational forces of a given star act upon the star’s unknown UEG mass mu of the sun, from which the UEG own mass (ignoring any small opposing gravitation from constant γ could be deduced. This would provide a use- nearby star(s)), whereas in the later the gravitational ful estimation for the UEG constant γ, which may be forces from one partner star acts upon the mass of the verified with estimation from the particle physics model other of the binary system. The former is a spherically [1]. This would independently support the UEG theory symmetric problem, whereas the later is not. The exist- as well as the value of the UEG constant, for the stellar ing models in [3, 7] are based on conventional Newtonian models. gravity, where the the same gravitational mass is used The expressions of the MLR, currently existing based when the star is either a source or target of gravitation, on the conventional Newtonian gravity, is introduced in which is equal to the inertial mass of the star. Conse- section II. This is followed by presentations of the two quently, the same unique mass term, equal to the iner- new models including the UEG effects. The spherically tial mass of the star, is used in the two existing MLR symmetric problem of the stellar energy-source model models of [3, 7]. However, due to the basic differing na- would be presented first in section III. The relatively sim- tures of gravitation acting in the two MLR models, as ple and definitive results from this study would then be explained above, one might be inclined to expect that the used together with the required constraints for the sun, presence of any additional gravitational forces due to the in order to analytically estimate the UEG constant γ. new UEG theory would in general result in two different This would be followed by modeling of the gravitation “equivalent-mass” terms for the two models, leading to in a binary-star system in section IV, based on the UEG apparent inconsistency in the UEG theory. However, the theory. A general treatment applicable for any separa- “equivalent-mass” terms in the two MLR models, based tion between the partner stars, and any level of lumi- on the new UEG theory, would be shown in this pa- nosity and associated UEG mass of each star, would be per to end up with approximately the same functional presented. This would require numerical integration for trend and magnitude, thus resolving the apparent incon- evaluating the gravitational force between the two stars. sistency. This would be a significant result, definitively In addition, a useful limiting situation when the effective validating the UEG theory, as extended to stellar nuclear gravitational force of each star is much larger than its in-

45 N. Das, A UEG Theory of Nature, 2018 4-3 ertial mass, will be presented. The gravitational force in or (me - L) relationship, according to the above equiv- the limiting case can be evaluated using simple analytical alence. The actual mass m (inertial) is expected to be formulas, which may be compared with the results from in general diﬀerent from the mg or me. Therefore, an the general derivation for validation. The results for new actual MLR (mass (actual, inertial)-luminosity relation) MLR deduced from the energy-source and the binary- is also expected to be diﬀerent from (1), as derived and star models would be compared with each other, as well presented in section V. cross-checked with the parameters of the sun in section V, followed by general discussions and conclusions from the study in section V.

II. STELLAR MASS LUMINOSITY RELATION (MLR) BASED ON NEWTONIAN GRAVITATION

The relationship between stellar mass, m, and lumi- III. ENERGY SOURCE MODEL FOR THE nosity, L, as it currently exists to date, is expressed as MASS LUMINOSITY RELATION, USING [4]: CONVENTIONAL GRAVITY AND UEG THEORY

L m 2.3 = 0.23( ) , m < 0.43m0, L0 m0 Let us first consider a derivation based on a conven- m 4 tional Newtonian model, but keeping in mind distinct = ( ) , 0.43m0 < m < 2m0, m0 contributions that may need be modified when the UEG m 3.5 effects are included in a new model. Note that there are = 1.5( ) , 2m0 < m < 20m0, m0 actually two mass parameters that as a product would m contribute to the energy or light output in a star: one = 3200( ), m > 20m0. (1) m0 mass parameter is the source of gravity, and the other is the target mass on which gravity is acting upon. How- where the parameters with a subscript 0 are associated ever, in a Newtonian gravity model, the two gravitational with the sun, which is the only star in our solar system. mass parameters happen to be equal to each other and This existing MLR was deduced from measurements of are equal to the inertial mass. Let us take for granted binary stars based on stellar dynamics using Newtonian that the existing Eddington’s model [4, 7] (1) for the gravitation, and was independently supported by a the- mass-luminosity relation (MLR), which expresses the lu- oretical model of the stellar energy source based on bal- minosity L as a function L(m) of the mass m, is in prin- ancing the pressure of Newtonian gravitation with the ciple correct, assuming only the Newtonian gravity is ap- thermal pressure due to nuclear fusion. According to the plied without any UEG effect. Let us rewrite the function equivalent modelings presented in the following sections, L(m) in the form of its inverse function m(L), the square the mass m in (1) would be equal to the equivalent grav- of which is the mass-square function m2(L). When the itational mass mg = mge in the orbital dynamics model UEG effects are added, let us define an equivalent-mass of section IV, or the equivalent mass me in the energy me, such that the new MLR would be equal to the Ed- source model of section III. Therefore, the solar mass m0 dington’s MLR when its mass term m is replaced by the in (1) refers to the equivalent gravitational mass mg0of me. Accordingly, the Eddington’s mass-square function 2 2 the sun, which is also equal to the equivalent solar mass m (L) is actuality the me(L) function when the UEG 2 2 me0 from the energy source model. It may be noted, the effects are added. Clearly, the me is equal to m when m0 = mg0 = me0 may be different from the actual solar only the Newtonian gravitation is included. Consider the mass (inertial) based on the UEG theory, as discussed in mass-squared function is a product of two mass terms, as the section I. discussed above. One of the mass terms, corresponding to The mg and me would be functions of both the con- the target mass of gravitation would remain unchanged ventional inertial mass m and the luminosity L of a star. with or without the UEG effect. This is because the Each of these equivalent masses is expected to be equal to UEG theory only changes the gravitational acceleration, the inertial mass m, when the UEG effects are excluded in which needs to be multiplied with the same inertial mass the modeling, keeping the existing MLR (1) unchanged in to get the gravitational force, just as in the Newtonian this case. On the other hand, when the UEG effects are case. The source-mass term would be sum of two parts, included, the luminosity derived from the following new one of which represents the Newtonian gravitation which models would be equal to that from the existing MLR remains unchanged as m. Whereas, the second part of (1), if the mass m in the existing MLR is replaced by the source mass is contributed due to the UEG effect, the equivalent mass, mg or me, of the respective models. and is expected to be proportional to the luminosity L The MLR (1) is shown in Fig.6, indicated as the (mg - L) or equivalently to its UEG mass mu.

46 N. Das, A UEG Theory of Nature, 2018 4-4

verified with the analytical results from the approximate reference. 2 0.5 me(L) = m(L)(m(L) + αmu(L)), me = [m(m + αmu)] . 2 2 α 2 me = m + αmmu ' (m + mu) , 2 E (r) = γW (r), α gu τ me ' (m + mu); mu << m. 2 γρvLR 3Gmu 2 Egu(r = 0) = γWτ (r = 0) = c = 2 , me = m(m + αmu) = m(m + mue) = mmge; R γρ R Gm L γL E (r = R) = γW (r = R) = vL = 1.5Gmu , Egu(r = d >> R) = u = γ , mu = . (2) gu τ 2c 2 d2 4πd2c 4πGc R Gm r Egu(r < R) ' u (3 − 1.5( )), (4) The validity of the equivalent model proposed above R2 R may be verified by deriving the average pressure in a star by including the UEG effects in addition to the conven- R tional Newtonian gravitation, and equating it with that R Pu(r) = ρvmEgu(r)dr by including only the conventional Newtonian gravita- r tion [7], when mass m in the later result is substituted ' Gmum [3(1 − ( r ) − 0.75(1 − ( r )2)], with the equivalent mass me, as per the above definition. (4/3πR4) R R This assumes that the luminosity is proportional to the R average pressure, in consistency of the Eddington’s model 1 R Gmum Gmum < Pu >= R Pu(r)dr = 1.20 4 ∼ 4 ; [7]. 0 (4/3)πR (4/3)πR With this goal, let us first derive the average pressure m L ρvm = 3 , ρvL = 3 . (5) due to only the UEG effect. For a typical star like the sun, (4/3)πR (4/3)πR the volume density ρ of luminosity, resulting in the vL Similar steps as above may be used for Newtonian total luminosity L, may be assumed to be uniform. The gravitation to obtain the acceleration function E (r), energy density of radiation produced by a volume element gm pressure function Pm(r), and average pressure < Pm >. of the energy source located at (r0, θ, φ), and observed at (r = z, θ = 0), may be integrated over the entire spherical volume of the star of radius R, to obtain the total energy Gmr Egm(r < R) = 3 , density Wτ (r). Due to spherical symmetry, the energy R R density would be independent of the θ and φ coordinates R Gm2 r 2 Pm(r) = ρvmEgm(r)dr = [0.5(1 − ( ) )], of the observation location, dependent only on its radial r (4/3)πR4 R distance r. R 1 R Gm2 < Pm >= R Pu(r)dr = 4 . (6) 0 4πR R π 2π 02 0 R R R ρvLr sin θdφdθdr Wτ (r) = 2 The average pressure < P > in the presence of both 0 4πc(r2+r0 −2rr0 cos θ) r =0 θ=0 φ=0 the Newtonian and UEG forces would be the sum of the 2 (r+r0) two terms < Pu > and < Pm >. Whereas, the average R 0 2 ρvL R R r 1 0 2 0 0 pressure for conventional Newtonian gravity is < P >. = 4c ( r )( t )dtdr ;(t = r + r − 2rr cos θ) m r0=0 t=(r−r0)2 As prescribed earlier for the equivalent modeling in (2), the equivalent mass m may now be expressed in terms 2 e ρ R 0 (r0+r) ρ 2 (R+r)2 = vL R ( r ) ln dr0 = vL [( R ) ln of the UEG mass mu and the inertial mass m. This may 4c r 0 2 4c 2r 2 r0=0 (r −r) (R−r) be compared with what we expected in (2), from which the equivalent UEG mass mue = αmu, and consequently R 02 ρ 2 2 + R 2r dr0] = vL [R + ( R −r ) ln R+r ]; the parameter α, may be deduced. The α is roughly es- 02 2 2c 2r R−r 0 (r −r ) timated to be 3.0 using analytical integration based on ρ R ρ R the reference approximation of Egu(r) in (4), but is accu- Wτ (r = 0) = vL ,Wτ (r = R) = vL . (3) c 2c rately calculated to be 3.60, using numerical integration The UEG acceleration Egu(r), directed towards the based on the rigorous expressions of Egu(r) and Wτ (r) in center, can now be expressed by multiplying the energy (3,4). density with the UEG constant γ [1], from which the pressure Pu(r) and average pressure < Pu > may be obtained as follows. For an approximate reference and simplicity < Pm>m=me =< P >=< Pm > + < Pu >, 2 2 2 of understanding, the energy density function Wτ (r) may Gme Gm 1.20Gmmu Gm Gmmu 4 = 4 + 4 ∼ 4 + 4 , be roughly approximated over the region r < R by lin- 4πR 4πR (4/3)πR 4πR (4/3)πR 2 ear interpolation of the values at r = 0 and r = R, which me = m(m + 3.60mu) = m(m + mue), α = 3.60. (7) can then be analytically integrated to obtain the pressure function and the average pressure. Numerical integration This calculation for the α is expected to be generally would be needed for accurate calculations, which may be valid for most typical stars, similar to our sun. For very

47 N. Das, A UEG Theory of Nature, 2018 4-5 high luminosity stars, as compared to the sun, the source (∼ 26%) difference from the estimate of α = 3.60 in (7). distribution of the luminosity may not be uniform as as- Further, using the accurate values of mu = 0.46me and sumed in the above analysis, but could be more con- m = 0.54me, the value of the effective mass that would centrated towards the center where the pressure can be be estimated using the estimated value of α = 3.60 in (7), p significantly higher. This may lead to a higher value for is equal to me(estimate) = 0.54(0.54 + 3.60 × 0.46)me = the α in these cases, as per the above modeling. 1.09me, which is within a small (∼ 10%) difference from the actual me. Any such reasonable difference between the estimated and actual magnitudes of the equivalent A. Estimating the UEG Constant from Energy mass may be accommodated (b) by using a more realis- Source Model for a Solar-Mass Star tic energy-source distribution that is different from the ideal uniform distribution assumed in the present model, The sun, which is the only star in our solar system, was or (b) by adjusting the magnitude of the MLR (1) de- used as the reference in the MLR of (1). Note that the duced from the energy source model of [4, 7] (Eddington’s mass of the sun that has been historically deduced from model) to be reasonably different from that from binary- measurement of orbital motions of all planets, in consis- star measurements, both of which we simply presumed to tency with that of our own planet earth, is actually the be identical. However, it is more significant to note the equivalent gravitational mass of the sun. Based on the functional form of the equivalent mass (7), which would UEG theory, introduced in the following section IV, the be shown in the following section IV to compare remark- sun may be considered an isolated body, for evaluation ably with an alternate relationship (19) derived from a of its gravitational force acting on the planets in the so- binary-star model, for large mu/m, that would provide a lar system. This is because the sun is an isolated star, strong validation for the UEG theory. and all the planets in the solar system are considerably Supported by the above estimations from the star- much lighter than the sun. Accordingly, the equivalent luminosity model, and to be supported even further by gravitational mass mg of the sun would be the sum of a UEG model of gravitation in a binary-star system in its inertial mass m and its UEG mass mu (as defined in the following section IV, it is particularly significant to 2). This mg = m + mu for the sun needs to be equated note the following consequence of the above results from to the equivalent mass me used here for energy source the UEG theory. The inertial mass of the sun might not modeling, as well as to the mg in the orbital modeling be what we have been believing [8, 9], estimated based in section IV, for all solar-mass stars, in full consistency on the Newton’s Laws of gravitation and motion [10, 11], with the MLR of (1). using observation of planetary motions, including orbital Applying the above solar condition me = m+mu in (7) motion of the earth around the sun. We now find, as per leads to calculation of the mu from the known value of the UEG theory, that the inertial mass m of the sun could 30 the solar gravitational mass mg = me = 1.989×10 kg [9]. actually be about half (m = 0.54me) of what is calculated This calculated value of mu can then be used in (2) to from the planetary motions based on the Newton’s Laws. estimate the UEG constant γ, given the solar luminosity The approximately other half (mu = 0.46me) is a result 26 L = L0 = 3.828 × 10 W [9]. of the new UEG force due to the sun’s light. In other words, we are being pulled by the light of the sun about as much as by the actual mass of the sun! This would 2 2 2 (m + mu) = m + mu + 2mmu = m(m + mue) be a significant discovery, where the new UEG theory 2 = m + 3.60mmu, mu = 1.60m, is shown to directly influence the gravitation in the solar system, and thus our common understanding of the basic m = m + m = (1 + 1/1.60)m , e u u nature of gravity that controls our own motion around mu = (1.60/2.60)me = 0.615me, m = 0.385me, the sun. γ = mu4πGc/L = 0.615me4πGc/L = 0.804 × 103 (m/s2)/(J/m3). (8) IV. UEG MODEL FOR EFFECTIVE This estimate compares reasonably close to an accu- GRAVITATIONAL MASS OF A STAR IN A rate calculation of γ = 0.6 × 103 (m/s2)/(J/m3) from a BINARY SYSTEM particle physics model [1], with a difference of about 33%. Considering that the present estimation is based on some A. Estimation of Distance Between Partner Stars simple assumptions and formulations of the UEG effects in an Observed Eclipsing Binary System on star luminosity, the above result is a reasonable support for the value of the UEG constant as well as for the The stellar mass-luminosity relation (MLR) is based UEG theory. With the accurate value of γ = 0.6 × 103 on measurement survey of mostly eclipsing binary stars, (m/s2)/(J/m3) from the particle physics model, the dif- where the partner stars are closely spaced from each ferent mass parameters of (8) may be back-calculated as other. Accordingly, in order to evaluate the MLR in rela- mu = 0.46me, m = 0.54me, mu = 0.85m, mue = 2.85mu, tion to the new UEG theory, it would be useful to analyt- with α = mue/mu = 2.85, which is within a reasonable ically estimate a typical or median value for the surface-

48 N. Das, A UEG Theory of Nature, 2018 4-6

FIG. 1. FIG. 2. to-surface distance δ between the two partner stars. For The magnitude of the attraction upon an elemental simplicity the two stars may be assumed to be identical, mass δm, at a given location of the target star, is to each of radius R. be determined by the energy density Wτ reaching at the As shown in Fig.1, the critical angle θ = θc between particular location due to radiation from the source star the orbital axis and the direction of observation, larger (presuming the target star was removed, or is transparent than which eclipsing would not be observed, may be ex- to the source-star’s radiation). When the CG approaches pressed in terms of the surface-to surface distance δ and the center of the target star (when mg → ∞, mg >> m), the star radius R. Assuming that orientation angle θ is we expect the total UEG force upon the target star to equally likely between zero and π/2, stars with smaller approach zero. In this limit, a mass-less source with any θc (which is equivalent to a larger δ/R ratio), are less non-zero luminosity (finite or infinite) can not exert a likely to be observed in a survey. A median probability non-zero force or acceleration upon a target of infinite P > 0.5 of observation would provide a useful estimate for mass. This condition may be empirically enforced by the separation factor δ/R in an observed eclipsing binary by redistribution of the energy-density function Wτ (r, θφ system into a new, effective energy-density function Wτe(r, θ, φ) with appropriate symmetry. For a spherically symmet- P (0 < θ < θ ) = 2θc , sin θ = 2R = D = 1 , ric target, this may be accomplished by simply using a c π c 2R+δ d δ 1+ 2R constant Wτe, which is an average of the Wτ (r, θ, φ) over δ the entire target sphere. A less restrictive, general ap- 2R = (csc θc − 1); proach would be to define the effective energy density π π θc ≥ 4 ,P (0 < θ < θc) > 0.5, δ ≤ 2R(csc 4 − 1) ' 0.8R, at a particular location as the average of the Wτ (r, θ, φ) δ = δm ' 0.4R. (9) over all θ and φ, for the same radial distance r of the given location. For further generality, we will choose an 0 effective energy density Wτe(r, θ ) which is cylindrically symmetric about the orbital axis (see Fig.2), calculated 0 0 0 B. General Calculation of the UEG Force as a by taking an average of the Wτ (r, θ , φ ) over all φ . Function of the Equivalent Gravitational Mass

0 0 0 0 −1 The additional gravitational force of attraction, as per Wτe(r, θ, φ) = Wτe(r , θ ), r = r, θ = cos (sin θ cos φ), the UEG theory, produced due to the light radiation from 2π 0 0 1 R 0 0 0 0 one star (source star) acting upon the other star (tar- Wτe(r , θ ) = 2π Wτ (r , θ , φ )dφ get star) in a binary system, would be directed towards φ0=0 a suitable center of gravity (CG). The distance δr (see 2π 1 R L Fig.2) of this CG from the center of the target star is = 2π 2 . (11) 0 4πc(r0 +d2+2r0d sin θ0 cos φ0) determined by the eﬀective gravitational mass mg of the φ =0 target star and the inertial mass m of the source star. The gravitational acceleration Egu due to the UEG effect may now be expressed using the Wτe and the UEG m = δr = δr ; m >> m, δr << R, d . (10) mg 2R+δ d g constant γ.

49 N. Das, A UEG Theory of Nature, 2018 4-7

Egu(r, θ, φ) = −rˆcEgu(r, θ, φ) = −rˆcγWτe(r, θ, φ) 2π = −rˆ 1 R Gmu dφ0, c 2π 2 2 0p 2 2 φ0=0 (r +d +2rd cos φ 1−sin θcos φ) γL mu = 4πGc . (12)

The total gravitational force F u towards the source star can be calculated by multiplying the −zˆ component of the acceleration Egu with mass density ρvm, and integrating over the target sphere. An effective gravitational mass mue associated with the UEG force F u may be defined, and related to the source star luminosity L or its associated UEG mass mu. The mu is the equivalent gravitational mass of the source star in the limit of operating in isolation, when the mg of the target star approaches zero or the distance δ approaches −d/2, which means cg FIG. 3. the effective CG used in the evaluation of the UEG force is at the center of the source star. The general formulas derived above can be simplified for this limiting case. For further simplicity of calcu- Gm m F u = −zFˆ u = −zˆ ue , lation, we would assume the energy density W to be d2 τ approximately uniform, equal to that at the center of R π 2π R R R 2 Fu = (−Egu(r, θ, φ) · zˆ)ρvmr sin θdφdθdr the target star. Under this assumption, the gravitational r=0 θ=0 φ=0 force due to a maximal spherical region around the center R π 2π of gravity with radius R−δr (spherical region with dashed R R R r cos θ+δr = [E (r, θ, φ) ρvm boundary, see Fig.3) would result in zero total force due gu p 2 2 r=0 θ=0 φ=0 r +δr +2rδr cos θ to symmetrical cancellation of contributions from its ele- 2 m mental parts. Contributions from only the thin boundary × r sin θdφdθdr], ρvm = . (13) (4/3)πR3 shell (see Fig.3) with thickness δR = δr(1 + cos θ) would be needed to calculate the total force. The expression (14) may be simplified under the above approximations, requiring integration only over the shell region of variable 2 2 R π 2π mue = Fud = d R R R [ 1 thickness δR = δr(1+cos θ), at r ' R. The only non-trivial mu Gmmu 3 2π (4/3)πR r=0 θ=0 φ=0 integration in (14) over θ may be analytically evaluated. 2π The resulting simple analytical expression for the limit- × R 1 dφ0] ing case would be useful for conceptual understanding, as 2 2 0p 2 2 φ0=0 (r +d +2rd cos φ 1−sin θcos φ) well as for approximate validation of the general results r cos θ+δr 2 for large mg. × p r sin θdφdθdr. (14) r2+δr2+2rδr cos θ π mue ' 3 R ( dR ) cos θ sin θdθ mu 2 R θ=0 C. UEG Force of Attraction for a Large Equivalent π = 3 R ( δr )(1 + cos θ) cos θ sin θdθ = δr Gravitational Mass 2 R R θ=0 = m ( d ), m = mum ( d ); δr, δR << d, R . (16) It may not be possible to evaluate the above general mg R ue mg R formulas analytically, requiring numerical integration for For a median distance δr= 0.4R, estimated for eclipsing any general values of the source mass m and the target close binaries in section IV A, the above limiting expres- equivalent gravitational mass mg. However, when the sion for the mue may be written as, mg is very large compared to the m, the CG would be close to the target star center, with the distance δr much smaller than the target radius R. m = mum ( δ+2R ) = 2.4 mum . ue mg R mg (17) Based on the general derivation of section IV B, the m = δr = δr ; δr << R, d, m >> m. (15) mg 2R+δ d g ratio mue/mu was computed using numerical integration

50 N. Das, A UEG Theory of Nature, 2018 4-8

FIG. 4. for different distances δcg = d/2−δr of the CG (as seen by established by equating mg2 = mge2 = mue2 + m2 and the source star) from the mid point between the source mg1 = mge1 = mue1 + m1. The process would be simpler and target stars, for a fixed value of the surface-to-surface when the two partner stars are identical, in which case distance δ between the two stars. The computed ratio the results of Fig.4 may be used for a final solution of mue/m are plotted in Fig.4 as a function of the normal- the mue of each star, by enforcing the additional condi- ized distance δcg/R, for selected values of the normalized tion mg = mge = mue + m for mutual balance. For this parameter δ/R. The computed results in the Fig.4 are condition, δr/R = m/mg = 1/(mue/m + 1) and therefore compared with the limiting values of the mue/mu, ana- δcg/d would be directly dependent on the ratio mue/m, lytically derived above in section IV C. The general and for a given parameter δ/R. Accordingly, for a given δ/R, the limiting plots in the Fig.4 are shown to validate each the Fig.4 essentially provides mue/mu as a function of other when δr approaches small values (δcg/d approaches mue/m, from which mue/m or mge/m = mue/m + 1 can 0.5), or equivalently when the target-star’s gravitational be deduced for different values of mu/m. These results mass mg is much larger than the source-star’s inertial are shown in Fig.5, for selected values of δ/R. These re- mass m, as expected. The effective UEG mass mue of sults are verified with those similarly deduced from the the source star approaches its maximum value mu when limiting expressions of mue/mu in (16) applicable for suf- the CG moves to the center of the source star, or δcg/d ficiently large mue/m, as follows. approaches −0.5, as expected. The mue starts to drop The limiting expression for the mue in (16) is approxi- significantly lower than the mu after the CG moves be- mately equal to the mge = mue+m because mue/m >> 1, yond the mid point between the source and target stars or mue >> m. Accordingly, (16) may be solved for the (δcg > 0), closer toward the target star. Note that the symmetry condition mue ' mge = mg, as prescribed ear- total effective gravitational mass mge of the source star lier for a binary star with two identical partners. is the sum of the inertial mass m and the effective UEG mass m . ue q m = m + m ' d m m , In a binary system, the same normalized plots of the g ue R u m q Fig.4 would be applicable to model force from the first g = mue + 1 ' dmu . (18) upon the second star, as well as from the second upon m m Rm the first. In the former case, the m , m , m and m ue u ge For the estimated δ = 0.4R in (9,17), we get, would be associated with the first star (referred to with a subscript 1), but the mg would be associated with the second star (referred to with a subscript 2), where as for √ mg = mue + m ' 2.4mum , the later case the associations would be reversed. The q mg mue 2.4mu final solutions for the mue1, mge1, mue2, mge2 may be m = m + 1 ' m , δ = δm = 0.4R. (19)

51 N. Das, A UEG Theory of Nature, 2018 4-9

FIG. 5.

Note that the above limiting relation of (19) is in similar the G, and is different for two specific bodies dependent form as derived from the energy source model in section on their individual radiation, inertial mass and separa- III, (7), for large mu/m. This is a strong correlation tion distance. This is presented in Appendix A, drawing between the two models, indicating a strong validation particular attention to interesting special conditions that of the associated UEG theory. The factor d/R in (18) may arise in section A 2. is functionally equivalent to the factor α deduced in (7). The expected value of α = 2.85 in (7) (see section III A) is somewhat larger than the estimated factor (d/R) ' 2.4 in V. DERIVATION AND VALIDATION OF THE (19). Note that the factor (d/R) was estimated in section NEW MLR IV A, (9), based on a visual condition of orbital eclipsing where the R is the visual radius of a star, whereas the re- In accordance with our equivalence modeling, the lation (18) was deduced from orbital dynamics where the mass-term in the MLR of (1) is actually equal to the R is the core radius of the star. Accordingly, the actual m for the orbital model of section IV, or the m for the (d/R) for use in (18,19) should be somewhat larger than g e energy-source model of section III. And, the luminosity the estimate of 2.4 from (9), which would be consistent L in the MLR (1) is proportional to the m as defined in with the α = 2.85 expected from the energy source model. u (2). In other words, the MLR of (1) actually provides the Further, the above expected value of α = 2.85 from sec- relationship between (m , m ) and m . This data, to- tion III A is valid only for a typical star like sun, with g e u gether with the theoretical relationship of Fig.5 between variations around this value for high and low intensity m /m = m /m − 1 and m /m, may be used to deduce stars, which may be expected to track similar variations ue g u a new MLR (mass(actual inertial)-luminosity relation), of the (d/R) factor, leading to the expected confirmation based on the orbital model of section IV, as plotted in between the two models. Fig.6 for d/R = 2.85. It may be noted, for a general case of gravitation be- Similarly, an alternate new MLR may be derived, us- tween two different bodies, and therefore for the special ing the theoretical relationship (7) between the equiva- case of two identical bodies as well, the basic theory of lent mass me and mu, based on the energy-source model Newton’s universal gravitation may need to be reviewed of section III, and the (mg, me) to mu relationship of the and revised in consistency with Newton’s laws of mechan- MLR (1). This new MLR is plotted in Fig.6 for the pa- ics, when one or both of the bodies are radiating. The rameter α = 2.85, for comparison with its alternate MLR UEG effects due to radiation as modeled above may be derived from the orbital model. As discussed earlier in represented in terms of a revised equivalent gravitational section IV, the parameters α and (d/R) from the two constant Gu, substituting for the universal constant G for models are functionally equivalent, and are estimated to non-radiating bodies. The Gu is in general different from have comparable values. The two new MLR’s from the

52 N. Das, A UEG Theory of Nature, 2018 4-10

FIG. 6. two independent models are seen to closely follow each The parameter α is borrowed from the energy source other, but somewhat deviate in the region of solar-mass model of (7), which is equivalent to the factor (d/R) from stars. Such agreement between the functional trends of the orbital model of (18), and is expected to be about 2.85 the two models validate the two models as well as the for solar-mass stars with possible variation toward larger associated UEG theory, over the entire range of low to values for more luminous stars, as discussed in the end high-luminosity stars. Note that the actual inertial mass of section IV. The parameters m0 = me0 = mg0, L0 and deduced in a new MLR in Fig.6 is significantly less than mu0 refer to the solar mass (equivalent gravitational, not what was believed (mg or me) based on the existing MLR inertial), luminosity and UEG mass, respectively, and the of (1) [4, 7]. Interestingly, for the mid-region of the MLR factor 0.46 represents the expected ratio mu0/me0. of Fig.6, the inertial mass m reduces for increasing luminosity, which may appear counter-intuitive based on a energy-source model [7] using Newtonian gravitation. VI. DISCUSSION AND CONCLUSIONS The significantly different trends of the new MLR, as compared to the existing MLR (1), may prompt review The UEG theory applied for stellar dynamics in a bi- of existing models of stellar evolution [12], which is be- nary star system is found to be consistent with that for yond the scope of the present work. stellar energy-source model, as per comparison of the re- The new MLR may be expressed in approximated an- sulting mass-luminosity relation (MLR) with the existing alytical forms for high luminosity, using the relation be- MLR [3, 4, 7] from orbital measurement of binary stars tween the normalized variables (mg/m0, me/m0) and as well as from the Eddington model of stellar energy- L/L0 = mu/mu0 from (1) and the limiting relationship source. The “mass” in the existing MLR was assumed to of (7,19) similarly normalized. For low luminosity with be the inertial mass of a star which is equal to the gravi- negligible mu, the new MLR would remain approximately tational mass as per the conventional Newtonian gravity, unchanged from the existing MLR (1). but it needs to be modified as per the UEG theory in different manners for the binary-star dynamics and for the stellar energy-soure model. However, the associated L m 2.3 ' 0.23( ) , m < 0.43m , “equivalent-mass” parameters in the two cases happen to L m 0 0 0 exhibit the same functional trend, with respect to actual 2.0 m −( 3.5 ) L inertial mass and luminosity (or the UEG mass) of the ' (1.5)−( 1.5 )(0.46α ) 1.5 , 16 < < 64000, m0 L0 star. Accordingly, the existing MLR derived from the m L binary-star measurement and the Eddington’s energy- ' (3200)2(0.46α ), > 64000. (20) m0 L0 source model were seen to agree with each other. This

53 N. Das, A UEG Theory of Nature, 2018 4-11 coincidence historically removed any doubt about the va- is a significant development, revising the Newton’s law of lidity of the existing theories of [3, 4, 7]. We now know universal gravitation and Einstein’s principle of equiva- that mass term in the existing MLR is not really the ac- lence, extended to gravitation between general radiating tual mass (inertial) as has been assumed for long. The bodies. new MLR between the star’s actual inertial mass and the Starting with the remarkable success of the UEG the- light output is derived and plotted using the new UEG ory in particle physics [1, 2], the further validation of theory. As per the new theory, the inertial mass of the the UEG theory in the present work of stellar orbital sun or any other solar-mass star is actually about half and energy-source physics to model the MLR, should of what has been believed all along! The sun’s gravita- now establish significant confidence in the theory, unify- tional force acting upon us on the earth is contributed ing its application in the small (elementary particles) as in approximately equal parts by its inertial mass as by well as large (solar system and binary stars) dimensions, the UEG force due to its luminosity. This is a signifi- and spherically symmetric (elementary particle and an cant new finding, fundamentally changing our common isolated single star) as well as asymmetric (binary star) understanding of the basic nature of gravity. structures. The effective gravitational mass of an individual star in a binary system, consisting of two stars of comparable mass and light intensity, would in general be different from that when the star operates as a single isolated Appendix A: UEG Theory of Gravitation for star applying gravitational force on a nearby planet of General Radiating Bodies, and Conservation of significantly smaller mass, when the additional UEG ef- Momentum and Energy fects are included. However, for observed binary systems with approximately two solar-mass stars, the equivalent gravitational mass in the binary-star system happened to The UEG theory of gravitation in a binary system, as be approximately equal to that of an isolated solar-mass modeled in this paper, expresses the gravitational accel- star (equivalent gravitational mass of sun in our solar sys- erations in terms of new effective gravitational masses tem), as well as equal to the equivalent mass needed in mg1 = m1 + mue1 and mg2 = m2 + mue2 of the two an energy-source model to produce its light output close bodies in the system. This may lead to review of differ- to the solar luminosity. This coincidence of the data for ent concepts of mass, which may determine not only the the solar-mass stars in the existing MLR with the ob- gravitational field produced by a body, but also the me- served gravitation of the sun in our solar system, also chanics of the body’s motion in terms of its inertial mass, helped in removing any further doubt in the validity of momentum and energy. In the process, we may need to the Newtonian gravitation. This, together with the other identify the mass of a given body in distinct forms, which coincidence of the two MLRs mentioned earlier, may be may or may not be equal under general conditions of a considered interesting “conspiracies of nature” in the his- radiating body. tory of science, which possibly allowed to hide the new The gravitational mass of a body, which acts like a UEG theory of gravitation without suspicion for so long, source or cause of the gravitational field the body pro- till now! duces in the surrounding medium, may be referred to as Further, Einstein’s equivalence principle in the general the source gravitational mass, mg. As per the UEG the- theory of relativity [6], together with Newton’s third law ory of binary stars, the mg = m + mue is the source grav- of action-reaction equality [10], can be shown to require itational mass of a radiating body, which is clearly dif- the gravitational mass of a source body to be equal to its ferent from that, mg = m, without the radiation. As per inertial mass, with the gravitational constant G assumed conventional theory of gravitation and mechanics (New- be universally applicable to all bodies and locations. This tonian or relativistic), in the absence of any radiation, fundamental requirement seemed to rule out possibility a unique mass parameter m defines not only the body’s gravitational field but also its inertial as well as internal of the source gravitational mass of a radiating body to 2 be any different from its inertial mass, also contributing energy (E = mc ). In the presence of radiation, we need to hiding the possibility of an additional UEG force for to review if the change in the body’s source gravitational radiating bodies, without suspicion, as discussed above. mass mg = m + mue 6= m may also change the body’s However, all issues are shown in appendix A to be re- inertia or energy. If any such change may violate certain solved, in full consistency with fundamental mechanical fundamental principles, the UEG theory may have to be principles, if an equivalent gravitational constant Gu is properly revised. allowed to be in general different for different pairs of As per the UEG model, the mue is the additional gravitating bodies accounting for any additional UEG source gravitational mass due to radiation from a given force due to radiation. This may limit the scope of the body, as seen by the other body in a particular binary 0 0 equivalence principle of the general relativity [6] to grav- system. The total gravitational acceleration, a12 or a21, itation only between non-radiating bodies in a strictly produced as per the new UEG theory by one body, 1 or “free-space” medium with no radiation-energy content, 2, and experienced at the location of the other body, 2 which the principle was fundamentally intended to. This and 1, respectively, is expressed as (see Fig.7):

54 N. Das, A UEG Theory of Nature, 2018 4-12

0 0 The accelerations a21 and a12 experienced by the bod- 0 0 ies 1 and 2, and the respective velocities v1 and v2 along the individual circular orbits, may be expressed in terms of their common angular velocity ω and the radial dis- 0 0 0 0 tances r1 and r2. The v1 and v2 are directed opposite with respect to each other.

0 2 0 0 2 0 a12 = ω r2, a21 = ω r1, G(m +m ) ω2(r0 + r0 ) = ω2r = a0 + a0 = g1 g2 , 1 2 12 21 r2 r G(m +m ) ω = g1 g2 , r3 m a0 Gm m P = m v0 = m ωr0 = g2 12 = g1 g2 , 2 g2 2 g2 2 ω r2ω m a0 Gm m P = m v0 = m ωr0 = g1 21 = g1 g2 , 1 g1 1 g1 1 ω r2ω P = P , P = m v0 , P = m v0 , FIG. 7. 1 2 1 g1 1 2 g2 2 P 1 = −P 2, P = P 1 + P 2 = 0 . (A3)

It is shown that total momentum P = P 1 + P 2 of the binary system would be conserved as zero, if the mo- 0 Gmg1 0 Gmg2 mentum P i, i = 1, 2, of an individual body is defined as a12 = 2 , a21 = 2 , r r P i = mgivi, not P i = mivi as conventionally defined in mg1 = m1 + mue1, mg2 = m2 + mue2. (A1) Newtonian mechanics. In other words, the inertial mass mI of a radiating body may no longer be equal to its Useful limiting conditions may be recognized. The mue conventional inertial mass m, but could now be equal to of a radiating body would be less than or equal to its max- its source gravitational mass mg. Accordingly, the UEG imum value mu, which would occur when the radiating theory would not only change the source gravitational body operates effectively in isolation, as the most dom- mass mg, but also the inertial mass mI of a radiating inant body with negligible or no gravitational influence body to be equal to mI = mg = m + mue. The inertial from any surrounding body. In the other limit, mue of mass mI would be equal to the conventional inertial mass the radiating body would be zero, and consequently the m, only when there is no radiation (mu → 0). source gravitational mass of the body would be equal to its Newtonian mass (mg = m + mue = m), when it operates in the presence of a nearby dominant body, with the mI1 = mg1 = mue1 + m1 6= m1; Newtonian mass of the source body much smaller than mI1 = m1, mue1 = mu1 → 0. that of the dominant body. Further, when the two bod- mI2 = mg2 = mue2 + m2 6= m2; ies are sufficiently far apart from each other, the m of ue m = m , m = m → 0. (A4) each body would be approximately equal to their respec- I2 2 ue2 u2 tive maximum values, mu. To be consistent with the conserved momentum, the gravitational force may also have to be defined as the product of the corresponding gravitational acceleration mue2 → 0 (mu2 6= 0), mg2 → m2, mue1 → mu1, and the new inertial mass mI , not the conventional mass m → (m + m ); m >> m . m. With this new definition of the gravitational force, g1 1 u1 1 2 0 the vector force, F , produced by one body acting upon m → m , m → m ; r → ∞. (A2) 12 ue1 u1 ue2 u2 the other would be equal in magnitude, but oppositely 0 directed, to that, F , when the source and target bodies The above accelerations (A1) may be expressed in spe- 21 are interchanged. This would satisfy the Newton’s third cial coordinates (primed) (see Fig.7), which may be re- law of action and reaction, resulting in the total force in ferred to as the UEG coordinates of the particular bi- the system equal to zero, as should be expected. nary system, with reference origin located between the 0 0 two bodies, at distances r1 and r2 from the bodies 1 and 2, respectively. This would be different from the coor- 0 0 Gmg1mg2 F = mg2a = , dinates (un-primed) used in a conventional modeling of 12 12 r2 the binary system using Newtonian gravity, where the 0 0 Gmg2mg1 F = mg1a = , respective distances of the reference origin would have 21 21 r2 0 0 0 0 0 0 been r1 = r × m2/(m1 + m2) and r2 = r × m1/(m1 + m2). F12 = F12, F 12 = −F 21, F 12 + F 21 = 0. (A5)

55 N. Das, A UEG Theory of Nature, 2018 4-13

Clearly, the Newton’s third law would not work if the of the two pairings. Clearly, such non-unique values of mass of the target body were its conventional mass m, the momentum and energy for the same particular body for radiating bodies with m 6= mg in general, resulting in would not be fundamentally sensible, warranting suitable a total non-zero force for the total system, which would revision of the UEG model in order to reestablish order also violate the principle of conservation of momentum. and consistency. In order that the momentum or energy of a given body be uniquely defined and conserved, they must be propor- Gm m Gm m 0 0 g1 2 0 0 g2 1 tional to the body’s conventional mass m which remains F12 = m2a12 = 2 ,F21 = m1a21 = 2 , r r unique under general conditions. This would be the case, 0 0 0 0 0 0 F12 6= F12 F 12 6= −F 21, F 12 + F 21 6= 0. (A6) if we require the gravitational force between two bodies to be proportional to the mass m of each body. We would The consistency of momentum and its conservation 0 0 0 maintain the same relative acceleration a = a = a12+a21 may be extended to the kinetic energy of the body, so between the two bodies in a binary system, defined in a that the energy may also be conserved as would be de- new coordinate system (unprimed), such that the angular sired. Accordingly, the kinetic energy may also need speed ω remains unchanged. In other words, the results to use the new inertial mass mI . That is, the ki- of orbital periodicity from the original UEG model still netic energy KE (non-relativistic) would be equal to remain valid through the following proposed revision. 2 2 2 KE = (1/2)mI v = (1/2)(m + mue) 6= (1/2)mv . This may lead to a fundamental dilemma. Extending the treatment of energy to special relativity, this may lead to 0 0 0 G(mg1+mg2) 2 a = a12 + a21 = 2 , a fictitious rest energy of E0 = mI c , which is different r 2 Gum1m2 from the actual rest energy E0 = mc of the body when F12 = 2 = F21, the radiation is turned off. This is contrary to the un- r F G m F G m a = 12 = u 1 , a = 21 = u 2 , derstanding, that the intrinsic energy of the body should 12 m2 r2 21 m1 r2 not be different, if the radiation is suddenly turned on G (m +m ) a = a + a = u 1 2 , (A7) or off. However, the above new definitions for the gravi- 12 21 r2 tational mass, inertial mass, momentum, kinetic energy, as well as the rest mass (though appears non-physical), could still be mathematically consistent with each other, 0 as shown in the above equations, in reference to the local ω = a = a , a = a0, UEG reference coordinates (primed coordinates). The r2 r2 above dilemma of rest energy may perhaps be ignored, mg1+mg2 m1+mue1+m2+mue2 Gu = G m +m = G m +m . (A8) by considering the rest mass simply as a reference value, 1 2 1 2 and any difference between the total and the reference The new model would satisfy all the Newton’s law’s rest energy may still be consistently used for modeling of motion, as well as conservation of momentum and and “book-keeping” of the kinetic energy. energy, with reference origin for orbital motion of the two stars same as that from Newtonian gravitation (r1 = r×m /(m +m ), r = r×m /(m +m )). However, each 1. Revised UEG model for General Radiating 2 1 2 2 1 1 2 Bodies pair of bodies would now be associated with a different equivalent gravitational constant Gu, which is different from the Newtonian gravitational constant G. This is a The above modeling of gravitation and inertia may be significant new development, which may warrant review consistently used, as discussed above, but only in a hypo- of orbital motions of two- and general multi-body sys- thetical situation of having the only two bodies in an ideal tems, which may include radiating (stellar) and/or non- empty space, in complete isolation from any other bodies. radiating (dark star or planets) elements. The model may be untenable in a real physical situation with other surrounding bodies. If the above model is followed for a general multi-body system, each pairing of the multi-body system would be associated with two inertial 2. Revised UEG Model for Gravitation in a Binary System Under Special Conditions masses for the two individual members of the pairing, but any particular body would in general carry a different inertial mass when it is associated with a different pairing. We may evaluate the equivalent gravitational constant Consider an arbitrary three-body system, where any two Gu, under useful special conditions: pairings of the system would share a common member. The common member would be associated with a differ- Case I: For the ideal case of a binary system with equal ent inertial mass, and therefore different momentum and partner stars, the magnitudes of all the accelerations a12, 0 0 energy for a given velocity, in modeling the gravitational a21, a12 and a21, in the primed as well the unprimed force it experiences from the other two different bodies coordinates would be equal.

56 N. Das, A UEG Theory of Nature, 2018 4-14

darker body, would be normally assumed (incorrectly) to be associated with the more luminous body. Although we m+mue Gu = G m , mue1 = mue2 = mue, m1 = m2 = m, establish only the limiting condition, as stated above (see 0 0 (A12)), similar inversion conditions would apply also for a12 = a21 = a = a 12 21 any general unequal binary-star systems, where the more Gm G(m+m ) = g = ue = Gum . (A9) luminous star has lower mass. In light of this signiﬁcant r2 r2 r2 new result, some old controversies in astronomy such as Case II: For a binary star system where the individual the Algol Paradox [13, 14] in close binary systems and stars are suﬃciently far apart, using the limiting condi- the mystery of the Sirus star system [15, 16] may need tion of (A2) in (A7,A8), we have, to be reevaluated.

m +m +m +m Gu = G 1 u1 2 u2 , Gm m1+m2 G ' ue1 , m >> m >> m , m = 0, u m2 ue1 2 1 ue2 m = m , m = m , r → ∞. ue1 u1 ue2 u2 (A10) G m Gm Gm a = u 2 ' ue1 >> 2 , 21 r2 r2 r2 Case III: In the solar system, the gravitational accel- Gum1 Gmue1m1 Gmue1 eration between the sun (body 1) and any of its plan- a12 = 2 ' 2 << 2 = a21. (A12) r m2r r ets (body 2) may be modeled by considering the sun the most dominant body, as per the limiting condition (A2). In this case, the Newtonian gravitational con- Case V: Consider gravitation between a dominant mas- stant G may be substituted by an effective gravitational sive body (body 1) with no or very little radiation, and constant Gu, which is larger than the G by the factor a radiating body (body 2) of relatively small inertial or UEG mass compared to the body 1. This situation would mg1/m1 = (m1 + mu1)/m1. be applicable to a planet like our earth, with a satellite or moon which may be naturally or artificially lighted. m1+mu1 mg1 Gu ' G m = G m , mue1 = mu1, m1 >> m2, The gravitational force and acceleration in this case may 1 1 be adequately modeled by using the conventional New- (m + m ) = m >> m = (m + m ), 1 ue1 g1 g2 2 ue2 tonian gravitation, with Gu ' G. This is independent of G m Gm m Gm a ' u 2 = g1 2 6= 2 , the light radiation from the satellite or moon, even if its 21 r2 m r2 r2 1 UEG mass mu2 or mue2 is significant compared to its Gum1 Gmg1 inertial mass m2, or equivalently its source gravitational a12 ' 2 = 2 . (A11) r r mass mg2 = m2 + mue2 is significantly larger than its inertial mass m , as long as the m , m , m and m Case IV: In contrast to the case III, the gravitation be- 2 u2 ue2 2 g2 are sufficiently smaller compared to the mass m of the tween a dominant luminous star (body 1) and a dominant 1 dominant body. In other words, the source gravitational massive but dark star (dark body 2) with little or no ra- mass m , which should have proportionately increased diation, where m << m , m >> m , m , would lead g2 1 2 ue1 1 2 the gravitational force as per the unrevised UEG model in to an interesting situation. The dark star would exert (A5), would no longer determine the gravitational force much more acceleration on the luminous star, compared in the revised model of (A7,A8), under the present condi- to that expected from its mass alone as per Newtonian tions. Instead, the gravitation is modeled by the simple gravity. It is as if the dark star has “acquired” the UEG Newtonian gravitation, with the force proportional to the mass m of the luminous star. Whereas, the luminous ue1 inertial mass m not the source gravitational mass m . star would exert much less acceleration on the dark body, 2 g2 compared to that expected from its large effective UEG mass mue1. This concept may be referred to as “inversion.” This results in complete opposite effect from what Gu ' G, m1 >> m2, mu2, mue2; would be expected from a conventional stellar model, m = m << m . (A13) where a larger mass (source gravitational), and there- u1 ue1 1 fore a larger gravitational acceleration exerted upon the

[1] N. Das, “A New Unified Electro-Gravity (UEG) The- ory of Nature,” (2018). ory of the Electron,” Paper #1, pp.4-13, in “A Unified [3] G. P. Kuiper, Astrophysical Journal 88, 472 (1938). Electro-Gravity (UEG) Theory of Nature,” (2018). [4] Wikipedia, “Mass-Luminosity Relation,” http://en. [2] N. Das, “A Generalized Unified Electro-Gravity (UEG) wikipedia.org/wiki/Mass-luminosity_relation, Re- Model Applicable to All Elementary Particles,” Paper trieved (2017). #2, pp.14-30, in “A Unified Electro-Gravity (UEG) The- [5] O. Y. Malkov, Astronomy and Astrophysics 402, 1055

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(2003). [11] S. I. Newton and S. Hawking, Principia (Running Press, [6] A. Einstein, Annalen der Physik 354, 769 (1916). 2005). [7] S. Lecchini, How Dwarfs Became Giants (Bern Studies in [12] M.Salaris and S. Cassisi, Evolution of Stars and Stellar the History and Philosophy of Science [c/o Institut fur Populations, Vol. 88 (John Wiley and Sons, 2005). Philosophie, G Grasshoﬀ], 2007). [13] Wikipedia, “Algol Paradox,” http://en.wikipedia. [8] Wikipedia, “Sun,” http://en.wikipedia.org/wiki/ org/wiki/Algol_paradox, Retrieved (2017). Sun, Retrieved (2017). [14] I. Pustylnik, Astronomical and Astrophysical Transac- [9] D. R. Williams, “Sun Fact Sheet. NASA Gooddard tions 15, 357 (1998). Space Flight Center,” http://nssdc.gsfc.nasa.gov/ [15] D. Benest and J. L. Duvent, Astronomy and Astrophysics planetary/factsheet/sunfact.html, Retrieved (2017). 299, 621 (1995). [10] S. I. Newton, Principia: Mathematical Principles of Nat- [16] Wikipedia, “The Sirius Mystery,” http://en. ural Philosophy. I. B. Cohen, A. Whitman and J. Bu- wikipedia.org/wiki/The_Sirius_Mystery, Retrieved denz, English Translators from 1726 Original (University (2013). of California Press, 1999).

58 5-1

Uniﬁed Electro-Gravity (UEG) Theory Applied to Spiral Galaxies

Nirod K. Das Department of Electrical and Computer Engineering, Tandon School of Engineering, New York University, 5 Metrotech Center, Brooklyn, NY 11201 (Dated: May 9, 2018) The unified electro-gravity (UEG) theory, which has been successfully used for modeling elementary particles, as well as single and binary stars, is extended in this paper to model gravitation in spiral galaxies. A new UEG model would explain the “flat rotation curves” commonly observed in the spiral galaxies. The UEG theory is developed in a fundamentally different manner for a spiral galaxy, as compared to prior applications of the UEG theory to the elementary particle and single stars. This is because the spiral galaxy, unlike the elementary particles or single stars, is not spherically symmetric. The UEG constant γ, required in the new model to support the galaxies’ flat rotation speeds, is estimated using measured data from a galaxy survey, as well as for a selected galaxy for illustration. The estimates are compared with the γ derived from a UEG model of elementary particles. The UEG model for the galaxy is shown to explain the empirical Tuly-Fisher Relationship (TFR), is consistent with the Modified Newtonian Dynamics (MOND), and is also independently supported by measured trends of galaxy thickness with surface brightness and rotation speed.

I. INTRODUCTION based on the physical asymmetry of the spiral galaxy. The UEG field may be defined in proportion to the redistributed, effective energy density, so that the field may Rotation curves of spiral galaxies [1] have been sus- satisfy certain basic requirements and be self-consistent pected not to confirm to gravitational forces due to galax- when applied to general problems. ies’ visible mass as per the Newton’s law of gravitation, which is known to work well in our day-to-day experi- The required UEG constant γ of proportionality, be- ence on earth as well for planetary orbits in our solar tween the UEG field and the associated effective energy system. In order to explain the observed rotation curves, density, may be deduced from the new UEG model using it has been proposed and long believed that there is sig- measured data from galaxy survey as well as data for se- nificant amount of invisible “dark matter” surrounding lected individual galaxies. The results may be compared almost all spiral galaxies. There was no other existing with the UEG constant deduced from a UEG theory of theory which could explain the rotation behavior in a elementary particles, for validation or verification of the satisfactory manner, although modification of the laws new UEG model. The functional trends established from of Newtonian dynamics has been proposed [2]. Recently, the new UEG model may be compared, for validation a new unified electro-gravity (UEG) theory is established, of the model, with those from the empirical Tully-Fisher which has been successfully applied to model elementary Relation (TFR) [6] and the Modified Newtonian Dynam- particles [3, 4], where a new gravitational force, propor- ics (MOND) model [2, 7]. The trends predicted from tional to electromagnetic energy density, is introduced. the UEG model would explicitly depend upon the spi- This UEG theory has also been extended to model en- ral galaxy’s aspect ratio (ratio of the scale lengths in ra- ergy generation in single stars [5], which are spherically dius and thickness), because the new model is formulated symmetric bodies like the elementary particles. However, based on the spherical asymmetry of the galaxy. This is the theory needed some basic modification when it was distinct from the the MOND model, where there may not extended to model orbiting of a binary-star system [5], be such definitive interrelation between the galaxy’s as- in order to accommodate the spherical asymmetry of the pect ratio and the rotation speed. The functional depen- binary system. In this paper, the UEG theory would be dence of the galaxy’s aspect ratio on the surface bright- applied to a spiral galaxy, which is another different non- ness and rotation velocity, as required for the UEG galaxy spherical body. The energy density due to star lights in model to reproduce the rotation curves, may be com- the galaxy would contribute to a new gravitational force, pared with available measurements, for another indepen- which could support the observed stellar rotation around dent validation of the basic UEG galaxy model. the galaxy. A constant rotation speed beyond certain ra- The formulation of the force-field in the UEG model dial distance would require a 1/r-dependent gravitational of a spiral galaxy, which is a non-spherical body, is ex- acceleration, in the given region. When the UEG theory pected to be distinct from that for an elementary particle is properly modified for the non-spherical structure of or an isolated star [3, 5], which are spherical structures. a spiral galaxy, the required 1/r-dependent acceleration The galaxy’s UEG force field is defined in proportion to may result, although the stellar light radiation from the an effective distribution of energy density, not the actual galaxy exhibit an approximate 1/r2 dependence, in the energy density of stellar radiation as was the case for given region. This is possible, because the energy density the spherical structures. The effective energy density is of the actual light radiation may need to be redistributed, obtained by suitable redistribution of the galaxy’s light

Copyright # TXu 2-128-436, c 2018 by Nirod K. Das. All Rights Reserved. 59 N. Das, A UEG Theory of Nature, 2018 5-2 radiation, in proportion to the distribution of the Newto- ery location, unlike that of a spherical structure which nian gravitation potential of the galaxy. The divergence is radially directed at every location. Assuming an ideal of the resulting UEG force field surrounding the galaxy disk structure for a spiral galaxy, which is independent would be equivalent to having a fictitious “dark-matter” of the azimuth (φ) coordinate, the Newtonian gravita- distribution, which may be needed in order to explain the tional field may be shown to consist of only the radial observed rotation behavior of the spiral galaxies, as well (r) and elevation (θ) components, with no φ-component. as formation and evolution of the galaxies, on the basis Like the Newtonian gravitation field, the UEG field for of the conventional Newtonian gravitation [8]. Beyond a a spiral galaxy may not be required to be strictly radial sufficiently large radial distance from the galactic center, in direction, at all general locations. And, the UEG field the galaxy would “look” like a point source with a spher- for the galaxy may ideally be directed along the galaxy’s ically symmetric distribution of the Newtonian potential, Newtonian gravitational field with the r− and θ− com- and with a 1/r2 dependence of its light intensity. In this ponents. far region the radial UEG field would also be spherically We may assume the UEG field to be energy- symmetric, and therefore the field would be directly pro- conservative, which is defined as the gradient of an as- 2 portional to the 1/r -dependent light’s energy density, sociated potential function. The desired non-radial (θ-) without any need for redistribution of the energy density component of the UEG field for a spiral galaxy, as dis- as per the proposed model. This spherically symmetric, cussed above, would require the potential function, and 2 1/r -dependent radial UEG field in the far region is asso- accordingly the field, to maintain a gradient or a deriva- ciated with zero field divergence, and therefore with no tive along the θ−direction. In other words, the distribu- dark matter. In contrast, the region sufficiently close to tion of the potential or the field on a spherical surface the center would in general be associated with a strong would be non-uniform in the θ variable. Such spheri- divergent UEG field, and therefore with a heavy dark- cal asymmetry in the galaxy’s UEG field or potential is matter distribution. This region of heavy dark-matter in distinct contrast to the UEG field or potential for a presence would at least include the smallest spherical re- spherical body, which is uniform on any spherical sur- gion which encloses most of the galaxy’s mass and light face. We may define the UEG field for a spiral galaxy, on sources, and may extend much farther. any spherical surface of a given radius, to be distributed Section II presents the theoretical concepts and an an- in proportion to the galaxy’s Newtonian potential on the alytical formulation of the theory. The results for flat spherical surface. This would ensure the gradients of the rotation velocity deduced from the model are validated UEG and Newtonian potentials in the θ−direction, or with measured data for a galaxy survey as well as for an equivalently the θ−components of the respective fields, individual galaxy, in sections III, IV. The Tully-Fisher to be in proportion to each other at all points on the Relation (TFR) and the Modified Newtonian Dynamics surface of the given radius. Further, the gradients of the (MOND) model are studied in section V, in relation to two potentials in the radial (r−) direction, and there- the present UEG galaxy model, for further validation of fore the r−components of the respective fields, may be the model. This is followed by discussion and general assumed to be proportionate to each other, at least in conclusion from the study. terms of their general functional trends on a first-order basis. Accordingly, the essentially proportionate components of the two fields would ensure the UEG and the II. THEORY Newtonian fields to be directed approximately parallel to each other, at all locations, which may be desired as A. The Basic Concept discussed earlier. The above model may be formulated by having the radial UEG field to be proportional to a As per the UEG theory, there exists a new gravita- suitable distribution of an effective energy density, with tional force-field which is dependent on the electromag- the UEG constant γ [3, 4] as the constant of proportion- netic energy density. For a simple spherical body, the ality. The effective energy density at any given location is new UEG field at any particular location is directly pro- defined by redistribution of the actual energy density of portional to the energy density at the given location, and the galaxy’s stellar radiation on a spherical surface pass- is directed toward the center of the body [3–5]. Such sim- ing through the location, in proportion to the galaxy’s ple, direct relationship between the UEG field and the en- Newtonian potential on the spherical surface. The redis- ergy density may not be valid for a general non-spherical tribution would maintain the total integral of the actual structure, with a non-spherical light distribution. Cer- and effective energy densities on the spherical surface to tain additional conditions may be established for a UEG be equal to each other, which is a definite measure of force-field, which could be implicit in, or consistent with, the equivalent UEG mass (dark-mass) enclosed inside the the simple relationship for a spherical structure. But, sphere. the additional conditions may have to be explicitly ap- An additional fundamental condition may need to be plied for a general non-spherical structure. The Newto- enforced in any general UEG field. It would be reason- nian gravitational field in a spiral galaxy structure is not able to require the total UEG force due to a general dis- directed radially toward the center of the galaxy at ev- tribution of energy density, produced due to the gen-

60 N. Das, A UEG Theory of Nature, 2018 5-3 eral distribution of its associated sources internal to a massive body, to be zero. Otherwise, a non-zero total force produced by a general source internal to a particular body, acting upon the given body itself, would not be fundamentally sensible. The above UEG model, as specifically proposed for a spiral galaxy, may be verified to enforce this basic condition of having zero total force. The azimuthal (φ) symmetry we assumed for an ideal spiral galaxy would ensure the total force to be zero, as required. However, it may also be ensured that this condition of zero total force may not be evidently violated when the UEG model is extended for a more general structure. It may be argued, that the definition of the UEG field for a spiral galaxy, as proposed above and implemented in the following section II B, is perhaps not the unique or best way to define the desired field. For example, the UEG field could have been defined in direct proportion to the actual energy density, and been non-radially di- FIG. 1. rected along the galaxy’s Newtonian gravitation field as may be desired. Such a UEG field may also be shown to satisfy the above required condition of having zero total Newtonian gravitational field of any general mass distri- force. The actual energy density due to the star light bution, acting upon its own mass distribution, the pro- in a disk galaxy is ideally independent of the azimuthal posed UEG field would similarly establish the condition angle φ. All components of the above alternate UEG of zero total force, at least on a first order basis. field for the disk galaxy, expressed in direct proportion As suggested above, the proposed UEG model, as for- to the galaxy’s φ-independent actual energy density, may mulated in the following section (II B), may not be the also be shown to produce a total zero force when the field most rigorous form of the UEG theory for general appli- acts upon an ideal φ-independent mass distribution of the cations, even for the specific application to spiral galax- galaxy. However, this alternate model would evidently ies. The model is intended as a first-order working hy- fail to produce the required zero total force for a general pothesis for the specific study of the flat rotation curves condition, if either the mass or the light distribution were in spiral galaxies. However, the proposed UEG model is φ-dependent. developed as a valuable theoretical framework, which sat- Such an alternate model, or any other similar propo- isfies expected fundamental conditions, ensures compat- sition which would lead to such evident invalidity when ibility with all prior successful applications of the UEG extended to a general situation, is rejected. Whereas, theory in [3–5], while it foresees no evident contradiction the original UEG field as proposed earlier and formu- for a general application. That is a significant scientific lated in the following section II B, particularly when the objective, to support future generalization of the UEG mass distribution is maintained to be symmetric in the theory towards a rigorous and complete theory. azimuth (φ) coordinate, may be verified to properly enforce the required condition of having zero total force, for any general distribution of the radiation energy den- B. Analytical Model sity. This would be based on the symmetry that is maintained in the proposed re-distribution model of en- The light radiation from a spherically distributed ergy density, which would result in a φ−independent source, like a single isolated star for example, exhibits effective energy density, in proportion to the New- a 1/r2 dependence of its radiation energy density with tonian potential of the φ−independent mass distribu- radial distance r, external to the spherical source. Such tion, and therefore a proportionate φ−independent UEG 1/r2 dependence of radiation may also be seen for a non- field. The φ−independent field, acting upon the ideal spherical source, in an approximate form, outside of a φ−independent mass distribution, would produce the re- spherical region of certain threshold radius. For a spi- quired total zero force. On the other hand, if the mass ral galaxy, such a spherical region may be identified with distribution was not ideally symmetric in the φ coordi- a threshold radius equal to the galaxy’s scale radius R. nate (with or without a φ−symmetry of the energy den- This means, the radiation of the galaxy establishes an sity), the required condition of zero total force is expected approximate spherical symmetry beyond the radius R to be closely established. The proposed UEG field tries A spherical source is defined by spherical equi-potential to closely mimic the Newtonian gravitational field of the surfaces, which means all points on a spherical surface of given mass distribution, as discussed before. Therefore, radius r have the same potential. In contrast, the spiral like the zero total force which is guaranteed from the galaxy may be represented as a thin disk of an average

61 N. Das, A UEG Theory of Nature, 2018 5-4 thickness z0, with the z0 much smaller than its disk ra- symmetric manner in the region, as if it radiates from a dius ∼ R. The equi-potential surfaces (as per Newtonian point source at the galaxy center. The total luminosity gravity) for the disk structure would be thin disk-like sur- may be expressed using the surface density µ, which may faces in the vicinity enclosing the source disk (see Fig.1). be modeled with an exponential profile with amplitude Such equi-potential surfaces exhibit spherical asymmetry µ0 and scale radius R. inherent in the disk structure, and such asymmetry in the Newtonian potential distribution may effectively extend 2 L µ R µ −r/R well beyond the scale radius R. This is unlike the light’s Wτ (r) ' = 0 ,Wτ (r = R) ' 0 , µ(r) = µ e , 4πr2c 2r2c 2c 0 energy density discussed above, which establishes a fairly ∞ ∞ R R −r/R 2 spherical symmetry beyond the galaxy’s scale radius. L = µ(r)2πrdr = µ0e 2πrdr = 2πµ0R . (3) Now, consider a spherical surface of radius r, with a 0 0 common center as the disk galaxy, as shown in Fig.1. The approximate energy density Wτ at r = R can The distribution of the Newtonian gravitational poten- then be related to the light surface density µ at r = R, tial on this surface would in general be non-uniform, with with e/(2c) as the proportionality factor. For conve- stronger potential values near the plane of the disk over nience of reference, the effective energy density function a constant thickness ∼ z (independent of r), and weaker 0 Wτe(r > R) may be defined proportional to an equivalent values in the rest of the spherical surface. As a first-order effective surface density function µe(r), with the same model, one may approximate the potential distribution to above factor e/(2c) of proportionality. Using the relation be uniform over its strong region of area ∼ 2πrz (Fig.1), 0 (2) between the Wτe function and Wτ (r = R) in the pro- and be negligible over the rest of the spherical surface. posed definition, the effective surface density function µe A uniform energy density W of light radiation over the τ may be related to the actual surface-density function µ. surface may be redistributed in proportion to the potential distribution, as approximated above, resulting in µ a stronger effective energy density Wτe near the galaxy 0 eµ(r=R) Wτ (r = R) ' 2c = 2c , plane. The radial UEG force is proposed to be propor- eµe(r) R tional to this effective energy density Wτe, not the actual Wτe(r > R) = 2c = Wτ (r = R) r eµ(r=R)×R energy density Wτ . In accordance with the above princi- ' = ea , ple, the two energy densities would in principle be equal if 2cr 2cr a µ(r=R)×R the potential was spherically symmetric, with a uniform a = µ(r = R) × R, µe(r) = r = r . (4) value everywhere on the spherical surface of Fig.1. The effective surface density function µe(r) may be viewed as a 1/r-functional fit to the actual surface sur- 2 [Wτ (r) × 4πr ] = [Wτe(r) × (∼ 2πrz0)], face density function µ(r), such that they are equal to r each other at r = R. As mentioned above, the surface Wτe(r) ∝ z × Wτ (r); 0 density function µ(r) is modeled as an exponential distri- 1 1 Wτ (r) ∼ ,Wτe(r) ∼ , r > R. (1) bution with an amplitude µ and a scale radius R. The r2 r 0 amplitude a of the µe distribution may be related to the 2 The original energy density Wτ with a ∼ 1/r depen- parameters µ0 and R. Consequently, the total luminosity dence would transform into an effective energy density L in (3) may be expressed in terms of the parameters a Wτe with a ∼ 1/r dependence on the galaxy plane. and R. The gravitational potential distribution would exhibit closer spherical symmetry as one approaches towards the µ(r) = µ e−r/R; µ(r = R) = a = µ e−1, µ = ea , center, resulting in the effective density Wτe to be close 0 R 0 0 R 2 to the actual energy density Wτ in the central region. L = 2πµ0R = 2πeaR. (5) Accordingly, as a first-order estimate, the effective and actual energy densities may be assumed to be equal to If the amplitude µ0 is maintained to be approximately each other for r < R. Based on this assumption and constant, then a would be proportional to R, or equiva- the above modeling (1), the effective and actual energy lently the luminosity L would be proportional to a2. This densities may be expressed as follows. may be the case for a large group of high surface brightness (HSB) galaxies, which were believed to confirm to the Freeman’s Law [9] of having an approximately con- Wτe(r) = Wτ (r), r < R; stant central brightness µ0. R2 Wτ (r) = Wτ (r = R) , r2 R µ ∼ constant (Freeman0s Law, HSB Galaxy), Wτe(r) = Wτ (r = R) r , r > R. (2) 0 a ∝ R,L ∝ a2. (6) The energy density Wτ for r > R may be approximated using the total luminosity L and the speed of light c, and The radial UEG field Egu may now be expressed pro- assuming that the total light radiates in a spherically portional to the equivalent energy density Wτe, with the

62 N. Das, A UEG Theory of Nature, 2018 5-5 constant of proportionality equal to the UEG constant γ. survey [10]. As suggested above, the data point is located The potential function associated with the above radial approximately at the statistical center of the survey sam- field could be obtained by integrating the field in the ra- ples. dial variable r, from which the θ component of the field may also be derived (in principle) as the θ-derivative of the potential function. However, we are interested here (I-band data): only on the radial UEG field, which completely deter- 10.4 36.4 5.2 mines the orbital acceleration on the central plane of the L = 10 L0 = 3.864 × 10 W, v = 10 m/s, galaxy, because the θ− component of the UEG field on R = 100.5kpc = 100.5 × 3.086 × 1019m, this plane would be zero. The magnitude E of the ra- 1.5 gu γ(I-band) = γ = 4π×3×3.086×10 dial UEG field on the central galaxy plane would be equal I 3.864 2 3 -2 -3 to the orbital acceleration v /r. The Egu (for r > R) is = 0.95 × 10 [(ms )/(Jm )]. (9) proportional to the effective surface density µe(r) = a/r, having the same 1/r dependence as the orbital accelera- Similarly, we estimate the γ from the K-band measure- tion. Accordingly, the rotation velocity v would exhibit ment of [10]. Note that an effective radius, R , is pro- 2 e a “flat” behavior for r > R, with v equal to the constant vided in [10] for the K-band measurements. The effective amplitude ‘a’. radius, defined as the radius of a sphere that encloses half of the total luminosity, would be 1.678 times the scale ra- ¯ γeµe dius R used in our modeling, assuming an exponential Egu = −rEˆ gu = −rγWˆ τe = −rˆ 2c , light profile. γeµe(r) γea v2 Egu(r) = 2c = 2cr = r , 2 γea v = 2c , r > R. (7) (K-band data): Combining (7,5), the luminosity L may be expressed in 10.8 36.8 5.2 L = 10 L0 = 3.864 × 10 W, v = 10 m/s, terms of the velocity v, radius R, and the UEG constant 0.6 0.6 19 Re = 10 kpc = 10 × 3.086 × 10 m,R = Re/1.678 , γ. 4π×3×3.086×101.2 γ(K-band) = γK = 3.864×1.678 3 -2 -3 4πRv2c 4πRv2c = 0.28 × 10 [(ms )/(Jm )]. (10) L = 2πeaR = γ , γ = L . (8) Accordingly, the UEG constant γ may be estimated Measurements in the K-band overestimates the lumi- from (8) using measured values of the L, v and R, avail- nosity and the energy density, leading to underestimation able from a galaxy survey [10]. Alternatively, the ampli- of the γ. On the other hand, measurements in the I-band tude a for the effective surface density µe(r) may be esti- underestimates the energy density, leading to overestima- mated directly from a measured surface-brightness pro- tion of the γ. Accordingly, the above results estimate a file µ(r) for a selected individual galaxy, and then the useful range for the value of the γ, which is consistent γ be estimated using the a and the measured flat ro- with the value of the γ = 0.6 × 103 (ms−2)/(Jm−3) de- tation velocity v, as per (7,4). The estimation directly duced from the UEG model [3] of elementary particles. using measured data of an individual galaxy would complement the estimation from the galaxy survey, providing an explicit illustration of the UEG model. However, the 0.28 × 103 < γ < 0.95 × 103[(ms-2)/(Jm-3)], estimation using an averaged data from the galaxy survey 3 -2 -3 can, in principle, be more reliable than that using data γ = 0.6 × 10 [(ms )/(Jm )]. (11) for individual galaxies. Inaccuracies from astronomical measurements of individual galaxy parameters, as well as The best estimate for γ is assumed to be the average uncertainty due to deviation of individual galaxy charac- of the two estimates in the I− and K− bands. teristics from any ideal theoretical assumptions, can often be significant. The resulting inaccuracy or uncertainty in the estimation of the γ is expected to be minimized by us- (γI +γK ) 3 -2 -3 ing an “average” or a central data point among a survey γ ' 2 =0.62 × 10 [(ms )/(Jm )]. (12) of large number of sample galaxies. The above estimate closely agrees with the γ from the particle model [3]. Considering that we used a first-order III. ESTIMATION OF γ USING MEASURED approximation in the UEG modeling of (1,2), such agree- DATA FROM GALAXY SURVEY ment is remarkable. This means that the ideal conditions we assumed in the first-order UEG modeling of (1,2) are We first estimate the γ based on (8), using an average remarkably valid for the central data point of [10] used data point from the I-band measurement of the galaxy in our estimation.

63 N. Das, A UEG Theory of Nature, 2018 5-6

FIG. 2.

IV. ESTIMATION OF γ USING MEASURED DATA OF AN INDIVIDUAL GALAXY 2 7 2 6 γ = v ×10 = v ×10 [(ms-2)/(Jm-3)] (Visible), s0d×6.61×4.6 s0d×3.04 Measured data for the surface brightness distribution 2 6 γ = ∆u×v ×10 [(ms-2)/(Jm-3)] (U-Band), µ(r) of a specific galaxy is first properly fitted with an s0d×3.04 exponential, and then an effective surface brightness dis- ∆ ×v2×106 γ = k [(ms-2)/(Jm-3)] (K-Band); tribution µe(r), as defined in (4). The data using mixed s0d×3.04 units, such as magnitude, arcsec, light-years, may be con- v(105m/s), d(MLyr), verted to suitable standard units. The µe distribution (4.74−3.28)/2.5 can then be related to the rotation velocity v using (7). ∆k = 10 = 3.84 = K-Band correction factor, (4.74−5.56)/2.5 ∆u = 10 −13 eµe s0×6.61×10 3 = 0.47 = U-Band correction factor. (14) Wτe = 2c = r J/m , 2 UEG Acceleration(m/s ) = Egu Using the U-band (assumed ' U’-band) surface- γs ×6.61×10−13 brightness data [11] for the galaxy NGC-2403, presented = γWτe = 0 r in Fig.2, we estimate the amplitude parameter s0 = 32.9. v2×1010 v2 5 This parameter, together with the galaxy’s distance d = = r = 6 ; v(10 m/s), m r×d×4.6×10 11.4MLyr [12] and flat rotation velocity v = 1.35×105m/s 2 4 2 3 1(lin-mag/arcsec ) = 1.46 × 10 (W/m ), [13], would provide an estimate for the γu = 0.75 × 10 −2 −3 r(arcsec)=r × d × 4.6 × 1016(m) (ms )/(Jm ), using the above relation (14). Similarly, using the K-band data [14] for the same galaxy NGC- = r (m), at distance d(MLyr). (13) m 2403, presented in Fig.3, we estimate the amplitude parameter s0 = 430. This would provide an estimate for the 3 −2 −3 The UEG constant γ is deduced using the amplitude γk = 0.47×10 (ms )/(Jm ), using (14). An average of a, or its equivalent parameter s0, of the effective surface these two estimates for the γ would lead to the best esti- 3 −2 −3 brightness distribution µe(r), the flat rotation velocity v mate for the γ = 0.61×10 (ms )/(Jm ) from the avail- and the distance d of the galaxy. Suitable correction fac- able data for the galaxy NGC-2403. This is close to the tors may be needed to relate the K- and U-band measured γ = 0.62 × 103 (ms−2)/(Jm−3) deduced from the galaxy magnitudes to a common reference of solar bolometric survey in (12) or the γ = 0.60 × 103 (ms−2)/(Jm−3) from magnitude of 4.74. This assumes the solar magnitudes in particle model [3]. Such remarkable agreement implies the K- and U-bands are 3.28 and 5.56, respectively. that any deviation from the basic model of (1-4) due to

64 N. Das, A UEG Theory of Nature, 2018 5-7

FIG. 3.

differences in the surface brightness µ0 (see section V)) of the individual galaxy NGC-2403 from the “average” 2 2πe2a2 8πv4c2 ea galaxy used in the estimation (12), is minimal. The µ L = 2πµ0R = µ = 2 , µ0 = R , 0,k 0 µ0γ 2 are estimated to be roughly equal to 16.75 (mag/arcsec ) L ∝ v4 (TFR), in both cases ([10], Fig.3), which is consistent with the 0 above expectation. µ0 ∼ constant (Freeman s Law, HSB Galaxy). (16) However, the Freeman’s Law is no longer believed to strictly valid, and galaxies are measured to exhibit a broad range of amplitudes µ0 covering variations among NGC-2403: the HSB galaxies as well as extending to low surface 3 -2 -3 brightness (LSB) galaxies with lower values of µ . For γu=0.75 × 10 (ms )/(Jm ) (U-Band), 0 a general treatment to closely model the variation in the γ =0.47 × 103(ms-2)/(Jm-3)(K-Band), k amplitude µ0, we may introduce a new parameter α for γ = (γv + γk)/2 fitting the 1/r profile of µe with the exponential profile = 0.61 × 103(ms-2)/(Jm-3) (Best Estimate). (15) of µ in (4). The unit reference value of α is expected to apply for an “average” HSB galaxy, as assumed in the basic model of (4) and in the estimations of (12,15). The µe may be adjusted to a smaller or larger value, relative to the µ(r = R), with a proportional adjustment of the V. THE TULLY-FISHER RELATION (TFR) parameter α, which would represent a smaller or large AND THE MODIFIED NEWTONIAN value of the UEG force, respectively, as per (7). DYNAMICS (MOND) MODEL, DERIVED FROM The variable factor α is accommodated in the gravita- THE UEG MODEL tional potential model of (1,2), Fig.1, by recognizing the galaxy thickness z0 to be an active variable, like the scale Combining (5,7) and assuming an approximately con- radius R or the surface brightness µ0, for parametrization stant µ0, a Tully-Fisher Relation (TFR) [6] may be de- of galaxy characteristics. In the potential model of (1), duced, where the total luminosity L would be propor- an approximately uniform (spherically) potential would tional to the fourth power of the flat rotation velocity v. be established for all radial distances less than a variable As mentioned before, the above condition of an approx- threshold radius Rt, dependent on a variable thickness imately constant µ0 is satisfied by a large group of high z0, not less than the ideal fixed threshold radius r = R surface brightness (HSB) galaxies that were believed to assumed in (2). Accordingly, the effective energy density confirm to the Freeman’s Law [9]. Wτe would match with the actual energy density Wτ for

65 N. Das, A UEG Theory of Nature, 2018 5-8 all the radial distances less than the variable threshold proportional to the square-root of the surface brightness radius, not the ideal reference threshold r = R assumed µ0. This general trend, of having the normalized galaxy in (2). Consequently, the Wτe(r = R) would no longer thickness z0/R to be smaller for a lower surface bright- be equal to Wτ (r = R) as ideally assumed in (2), but ness µ0, may seem to be a sensible characteristic. The now be equal to αWτ (r = R), with the variable factor α specific required relationship between the galaxy thick- proportional to the normalized galaxy thickness R/z0. ness and the surface brightness may be compared and The model of (1,2) may be revised as follows, as ex- verified with the measured data in [16]. plained above. Using the above required relationship (19) between the µ0 and the normalized scale z0/R in (18,7) would trans- late to another galaxy scaling relationship between the r r R 1 1 Wτe ∝ Wτ × = Wτ × × ; Wτe ∼ ,Wτ ∼ . z0 R z0 r r2 absolute thickness z0 (not normalized to R) and the flat R R rotation velocity v. Wτe(r) ∝ Wτ (r = R) × × , r > R; r z0 W = W , r < R ∝ z . (17) τe τ t 0 2 ea 2 γea γµ0αR (µ0α )R µ0 = , v = = = ∝ z0, Using the above revisions and (3), the relation (4) be- αR 2c 2c 2cα µ α2 = constant, α ∝ R . (20) tween the surface density µ and effective surface density 0 z0 µe, and the resulting expression for the luminosity L (5) using (7), may also be revised. Accordingly, the galaxy thickness z0 is required to be proportional to the square of the flat rotation velocity v. This required relationship is clearly verified from the a µ(r=R)R R measured data of [16]. It is significant to note that the µe(r) = r = α × r , α ∝ z , 0 above two required relations (a) between the galaxy nor- a = α × µ(r = R) = αµ e−1, µ = ea , R 0 0 αR malized thickness z0/R and the surface brightness µ0, and 2 2πe2a2 8πv4c2 (b) between the thickness z0 and the flat rotation veloc- L = 2πµ0R = 2 = 2 2 . (18) α µ0 α µ0γ ity v, are independently predicted from the UEG model of (17,18), based on the observed TFR (19,16), but could The TFR (16), which was established based on the sim- not have been anticipated either from the TFR of [6] or ple assumption of an approximately constant µ0, would the MOND [2, 7]. Verification of the above predictions still be valid for a range of different surface brightness µ0, from [16] is a significant development, which strongly val- 2 if µ0α in (18) is approximately a constant. This condi- idates the new UEG model of (1,17), as applied to the tion, of having a larger value of the α for a lower µ0, non-spherical structure of a galaxy. means there would be relatively more contribution from the UEG force as the surface brightness µ0 reduces. This trend better represents observed characteristics among A. Refinement in the Tully-Fisher Relation the HSB galaxies, extending to LSB galaxies as well. The higher UEG contribution for a lower surface brightness Some refinement in the above TFR (19) may be µ0 would be equivalent to having relatively more “dark needed, in order to confirm to the measured data [6, 10] matter” contribution for a LSB galaxy [15], as per the more accurately, where the luminosity seems to be pro- current dark-matter paradigm. portional to a smaller exponent (than the ideal value of 4 in (19)) of the velocity v. This trend may be empirically established from (19) by having the factor µ α2 L ∝ v4 (MOND, TFR), 0 to be weakly dependent on the velocity v (proportional z 2 µ α2 = constant, α ∝ √1 ; α ∝ R , µ ∝ ( 0 ) , to a relatively small exponent of v), instead of the ideal 0 µ0 z0 0 R constant factor µ α2 suggested above. This may be rep- α (LSB Galaxy) > α (HSB Galaxy) ∼ 1, 0 resented by suitable refinement in the required relation in > Dark Matter (LSB) Dark Matter (HSB), (19) between the galaxy normalized thickness z0/R and z0 z0 the surface brightness µ . R (LSB) < R (HSB). (19) 0 The above TFR of having the luminosity proportional 2 b to the fourth power of the velocity v, is also consistent µ0α ∼ v , 0 < b < 0.5; with prediction from an alternate model using a modified L ∼ v4−b = vd, 3.5 < d < 4 . (21) Newtonian dynamics (MOND) [2, 7]. As derived in (17,18), the parameter α, which propor- However, this refined TFR does not confirm to the tionately represents the equivalent distribution Wτe or MOND, where the luminosity is definitively required to µe, is proportional to the normalized galaxy scale R/z0. be proportional to the fourth power of the velocity v. It 2 Accordingly, the condition (19) of a constant factor µ0α , is not clear if the above refinement (21) is really funda- required for the validity of the TFR or MOND, would mental or is simply due to selection bias in the measure- be satisfied if the normalized scale parameter (z0/R) is ments of [6, 10], resulting in a limited range in the data

66 N. Das, A UEG Theory of Nature, 2018 5-9 over which the exponent d is estimated with a smaller Accordingly, for a given surface luminosity µ0, a larger value d < 4. value of the baryonic mass density Ab is expected to result The total luminosity and surface brightness profile are in a tighter confinement of the gravitational potential usually proportional to the total baryonic mass and its near the galaxy surface (smaller z0), resulting in a larger mass distribution, respectively, in which case the TFR α. The two refinements (21,22) may need to be studied would work as well if the luminosity is interchanged together, which may be associated with interdependent with the baryonic mass. The proportionality between and/or mutually compensating physical effects. the baryonic mass and the luminosity may not, however, strictly extend to all LSB galaxies, having smaller lumi- VI. CONCLUSION nosity and rotation velocity. In this case, the measured data follow a TFR more accurately, if the total baryonic mass Mb is used in the relation (19,21), instead of the total luminosity L. The revised relation is referred to as the Baryonic Tully-Fisher Relation (BTFR) [17]. The baryonic mass Mb would be proportional either to the fourth power or to a smaller exponent of the velocity, if The estimate of the UEG constant γ from measured the baronic mass substitutes the luminosity in the TFR data from a galaxy survey [10], based on the new UEG versions (19) or (21), respectively. The former version model, agrees well with an accurate value derived from of the BTFR is consistent with MOND which, to fun- the UEG model of elementary particles [3, 4]. This is damentally begin with, relates the baryonic mass to the based on a statistically average data point from the sur- fourth power of the velocity v. vey samples. Direct analysis of measured brightness pro- The deviation from the original TFR may be partly file and rotation curve of a specific selected galaxy is also attributed to the larger contribution to the rotation ve- illustrated to provide a similar estimate for the γ, that locity v from the Newtonian gravity due to the propor- is consistent with the estimate from the galaxy survey. tionately larger regular mass (baryonic), in the lower- Further, the UEG galaxy model confirms to the TFR luminosity LSB galaxies. More significantly, the revised [6, 17] for varying range of galaxy amplitudes, and is trend may be empirically accommodated by properly ad- consistent with results from a modified Newtonian dy- justing the parameter α in (18) to be dependent on both namics (MOND) [2, 7] model. The required condition the surface brightness µ0 and an equivalent baryonic sur- for the agreement between the UEG model, TFR and MOND is supported by measured relations of the galaxy face mass density Ab of the galaxy. This would be consistent with the basic principles of the present UEG model thickness with the surface brightness and the rotation ve- in (1- 5,17), where the gravitational potential function locity [16], which may be considered as an independent that determines the redistribution of the energy density validation of the UEG model. The above studies strongly Wτ into the effective density Wτe (see Fig.1) may be rec- support validity of the new UEG model, established for ognized to depend upon both the Newtonian gravitation the non-spherical structure of a disk galaxy. The UEG (related to mass profile) as well as the UEG field due to theory is intended to serve as a theoretical substitute for the light profile of a galaxy. However, more specific phys- the current “dark-matter” hypothesis. ical explanation behind such an empirical trend, leading to the preference of the baryonic mass over the luminosity in the BTFR, is at this point unclear, and is beyond the scope of the present work. The UEG theory, which has been successfully applied for elementary particles [3, 4] as well as single and binary 4 2 Mb 8πv c Mb stars [5], and is now supported as well for galaxy model- Mb = L × L = 2 2 × L α µ0γ ing, may provide a new unified theoretical paradigm for 8πv4c2 Mb Ab a broad range of physical concepts, covering both small = 2 2 2 , L = µ , α (µ0 /Ab)γ 0 and large size scales of nature, and spherically symmetric 4 2 2 Mb ∝ v , α × (µ0 /Ab) = constant. (22) as well as asymmetric structures.

[1] V. Rubin, N. Thonard, and J. W. K. Ford, Astrophysical [4] N. Das, “A Generalized Unified Electro-Gravity (UEG) Journal, Part 1 238, 471 (1980). Model Applicable to All Elementary Particles,” Paper [2] M. Milgrom, Astrophysical Journal, Part 1 270, 365 #2, pp.14-30, in “A Unified Electro-Gravity (UEG) The- (1983). ory of Nature,” (2018). [3] N. Das, “A New Unified Electro-Gravity (UEG) The- [5] N. Das, “Unified Electro-Gravity (UEG) Theory Applied ory of the Electron,” Paper #1, pp.4-13, in “A Unified to Stellar Gravitation, and the Mass-Luminosity Rela- Electro-Gravity (UEG) Theory of Nature,” (2018). tion (MLR),” Paper #4, pp.44-58, in “A Unified Electro-

67 N. Das, A UEG Theory of Nature, 2018 5-10

Gravity (UEG) Theory of Nature,” (2018). NGC-2403,” http://ned.ipac.caltech.edu, Retrieved [6] R. B. Tully and J. R. Fisher, Astronomy and Astro- (2017). physics 54, 661 (1977). [13] K. Begeman, PhD Thesis, University of Groningen, [7] M. Milgrom, Astrophysical Journal, Part 2 270, 371 Netherlands (2006). (1983). [14] J. Munoz-Mateos, A. G. de Paz, S. Boissier, J. Zamorano, [8] A. Borriello and P. Salucci, Monthly Notices of the Royal T. Jarrett, J. Gallego, and B. F. Madore, Astrophysical Astronomical Society 323, 285 (2001). Journal 658, 1005 (2007). [9] K. C. Freeman, Astrophysical Journal 160, 811 (1970). [15] W. J. G. de Blok and S. S. McGaugh, Monthly Notices [10] S. Courteau, A. A. Dutton, F. C. van den Bosch, L. A. of the Royal Astronomical Society 290, 533 (1997). MacArthur, A. Dekel, D. H. McIntosh, and D. A. Dale, [16] D. Bizyaev and S. Kajsin, The Astrophysical Journal Astrophysical Journal 671, 203 (2007). 613, 886 (2004). [11] B. M. H. R. Weavers, P. C. van der Kruit, and R. J. [17] S. S. McGaugh, J. M. Schombert, G. D. Bothun, and Allen, Astronomy and Astrophysics Supplement Series W. J. G. de Blok, The Astrophysical Journal 533, L99 66, 505 (1986). (2000). [12] NED, “Nasa Extragalactic Database, Results for

68 6-1

A Uniﬁed Electro-Gravity (UEG) Model as a Substitute for Super-Massive Black Holes (SMBH) at Galactic Centers

Nirod K. Das Department of Electrical and Computer Engineering, Tandon School of Engineering, New York University, 5 Metrotech Center, Brooklyn, NY 11201 (Dated: May 9, 2018; Revised April 30, 2019) The UEG model for rotation in a disk galaxy is extended in this paper to model gravitation in a galaxy’s central region. The strong gravitational eﬀects at the centers of many galaxies, observed in the form of fast stellar orbits or velocity dispersion near the center, may be explained using the UEG theory, without having to invoke the hypothesis of super-massive black holes (SMBH).

I. INTRODUCTION bution of energy density, which is obtained by proper redistribution of the light density in the galactic core. The energy re-distribution is in proportion to the density of a The UEG theory of [1, 2] was extended in [3] to suc- non-spherical mass distribution, that is expected to exist cessfully model rotation around disk galaxies, without in the central core region, possibly in the form of galac- need for any “dark-matter”. The UEG theory for the tic jets or bars that might originate from this region. disk region of a disk galaxy may be extended as well into The UEG theory, in its simple form in the presence of the galaxy’s bulge, in order to model gravitation near a uniform or spherically symmetric mass distribution, or the galactic center. This may provide a substitute the- with suitable adjustment to account for any non-spherical ory for the super-massive black holes (SMBH), that are mass distribution in a central region, would establish a currently hypothesized to exist at the centers of most suitable M-sigma relation [5, 7] between the equivalent galaxies. In the absence of an alternate scientific theory, mass of the SMBH in the central nucleus and the veloc- the SMBH hypothesis is invoked in order to explain fast ity dispersion or rotation velocity in the galaxy’s bulge. stellar orbits that is directly observed near the central This includes both disk as well as elliptical galaxies. nuclear region of our own Milky Way galaxy [4], as well as indirectly deduced from Doppler measurements in the central region of many other galaxies [5]. The stars in II. THEORY the central galactic region of our own Milky Way galaxy are known to be significantly brighter, on average as well as in individual terms, as compared to the stars in the We will follow the UEG theory [3] that was developed galaxy’s outer periphery [6]. Such distribution of bright to model gravitation in the external disk region of a spiral stars in the Milky Way galaxy is likely to extend to other galaxy, with reasonable assumptions for analytical sim- galaxies. The UEG field due to the light radiation from plicity and extension of the model into the galactic bulge. a reasonable number of individual bright stars, or due to In the most of the internal region of the bulge, the effec- integrated light intensity over a suitable volume around tive energy density Wτe would be equal to the actual the central region, may support the high magnitude of energy density Wτ (r), which is assumed to be approxi- acceleration associated with the fast stellar motions near mately uniform, produced due to a uniform distribution the galaxy centers. of stellar sources. However, the uniformity of the effective energy density may not be valid in a core region In addition to the high magnitude of acceleration, the of radius r , where the effective energy density may be fast motions in the galactic centers are also deduced to 0 modeled differently. follow conventional Keplerian orbits [4]. Accordingly, in addition to supporting the high magnitudes of acceleration, the proposed UEG field would also be required a Wτe(r) = Wτ (r) = Wτ0 = , r0 < r < R0; to justify a 1/r2 dependence of the field with respect to R0 the radial distance r from the galaxy center, in order to b = , r < r0; support the Keplerian orbits. The required 1/r2 depen- r2 2 dence of the UEG field near the nucleus could be sup- b a ar0 2 r0 = α0R0, = , b = = aα0R0. (1) ported by presence of a single star or a few stars with r2 R0 R0 sufficiently high total luminosity, at or very close to the 0 nucleus of the galaxy. Farther from the nucleus, the same The effective energy density Wτe in the core region is 1/r2 dependence of the UEG field is also assumed to be modeled with a 1/r2 variation, as discussed later in the effectively maintained, produced due to radiation from section III. This results in a proportional radial compo- 2 the high concentration of stars in the region, in presence nent Eur of the UEG acceleration with the 1/r variation, of high extinction. Alternatively, the required 1/r2 de- where the constant of proportionality is the UEG con- pendence may be explained using a suitable UEG model, stant γ. This would be equivalent to having a mass m0 where the UEG field is determined by an effective distri- as per the Newtonian gravitation. In the rest of the bulge

Copyright # TXu 2-128-436, c 2018 by Nirod K. Das. All Rights Reserved. 69 N. Das, A UEG Theory of Nature, 2018 6-2 the UEG acceleration is proportional to the effective en- proportional to the Lb: larger is the total luminosity of ergy density, which is equal to the actual energy density the star lights in the galaxy’s bulge, more likely there Wτ (r) = Wτ0. The uniform UEG acceleration in this re- would be bright stars of proportionately larger luminos- gion would be equal to the circular orbital acceleration ity to be found in the core region inside the bulge, to 2 2 v /R0 at the boundary of the bulge. We assume that the support the 1/r -dependent UEG field in the core re- central velocity dispersion σ in the bulge is equivalent gion. Based on this assumption Lc ∝ Lb, we would have to the circular velocity v at the boundary of the bulge b ∝ R0a. Using this result in the model (2), it may also r = R0. Accordingly, the equivalent mass m0 for the core be reasonable to assume that the core radius r0 is pro- region may be related to the flat rotation speed v, based portional to the bulge radius R0, which is equivalent to on the model (1). having the parameter α0 to be a constant. 2 Under the above conditions (R0 ∝ v , Freeman’s Law, and constant α0, Lc ∝ Lb), the M-Sigma relation [5] γb Gm γb Eur(r < r ) = γWτe(r < r ) = = 0 , m = , would be established, where the equivalent SMBH mass 0 0 r2 r2 0 G M = m0 is shown to be proportional to the fourth power γa v2 Eur(r = R0) = γWτe(r = R0) = γWτ0 = = , of the velocity dispersion or the flat rotation velocity R0 R0 2 2 σ = v [7]. With a reasonable variation of the above model, v = γa = γWτ0R0, b = aα0R0, where the r0 is proportional to the 5/4 power of the R0 v2α2R 0.25 2 2 0 0 (α0 ∝ R0 ), an empirically more accurate version of the m0G = v α0R0, m0 = G . (2) M-sigma relation could be established, where the m0 is proportional to the fifth power of the velocity v [5, 7]. We may also assume that the average surface brightness µ0 ' µ(r), or its associated energy density Wτ0, is constant, as a first-order approximation for simplicity. M-Sigma Relation : This would include a large group of galaxies, which in m ∝ v4,R ∝ v2(Freeman’s Law), α ∼ (constant), case of disk galaxies would confirm to the Freeman’s Law 0 0 0 5 2 0.25 1.25 of High Surface-Brightness (HSB) galaxies [8]. With this m0 ∝ v ,R0 ∝ v , α0 ∝ R0 , r0 ∝ R0 . (3) assumption, from (2) the bulge radius R0 may be shown to be proportional to the square of the rotation veloc- The core radius r0 may be estimated for the Milky Way 2 galaxy. ity v (R0 ∝ (v /Wτ0)). This is similar to the relation between the disk scale length R of a disk galaxy and its flat rotation speed v in the outer disk region, when For Milky-Way: the galaxy’s surface brightness is approximately constant 6 (Freeman’s Law) [3]. This also leads to a Tully-Fisher m0 = 4 × 10 Msun, 30 Relation (TFR) for the disk galaxies, where the galaxy’s Msun = 1.989 × 10 kg, luminosity is proportional to the fourth power of the ve- v = 100 km/s, locity v, as shown in [3]. The equivalent relation, the 19 R0 ∼ 10000 Lyr = 9.4 × 10 m (4) Faber-Jackson Law [9], may be established for an ellipti- −11 2 2 cal galaxy. G = 6.67 × 10 Nm /kg , r q Any deviation from the ideal trend, due to deviation m0G 4×1.989×6.7 −2 −2 α0 = 2 = 9.4 × 10 = 2.38 × 10 , of the UEG acceleration Eur in (2) as the surface bright- v R0 ness deviates from an ideal reference value, is assumed r0 = α0R0 ∼ 238 Lyr = 73 parsec. (5) to be empirically balanced by suitable counter-effects in the UEG acceleration. This may include dependence of This radius is comparable to the size of the Milky the UEG acceleration with the bulge’s aspect ratio in Way’s central region, where high concentration of bright mass or light distribution (or effective radius), having a stars have been observed [6]. Such bright stars in the counter dependence with the surface brightness. Similar central region of the Milky Way, and presumably in the countering UEG effect due the thickness of a disk galaxy central region of other galaxies as well, would provide was discovered in the [3], in the modeling of the galaxy’s the required high energy density and the associated high rotation curve. Any further specific study in this direc- acceleration, as anticipated in the proposed UEG model. tion, particularly for an elliptical galaxy, is beyond the scope of this work. An equivalent luminosity Lc of the star lights enclosed III. UEG MODEL IN THE CORE REGION 2 inside the core of radius r0, which produces the 1/r - 2 2 dependent effective energy density Wτe = b/r in the core The 1/r dependence of the UEG acceleration in the 2 region, would be equal to Lc = 4πr0cWτe(r = r0) = 4πbc. galactic core region, as assumed in the above UEG model The c is the speed of light in free space. Similarly, an (2), may be established in following possible ways: equivalent luminosity Lb in the total bulge region of ra- (I) A single bright star, or few stars closely together, of 2 dius R0 would be equal to Lb = 4πR0cWτe(r = R0) = sufficiently high total luminosity, is(are) assumed to exist 4πR0ac. It may be reasonable to assume that the Lc is at or near the nucleus. The bright star(s) may yet to be

70 N. Das, A UEG Theory of Nature, 2018 6-3 discovered in the nuclear region of the Milky Way, likely undetectable due to high extinction in its radiation spectrum. Or, the luminosity of one or a few of the known stars near the nucleus of the Milky Way are actually sufficiently bright, but their luminosities have been under- estimated. Such bright sources at or near the nucleus would exhibit the conventional 1/r2-dependent light radiation, like a central point source, which would support the 1/r2-dependent UEG acceleration of sufficient magnitude, as required. After all, stars with luminosity close to 8 million solar luminosity, needed to emulate the central SMBH of the Milky Way as per the UEG theory, have been discovered elsewhere in our galactic neighborhood (star R136a1 has 8 million solar luminosity [10]). Stars of comparable luminosity have also been discovered in the Milky Way within ∼ 1pc of the galactic center (WR102ka and WR102c have about 3 million solar luminosity each [6]). It is quite likely that a single star or a few stars with the required total high luminosity (comparable to FIG. 1. one R136a1 or three WR102ka-type stars), but radiating mostly in the visible spectrum, is(are) already present in or near the nuclear region, but remain(s) undetected gion, is similar in principle to the UEG model of [3] for due to the high extinction in the visible spectrum. In the the non-spherical mass distribution in the disk region of a core region, farther away from the nucleus, the same 1/r2 disk galaxy. Particular differences are in the specific mass dependence of the UEG acceleration is also assumed to distribution and the region of interest in the two cases. be maintained, effectively, determined by distribution of The UEG field in the thin disk region, which is the re- star radiation and extinction characteristics in the region. gion of higher mass(baryonic) concentration as compared It may be mentioned, a star could attain the required to the large external halo, supports the flat rotation be- high luminosity (8 million solar luminosity) with mass havior in the disk. Whereas, the UEG field in the region comparable to only a solar-mass, as per the UEG model outside of a narrow jet in the galactic core, which is the of stars [11]. It is feasible to attain even much higher region of lower mass concentration as compared to the stellar luminosity, with only reasonably larger level of narrow jet, supports the fast stellar motion in the galac- stellar mass, without any un-physical requirement of tic core. mass or mass concentration. Presence of such brighter The required 1/r2 dependence of the acceleration is stars would be needed, as per the UEG theory, to sup- achieved in the region outside a small nucleus of radius port much higher acceleration, or equivalently to emulate rn at the galactic center, and external to a narrow conical much heavier SMBH, measured at the centers of other jet of solid angle ∆Ω = 4π/N (or a combination of more galaxies. than one jet of the same total solid angle), likely with The theory in the above first scenario is based on a N >> 1. This is achieved by establishing in the region 2 simple UEG model [1], where the UEG acceleration is an effective energy density Wτe(r) with the 1/r varia- proportional to the energy density of radiation, assuming tion, while the actual energy density, Wτ , of the light is that the mass distribution in the core region is approxi- assumed to be approximately uniform in the vicinity of mately uniform or spherically symmetric. the central nucleus (see Fig.1). This would be possible if (II) In a second scenario, the UEG theory may need to the mass density in the jet region is larger than the sur- be applied differently in the galactic core, taking into rounding region by a suitable jet density factor β(r) > 1, account suitable non-spherical mass distribution in the which is assumed to be physically realistic for formation of the jets. core region (r < r0), possibly forming jets or handle-bars. The new model can provide the required 1/r2-dependent acceleration, while the actual energy density of stellar W (r)(4π − ∆Ω) + W (r)β(r)∆Ω = 4πW (r), light is approximately uniform in the central nuclear re- τe τe τ W (r)N W (r) = τ , ∆Ω = 4π . gion. This possibility is theoretically interesting, and τe (N−1)+β(r) N (6) could be physically quite likely, considering that galaxies are known to exhibit jet or handle-bar formations near The energy density of stellar radiation is approxi- the galactic center, and such non-uniform mass forma- 0 mately uniform in a sub-core region r < r0, with suf- tions may originate from the nuclear region as assumed ficiently large magnitude. This maybe produced by a in the present scenario. high-density distribution of luminous stars in the sub- It may be noted, that the above proposed model for core region. As mentioned, the stars may possibly be a non-spherical mass distribution in the galactic core re- radiating in the visible or ultraviolet region, and may

71 N. Das, A UEG Theory of Nature, 2018 6-4 remain undetectable due to high extinction in the galac- region of many galaxies, including the Milky Way. With tic center. Due to the high extinction, the energy den- suitable simplifications, the UEG model deduces the em- sity may sharply drop just outside of the sub-core region pirical M-Sigma relation, thus validating the proposed 0 (r0 < r < r0), from its high uniform value Wτn in the model. The hypothesis that there is a SMBH at the 0 sub-core region r < r0, to its ideal constant value Wτ0 center of many galaxies may no longer be necessary, in outside the core region R0 > r > r0. order to explain the strong central accelerations observed in the galaxy core. Instead, the hypothetical SMBH is 0 emulated in the form of an equivalent UEG acceleration Wτ (r) = Wτn, r < r0; of sufficiently high magnitude, and with the required 1/r2 Wτn − Wτ0 0 dependence for the Keplerian motion. ' Wτ0 + 0 (r0 − r), r0 < r < r0; r0 − r0 = Wτ0, r0 < r < R0. (7) The strong magnitude of the UEG force is possible From (6,7,2), the density factor β(r) would be unity at 2 due to star lights from sufficiently bright stars that are the nucleus r = rn, and have a r dependence towards 0 assumed to exist, in the form of a few individual bright the sub-core boundary r = r0 if the sub-core radius is 0 stars or in an integrated form due to distributed stars, in much larger than the nucleus (r0 >> rn). The density the galactic core of most galaxies. This is consistent with factor would have a reasonable value (1 < β < N) in the the presence and distribution of bright stars in the core region r0 < r < R0 outside the core, which may ideally region of our own Milky Way galaxy. The high UEG ac- be assumed to be a constant β0. In the transition region, 0 celeration is possible without any un-physical, high mass r0 < r < r0, the density factor would have a suitable concentration at the galactic nucleus, as might be re- variation as per (6), related to the energy density Wτ (r) 2 quired based on the conventional Newtonian gravitation. of (7) in the transition region and the 1/r -dependent The un-physicality of the mass concentration, as per the effective energy density Wτe(r) assumed in the SMBH Newtonian gravitation, required to support the fast stel- model of (2). lar motion very close to the galactic center, is often invoked to “predict” the presence of SMBH at the centers of most galaxies, which may no longer be justified. Further, Wτe(r = rn) = Wτ (r = rn) = Wτn, β(r = rn) = 1; a recent observation of radio signals generated from fast- Wτe(r0 < r < R0) ' Wτ (r0 < r < R0) = Wτ0, moving charged particles much closer to a galaxy’s center β(r0 < r < R0) = β0, 1 < β0 << N; [12], which apparently suggested an “event-horizon” of a 2 2 Wτnr Wτ0r b presumed central SMBH, could as well be explained in Wτe(r) = n = 0 = , rn < r < r ; (8) r2 r2 r2 0 terms of the high central acceleration due to radiation from the bright central stars, as per the UEG theory. This would neither require a super-massive body, nor a W (r)N W (r) ' τ , β(r) >> N, black hole, but simply bright stellar (or quasi-stellar) ob- τe β(r) jects at the galactic center, having relatively small mass W r2 W (r) ' WτnN = τn n , (Newtonian, baryonic) as per [11], which exhibit black- τe β(r) 2 r hole-like characteristics with a large UEG-based central 0 r2 β(r0 > r >> rn) = N 2 >> N. (9) acceleration. The bright stellar light coming from the rn galaxy’s center, presumed in the UEG theory, would ac- Note that a SMBH has not been located at the center of tually contradict presence of a central black hole from each and every galaxy. Accordingly, the above proposed which the light supposedly should not have escaped. UEG models are not likely to be applicable universally for all galaxies. One model may be more likely than the other in an elliptical or a disk galaxy. For galaxies The UEG theory was successfully applied to elemen- that are associated with a central SMBH, either of two tary particles [1, 2] as well as single stars [11], which proposed models, or possibly a combination of parts of are ideal spherical structures. The UEG theory was also the two models, may be generally applicable, leading to successfully applied with suitable modification to model the M-Sigma relation (3) for these galaxies. non-spherical structures such as a binary star [11], as well as the disk region of a spiral galaxy [3]. The UEG model of [3] is now extended to model gravitation in the IV. CONCLUSION galactic core, which may not consist of spherically uniform mass distribution, as a substitute for the SMBH The UEG theory, either in its simple form [1, 2] in a hypothesis currently believed. The UEG theory, with uniform material environment, or in a modified form [3] suitable extensions to general problems, may provide a in a non-uniform environment of narrow jets or handle- unified paradigm to model diverse physical phenomena bars, can successfully model the strong stellar accelera- in small as well as large size scales, in structures with tion, with a Keplerian orbit, that is observed in the core ideal spherical as well as non-spherical distributions.

72 N. Das, A UEG Theory of Nature, 2018 6-5

[1] N. Das, “A New Unified Electro-Gravity (UEG) The- tronomy and Astrophysics 486, 971 (2008). ory of the Electron,” Paper #1, pp.4-13, in “A Unified [7] Wikipedia, “M-Sigma Relation,” http://en.wikipedia. Electro-Gravity (UEG) Theory of Nature,” (2018). org/wiki/M-sigma_relation, Retrieved (2013). [2] N. Das, “A Generalized Unified Electro-Gravity (UEG) [8] K. C. Freeman, Astrophysical Journal 160, 811 (1970). Model Applicable to All Elementary Particles,” Paper [9] S. M. Faber and R. E. Jackson, The Astrophysical Jour- #2, pp.14-30, in “A Unified Electro-Gravity (UEG) The- nal 204, 668 (1976). ory of Nature,” (2018). [10] P. A. Crowther, O. Schnurr, R. Hirschi, N. Yusof, R. J. [3] N. Das, “Unified Electro-Gravity (UEG) Theory Applied Parker, and S. P. G. H. A. Kassim, Monthly Notices of to Spiral Galaxies,” Paper #5, pp.59-68, in “A Unified Royal Astronomical Society 408, 731 (2010). Electro-Gravity (UEG) Theory of Nature,” (2018). [11] N. Das, “Unified Electro-Gravity (UEG) Theory Applied [4] A. M. Ghez, S. Salim, S. D. Hornstein, A. Tanner, J. R. to Stellar Gravitation, and the Mass-Luminosity Rela- Lu, M. Morris, E. E. Becklin, and G. Duchene, The tion (MLR),” Paper #4, pp.44-58, in “A Unified Electro- Astrophysical Journal 620, 744 (2005). Gravity (UEG) Theory of Nature,” (2018). [5] L. Ferrarese and D. Merritt, Astrophysical Journal 539, [12] T. E. H. T. Collaboration, The Astrophysical Journal L9 (2000). 875 (2019). [6] A. Barniske, L. M. Oskinova, and W. R. Hamann, As-

73 7-1

The Uniﬁed Electro-Gravity (UEG) Theory Applied to Cosmology

Nirod K. Das Department of Electrical and Computer Engineering, Tandon School of Engineering, New York University, 5 Metrotech Center, Brooklyn, NY 11201 (Dated: May 9, 2018; Revised April 30, 2019) The Uniﬁed Electro-Gravity (UEG) theory is extended for the unique conditions of cosmology, which may support a possible reversal of the current expansionary phase of the universe, explain the current accelerated expansion of the universe without need for any dark energy, and also explain the signatures of the baryon acoustic oscillation (BAO) in the cosmic microwave background (CMB) and in the correlation function of galaxy distribution, without any dark matter. UEG eﬀects due to the the CMB radiation in the recent universe, and in the ionized environment before recombination, as well as those due to anticipated star lights in the future universe, are modeled with suitable cosmological assumptions. This may provide a new theoretical paradigm, which can potentially answer some of the most fundamental questions in cosmology today.

I. INTRODUCTION We attempt in this paper to answer some of the basic questions, based on the UEG theory, with suitable The Unified Electro-Gravity (UEG) Theory has been assumptions and extensions to accommodate unique cos- successfully applied to model elementary particles [1, 2], mological conditions, which might not have been encoun- quantum mechanics [3], stars [4] and galaxies [5]. In the tered in the other problems [1]-[5] solved by the UEG the- simplest form, the UEG theory introduces a new gravi- ory. We would assume an ideal homogeneous, isotropic tational field in proportion to the energy density of ra- universe, and propose a cyclic universe that anticipates diation [1], with the UEG constant γ as the constant of future events. The present expansionary state of the uni- proportionality. This simple form applies only for small verse is assumed to have been adjusted over repeated levels of energy density, having an ideal spherical sym- cycles in the past, such that together with the future an- metry. Suitable modification is needed to model non- ticipated events it would result in a reversal of its current spherical distributions of energy density in a binary star expansion, leading to a complete cyclic process. Any en- [4], and a spiral galaxy [5], or for higher levels of energy density associated with the CMB radiation, or with ergy density seen in elementary particles [2]. Based on any present and future star lights, would produce new the past successes of the UEG theory under the diverse UEG forces. These new forces, in addition to the New- conditions, we may expect that the UEG theory, with tonian gravity due to conventional matter content of the suitable extensions to account for the unique conditions universe, would constitute the complete physical basis for of cosmology, would help to answer different unresolved the current expansion, possible future contraction, and questions in cosmology today. A fundamental question the geometry of the universe, without need for any ficti- may be opened: is the big bang just a fortuitous one- tious dark energy or dark matter. The new UEG model time event for our universe, or it could be only one of the could potentially explain important cosmological obser- natural sequence of big bounces in a cyclic universe (e.g. vations, such as (a) the supernova distance-redshift mea- [6]), with periodic expansion and contraction? The an- surements [7, 8] without any dark energy or dark matter, swer to this question would hinge on a satisfactory theory and (b) the basic BAO signature in the CMB [10, 11] that could support reversal of the current expansion of without any dark matter, as well as (c) support a possi- the universe, back to a contracting phase, to be followed ble reversal of the current expansion of the universe, in by a natural big crunch and a bounce, which the cyclic order that the universe can contract and then cycle back model would presume. Another basic question would be, to maintain a periodic process. The regime of big-bang what is the physical basis for the apparent accelerated ex- nucleo-synthesis (BBN) [13, 14], when nuclei of light ele- pansion [7, 8] of the current universe? Any new physics is ments are believed to have been synthesized in the early expected to emulate the hypothetical dark energy, which universe, is assumed to be unaffected by the UEG theory. is invoked by scientists today to explain the accelerated This is possibly because in the BBN regime the photons expansion [7, 8]. The new physics is also expected to in the cosmic radiation would remain tightly coupled to emulate the hypothetical dark matter, which apparently the highly ionized material environment, which may not explains the signature of the baryon acoustic oscillations contribute to any significant UEG forces. In addition, (BAO) in the cosmic microwave background (CMB) [9– in this regime when the universe is dominated by radia- 11] as well as in the large-scale correlation function of tion, any UEG effect may have been highly diluted over galaxy distribution [12]. With the success of the UEG a larger effective region of universe far beyond the ob- theory in modeling the flat rotation curves in spiral galax- servable universe. Accordingly, all successful predictions ies [5], without any need for the hypothetical dark mat- of the BBN may remain largely unaffected by the new ter, the current theory for the BAO and CMB signatures UEG theory. based on the dark matter may no longer be tenable. All the above proposed physics is expected to be based

Copyright # TXu 2-128-436, c 2018 by Nirod K. Das. All Rights Reserved. 74 N. Das, A UEG Theory of Nature, 2018 7-2 on a UEG theory that is applicable at a relatively low the observable universe r = R, attributed to the Newto- level of energy density. The theory in principle may be nian gravitation. The UEG acceleration au due to the extended in the regime of high energy density, as was the CMB radiation is about half of the Newtonian acceler- case for modeling elementary particles, to model very ation ag at r = R due to matter (baryonic) content of early universe. The UEG theory may be extended to the the universe. The au is a fundamental parameter, which highest level of energy density, beyond the levels applica- would shape any cosmological model based on the UEG ble to model elementary particles, where the gravitation theory. may transition from its usual attractive to a new repulsive nature. This reversal of gravitation could provide a −11 2 definitive physical basis for the reversal from a previously ag = (0.049) × a0 = 5.179 × 10 m/s , contracting (big crunch) to the currently expanding (big au = (2.5/5.179)ag = 0.483ag. (3) bounce) universe, supporting an inflation-like [15] or any other suitable form of expansion (or contraction) in the transitional phase. B. UEG Acceleration Due to any Present or We have used available cosmological parameters for Future Star Lights different estimations in this paper, that may have been adopted from the Wilkinson Microwave Anisotropy The Newtonian acceleration ag may be directly ex- Probe (WMAP) 9 year results [10] or from early Planck pressed using Newton’s law of gravitation, in terms of 2 Mission (2013) results [11], as appropriate and adequate the energy density w = ρvc associated with the matter for particular purposes. All parameters are specified, for density ρv. proper interpretation of results with respect to variation of the parameters. Any variations based on more recent 4πGρvR 2 results of the Planck Mission (2015) [16] may not mate- ag = 3 = 1.36w, w = ρvc , rially change the results or conclusions from the study. G = 6.67 × 10−11(m3/s2)/kg, R=46.3 BLy = 4.409 × 1026s. (4) II. BASIC THEORY About 3/4-th of the matter content of the universe is made of hydrogen [13, 18], and only a negligible percent- A. UEG Acceleration Due to the CMB Radiation age of the hydrogen have been used for hydrogen fusion in the stars. Most of the hydrogen content remain un- The UEG acceleration au, associated with the energy used outside of the stars in inter-stellar and intergalactic density of the current CMB radiation at temperature T = space, waiting for possible right conditions to locally col- 2.7250K may be estimated, assuming a nominal value of lapse and light up in the form of stars and galaxies of 3 2 3 the UEG constant γ0 = 0.6 × 10 (m/s )/(J/m ) derived the future. If we ideally allow all the hydrogen to, at from a UEG model of elementary particles [1, 2]. once, form light radiation through hydrogen fusion today, the density of the light radiation would be about 2 0.7% of the energy density [19] (3/4)w = (3/4)ρvc associ- a = γ W = γ (4σ/c)T 4 = 2.5 × 10−11 2, u 0 τ 0 m/s ated with the hydrogen mass density (3/4)ρv. The UEG 3 2 3 0 γ0 = 0.6 × 10 (m/s )/(J/m ), acceleration au produced by this star radiation may be σ = 5.670 × 10−8W/(m2K4), T=2.7250K. (1) expressed in terms of the w, and then compared with the Newtonian acceleration ag of (4). The σ is the Stefan-Boltzman constant, and the c is the speed of light in empty space. The UEG acceleration 0 au may be compared with the cosmological acceleration au = w × 0.007 × γ0 × 0.75 = 3.15w, 0 a0 at the boundary of the observable universe of radius au = (3.15/1.36)ag = 2.3ag = (2.3/0.483)au = 4.8au, R ' 46.3 GLy [17], approximated using the current value 0 −11 −10 2 au = 4.8 × 2.5 × 10 = 1.2 × 10 m/s . (5) of the Hubble constant H ' 67.8 (km/s)/Mpc. 0 The UEG acceleration au due to light radiation of the possible future stars is 2.3 times the Newtonian accelera- a = 1 H2R = 1.057 × 10−9m/s2, 0 2 tion ag at r = R due to matter (baryonic) content of the H = 67.8 (km/s)/Mpc, R=46.3 BLy, universe. Like the UEG acceleration au = 0.483ag due to 0 Mpc=3.0857 × 1022m, 1 BLy=9.4607 × 1024m. (2) the CMB radiation, the au = 2.3ag = 4.8au due to the future star lights is also a fundamental parameter, which The au is smaller than the a0 by about a factor of 40. would shape any cosmological model based on the UEG Assuming the matter (baryonic) density of the universe theory. Assuming that only a negligible fraction of the to be 4.9% of the critical density to maintain the current total primordial hydrogen has so far been used in all the expansion rate associated with the Hubble constant H0, stars, the UEG acceleration due to all the current star 0 we may estimate the acceleration ag at the boundary of lights is negligible compared to the au or the ag.

75 N. Das, A UEG Theory of Nature, 2018 7-3

C. Equivalent UEG Mass and Energy for mass M enclosed in the sphere of radius r = R would be Cosmological Modeling related to the Mu by the same ratio between the respective mass densities ρv and ρuv. 0 The UEG accelerations au and au would be uniform everywhere, in proportion to the associated uniform en- R R 0 R ρu0 r 2 2 R r 2 2 ergy densities Wτ and Wτ , respectively, as per the UEG Mu = ( r )(1 − R ) 4πr dr = ρuv(1 − R ) 4πr dr theory applied in a simple form [1]. In contrast, the ac- 0 0 celeration a due to the Newtonian gravitation increases 4π 2 4π 3 5ρu0 5au g = 12 ρu0R = 30 ρuvR , ρuv = 2R = 4πGR , linearly with distance, assuming a uniform mass den- R sity. Accordingly, the simple UEG model would produce G R ρu0 2 au = 2 ( r )4πr dr = 2πGρu0, (6) much larger acceleration at smaller distances, compared R 0 to the Newtonian acceleration, leading to a possible non- uniform expansion which would be clearly incompatible 3a ρ = g , ρ = 5auρv = (5/3) × 0.483ρ = 0.8ρ , with the fundamental assumption of a uniform, isotropic v 4πGR uv 3ag v v universe. The simple UEG theory may have to be prop- R R r 2 2 4π 3 erly revised for cosmology, requiring basic UEG parame- M = ρv(1 − R ) 4πr dr = 30 ρvR ,Mu = 0.8M. (7) ters to be properly redistributed in proportion to the re- 0 spective parameters from the Newtonian gravity. Equiv- 0 Similarly, the equivalent uniform mass density ρuv as- alently, there may be some new physics at cosmological 0 sociated with the UEG acceleration au, and its weighted scale, which would transform the basic non-uniform ex- 0 mass Mu may also be expressed, and compared with re- pansion due to the UEG gravitation into the expected spective material parameters ρv and M. uniform expansion, leading to effectively the same results as the redistribution model suggested above. 5a0 ρ We will follow a redistribution model for the UEG the- ρ0 = u v = (5/3) × 2.3ρ = 3.83ρ ,M0 = 3.83M.(8) uv 3ag v v u ory, which would confirm to the fundamental assumption of a uniform or homogeneous universe. Basic UEG pa- Using the same weighting factor used above for calcu- rameters, such as equivalent UEG mass and energy, may lating equivalent cosmological mass parameters, we may be redistributed in proportion to the respective quantities also find equivalent kinetic energy parameters of expan- 0 expected from the Newtonian gravity, such that certain sion Wu, Wu and W , associated with the UEG masses 0 total measure of the UEG parameters are conserved. Mu, Mu and the material mass M, respectively. A simple objective measure of conservation may be to ensure the total integration of a UEG parameter over R the volume of the observable universe of radius r = R R 1 2 ρu0 r 2 2 Wu = 2 v r (1 − R ) 4πr dr to remain fixed. Here, the volume of the observable uni- 0 verse is a naturally objective region. However, the above R R 1 2 ρu0 r 2 2 measure of conservation would truncate the integration = 2 (Hr) r (1 − R ) 4πr dr 0 of the conserved parameter abruptly at r = R, which 4π 2 4 1 2 1 2 2 may seem arbitrary. Instead, conserving a weighted inte- = 120 H ρu0R = 10 (HR) Mu = 10 (H × 46.3BY ) Muc gration over the observable universe, with the weighting 1 67.8 −19 18 2 2 2 = ( × 10 × 1.46 × 10 ) Muc = 1.048Muc , factor at a location proportional to the red-shift factor 10 3.0587 0 0 2 associated with the location, may be physically meaning- Wu = 1.048Muc . (9) ful. The weighting factor would gradually de-emphasize The relationship in (6) between M and the UEG mass- the conserved integrand from its reference unit value at u density coefficient ρ is used in the above derivation. the center r = 0, to zero at the edge of the observable u0 Similarly, universe. The equivalent mass density associated with the uniform UEG acceleration au has a ρ /r distribution. This R 0 W = R 1 v2ρ (1 − r )24πr2dr may be redistributed with a uniform mass density ρ , 2 v R uv 0 such that the total mass M integrated over the observ- u R able universe with a weighting function (1 − r/R)2 is con- R 1 2 r 2 2 = 2 (Hr) ρv(1 − R ) 4πr dr served. The selected weighting function may be shown 0 4π 2 5 1 2 1 2 2 to be the red-shift factor for an ideal universe with a = H ρvR = (HR) M = (H × 46.3BY ) Mc critical material density, which is assumed to be approx- 210 7 7 1 67.8 −19 18 2 2 imately valid for the proposed redistribution. The equiv- = 7 ( 3.0587 × 10 × 1.46 × 10 ) Mc alent UEG mass density ρuv may now be compared with = 1.496Mc2. (10) the material density ρv, which is related to the Newto- 0 nian acceleration ag at r = R. The relation (3) between The Wu, Wu and W are the kinetic energies of the the au and ag may be used here. The weighted material current universe at the expansion velocity v, associated

76 N. Das, A UEG Theory of Nature, 2018 7-4

0.5 with a critical mass density ρvc. It may be useful to acceleration. The α dependence of the horizon in ex- 0 find the corresponding kinetic energies,√Wu0, Wuo and cess of the scale factor is valid for an ideal condition of W0, if the expansion velocity were v0 = 0.049 × v. The a matter-only, flat universe, but is assumed to be ap- v0 is the expansion velocity associated with the current proximately valid in the present model, representing a material density ρv, which is about 4.9% of the critical small exponent in the UEG acceleration in addition to 4 mass density ρvc. the primary 1/α dependence. 0 The total kinetic energy W0 associated with the threshold velocity v0 may be expressed as, 2 0 Wu0 = 0.049Wu = 1.048Muc × (0.049), 0 0 0 2 Wu0 = 0.049Wu = 1.048Muc × (0.049), 0 0 0 Wu+Wu 2 W0 = 2.32(W0 + Wu0 + W u0) = 2.32W0 × (1 + ) W0 = 0.049W = 1.496Mc × (0.049), W 0 2 2 1.048(Mu+Mu) v0 = v × (0.049) . (11) = 2.32W0 × (1 + 1.496M ) = 2.32 × 0.049W × (1 + 3.24) 2 III. A UEG MODEL IN ANTICIPATION OF A = 0.482W = 0.73Mc . (13) FUTURE CONTRACTION OF THE UNIVERSE Now, the total energy W2 available in the future universe at the threshold of possible contraction, after the As mentioned, the universe may be anticipating fu- ideal star burst phase, may be calculated. This is ob- ture star light due to fusion of existing hydrogen con- 0 tained by adding the threshold kinetic energy W0 to the tent, mostly unused to date. If the star burst ideally equivalent mass-energies of the Newtonian mass M, and happens today, all at once, the total mass content would 0 0 of the UEG masses Mu, Mu due to the CMB radiation be M + Mu + Mu, consisting of the UEG masses Mu and 0 and star lights, respectively. Similarly, the total energy Mu due to CMB radiation and star light, in addition to W1 in the current universe may be obtained, by adding the material mass M. This would be associated with a 0 the kinetic energies W and Wu to the mass-energies as- critical expansion velocity v0, which may be related to sociated with the Newtonian mass M and the UEG mass the velocity v defined in (11). The critical velocity v0 0 0 Mu due to the CMB radiation. is the threshold velocity less than which eventual contraction would be possible. We assume that that UEG 4.5 0 2 0 acceleration would reduce as 1/α , as the scale factor α W = (M + Mu + Mu)c + W 2 2 0 increases, in contrast with a 1/α variation for the New- 2 2 2 tonian acceleration. Integration of the acceleration with = (M + Mu)c + 3.83Mc + 0.73Mc , 2 the scale factor would be proportional to the the respec- W1 = (M + Mu)c + W + Wu 2 2 2 tive contributions to the squared critical velocity, which = (M + Mu)c + 1.496Mc + 1.048 × 0.8Mc , would be associated with integration coefficients 1/3.5 3 2 3 W2 > W1, γ = γ0 = 0.6 × 10 (m/s )/(J/m ). (14) and 1, respectively. This fundamentally assumes that the equivalent mass/energy of the UEG field due to radiation Note that the W2 is greater then the W1, which means is modeled as pressure-less, unlike the mass/energy of the excess kinetic energy in the current universe may not conventional radiation which is associated with radiation be enough to guarantee continuation of the expansion. pressure. Otherwise, the above integration coefficient for The above calculations assume a nominal value of γ = γ0. the UEG contribution would have been 1, the same as We may trace the above calculations with γ = αγ0, and that for the Newtonian gravitation due to conventional find the required γ for the current universe to lead to matter. a future universe just at the threshold of possible contraction, as per the ideal model, by solving a quadratic 0 equation of α. 0 2 Mu+Mu 1 2 v0 = (1 + M × 3.5 )v0 ρ +ρ0 = (1 + uv uv × 1 )v2 = 2.32v2. (12) ρv 3.5 0 0 W2 = W1, γ = αγ0, 1.496Mc2 + 1.048 × 0.8αMc2 The 1/α4.5-dependence of the UEG acceleration, as- 2 2 sumed above, may be explained as follows. As the uni- = 3.83Mc α + 0.049 × 1.496(1 + 1.32α)(1 + 3.24α)Mc , verse expands, the energy density of radiation would re- 1 + 0.56α = 2.56α + 0.049(1 + 1.32α)(1 + 3.24α), 4 duce with a 1/α variation, which would directly con- 0.21α2 + 2.22α − 0.951 = 0, √ tribute to the reduction of the UEG acceleration. The −2.22+ 4.93+0.8 horizon of the observable universe is assumed to expand α = 0.42 = 0.41, 0.5 3 2 3 in excess of the scale factor, proportional to α , which γ > αγ0 = 0.41γ0 = 0.25 × 10 (m/s )/(J/m ). (15) may contribute to an additional factor of 1/α0.5 in the reduction of the UEG acceleration, according to the re- Essentially, the above model predicts a lower limit of distribution model of section II C to determine the UEG the γ > 0.25 × 103 (m/s2)/(J/m3), in anticipation of a

77 N. Das, A UEG Theory of Nature, 2018 7-5

2 2 future contraction that would lead to a cyclic universe. time. The ratios W/M = 1.496c and Wu/Mu = 1.048c The predicted lower limit is consistent with the the γ = γ0 between the kinetic energies and the respective masses deduced from a UEG model of elementary particles [1, 2]. for the conventional gravity and the UEG ﬁeld, respec- Conversely, if the value of the γ is given to be equal to tively, as derived in (9,10) for the current universe, would γ0, the above model may be extended into the future apply for all scale factors for a unit normalized squared- for estimation of an eﬀective timing for the anticipated velocity (normalized to a unit value for the current uni- star-burst event in the future. verse at r=R). Accordingly, the excess kinetic energies W∆ and Wu∆ may be expressed in terms of the associated masses M and Mu, respectively, proportional to the 2 IV. A UEG MODEL FOR THE ACCELERATED excess squared-velocity v∆ EXPANSION OF THE UNIVERSE

2 2 2 2 We will model the expansion velocity v as it changes W∆(z) + Wu∆(z) = 1.496M(z)c v∆(z) + 1.048Mu(z)c v∆(z) with the scale factor α < 1, or its associated redshift = W∆(z = 0) + Wu∆(z = 0) 2 2 2 2 factor z > 0, of the universe. The velocity v may be nor- = 1.496M(z = 0)c v (z = 0) + 1.048Mu(z = 0)c v (z = 0), malized with its unit reference equal to the total velocity ∆ ∆ 1.496M(z=0)+1.048M (z=0) v2 (z) = v2 (z = 0) × u of the current universe, at time t = t0. ∆ ∆ 1.496M(z)+1.048Mu(z) 1+0.7Mu(z=0)/M(z=0) = (1 − Ω − Ωu) × . (18) b 1+0.7Mu(z)/M(z=0) v = α˙ = α˙ , α˙ = dα , α˙ (t=t0) H0 dt The relationship (7) between the Mu and M in the 2 2 2 2 1 v (z) = vg(z) + vu(z) + v∆(z), α = 1+z . (16) current universe, and the associated relationship (17) between the Ωb and Ωu, may be used in the above ex- The squared-velocity v2 may be expressed consisting pression. The conventional mass M in the observable 2 2 2 2 of three parts. The first two parts, vg and vu, are con- universe, which is associated with a 1/α = (1 + z) de- tributed from the critical velocities that could be sup- pendence of its acceleration due to Newtonian gravity, ported by the Newtonian gravity and the new UEG field, remains constant with the scale factor α. Whereas, the respectively. The contribution from the Newtonian grav- equivalent UEG mass Mu, which is associated with a 4.5 4.5 ity of conventional matter is Ωb(1+z), where Ωb = 0.049 is 1/α = (1 + z) dependence of the UEG acceleration, the fractional density of the conventional matter, with re- would change with a 1/α2.5 = (1 + z)2.5 dependence. spect to the critical density necessary to support the observed expansion of the current universe. The functional 2 1+0.7×0.8 dependence of the UEG contribution v2 was explained in v (z) = (1 − 1.23Ω ) × u ∆ b 1+0.7×0.8(1+z)2.5 section III. = Ω × 1.56 , ∆ 1+0.56(1+z)2.5 2.5 2 M(z) = M(z = 0),Mu(z) = Mu(z = 0)(1 + z) , vg(z) = Ωb(1 + z), 3.5 Mu(z = 0) = 0.8M(z = 0), 2 3.5 ρuv (1+z) vu(z) = Ωu(1 + z) = ρ Ωb , v 3.5 Ω∆ = 1 − Ωb − Ωu = 1 − 1.23Ωb = 0.94 . (19) ρuv 1 1 Ωu = ρ × 3.5 Ωb = 0.8 × 3.5 Ωb = 0.23Ωb. (17) v Combining (19,17) in (16), the total normalized veloc- 2 ity v(z) may be expressed. The third term v∆ in (16) is the contribution of the 2 excess velocity v∆. The v∆ is the total squared-velocity in excess of the first two terms that are critically sup- 2 3.5 v (z) = Ωb(1 + z) + Ωu(1 + z) ported by the Newtonian gravity and the UEG field, re- 1.56 + (1 − Ωb − Ωu)[ ] spectively. The excess squared-velocity is 1 − Ωb − Ωu 1+0.56(1+z)2.5 at t = t , and is required to change with the scale fac- 0 = 0.049(1 + z) + 0.011(1 + z)3.5 tor with the following basic condition. The kinetic en- + 0.94[ 1.56 ]. (20) ergy associated with the excess velocity, enclosed inside 1+0.56(1+z)2.5 a co-moving spherical volume, with its radius changing in proportion with the scale factor, with a current reference The above derivations assumes a specific set of avail- radius equal to the current horizon distance, is required able cosmological parameters, with the Hubble constant to be conserved independent of the scale factor. The ki- H0 = 67.8 (km/s)/Mpc and Ωb = 0.049. Any variations netic energy over any given spherical volume is defined due to changes of the parameters may be similarly traced such that the ratio of the kinetic energy to the mass in by introducing additional factors for the different parts, the volume at a given time, for a unit reference velocity in terms of fractional changes of the appropriate param- at the spherical boundary, is equal to that over the en- eters, which may be expressed in terms of the fractional tire volume of the observable universe, at the particular change h67.8 of the measured Hubble constant H0 with

78 N. Das, A UEG Theory of Nature, 2018 7-6 respect to the best value of H0 = 67.8 (km/s)/Mpc cur- are plotted in Fig.3 in relative magnitudes with respect rently available. to a nearly empty universe with ΩM = 0.2, ΩΛ = 0, which are consistent with measurements of high-z supernovae from [7]. The theoretical results from the UEG 2 −2 −1 3.5 v (z) = Ωbh67.8(1 + z) + Ωuh67.8(1 + z) and standard models over a larger range of redshift are −2 −1 1+0.56h67.8 shown in Fig.4, which maybe similarly compared with + (1 − Ωbh67.8 − Ωuh67.8)[ 2.5 ] 1+0.56h67.8(1+z) results from [21] that include measurements of gamma- ray bursts (GRB). In summary, the UEG model with no = 0.049h−2 (1 + z) + 0.011h−1 (1 + z)3.5 67.8 67.8 dark matter or dark energy is shown to be consistent with −2 −1 1+0.56h67.8 + (1 − 0.049h67.8 − 0.011h67.8)[ 2.5 ], measurements of high-z supernovae, and possibly GRBs, 1+0.56h67.8(1+z) emulating the standard model with the hypothetical dark H0 matter and dark energy. That is a remarkable develop- h67.8 = −1 −1 . (21) 67.8 kms Mpc ment. It may be noted that, unlike the conventional matter The normalized squared-velocity v2 as derived above which is associated with a definitive mass density, the is compared in Fig.1 with those from the standard cos- equivalent UEG mass density is not a definitive quan- mological model, where v2 = Ω (1 + z) + Ω /(1 + z)2 + M Λ tity, but is modeled in relation to the horizon distance of (1 − Ω − Ω ). The acceleration in the standard model M Λ the observable universe, as per the redistribution model due to the dark-energy term, with its fractional density developed in section II C. The above derivations use an Ω , is emulated in the UEG model without any need Λ approximate formulation for the dependence of the hori- for the hypothetical dark-energy. Further, only the con- zon distance of the observable universe on the scale fac- ventional matter with the fractional density Ω is used b tor. We assume that the horizon distance expands with in the UEG model, without need for any additional dark a α0.5 dependence as an excess factor, multiplied to the matter. Whereas, the standard model uses the mass frac- scale factor α of normal expansion of the universe. This tion of Ω = Ω + Ω , which includes the density Ω M b dm b dependence is the ideal case for a matter-only universe. of the conventional baryonic mass and additional den- It is used in the derivations only as an effective depen- sity Ω of dark matter. Unlike the mass M due to dm dence of the horizon, as a good reference, which may work the conventional baryons or dark matter, the equivalent in an overall average sense over a range of the scale fac- UEG mass M is reduced as the universe expands. This u tor. A more accurate formulation may require the above results in an effective outward acceleration as the uni- dependence to be modeled with a variable exponent of verse expands, so that the kinetic energy in the observ- the scale factor, that varies as a function of the scale able universe due to the excess velocity is conserved, as factor. Refinement of the reference analysis may be pos- the UEG model requires. This effective acceleration is sible by estimating the exponent under different specific seen only in the current and recent past of the universe conditions, and revising the associated formulations ac- (z < 1 from Fig.1) when the excess velocity is signifi- cordingly. However, it may require a involved numerical cantly larger than the critical velocity that can be sup- computation process, in order to rigorously model the ported by the conventional and UEG masses. In the past, horizon distance and incorporate it into the UEG model. the UEG mass Mu was larger, and consequently the excess velocity to conserve the excess kinetic energy was smaller. In sufficient past, the smaller excess velocity, as compared to the critical velocity in the current universe, V. A UEG MODEL FOR THE ACOUSTIC was even much smaller than the larger critical velocity HORIZON AND DARK MATTER BEFORE that could be supported at the time. This would result in RECOMBINATION having the effective outward acceleration associated with the smaller excess velocity, due to the kinetic-energy con- Based on the comparisons with the standard model, servation discussed above, to be much smaller than the in consistency with the current observations, the above normal gravitational deceleration (inward acceleration) model for the UEG mass based on the size of the ob- in the sufficient past associated with the larger critical servable universe appears to be reasonably valid in the velocity supported at the time. Accordingly, the expan- recent universe, covering redshifts of the order of z ∼ 10 sion of the universe was effectively decelerating in the possibly even larger. However, for much larger redshifts sufficient past (z > 1 from Fig.1), while it is accelerating questions may arise about the fundamental validity of the only currently and in recent past (z < 1 from Fig.1), as redistribution model of section II C, based on the size per the UEG model, which would be consistent with the of the observable universe. The radial distance of the standard model with ΩΛ ∼ 0.76. observable universe may be referred to as the “matter The luminosity distance DL [20] derived from the ve- horizon”, which is the farthest distance the matter pro- locity function v(z) is plotted in Fig.2, which shows the duced in the earliest universe would appear to be located UEG model with Ωb = 0.49 is comparable to the standard to a current observer. The universe is assumed to be model with ΩΛ = 0.76, for z < 2. The results from the expanding with a gravitational deceleration (or accelera- UEG model and the standard model with the ΩΛ = 0.76, tion toward the reference origin), due to the conventional

79 N. Das, A UEG Theory of Nature, 2018 7-7

FIG. 1.

FIG. 2. attractive form of gravity, since the time of the earliest level of energy density. The horizon associated with this matter formation. However, before the era of matter for- gravitationally repulsive phase of accelerated expansion, mation, the earliest radiation-only universe might have which may be referred to as the “radiation horizon,” is undergone an inflationary phase [15] with gravitational expected to be much farther than the “matter horizon”. acceleration (away from the reference origin), possibly It is conceivable that the effective UEG mass may have following a big crunch, and supported by a form of re- to be modeled differently for different scale factor of ex- pulsive gravity based on a UEG theory at the highest pansion, based on the matter horizon, or the radiation

80 N. Das, A UEG Theory of Nature, 2018 7-8

FIG. 3. horizon, or possibly a combination of the both, depend- be valid just after and sufficiently during the recombi- ing on the relative matter and radiation contents of the nation process. However, before the recombination the universe. In the current universe, which is matter domi- universe was ionized and opaque to radiation. Therefore, nated, the conventional matter horizon appear to deter- unlike in the current universe, there would not have been mine the UEG physics accurately. However, as one gets any CMB radiation available before the recombination close to the scale of recombination (z ∼ 1100), the radi- to produce UEG forces. Accordingly, the effective UEG ation density would be comparable to the matter (bary- mass density sufficiently before the recombination would onic) density, in which case the UEG model applicable in have been zero, but it increased to be effectively about the current universe may not be valid. 5.47 times the baryonic matter density after the recom- The UEG model in the time frame around the recombination, which may be considered a relatively abrupt bination may require the knowledge of the radiation hori- process. This is similar to the possible increase of UEG zon, which is not available without a definitive model of effects in the current universe due to anticipated future the inflationary phase. However, considering that the ra- star lights, modeled in section III. As the similar case in diation horizon would be much farther than the matter the current universe, the expansion velocity before the horizon, it may be assumed that the effective UEG mass recombination would have been larger than the critical in this time frame would be much smaller than that mod- velocity that could be supported by the baryonic and eled using the matter horizon, as it was done in the UEG radiation mass densities, in anticipation of the increased model for the current universe. The effective UEG mass UEG mass and associated kinetic energy after the recom- at the recombination may emulate the hypothetical dark bination. We will follow a formulation similar to that in matter in the standard model. Accordingly, we may as- section III, in order to find the expansion velocity before sume the effective UEG mass enclosed inside the matter the recombination. universe, or its associated matter density, to be larger For convenience, all parameters used in the following than the respective baryonic parameters by a factor of modeling will refer to the recombination time scale as Ωdm/Ωb = 0.268/0.049 = 5.47, as a reference, effective the current reference with zero redshift or unit scale fac- over the duration of recombination. The Ωb and Ωdm tor, and the final results may be properly scaled back are the baryonic and dark matter fractions of the cur- to the actual current universe as needed. Distinct from rent universe, respectively, used in the standard model. the actual current universe, the universe at the recom- The ratio of the baryonic and dark matter is assumed to bination would have appreciable mass density of radia- remain constant over all different time scales, in accor- tion (photon) and neutrino, as compared to the baryon dance with the dark-matter characteristic of the standard mass density, and all parameters associated with the con- model. ventional photon radiation would be referred to with a The above effective UEG mass density is assumed to subscript r0, and those associated with total relativistic

81 N. Das, A UEG Theory of Nature, 2018 7-9

FIG. 4.

particles (photon and neutrino) with a subscript r. v0 is the baryonic and radiation masses, and any equivalent the reference critical velocity supported by the baryonic UEG mass due to the radiation at the recombination, 2 matter content, and the associated kinetic energy of the may be expressed in terms of the v0. As discussed above, baryons is W0. Unlike the matter horizon, the radiation the squared-velocity due to the UEG mass would require horizon can be shown to expand proportional to the scale a factor 1/3. This is the factor needed in the derivation factor. Assuming that the UEG model at the recombina- of the squared-velocity, implemented as the integration tion would be based dominantly on the radiation horizon, of the 1/α4-dependent UEG acceleration with the scale as discussed earlier, all special adjustments (with 0.5 ex- factor. ponent of scale factor used for modeling in the current universe) to account for variation of the horizon distance may not be needed for modeling in the recombination 2 phase. Accordingly, the UEG acceleration, associated v0 = (1 + Mr + Mu × 1 )v2 = (1 + ρrv + ρuv × 1 )v2 critical squared velocity which is obtained by integration 0 M M 3 0 ρv ρv 3 0 Ωr Ωdm 1 2 of the UEG acceleration with the scale factor α, and the = (1 + Ω + Ω × 3 )v0. (22) effective UEG mass or mass density, would have a 1/α4, b b 1/(3α3), and 1/α2 dependence for modeling in the recombination phase. This is in contrast with the 1/α4.5, Ω /Ω = 5.47 1/(3.5α3.5), and 1/α2.5 for the respective dependencies The dm b is the ratio of the dark and bary- used for modeling in the current universe. Further, the onic masses, whose value is maintained independent of distance to the radiation horizon, which is assumed to the scale factor, and Ωr/Ωb = 25/12 is the ratio of the radiation (photon) and baryonic masses at the scale of be the effective horizon for the UEG modeling in the 0 recombination phase, is much farther than the conven- recombination. The kinetic energy of expansion W0 after tional matter horizon. Therefore, in the recombination the recombination may be expressed in terms of the kinetic energy W0 of the baryonic matter associated with phase, the effective kinetic energy due to the expansion 0 velocity, which linearly increases with distance, would be the reference velocity v0. The W0 is also approximately much larger than the mass-energy enclosed by the effec- equal to the total energy W2 after the recombination. tive horizon. Accordingly, different mass-energies may The same relationships (9,10) between the energy W and be ignored in the computation of the total energy en- mass M for the baryonic mass, and the Wu and Mu for closed by the horizon, for the present derivation in the the UEG mass (equivalent to dark matter content at re- recombination phase. combination), in the current universe is also used here as a rough estimate, although the associated redistribution 0 2 The squared-velocity v0 just after recombination, weighting factor (see section II C) may not apply as well which is the total critical squared-velocity supported by at the recombination phase.

82 N. Das, A UEG Theory of Nature, 2018 7-10

FIG. 5.

0 2 0 2 0 v0 Mr 1.048Mu v0 Ωr 1.048Ωdm W = W0 × (1 + + ) W = W , W × (1 + + ) 0 v2 M 1.496M 2 1 2 0 Ω 1.496Ω 0 v0 b b 2 2 + v0 1.048Ω v (z=0 ) Ωr 0 Ωr dm 0 = W0 × (1 + ), = 2 W0 × (1 + Ω + 1.496Ω ),W2 = W0. (23) v2 Ωb v0 b b 0 Ωr Ωdm 1 Ωr 1.048Ωdm (1 + Ω + Ω × 3 )(1 + Ω + 1.496Ω ) The squared-velocity v2(z) may be expressed as a func- b b b b v2 (z=0) tion of the redshift z > 0, with reference zero redshift Ωr ∆ Ωr = (1 + Ω + 2 )(1 + Ω ), at the recombination. It consists of two principal parts, b v0 b which are the critical squared-velocities supported by the Ω 1.048Ω 2 (1+ Ωr + dm × 1 )(1+ Ωr + dm ) baryonic and radiation contents, and the additional term v (z=0) Ω Ω 3 Ω 1.496Ω ∆ = b b b b 2 2 Ωr v∆ in excess of the principal parts. The total energy v0 (1+ ) + Ωb W1 just before recombination (z = 0 ), which is approx- − (1 + Ωr ). (26) imately equal to the total kinetic energy as discussed, Ωb may be expressed using the v2(z = 0+). The sum of the kinetic energies W∆(z) and Wr∆(z) of the baryonic and radiation masses, respectively, associ- 2 2 2 Ωr 2 2 v (z > 0) = v0(1 + z) + v0 Ω (1 + z) + v∆(z), ated with the excess velocity v∆(z) may be required to be b conserved for all redshifts. This would lead to expressing v2(z = 0+) = v2 + v2 Ωr + v2 (z = 0), (24) 2 0 0 Ωb ∆ the z-dependence of the squared-velocity v∆(z), in terms 2 of the v∆(z = 0) solved above. Using this result in (24) would provide a complete expression for the v2(z > 0) 2 + v (z=0 ) Mr before recombination. W1 = 2 W0 × (1 + M ) v0 v2(z=0+) = W × (1 + Ωr ). (25) v2 (z) 2 0 Ω W (z) + W (z) = ∆ (W + W Ωr (1 + z)) v0 b ∆ r∆ 2 0 0 Ω v0 b 2 = W∆(z = 0) + Wr∆(z = 0) The excess squared-velocity v∆(z = 0) just before re- v2 (z=0) combination may be solved by enforcing energy conser- ∆ Ωr = 2 (W0 + W0 Ω ), (27) vation with W2 = W1. v0 b

83 N. Das, A UEG Theory of Nature, 2018 7-11

where Ωr0/Ωb = 15/12 is the ratio of radiation (pho- (1+ Ωr ) ton) and baryon mass densities at the recombination. Ω 2 2 b The values of the rs from the UEG model with vari- v∆(z) = v∆(z = 0) Ω . (28) (1+ r (1+z)) able UEG mass-density, defined by the parameter η Ωb = (ρuv/ρv)/(Ωdm/Ωb), are compared in Fig.6 with those In the above derivation we ideally assumed the ratio of from the standard model with the fixed parameter the UEG and baryonic masses or the respective mass den- Ωdm/Ωb = 5.47 for the dark matter. The reference sities Mu/M = ρuv/ρv to be equal to Ω /Ω , abruptly dm b value for the parameter η = 1 corresponds to ρuv/ρv = after the recombination, and equal to zero in the ionized Ωdm/Ωb = 5.47. The results show that the UEG model environment before the recombination. However, in re- using only baryon and radiation contents, and UEG ef- ality the UEG effect would gradually transition as the fect of the radiation, but without any dark matter, would ionization changes during this phase. In order that this emulate the standard model that includes the baryon and transitional UEG effect emulates the effect of the dark radiation, and additional dark-matter, with a reasonable matter of the standard model, which is maintained at adjustment of the parameter η ∼ 1.75, η > 1, as we ex- its constant value throughout the phase, the above ratio pected. of the UEG and baryoninc masses may have to be suffi- The agreement of the sound horizon rs from the stan- ciently larger than the ratio Ω /Ω by a factor η > 1. dm b dard model with the UEG model for η ∼ 1.75 means that With introduction of this factor, the complete function all signatures of the baryon acoustic oscillation (BAO) in v2(z > 0) is expressed as follows. the current universe (in the CMB [10, 11] and in the correlation distance of galaxy density [12]) predicted by the v2(z > 0) = v2(1 + z) + v2 Ωr (1 + z)2 + v2 (z) standard model, that are based on the horizon distance 0 0 Ωb ∆ rs as a reference “ruler”, would be emulated in the UEG = v2(1 + z) + v2 Ωr (1 + z)2 0 0 Ωb model for the given η. In addition, this value of η ∼ 1.75, Ω 1.048Ω which is reasonably larger than unity, means that the (1+ Ωr +η dm × 1 )(1+ Ωr +η dm ) 2 Ωb Ωb 3 Ωb 1.496Ωb Ωr UEG mass during the transition phase of recombination + v0[ − (1 + )] (1+ Ωr ) Ωb could potentially emulate the effective gravity of the dark Ω b matter in acoustic oscillations before the recombination. (1+ Ωr ) Ω Therefore, the signature of the dark matter in the CMB × [ b ]. (29) (1+ Ωr (1+z)) could also be replicated by the UEG effects. The char- Ωb acteristics of the BAO in the ionized environment before For comparison, the squared-velocity function v2(z) recombination [22] needs to be reviewed, by properly in- from the standard model is expressed in the following cluding the new UEG effects, in order to make a more form. definitive evaluation. In any case, the present results in- dicate that all essential signatures of the BAO observed Ω in the current universe could be potentially explained by v2(z > 0) = v2(1 + z) + v2 dm (1 + z) 0 0 Ωb the UEG theory without need for any dark matter, in consistency with the standard model predictions that re- + v2 Ωr (1 + z)2, (30) 0 Ωb quire the hypothetical dark matter. That is a significant development. where Ωr/Ωb ∼ 25/12 is the ratio of the radiation (photon and neutrino) and baryon masses at the time of recom- Furthermore, a somewhat different value of the Hubble bination. constant H0 6= 67.8(km/s)/Mpc, that can be consistent 2 2 with the CMB observations as well as recent measure- Fig.5 shows the normalized velocity function v (z)/v0 of (29) for different values of the parameter η, that are ment of the H0 in the local universe [23, 24], could also compared with the corresponding function (30) from the be accommodated in the present UEG theory, by suitable standard model. adjustment of the parameter η. This is significant as well, The sound horizon distance rs at the recombination, in order to overcome any tension between the CMB sig- scaled back in the current universe, may be derived using nature and the local H0 measurement, which is getting the v(z)/v0 functions of (29,30), in terms of the Hubble increasingly difficult to resolve based on a conventional constant H0 = 67.8 (km/s)/Mpc and the fractional mat- dark-energy model using a cosmological constant [23, 24]. ter content Ωb = 0.049 in the current universe, the scale factor αc = 1/(1 + zc) = 1/1100 at the recombination, and the sound speed cs(z) in the photon-baryon plasma. VI. DISCUSSION AND CONCLUSION

∞ c (z)dz The UEG theory, originally developed for modeling el- r = √ 1 R s , s [v(z)/v ](1+z) H0 0.049(1+zc) z=0 0 ementary particles [1, 2], is applied under special condi- c tions to cosmology. The new theory explains the accel- cs(z) = s , (31) 3Ω erating expansion of the recent universe, consistent with 3(1+ b 4Ωr0(1+z) supernova measurements [7, 8]; explains the expansion of

84 N. Das, A UEG Theory of Nature, 2018 7-12

FIG. 6. the universe before recombination, consistent with mea- CMB radiation, or with a uniform distribution of star sured signatures of BAO in galaxy distributions [12] and lights produced from a uniform galaxy distribution. The in the CMB [10, 11]; could resolve a potential tension be- equivalent mass associated with the uniform UEG accel- tween different measured values of the Hubble constant, eration, based on a simple UEG theory, can be shown to consistent with the CMB signatures as well as recent local be non-uniform in distribution. This would be in con- measurements of the H0 [23, 24]; and the theory supports tradiction with a uniform mass distribution, with asso- a future contraction of the universe presumably leading ciated gravitational acceleration linearly increasing with to a cyclic process. The theory is based on gravitation of radial distance, that would be expected in consistency conventional matter in the universe, with additional ac- with the basic cosmological assumption of a homogeneous celeration and equivalent mass due to new UEG effects universe. In order that the UEG theory be consistent of the CMB radiation and any future star lights, without with the basic cosmological assumption, suitable adjust- need for hypothetical dark matter or dark energy. As per ment in the simple UEG theory is needed. A new model the UEG theory, the energy density associated with any is proposed and implemented in the paper, in the form of radiation, such as the CMB radiation or star lights, would an effective UEG model for cosmology. Based on the suc- produce new gravitational acceleration. The new UEG cess of the proposed approach, validated in consistency acceleration is modeled in terms of an equivalent mass with measured observations, the effective model might distribution, just like conventional gravitational acceler- be accepted as a fundamental new UEG theory for cos- ation is modeled in terms of a conventional mass distri- mology. Alternatively, some new physical process in the bution as its source. The UEG mass distribution may cosmological scale may allow readjustment of the expan- also be treated like any distribution of conventional in- sion and mass distribution of the universe, in response to ertial mass, and accordingly be associated with its rest the uniform UEG accelerations supported by the simple as well as kinetic mass-energy based on special relativity. UEG theory, which may ultimately lead to the same out- These are significant new understandings on the basic come as predicted from the effective UEG model. Both principles of gravity, mass and energy, in the context of the above possibilities would be theoretically equivalent the new UEG theory. to each other. The above new UEG effects are to be adopted in con- The effective UEG model proposes suitable redistribu- sistency with the basic cosmological assumption of a ho- tion of the UEG mass and energy enclosed inside a spher- mogeneous and isotropic universe. However, the UEG ical volume, defined by a “particle horizon” associated theory, when applied in its simple form, would lead to a with first creation of conventional matter, or possibly uniform acceleration independent of the radial distance by a “radiation horizon” (much farther than the particle from a center of observation. The uniform acceleration horizon) associated with the early radiation-dominated is associated with a uniform energy distribution of the inflationary universe, or even by a suitable combination

85 N. Das, A UEG Theory of Nature, 2018 7-13 of the above two horizons based on the mass and ra- proposed UEG model could be implemented in principle. diation content of the universe at a given time. The For analytical simplicity we may have assumed certain particle or radiation horizon would carry fundamental functional variation for the horizon distance of the ob- significance in the proposed UEG theory of cosmology, servable universe, at different scale lengths or associated having critical theoretical as well as philosophical impli- red-shifts. These assumptions may be effective in an av- cations. The effective UEG model in the present work erage sense over a range of scale lengths, with adequate is implemented using a physically reasonable redistribu- validity for the specific results presented. For accurate re- tion model, that may be adequate for certain objectives. sults in general applications, the functional variation of However, further theoretical or observational develop- the horizon distance may have to be accurately tracked, ment may be needed for a more definitive and rigorous using a more involved numerical computation. UEG redistribution model. In addition, the physics of the With evident success in answering some of the key BAO [22] may need to be re-evaluated based on the pro- questions in cosmology today, as presented in this paper, posed UEG model, in order to properly account for any the UEG theory may provide a new theoretical frame- UEG effects on the CMB signature. The UEG effects in work for any future advancement in physical cosmology. the ionized environment before recombination could po- With successful prior application of the UEG theory in tentially emulate effects of the hypothetical dark matter, particle physics [1, 2], quantum mechanics [3], and stel- as we have assumed in section V. A focused study of the lar and galactic modeling [4, 5], the present extension BAO physics, based on the new UEG theory, would be of the UEG theory to cosmology may help to establish needed for a rigorous understanding and analysis, beyond a complete, unified theory of physics, fundamentally in- the scope of the present work. tegrating gravity, electromagnetics, as well as quantum Aside from the possible fundamental advancement in mechanical concepts, with validity in in the smallest (el- the proposed UEG theory of cosmology, as discussed ementary particles) to the largest (cosmology) domains above, some analytical or numerical refinement in the of the nature.

[1] N. Das, “A New Unified Electro-Gravity (UEG) The- [11] P. A. R. Ade, N. Aghanim, C. Armitage-Caplan, M. Ar- ory of the Electron,” Paper #1, pp.4-13, in “A Unified naud, et al., “Planck 2013 Results, Planck Collaboration, Electro-Gravity (UEG) Theory of Nature,” (2018). XVI: Cosmological Parameters,” Astronomy and Astro- [2] N. Das, “A Generalized Unified Electro-Gravity (UEG) physics, Special Feature, Vol. 571 (A16), p.1-66, Novem- Model Applicable to All Elementary Particles,” Paper ber (2014). #2, pp.14-30, in “A Unified Electro-Gravity (UEG) The- [12] D. J. Eisenstein, I. Zehavi, D. W. Hogg, R. Scoccimarro, ory of Nature,” (2018). et al., Astrophysical Journal 633, 560 (2005). [3] N. Das, “Unified ElectroGravity (UEG) Theory and [13] G. Steigman, Annual Review of Nuclear and Particle Sci- Quantum Electrodynamics,” Paper #3, pp.31-42, in ence 57, 463 (2007). “A Unified Electro-Gravity (UEG) Theory of Nature,” [14] R. A. Alpher, H. Bethe, and G. Gamow, Physical Review (2018). 73, 803 (1948). [4] N. Das, “Unified Electro-Gravity (UEG) Theory Applied [15] A. H. Guth, Physical Review D 23, 347 (1981). to Stellar Gravitation, and the Mass-Luminosity Rela- [16] P. A. R. Ade, N. Aghanim, M. Arnaud, M. Shdown, et al., tion (MLR),” Paper #4, pp.44-58, in “A Unified Electro- “Planck 2015 Results, Planck Collaboration, XII: Cosmo- Gravity (UEG) Theory of Nature,” (2018). logical Parameters,” Astronomy and Astrophysics, Spe- [5] N. Das, “Unified Electro-Gravity (UEG) Theory Applied cial Feature, Vol.594 (A13), p.1-63, October (2016). to Spiral Galaxies,” Paper #5, pp.59-68, in “A Unified [17] P. Halpern and N. Tomasello, Advances in Astrophysics Electro-Gravity (UEG) Theory of Nature,” (2018). 1 (2016). [6] P. J. Steinhardt and N. Turok, New Astronomy Reviews [18] H. Suess and H. Urey, Reviews of Modern Physics 28, 53 49, 43 (2005). (1956). [7] A. G. Riess, A. V. Filippenko, P. Challis, A. Clocchiatti, [19] E. Bohm-Vitense, Introduction to Stellar Astrophysics et al., The Astronomical Journal 116, 1009 (1998). (Ch.8) (Cambridge University Press, 1992). [8] S. Perlmutter, G. Aldering, G. Goldhaber, R. A. Knop, [20] S. M. Caroll, W. H. Press, and E. L. Turner, Annual et al., The Astrophysical Journal 517, 565 (1999). Review of Astronomy and Astrophysics 30, 499 (1992). [9] A. A. Penzias and R. W. Wilson, Astrophysical Journal [21] E. L. Wright, arXiv:astro-ph/0701584v3 (2007). 142, 419 (1965). [22] W. Hu, Annals of Physics 303, 203 (2003). [10] G. Hinshaw, D. Larson, E. Komatsu, D. N. Spergel, [23] A. G. Riess, S. Casertano, W. Yuan, et al., The Astro- et al., “Nine-Year Wilkinson Microwave Anisotropy physical Journal 861, 126 (2018). Probe (WMAP) Observations: Cosmological Parameter [24] A. G. Riess, L. M. Macri, S. L. Hoffman, et al., The Results,” The Astrophysical Journal, Supplement Series, Astrophysical Journal 826, 56 (2016). Vol.208 (2), p.19-43, October (2013).

Part-III: Derivation of the Maxwell’s Equations of Electromagnetic Theory from First Principles, which Supersede Newtonian Mechanics

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A New Approach to Teaching Maxwell’s Equations, as Derived from Simple Relativistic Transformation Principles: A Tutorial

Nirod K. Das

Department of Electrical and Computer Engineering Tandon School of Engineering, New York University, Five Metrotech Center, Brooklyn NY 11201 (Dated: August, 2016)

Abstract

In this paper we present a simple, rigorous approach to introduce electromagnetic theory, based on basic relations for relativistic space-time transformation, and a new general principle of charge invariance. Basic space-time transformation equations are introduced reasonably quickly, using two independent methods: one is based on mathematical deduction, and the other on a physical analysis. More elaborate interpretation of the space-time transformation equations, and any further advancement into relativistic mechanics, such as mass and force transformation, are not necessary in the entire development. The space-time transformation equations are then used to derive basic transformation relations for electric current and charge densities, which enforce continuity or conservation of the electric charge. This is followed by derivation of the transformation relations for the Gauss’ Laws for electric and magnetic ﬁelds, leading to the Ampere’s and Faraday’s Laws, respectively, which together constitute the Maxwell’s Equations. The material is intended to ﬁll an educational need to introduce modern relativity theory in teaching engineering electromagnetics. This is intended to provide an alternative interpretation as well as a complete ”derivation” of the Maxwell’s Equations, that can be adopted in an early, basic level of teaching. The approach would be suitable for a senior (even junior) undergraduate or an introductory graduate engineering class.

1. Introduction

1.1. Background. The concept of the Ampere’s Law relating magnetic field to conduction and displacement currents, and the concept of the Farady’s Law relating electric field to a time-varying magnetic field, were established historically through practical experimentation and gradual generalization. The general findings were formalized in suitable mathematical forms for analytical use, with only limited physical and philosophical rationalization. The Ampere and Faraday’s Laws, in their most generalized forms, constitute the Maxwell’s Equations [1], which are known to supersede all other basic circuit as well as field laws, such as Kirchoff’s Current Law (KCL), Kirchoff’s Voltage Law (KVL), Biot-Savart Law, and Gauss’ Laws of electric and magnetic fields. Such consistency with other pre-existing laws, as well as their success in understanding and predicting practical experiments, develop remarkable confidence in the Maxwell’s Equations. They have so far stood the test of time, and are now accepted to be valid with farthest universal reach, except possibly in the nuclear and quantum-mechanical domains. In spite of all the remarkable generality and success of the theory, the underlying concepts of the Maxwell’s Equations, or equivalently the Ampere and Faraday’s Laws, may appear quite “mysterious” to basic human intuition. Without some form of logical explanation of the principles, one has to accept them solely on the basis of confidence in the experimental observations, and their consistent validity to date. This may not be fully satisfactory and insightful. Later on, Einstein’s theory of special relativity was established as an independent physical principle [2, 3]. This is founded upon the basic assumption that the speed of light or any electromagnetic wave, propagating in a uniform free-space medium, is a fixed constant when measured from any reference frame. Further, if the reference frames are “unbiased” (referred to as inertial reference frames,) then the light would propagate with the fixed invariant speed, along any given straight-line path. The theory also assumes that any “fundamental” physical experiment, designed to establish a particular “fundamental” physical principle or parameter, would lead to the same conclusion when measured or observed from any of the inertial frames of reference. This led to new transformation equations, relating the space and time coordinates of two inertial reference frames, that are consistent with the basic assumptions in the principles of relativity. In this paper, we will use the new space-time transformation relations to deduce the Maxwell’s Equations from the Gauss’ Laws of electric and magnetic fields. The deduction is intended to serve as an alternate explanation or validation for the Ampere’s and Faraday’s Laws. In basic intuitive terms, the relativistic transformation principles may not be that easy to accept or understand, as compared with the standard non-relativistic principles one gathers through common, day-to-day experiences. However, with proper interpretation and logical understanding, they are arguably much less “mysterious” to reason than the principles of the Ampere’s and Fara- day’s Laws. Similarly, the Gauss’s Laws for electric and magnetic fields, which are fundamentally established in integral forms, at first may not seem to be readily acceptable in intuitive terms. However, they can be explained as logical and philosophical generalization from the basic, more intuitive concept of the Coulomb’s Law, and therefore accepted to be much less mysterious than the Ampere’s and Faraday’s Laws. Accordingly, the deduction of the Ampere’s and Faraday’s Laws, or equivalently the Maxwell’s Equations, from the Gauss’ Laws through the relativistic transformation is as well expected to be much less “mysterious.” Such a deduction would be much easier to logically comprehend, and therefore be intellectually satisfying, in contrast with simply accepting the Maxwell’s Equations based on experimental observation. In this sense, the deduction would be considered philosophically or logically fundamental. Traditionally, the basic level texts on engineering electromagnetics [4, 5, 6] have used the Maxwell’s Equations under different special, simplified conditions, or in the complete form, as the starting point to study various electromagnetic fields and radiation problems. However, connecting the Maxwell’s Equations to the relativity theory is not usually explored in the basic engineering texts. Simple relativistic treatment of the electromagnetic fields are available in some physics text books [7, 8] which are meant only to initiate an interesting, alternate mode of reasoning, but not really to provide a comprehensive development of the Maxwell’s Equations from the relativistic theory. More involved relativistic treatments of the Maxwell’s Equations at higher levels are available in [8, 9, 10], which can be mathematically and/or conceptually tasking, making them inaccessible

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to introductory, even some advanced level students. All the available approaches assume a descent level of prior study of the theory of special relativity, including mass, momentum and force transformation relations. That may be ﬁne for advanced physics students, but usually not for introductory - even advanced - level engineering students. For the varied reasons, the available approaches have been rarely used in introductory engineering classrooms. Useful contributions in in the general context of introducing special relativity to engineering electromagnetics, particularly for teaching at the undergraduate level, include [11]-[12].

1.2. Objective. It would be valuable for students of introductory engineering electromagnetics to study the theory of special relativity at some essential level, and learn its direct connection to the Maxwell’s Equations to provide a modern, alternative perspective of the electromagnetic principles. The new perspective would enrich the introductory learning process, leading to modern, creative thinking, possibly leading to insightful design applications as well. The potential benefits may justify introduction of the new teaching, only if the material can be presented economically in a simple, comprehensive manner, with minimal distraction into the involved mechanical principles of the special relativity. This is particularly considering the limited time and resources available for the engineering electromagnetics in a typical undergraduate electrical engineering curriculum. To this end, we develop comprehensive material to introduce the concepts of the special relativity at an early, basic level of teaching engineering electromagnetics. This is intended to provide an alternate interpretation as well as a complete “derivation” of the Maxwell’s Equations. We accomplish this with mathematical as well as conceptual simplicity and economy, so that the material could be widely adopted. Use of differential operators such as curl and divergence are even avoided in the main derivation. The summation or integral principles are used instead, which are mathematically simpler and intuitive, so that the material can be introduced in an earliest possible, basic level of teaching. We first introduce the space-time transformation equations of the relativity theory. They can developed reasonably quickly with minimal effort, using two independent approaches, which might appeal to diverse student mindsets. One is based on simple mathematical deduction, whereas the other is based on intuitive physical analysis. More elaborate interpretation of the space-time transformation equations, and any further advancement into relativistic mechanics, are not needed. The space-time transformation equations are then used to derive basic charge-current transformation relations, by enforcing continuity of the charge. This is followed by derivation of suitable transformation relations for the Gauss’ Laws for electric and magnetic fields, which leads to the Ampere’s and Faraday’s Laws, respectively. A simple transformation from the Gauss’ Laws to the Ampere’s and Faraday’s Laws is possible by using a new general kind of principle of invariance of charge, as measured or seen from different inertial reference frames (see section 6). As mentioned, this approach needs only the simple space-time relativistic transformation relations, and is not dependent upon more advanced concepts of relativistic mechanics, and associated transformation relations for mass, momentum and force. In contrast, the common theoretical approaches to develop or interpret Maxwell’s Equations from special relativity in [7]-[10] are based on a more restrictive principle of invariance for the source and test charges, and on the transformation relations for mass, momentum and force [13, 14]. In these approaches, the mechanical transformation relations need to be first developed from the principle of momentum and energy conservation, which must be assumed to be valid for any relative uniform velocity of the inertial frame of observation. This process could consume significant valuable time and attention in the initial development of the relativistic transformation relations, and understanding the involved mechanical concepts, which can be a major distraction from an introductory engineering electromagnetics class. Further, in contrast to the present proposed approach, the use of the mechanical transformation relations in the derivation of the Maxwell’s Equations often leads to significant mathematical complexity, requiring restrictive assumptions or intricate conceptual arguments. This makes the material inaccessible or unsatisfactory to basic as well as advanced level engineering, even physics students. Contributions in the general context of introducing special relativity to engineering electromagnetics include [11]-[12].

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2. Space-Time Transformation

Consider two frames of reference, as shown in Fig.1. The time and space coordinates of a particular event, as measured in one reference frame, are represented by four variables (x, y, z, t). The respective coordinates of the event, as measured in the second reference frame, are represented by primed variables (x0, y0, z0, t0), Accordingly, the ﬁrst and the second reference frames may be informally referred to as the “un-primed” and the “primed” reference frames or reference coordinates, respectively. The four variables may be referred to as the space-time coordinates of the event, and the event may be referred to as a particular space-time point in a coordinate system. Now, assume that the primed reference frame is moving with a uniform velocity V in the x direction, with respect the unprimed coordinates. Also assume that at time t = 0, the origins of the two coordinate systems are aligned. That is, at t = 0, the space coordinates x0 = y0 = z0 = 0 in the primed system coincide with the space coordinates x = y = z = 0 in the unprimed system. Further, at this instant the clock in the primed coordinate system is initialized with that of the unprimed coordinate system with t0 = 0.

Figure 1

2.1. Non-Relativistic Transformation. If one uses a conventional or “common-sense” coordinate transformation, referred to as non-relativistic transformation, the space-time coordinates of the event in the two reference frames can be be related as follows:

y = y0, z = z0, x = x0 + V t, t = t0. (1)

This transformation would also lead to relating any incremental changes in the coordinates, represented by a pre-ﬁx ∆:

∆y = ∆y0, ∆z = ∆z0, ∆x = ∆x0 + V ∆t, ∆t = ∆t0. (2)

Accordingly, the components of the velocity of the point in diﬀerent directions may also be related by dividing respective incremental distance by incremental time:

∆y ∆y0 v = = = v0 , y ∆t ∆t0 y ∆z ∆z v = = = v0 , z ∆t ∆t0 z ∆x ∆x0 + V 4t0 ∆x0 v = = = + V = v0 + V. (3) x ∆t ∆t0 ∆t0 x Notice that the two time variables t and t0 are assumed to be equal. This means, such standard transformation implicitly assume that the ﬂow of time is “absolute,” meaning it is independent of

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the relative motion between the frames. In addition, t is equal to t0, independent of the space coordinates of the event being observed. However, the above two assumptions do not have any concrete physical basis, and should be open to review and adjustment based on any advanced understanding or experimental observation.

2.2. Special Nature of Light, and Incompatibility with Non-Relativisitc Transforma- tion. In the non-relativistic transformation, we see that the x-directed velocity in the unprimed reference frame can be obtained from that in the primed reference frame by simply adding the relative velocity V between the frames. Accordingly, any x-directed velocity measured in one coordinate frame would not be equal to that measured in the other, for a non-zero value of the relative velocity V . Further, the magnitude of the total velocity, which is the speed, defined as the square- root of the sum of squares of the individual velocity components in the three directions x, y and z, would also not be equal when measured in the two reference frames. Now, let us apply the standard common-sense, non-relativistic transformation to a light signal propagating in free-space. As per the transformation rules discussed above, the speed of light in free-space would be measured with different results in the two reference frames. With some insight on the special nature of light, this conclusion may be brought to question, which would lead to invalidating the non-relativistic transformation for general applications. The speed of light is different from that of any common material substance such as a rock or a bullet. Light is not “thrown” off a light bulb like a rock thrown from someone’s hand or a bullet from a gun. Therefore, unlike a rock or a bullet, the speed of light can not be fixed as a given value with respect to its source, and the given speed could not be changed as one desires by employing more strength or energy of its source. Instead, light is generated as a wave from the bulb, more like a sound wave generated from a flute, or more like the water ripples generated when a rock is dropped in a lake. Once the wave is generated by its source, it is physically detached from its source and its propagation speed is supposed to be determined by the medium, independent of the source. However, unlike a mechanical wave, such as the sound or water wave, the light wave does not require a material medium for propagation. Light can propagate in free-space. It would bring challenges in the understanding, when light propagation in an ideal free-space is considered. The water wave propagates as a disturbance of the water particles, and the disturbance propagates with respect to the body of the water. Therefore, the body of the water, or equivalently the body of the solid ground around the lake with respect to which the water is assumed to be stationary, is the natural frame of reference. The speed of the water wave is derived from the material properties of the water, and is clearly defined with respect this natural frame of reference. Similarly, for the sound wave, the solid earth with respect to which the surrounding air is assumed to be stationary, is the natural reference frame. The speed of the sound wave is a known constant (as determined from the physical parameters of the air), clearly defined with respect to the earth or the surrounding stationary air. Once the speed of the water wave or the sound wave is definitively known, with respect to its natural, or “preferred,” reference frame as discussed, pointed in any specific direction, the wave velocity with respect to any other reference frame may be determined by using velocity transformation rule. In distinct contrast, an ideal free- space in which light propagates, as per the very philosophical nature of the free-space, does not have a natural or “preferred” reference frame. Unlike the water in a lake or the air surrounding earth, the free-space is not “attached” to any definite body of reference. Therefore, to begin with, there is no preferred reference frame with respect to which we can define and fix the speed of light in free-space. Accordingly, the standard procedure of starting with a known value of the speed, pointed in any specific direction, with respect to a naturally preferred reference frame, and then calculating the velocity with respect to other references using velocity transformation, may not be as evident for light propagation in free-space. In this process, if one would start by fixing a certain value of the speed to any particular reference frame, that may appear arbitrary without proper logical basis.

2.3. Theory of Relativity and the Speed of Light in Free-Space. We may, however, look at the above situation from a new perspective. From the earlier discussion, we understand that there is no naturally preferred reference frame for propagation of light in free-space. To put it a

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bit differently, all reference frames should be equally preferred for treatment of the propagation of light in free-space. Accordingly, if one may start with a fixed value for the speed of light in free-space in one reference frame, arbitrarily chosen, then there would be no reason why the same value could not have been fixed in any other reference frame. This is an important philosophical argument, leading to the conclusion that, whatever is the speed of light in an ideal free-space, it may have to be the same universal constant for all reference frames of measurement. Further, if the reference frame is also “unbiased,” then the light would propagate with the same speed along any given direction, with no biased preference to deflect the propagation direction to one side or other. This was postulated in Einstein’s theory of relativity as a universal principle. This conclusion is also established through practical experiments. The new theory of relativity would clearly contradict the principles of the non-relativistic transformation. The basic rule of non-relativistic transformation discussed earlier - that the speed of a given physical event would necessarily have unequal values, when measured in two reference frames moving with a non-zero relative velocity V - is clearly violated for light propagation in free- space. The transformation relations would then have to be revised in order to be consistent with the new theory of relativity. We will show in the following how this can be achieved, using two approaches: (1) mathematical analysis, and (2) physical experimentation. The first approach mainly uses mathematical arguments for the derivation of the new transformation relations. Whereas, the second approach derives the transformation relations based on a more intuitive understanding of a physical measurement process using light. The two approaches may be used equally effectively, or may be used in complementary ways, in order to develop an understanding of the new transformation relations. The new transformation relations, as derived in the following, are referred to as the relativistic transformation relations. In order that the new relativistic transformation relations are to be universal in scope, in scientific consistency, they should be applicable not just for light in free-space, but for all physical phenomena. They would have significant implications when applied to the mechanics of material bodies. However, when the relative velocity V is much smaller than the speed of light in free-space, which is usually the case in our “common-sense” world, the new relativistic relations are seen to approach the standard non-relativistic relations. Accordingly, the non-relativistic relations are accurate enough for common day-to-day observations, as normally expected. Physical mechanics, under general conditions of relative motion, is to be revised based on the new theory of relativity and transformation relations. The reader may be referred to textbooks on modern physics for further study on this topic. Development of mechanics under the revised relativistic conditions may have indirect implications to the electromagnetic theory, but it is not the goal of the present study. Here, we would focus on the applications of the relativistic transformation specifically to the development and understanding of the Maxwell’s Equations.

2.4. Relativistic Transformation: Mathematical Analysis. The two reference frames in Fig.1 are assumed to be moving with a relative velocity V in the x direction. The relative motion in the x direction is not expected to alter the y and z coordinates of an event, when seen from the two frames, assuming that the two frames and the free-space medium are equally unbiased in the y and z directions. That is, we have y = y0 and z = z0. This leaves the transformation of the only space coordinate along the x axis, and the time coordinate. Accordingly, in the following we may represent an event with only two variables, (x, t) or (x0, t0). Consider that the primed space-time coordinates (x0, t0) are linearly related to the respective unprimed coordinates (x, t).

x0 = ax + bt, t0 = px + qt, (4) where a, b, p and q are now unknown constants, to be determined. Let us mathematically see if transformation relations in the above form can be established, which would be consistent with the requirements of the relativity theory. The above relationships, to begin with, already satisﬁed

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the initial conditions we have assumed. From the above equations, we have x0 = 0 when t = 0 and x = 0. That means, at time t = 0, the origin of the unprimed coordinate system (x = 0) coincides with that of the primed coordinate system (x0 = 0). At the same instant, that is when x = x0 = t = 0, we would also have t0 = 0 from the above equations. In other words, the clock in the primed coordinate system located at x0 = 0 is initialized with t0 = 0, when the origins of the two reference frames coincide, which is the same instant when the clock in the unprimed coordinate system is also initialized to zero t = 0. In addition to the initial conditions, we need other basic conditions to be enforced in order for the transformation relations to be logically valid, and meet the physical constraints of the relativity theory. These additional conditions would determine the values of the constants in the transformation equations (4). At time t = 0 the origins of the two frames coincide. We also know, the origin of the primed coordinate system moves with a given velocity V as seen from the origin of the unprimed coordinate system. Accordingly, for x0 = 0, we have to have x = V t. Using this constraint in (4) we get,

0 b x = 0 = ax + bt, x = − a t = V t, b a = −V. (5) One basic expectation in the relativity transformation is that, if one observer sees a second observer to move with a velocity V , then the second observer should see the ﬁrst moving with a velocity −V . We just explained that the origin of the primed coordinate system moves with a velocity V as seen by an observer at the origin of the unprimed coordinate system. Therefore, the origin of the un-primed coordinate system, that is x = 0, should be seen by an observer at the origin of the primed coordinates to be moving with a velocity −V . Accordingly, for x = 0, we have to have x0 = −V t0 Using this constraint in (4) we get,

t0 = qt, x0 = bt; x = 0, 0 b 0 0 x = bt = q t = −V t , b q = −V. (6)

The above constraints we have enforced are quite elementary, and are not particularly unique to the relativity theory. What would be new or unique to the relativity theory is the following. Consider a point (x, t) in the unprimed coordinate system, which is seen as a point (x0, t0) in the primed coordinate system. Consider that the point is moving in the x direction with a speed equal to the ∆x speed of light c, as seen in the unprimed coordinates. That is, ∆t = c. Then, the speed of the same point, as measured by the primed coordinates, would also be equal to the speed of light c. That is, ∆x0 and ∆t0, corresponding to the incremental distance and time ∆x and ∆t, respectively, would ∆x0 be such that ∆t is also equal to the speed of light c. Using this condition in (4)

∆x0 = a∆x + b∆t, ∆t0 = p∆x + q∆t,

0 ∆x ∆x a∆x+b∆t a ∆t +b ∆t0 = p∆x+q∆t = ∆x , p ∆t +q ac+b 2 c = pc+q , pc + qc = ac + b, pc2 + (q − a)c − b = 0. (7)

Using (5,6) in (7)

pc2 = b. (8)

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For x = 0, we have t0 = qt. Accordingly, for x0 = 0 we should also expect t = qt0. This is a basic symmetry condition between times in the two coordinates. Using this symmetry condition in (4),

0 b x = 0 = ax + bt, x = − a t, 0 pb b t t = px + qt = − a t + qt = t(q − p a ) = q , 2 b q − pq a = 1. (9) Now, the constants a, b, p and q can be solved from the equations (5-9)

2 2 b2 2 2 V 2 a − pb = 1 = a − c2 = a − a c2 = 1, 1 a = q 2 = q, 1− V c2 −V b = −V a = q 2 , 1− V c2 −V b c2 p = c2 = q 2 , 1− V c2 0 x−V t x = ax + bt = α , −V x +t 0 c2 t = px + qt = α , q V 2 α = 1 − c2 . (10)

2.5. Relativistic Transformation: Physical Experimentation. We will use simple physical processes employing light signals in free-space to measure time and distance. A particular process using a light signal is measured from two diﬀerent reference frames. The results of measurement in the two frames can then be compared to establish transformation relations between the space and time coordinates of the two frames.

Figure 2

2.5.1. Time Transformation. First, let us measure time by bouncing a light through a ﬁxed distance ∆y in the y-direction. The round-trip delay experienced by the light to cover a ﬁxed distance may be used as a proportionate measure of the time. Consider such a time-measurement process conducted in the primed reference frame (at the origin x0 = 0). This same process is observed in the unprimed reference frame, and compared with another identical time-measurement process conducted at the origin of the unprimed reference frame. The relative velocity V in the x direction is assumed not to change the y dimension in the two reference frames. Accordingly, ∆y = ∆y0. Let 2∆t0 be the round-trip time taken by the light, as measured in the primed reference frame, to cover a reference distance ∆y = ∆y0.

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∆y = ∆y0 = c∆t0. (11)

This light-bouncing experiment conducted in the primed frame, when observed from the un-primed frame, would appear to take a longer route (see Fig.2). The round-trip time would be measured 2∆t, as compared with its local timing process conducted in the unprimed frame. Accordingly, ∆t is referred to as “equivalent” to ∆t0, as observed from the unprimed reference frame. As a deﬁnition, an incremental interval between a pair of time measurements, with given starting and end points, is referred to be “equivalent” or “simultaneous” to that between another pair, as observed from a particular reference frame, under the following condition. If the reference starting points of the two pairs coincide with each other (if they happen to occur simultaneously, or by introducing a suitable delay between their observations,) then the corresponding end points would also be observed in coincidence, or “simultaneously,” from the given frame. In reference to the Fig.2, one can now geometrically relate ∆t with ∆t0 in terms of the the speed of light c and the relative velocity V .

c2∆t2 = c2∆t02 + V 2∆t2, (12)

2 0 V 1 ∆t = ∆t(1 − ) 2 = ∆tα. (13) c2

The above geometric analysis assumes that the two reference frames are ideally unbiased, from which the light is observed to propagate along straight lines. Also, as per the theory of relativity, we have assumed the same value c for the speed of light as observed in either of the two reference frames. Equation (13) is a simple relationship between an incremental time ∆t0 measured at the origin of the primed coordinate system, and the corresponding incremental time ∆t as measured and observed to be equivalent in the unprimed coordinate system. Now, as we have assumed before, the times t0 and t are initialized to zero when the origin x0 = 0 coincides with the origin x = 0. With this initialization, the above proportionality relationship between the equivalent incremental times would apply to the corresponding reference times t0 and t. The time t0 measured at the origin x0 = 0 of the primed coordinates is determined to be equivalent or “simultaneous” to the time t as measured and observed in the unprimed coordinates. These equivalent times t0 and t are related as follows.

2 0 V 1 0 t = t(1 − ) 2 , x = 0. (14) c2

Note that the time t0 is specified in association with a particular location of measurement x0 = 0. This is important. If the particular location x0 at which the original experiment is conducted is changed, the relationship between the equivalent times t0 and t would be different. If the point x0 is fixed in the primed coordinate frame (independent of t0), but not at the origin x0 = 0, the relationship between the equivalent incremental times in (13) would still be valid, whereas the relationship (14) between the actual reference times t0 and t would not (requiring some time offset). Also, as we have stated, the above relationship assumes that the t0 and t are determined to be equivalent or “simultaneous” as observed from the unprimed reference frame. Instead, if the determination of equivalence or simultaneity is made as observed from the primed reference frame, the relationship would be different. For example, if the experiment is conducted at the origin of the unprimed coordinates, and the observation of simultaneity is made from the primed reference frame, it can be shown (by principle of symmetry between the frames) that the relationship between the equivalent times t and t0 can be expressed using (14), but with the variables t0 and t switched. These concepts of equivalence or simultaneity should be clearer from a complete space-time transformation relationship between (x, t) and (x0, t0).

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2.5.2. Space-Time Transformation. The above experiment established relationship between equivalent times t and t0 in the two frames of measurement. Now we will conduct another light experiment, using which we can measure the distance and time of a particular “event” as observed in the two reference frames. Let a light signal be initiated from the origin of the unprimed coordinate system at time t = t1. Let the light signal be aimed towards a particular event-point in space, as shown in the Fig.3, whose space-time coordinates (x, t) and (x0, t0) we are interested to determine as measured from the two reference frames. The signal passes through the origin of the primed reference frame at time t2, reaches the particular point of interest at time t3, reﬂects and returns towards the starting point, passes again through the origin of the primed reference frame on its way at time t4, and reaching ﬁnally at the origin of the unprimed reference frame at time t5. All the above times are as measured with respect to the unprimed coordinate system. The corresponding equivalent times measured at the origin of the primed reference frame can be obtained using the above relationship (14). The space and time coordinates (x, t) and (x0, t0), as measured with respect to the two reference frames, can be determined and related from the above timing information.

The distance x of the event is determined from the time delay t5 − t1. The time t = t3 of the event is determined as the average of t5 and t1.

c x = (t − t ) , (15) 5 1 2

x t − t t + t t = t − = t − 5 1 = 1 5 = t . (16) 5 c 5 2 2 3

From the above two equations, the measured x and t can be related to the initial time t1.

−x + ct = ct1. (17)

0 0 0 0 0 The distance x of the event is determined from the time delay t4 − t2. The time t = t3 = αt3 of 0 0 the event is determined as the average of t4 and t2.

c x0 = (t0 − t0 ) , (18) 4 2 2

x0 t0 − t0 t0 + t0 t0 = t0 − = t0 − 4 2 = 2 4 . (19) 4 c 4 2 2

0 0 0 0 The above (x , t ) are expressed in terms of t4 = t4α and t2 = t2α. Equation (17) relates (x, t) to 0 0 t1. Accordingly, in order to relate (x , t ) with (x, t), we may need to relate t2 and t4 with t1. This can be accomplished as follows. The distance covered by the light signal during time t2 −t1 is equal to distance V t2 between the reference frames at time t2. Solving from this information, t2 may be related with t1

t c c(t − t ) = V t , t = 1 . (20) 2 1 2 2 c − V

Similarly, the distance covered by the light signal during time interval t4 − t1 is equal to round-trip distance 2x of the event point minus the distance V t4 between the reference frames at time t4. Solving from this information, t4 may be related with t1 and x.

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2x + ct c(t − t ) = 2x − V t , t = 1 . (21) 4 1 4 4 c + V

Figure 3

Using (17) in (20,21), we can relate t2 and t4 with x and t .

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ct − x 2x + (ct − x) ct + x t = , t = = , (22) 2 c − V 4 c + V c + V

−2cV t + 2cx 2c2t − 2V x t − t = , t + t = . (23) 4 2 c2 − V 2 2 4 c2 − V 2

Use (23) in (18,19) to relate (x0, t0) to (x, t).

2 0 α (c x−c2V t)α x−V t x = (t4 − t2)c 2 = c2−V 2 = α , 2 t− V x 0 t2+t4 α(c t−V x) c2 t = 2 α = c2−V 2 = α . (24)

The above transformation relations (24), derived here using a complete physical analysis, are the same relations in (10) which were derived on a mathematical basis. Both are valid and complementary approaches to establish the relativistic transformation. Using the above transformation relations, it can be shown that a straight-line path of light propagation along any general orientation in space, as originally observed from one of the unbiased frames, would be seen with a deviated orientation in the other frame. The amount of the deviation would be dependent on the magnitude V of the relative velocity between the two frames. The relative orientation of the light path seen in the second frame, would obviously depend on the direction of the relative velocity V between the frames. Therefore, if the velocity V (magnitude and/or direction) varies in time, the resulting light path with its variable orientation in time would appear as a curved path, as seen in the second frame. This would contradict the unbiased nature of the two reference frames, that we assumed in the above analyses, which expects any observed light path to be strictly a straight line. Therefore, the unbiased nature of the two frames would require the relative velocity V between the frames to be constant in time, as a necessary fundamental condition.

3. Relativistic Transformation of Current and Charge Density

3.1. Current Along the Relative Velocity. Let us ﬁrst consider only an x-directed current. An experiment is conducted in the primed reference frame to verify the continuity of current and charge conservation (Fig.4). The experiment may be mathematically expressed as follows.

0 0 0 0 0 0 0 0 0 0 0 ∂4Q 0 ∂ρv 0 0 0 0 ∆τ [Jx(x + ∆x /2, t ) − Jx(x − 4x /2, t )]∆A = − ∂t0 ∆τ = − ∂t0 (x , t )∆τ ∆A∆x , 0 0 0 0 0 0 0 0 0 Jx(x +∆x /2,t )−Jx(x −4x /2,t ) ∂ρv 0 0 ∆x0 = − ∂t0 (x , t ), 0 0 ∂Jx ∂ρv ∂x0 = − ∂t0 . (25)

The same experiment is observed from the unprimed coordinate frame, as shown in the Fig.4. Let us ﬁnd relationships between the space (∆x, ∆x0) and time (∆t, ∆t0), (∆τ, ∆τ 0) increments observed in the two reference frames. The experiment was performed in the (x0, t0) coordinates with measurements done at the two ends of the volume x0 − ∆x0/2 and x0 + ∆x0/2 at the same time t0. This means ∆t0 = 0 at the two ends. In the (x, t) reference frame, the corresponding increments (∆x and ∆t) may be related with each other and with ∆x0 using relativistic transform relations (10,24).

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0 t−xV/c2 0 x−V t t = α , x = α , 0 4t−4xV/c2 2 4t = α = 0, 4t = 4xV/c , 0 4x−4tV 4x−4xV 2/c2 4x = α = α = α4x, 4x0 2 4x0 2 4x = α , 4t = 4xV/c = α V/c . (26)

The charge ∆Q and ∆Q0 in the two frames would be equal. We also assume that the transverse cross-section area ∆A is the same in both frames. Because the lengths ∆x0 and ∆x are diﬀerent when measured in the two frame, the same charge enclosed between the lengths would appear as diﬀerent charge densities. Let ρv0 be the charge density of the volume element as observed in the unprimed frame.

0 0 0 ∆Q = ∆Q = ρv∆A∆x = ρv0∆A∆x, 0 ∆x0 0 ρv0 = ρv × ∆x = ρvα. (27)

The time increment ∆τ 0 is an arbitrarily small interval between two instants of measurements performed at a given x0(∆x0 = 0). The corresponding time interval in the (x, t) reference frame is ∆τ = ∆τ 0/α. As per a basic principle of the theory of relativity, the outcome of the basic, physical experiment conducted in the primed reference frame at a given t0, ∆t0 = 0, and that observed from the other reference frame should be the same. The current continuity experiment, as observed in the (x, t) frame can be expressed as follows.

∆τ[Jx(x + ∆x/2, t + ∆t/2) − Jx(x − 4x/2, t − 4t/2)]∆A ∂∆Q ∂ρv0 = − ∂t ∆τ = [− ∂t (x, t)∆x∆A]∆τ,

∂Jx ∂Jx ∂Jx ∂Jx ∂ρv0 (J x + ∂x ∆x/2 + ∂t ∆t/2) − (J x − ∂x ∆x/2 − ∂t ∆t/2) = − ∂t (x, t)4x, ∂Jx ∂Jx V ∂ρv0 ∂x + ∂t c2 = − ∂t , R ∂Jx V ∂x ∂t + c2 Jx = −ρv0. (28)

When the velocity V = 0, we know ρv0 = ρv. Using this condition in the above equation, the integral term can be recognized as −ρv.

0 V ρv0 = ρvα = ρv − c2 Jx = ρv + ∆ρv, V ∂ρv ∂Jx ∆ρv = − c2 Jx, ∂t = − ∂x . (29)

ρv is the charge density the observer in the (x, t) frame normally estimates (at a given t, ∆t = 0) and ρv0 is the charge density the same observer would estimate in the transformed current continuity 0 0 experiment (at a given t , ∆t = 0). These two parameters diﬀer, with the diﬀerence ∆ρv dependent 0 0 on the current density Jx. If the current is charge free as seen in the (x , t ) frame, then the 0 V V transformed experiment would yield ρv0 = 0 = αρv = ρv − Jx c2 . This means ρv = Jx c2 . In other words, a charge free current in the (x0, t0) frame would look charged in the (x, t) frame. This is an important observation.

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Figure 4

3.2. Current in Arbitrary Direction. In the above derivation, for simplicity we have considered only x-directed current (along the relative velocity). If the current is along an arbitrary direction, with x, y and z components, the above derivation may be extended. It can be shown that the same relationship between the charge density and current would be valid for the general case as well. It may not be necessary to present all derivation steps for the general case with all current directions. We will provide a conceptual picture, and discuss necessary changes in the above derivation. For the general treatment, the volume element in the above experiment (Fig.4) may now be considered to be a rectangular box with dimensions ∆x, ∆y and ∆z in the three rectangular coordinates. The space-time coordinates for the center of the box is (x, y, z, t). The volume element would consist of six rectangular faces, defined by the coordinates (x ± ∆x/2, y, z, t ± ∆t/2), (x, y ± ∆y/2, z, t), and (x, y, z ± ∆z/2, t). The additional currents Jy and Jz would flow through the xz and xy faces, respectively. Let us consider changes to the equations (28) and (29), when the current continuity experiment is observed in the unprimed coordinates. The experiment is conducted in the primed reference frame (∆t0 = 0). Incremental lengths ∆y0 and ∆z0 in the y and z directions would be observed in the unprimed frame with equal lengths (∆y0 = ∆y and ∆z0 = ∆z), at the same instant (∆t = 0). This is in distinct contrast with the dimensions in the x direction, which are observed in the unprimed frame with incremental time differences. Accordingly, the time increment

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∆t would be zero for treatment of currents Jy and Jz in the xz and xy surfaces. With this in mind, the modiﬁed version of the equation (28) can be expressed as follows.

h ∆τ J x(x + ∆x/2, t + 4t/2) − Jx(x − 4x/2, t − 4t/2) ∆y4z + J y(y + ∆y/2, t) − Jy(y − 4y/2, t) ∆x4z i ∂∆Q ∂ρv0 + J z(z + ∆z/2, t) − Jz(z − 4z/2, t) ∆x4y = − ∂t ∆τ = [− ∂t ∆x∆y∆z]∆τ,

∂Jx ∂Jx V ∂Jy ∂Jz ∂ρv0 ∂x + ∂t c2 + ∂y + ∂z = − ∂t ,

R ∂Jx ∂Jy ∂Jz V ( ∂x + ∂y + ∂z )∂t + c2 Jx = −ρv0. (30)

When the velocity V = 0, we know ρv0 = ρv. Using this condition in the above equation, the integral term can be recognized as −ρv.

0 V ρv0 = ρvα = ρv − c2 Jx = ρv + ∆ρv, V ∂ρv ∂Jx ∂Jy ∂Jz ∆ρv = − c2 Jx, ∂t = − ∂x − ∂y − ∂z . (31)

Equation (31), derived here for the general case with current in arbitrary direction, establishes the same relationship (29) we had derived for the simple case with only Jx current. ∆ρv, which is the 0 diﬀerence between ρv0 = αρv and ρv, depends only on the Jx component of the current. The above general equation is a useful transformation relation between current and charge densities in two 0 0 0 0 reference frames. Note that ρv as observed in (x , t ) coordinates (∆t = 0) depends both on ρv and 0 J observed in the (x, t) reference frame. A similar expression can be written relating ρv to ρv and 0 0 J , by switching ρv and ρv, and replacing V by −V . This is equivalent to switching the frame in which the current continuity experiment is conducted and the frame in which it is observed.

V ρ α = ρ0 + J 0 . (32) v v c2 x

0 0 Substituting ρv in the above equation using (31), a relationship between Jx, ρv and Jx can be obtained.

2 2 0 c 0 c ρv V Jx = V (ρvα − ρv) = V (ρvα − α + Jx c2α ), 0 Jxα = Jx − ρvV. (33)

0 0 A similar expression can also be written relating Jx to Jx and ρv. This is obtained by replacing the primed with corresponding unprimed variables, and replacing V by −V . This is equivalent to switching the reference frames of measurement and observation.

0 0 Jxα = Jx + ρvV. (34)

The equations (31-34) constitute the relativistic transform relations between current and charge densities in diﬀerent reference frames. Out of the above four relations, any two can be used independently, from which the other two can be derived. The relations appear very much similar to the transform relations between space-time coordinates. The current density represents timing information, and the charge density represents dimensional information (length) for charge distribution.

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4. Relativistic Transformation of the Gauss’ Law for the Electric Field: The Ampere’s Law

Figure 5

4.1. Gauss’ Law for the Electric Field, in the Primed Coordinates. Consider the Gauss’ Law for the electric ﬁeld, applied in the primed coordinate system (x0, y0, z0, t0) to a closed surface S0.

X 0 0 0 Di · 4Si = ∆Q , (35) S0 where ∆Q0 is the total charge inside the surface S0. In reference to the Fig.5a, the above form of the Gauss’ Law may be applied to a closed surface S0 consisting of six faces of an elemental rectangular box of dimensions ∆x0 × ∆y0 × ∆z0. The equation (35) may be expressed using normal components of the ﬂux density Di at the center of the six faces of the box, and the internal charge 0 density ρv

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Figure 5. (a) Electric ﬁeld divergence (Gauss’ Law) experiment in the (x’,y’,z’,t’) reference frame, and (b) the same experiment as seen by an observer from the (x,y,z,t) reference frame.

0 0 0 0 0 0 0 0 0 0 0 0 (Dx3 − Dx1)∆y ∆z + (Dy4 − Dy2)∆x ∆z + (Dz6 − Dz5)∆y ∆x 0 0 0 0 0 = ∆Q = ρv∆x ∆y ∆z . (36)

The above summation form of the Gauss’ Law may also be expressed equivalently using derivatives 0 of the ﬂux density D at the center of the box.

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (Dx3 − Dx1)∆y ∆z + (Dy4 − Dy2)∆x ∆z + (Dz6 − Dz5)∆y ∆x = (ρv)∆x ∆y ∆z 0 0 0 ∂Dx 0 0 0 ∂Dy 0 0 0 ∂Dz 0 0 0 0 0 0 0 = ∂x0 4x ∆y ∆z + ∂y0 4y ∆x ∆z + ∂z0 4z ∆y ∆x = ρv∆x ∆y ∆z , 0 0 0 ∂Dx ∂Dy ∂Dz 0 ∂x0 + ∂y0 + ∂z0 = ρv. (37)

4.2. A Basic Gauss’ Law Experiment as Measured from Two Reference Frames. Let an observer in the primed coordinate system conduct an experiment to verify or validate the

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Gauss’ Law for the rectangular box, placed in a free-space medium. The observer would need to measure the normal component of the electric field Ei for each of the six faces, i = 1, .., 6, of the rectangular box, from which the corresponding normal component of the flux density Di = 0Ei can be calculated. The electric field and the flux density are fundamentally defined in terms of the force experienced by a given stationary charge q.

0 0 q 0 0 0 F = qE = D , D = 0 F . (38) 0 q Accordingly, in order to validate the Gauss’ Law, the observer may place a reference stationary 0 charge q at the center of each face of the rectangular box, and then measure the force F i experienced by the charge using an appropriate measurement technique. Now, the Gauss’ Law in the summation form of equation (36) may be expressed in terms of the measured normal components of the force 0 and the charge density ρv as follows.

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 q (Fx3 − Fx1)∆y ∆z + q (Fy4 − Fy2)∆x ∆z + q (Fz6 − Fz5)∆y ∆x 0 0 0 0 = ρv∆x ∆y ∆z . (39)

The above experiment in the primed coordinates is conducted at a given time t0. In other words, all the above force measurements are conducted simultaneously in the primed coordinate system, with time diﬀerence ∆t0 between experiments at diﬀerent faces equal to zero. Now, let the experiment be observed from the unprimed coordinate system. The force measurements on the six faces, performed simultaneously in the primed coordinates (∆t0 = 0), would not be observed simultaneously in the unprimed coordinates (∆t 6= 0). Using the relativistic transformation relations (10,24) one may relate the time (∆t) and location (∆x) parameters of observation in the unprimed reference frame. The timing would be independent of the y and z coordinates. In other words, ∆t would only depend on ∆x, not on ∆y or ∆z.

0 t−xV/c2 0 x−V t t = α , x = α , 0 4t−4xV/c2 2 4t = α = 0, 4t = 4xV/c , 0 4x−4tV 4x−4xV 2/c2 4x0 4x = α = α = α4x, 4x = α , y = y0, z = z0, 4y = 4y0, 4z = 4z0. (40)

0 Accordingly, the forces F i measured simultaneously in the primed coordinate system would be measured as F i in the unprimed coordinates at different times. The corresponding charge density, 0 0 0 as observed from the unprimed frame, would be ρv0 = ρv4x /4x = ρvα. This is shown in the 0 last section, and is different from the charge density ρv, normally measured in the primed reference 0 0 frame at a given time t , ∆t = 0. As derived in the last section, in equations (29,31), ρv0 would also be different from the charge density ρv one would normally measure in the unprimed coordinates at a given time t, ∆t = 0,

2 ρv0 = ρv + ∆ρv = ρv − JxV/c . (41)

As per the general principle of relativity, any “basic” experiment conducted in one coordinate system should lead to the same conclusion when measured or observed from another coordinate system, moving with an uniform velocity with respect to each other. The Gauss’ Law experiment, which establishes the basic deﬁning relationship for the charge, may be considered one such basic experiment. This assumes that a given charge, as fundamentally deﬁned by the Gauss’ Law, is relativistically invariant, which means the value of the charge is independent of the relative velocity

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of the charge and the observer. Accordingly, the results of measurement in (39) would lead to the following relationship between the forces F i and the charge density ρv

0 0 0 q (Fx3 − Fx1)∆y∆z + q (Fy4 − Fy2)∆x∆z + q (Fz6 − Fz5)∆y∆x 2 = (ρv + ∆ρv)∆x∆y∆z = (ρv − JxV/c )∆x∆y∆z. (42)

4.3. Recognizing the Need for a New Force Field. Let us first assume that the force on a reference charge in the unprimed frame is produced due to the electric force, in a conventional sense. With this assumption, the forces F i and the equivalent flux densities at different faces of the test box (see Fig.5) may be related to the flux density D at the center of the box (x, y, z, t) using derivatives. Remember that the flux densities in the above experiment at different locations are measured in the unprimed coordinates at different times, as per the space-time relations (40).

F = q D = q D (x − 4x/2, t − 4t/2) = q (D − ∂Dx ∆x/2 x1 0 x1 0 x 0 x ∂x ∂Dx q ∂Dx ∂Dx V − ∆t/2) = (D − ∆x/2 − 2 ∆x/2), ∂t 0 x ∂x ∂t c F = q D = q D (x + 4x/2, t + 4t/2) = q (D + ∂Dx ∆x/2 x3 0 x3 0 x 0 x ∂x ∂Dx q ∂Dx ∂Dx V + ∆t/2) = (D + ∆x/2 + 2 ∆x/2), ∂t 0 x ∂x ∂t c F = q D = q (D − ∂Dy ∆y/2), y2 0 y2 0 y ∂y F = q D = q (D + ∂Dy ∆y/2), y4 0 y4 0 y ∂y F = q D = q (D − ∂Dz ∆z/2), z5 0 z5 0 z ∂z F = q D = q (D + ∂Dz ∆z/2). (43) z6 0 z6 0 z ∂z Note that there is no time difference between the measurements at the center of the faces 4 and 2, and similarly those of faces 6 and 5. This is because they differ only in their y and z coordinates with respect to the center of the box. The time difference ∆t of observation at two locations is dependent only on the x coordinates of the locations. Now, using the force expressions of (43) in the equation (42), we get,

∂Dx ∂Dx V ∂Dy ∂Dz V ∂x + ∂t c2 + ∂y + ∂z = ρv0 + ∆ρv = ρv − Jx c2 . (44)

The above measurement for the Gauss’ Law for the electric field was originally performed in the primed coordinates, which was then observed from the unprimed coordinates. The experiment may also be duplicated in the unprimed reference frame itself. In this case, the measurements in the different faces of the box is to be performed simultaneously in the unprimed coordinates ∆t = 0. The results of this experiment may be obtained from the equivalent expression (37) in the primed coordinates, by simply substituting the primed variables by the corresponding unprimed variables. This would be the differential form of the Gauss’ Law in the unprimed rectangular coordinates.

∂D ∂D ∂D x + y + z = ρ . (45) ∂x ∂y ∂z v

Equation (44) would be clearly inconsistent with the equation (45) for general conditions with a non-zero current density J and/or a time-varying electric ﬂux density D. Let us examine the possible source of this inconsistency. We assume that the Gauss’ Law for electric ﬁeld applies universally in all reference frames. This means the equivalent expressions of the Gauss’ Law in (35-39) for the primed coordinates, and in (45) for the unprimed coordinates, must be correct.

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We also assume that the basic postulate of relativity, which requires that a validation experiment for a basic theory should lead to equivalent conclusions when conducted in one reference frame, and when the experiment is observed from another frame moving with a constant relative velocity. Accordingly, the deduction of the equations (42) through relativistic transformation of the Gauss’ Law in (39) must also be correct. This leaves equations (43,44) to be examined for any deficiency. The total forces on the reference charges, which are expected to satisfy the relationship (42), may not have been properly characterized in the equations (43,44). Note that the reference charges q in the experiment are assumed to be stationary in the primed coordinates, in which the experiment was originally conducted. This experiment in the primed coordinates is described by the force and field equations (35-39). These equations are based on the Gauss’ Law and on the definition of the electric field as the total force on a unit stationary reference charge. However, when the experiment is observed from the unprimed coordinates, the reference charges are no longer stationary, and are moving with a velocity V along the x direction. The total forces experienced by these moving reference charges in the unprimed coordinates have been established also using the electric fields through the equations (43,44). Now, the forces in (35-39), where the charges are stationary, are assumed to be valid, leading also to the validity of the force equation in (42). Whereas, those in (43,44), where the charges are moving, we suspect could be wrong. Therefore, one would conclude that the forces in (43,44) will have to be corrected with additional parts that must dependent on the velocity of the reference charge q. The essential conclusion is that, the force experienced by a moving charge may have to be characterized differently from that by a stationary charge. The total force on a stationary charge is determined as the product of the charge and the electric field, as per the very definition of the electric field. Having defined the force on a stationary charge this way, the total force on a moving charge may not also be characterized in the same way - as the product of the charge and the electric field - by simply ignoring possible contributions due to its velocity. Additional force effects that depend on the velocity of the charge may have to be introduced. Otherwise, the forces on an electric charge would be relativistically inconsistent and incomplete, as we have now encountered. This is an important basic finding.

4.4. Force Correction, the Magnetic Field, and the Ampere’s Law. As per the above discussion, we seek to adjust the forces in the equations (43,44), such that the total forces would satisfy (42), as well as be consistent with (45). Note that the diﬀerences between the equations (44) and (45) are linear with velocity V . This suggests, in order that the revised versions of (43,44) be consistent with (45), any new motion-dependent force that needs to be added to (43,44) should be linearly dependent on the charge velocity. Further, the new force might possibly be directed along or perpendicular to the charge velocity. However, if one would proceed by adding a new force directed along the charge velocity V , it can be shown not to resolve the above inconsistency. The new force ∆F will have to be directed perpendicular to the charge velocity. With the charge velocity along thex ˆ direction, this means ∆F may only have a non-zeroy ˆ orz ˆ component (4Fz 6= 0, 4Fy 6= 0), with itsx ˆ component equal to zero (4Fx = 0). Accordingly, in reference to the Fig.5b, we need to revise the normal-force equations (43,44) by adding new normal forces ∆Fy2, ∆Fy4, ∆Fz5 and ∆Fz6, respectively for the faces 2, 4, 5 and 6. These would be in addition to the conventional normal forces due to the electric ﬁeld. Whereas, no new normal force would be added for the faces 1 and 3. This is because, for these two surfaces the normal directions are ±xˆ, along which the new force would be zero (∆Fx = 0), as we explained.

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F = q D , x3 0 x3 F = q D , x1 0 x1 F = q D + ∆F , y4 0 y4 4 F = q D + ∆F , y2 0 y2 2 F = q D + ∆F , z6 0 z6 6 F = q D + ∆F , z5 0 z5 5 ∂Dx ∂Dx V ∂Dy 0 ∆x∆y∆z( ∂x + ∂t c2 ) + ∆x∆z[ ∂y ∆y + (∆Fy4 − ∆Fy2) q ] ∂Dz 0 V +∆x∆y[ ∂z ∆z + (∆Fz6 − ∆Fz5) q ] = (ρv − Jx c2 )∆x∆y∆z. (46)

Combining the equations (45,46), we get

∂Dx V 0 V ∆x∆y∆z ∂t c2 + ∆x∆z(∆Fy4 − ∆Fy2) q + ∆x∆y(∆Fz6 − ∆Fz5) = −Jx c2 ∆x∆y∆z, ∂Dx V 0 0 ∆y∆z( ∂t + Jx) c2 = (∆Fy2 − ∆Fy4) q ∆z + (∆Fz5 − ∆Fz6) q ∆y, 2 ∂Dx 0c ∆y∆z( ∂t + Jx) = [(∆Fy2 − ∆Fy4)∆z + (∆Fz5 − ∆Fz6)∆y] qV . (47)

For analytical convenience, one may deﬁne a new vector H, using which any additional force ∆F is indirectly expressed in terms of the charge velocity v as follows:

q 1 2 v¯ × H = ∆F = qµ0v¯ × H, µ0 = 2 . (48) 0c 0c

The force is deﬁned proportional to the cross product of the velocity vector with H. This would ensure that the magnitude of the force is proportional to the velocity, and the force is directed normal to the direction of the velocity, as per our expectation discussed earlier. The components of the new force that are normal to diﬀerent faces of the test box may be related to appropriate components of the new vector H.

∆F 2 = qµ0V xˆ × H2, ∆Fy2 =y ˆ · ∆F 2 = −qµ0VHz2,

∆F 4 = qµ0V xˆ × H4, ∆Fy4 =y ˆ · ∆F 4 = −qµ0VHz4,

∆F 5 = qµ0V xˆ × H5, ∆Fz5 =z ˆ · ∆F 5 = qµ0VHy5,

∆F 6 = qµ0V xˆ × H6, ∆Fz6 =z ˆ · ∆F 6 = qµ0VHy6. (49)

Now combine equations (47) and (49) .

∂Dx ∆y∆z( ∂t + Jx) = −Hz2∆z + Hz4∆z + Hy5∆y − Hy6∆y P = Hi · ∆li , (50) 4C

∂Dx X ∂D (J + )∆S = H · ∆l = (J + ) · ∆S. (51) x ∂t i i ∂t ∆C

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The loop ∆C refers to a closed rectangular loop parallel to the face 3 of the rectangular box in Fig.5b, passing through the center of the box. This is shown separately in the Fig.6. Further, the orientation of the path of the loop ∆C is such that the normal direction to the enclosed surface vector ∆S, as per the right-hand rule, is along the normal to the face 3 of the box (that is, along +ˆx).

Figure 6. Deduction of the Ampere’s Law from the Electric ﬁeld divergence (Gauss’ Law) experiment of Fig.5, as seen by an observer from the (x,y,z,t) reference frame.

We have found a relationship (51) for the new force ﬁeld H, such that the equation (42) is satisﬁed for the total forces, as observed in the unprimed reference frame. This means that the Gauss’ Law experiment in the primed coordinates would now be consistent or equivalent when observed from the unprimed coordinates, as fundamentally required. The orientation of ∆C, and the normal to the enclosed surface ∆S, used in the above derivation of (51) are established by the direction of the relative motion between the primed and unprimed frames of reference. We happened to have selected the direction of the relative velocity V to be alongx ˆ. In general, by doing the analysis with velocity V in an arbitrary direction, it may be recognized that the equation (51) would apply for an elemental closed loop ∆C, and its enclosed surface ∆S, having any arbitrary orientation.

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X ∂D H · ∆l = (J + ) · ∆S. (52) i i ∂t ∆C

The equation (52) is the Ampere’s Law expressed for an elemental loop. The new force field described by the vector H, which we have defined in (48), may be recognized as the magnetic field. The above relationship (52) may also be expressed using the curl operator.

X ∂D ∂D H · ∆l = (∇ × H) · ∆S = (J + ) · ∆S, ∇ × H = J + . (53) i i ∂t ∂t ∆C

The obvious implication of the derivation is that the Ampere’s Law no longer has to be established as an independent law. It can be derived as a direct consequence of the Gauss’ Law for the electric field, through relativistic transformation. Interestingly, the magnetic field, which relates to the electric current and flux density through the Ampere’s Law, may be interpreted as a new “synthesized” field. This synthesized field will have to be added in order that the Gauss’ Law for the electric field is generalized across reference frames through the principle of relativistic equivalence. This unique interpretation of the magnetic field, from the view point of the relativistic transformation, is a significant development.

5. Relativistic Transformation of the Gauss’ Law for the Magnetic Field: The Faraday’s Law

The new magnetic field H was introduced in order that the Gauss’ Law for the electric field is relativistically consistent. Now, analogous to the electric flux density and flux, we define a magnetic flux density to be B = µ0H, using which magnetic flux over a given surface can be determined. Further, like the Gauss’ Law for the electric field, we may establish a Gauss’ Law for the magnetic field in terms of the magnetic flux. This defines the total flux over a closed surface to the total charge enclosed by the surface. The charge in this case is referred to as the magnetic charge Qm from which the magnetic field is supposed to originate or diverge from. However, no magnetic charge has been found to exist naturally, in which case the magnetic charge used in the Gauss’ Law for the magnetic field would be zero. In any case, like the Gauss’ Law for the electric field, the Gauss’ Law for the magnetic field may also be considered to be the basic governing relationship for the magnetic field, which is relativistically invariant. Accordingly, let the Gauss’ Law for the magnetic field be “observed” from different relativistic frames, in order to see what additional relationship the magnetic field will have to satisfy so that it is relativistically consistent. We will closely follow the process employed in the last section for the Gauss’ Law for the electric field, but suitably extended to the Gauss’ Law for the magnetic field. This is presented in the following, leading to the Faraday’s Law as an additional relationship that must be established between the magnetic and electric fields.

5.1. Gauss’ Law for the Magnetic Field, in the Primed Coordinates. Consider the Gauss’ Law for the magnetic ﬁeld, applied in the primed coordinate system (x0, y0, z0, t0) to a closed surface S0.

X 0 0 Bi · 4Si = 0. (54) S0

As mentioned above, this is similar to the Gauss’ Law for the electric ﬁeld, except that the magnetic charge is assumed here to be zero for real-word phenomena. One may sometimes ﬁnd hypothetical or equivalent magnetic charges included in electromagnetic modeling. Such magnetic charges are used only for useful analytical manipulations, and should not be confused to be physically realistic.

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Figure 7

Such hypothetical magnetic charges can be included in the Gauss’ Law for analytical generality. In this case, the right side of the Gauss’ Law equation (54) is simply replaced by the total equivalent 0 0 magnetic charge ∆Qm inside the closed surface S . For all analytical considerations, the magnetic charge would be treated analogous to the electric charge. The magnetic charge would be assumed to be relativistically invariant. All derivations and relationships established for electric charge and current in the last sections, can also be equivalently extended for magnetic charge and magnetic current.

X 0 0 0 Bi · 4Si = ∆Qm. (55) S0

In reference to the Fig.7a, the above form (55) of the Gauss’ Law may be applied to a speciﬁc closed surface S0, consisting of six faces of an elemental rectangular box of dimensions ∆x0 ×∆y0 ×∆z0. For this case, the equation (55) may be expressed using normal components of the ﬂux density Bi at the center of each of the six faces of the box, and the equivalent magnetic charge 0 0 0 0 0 0 density ρmv at the center of the box. The magnetic charge ∆Qm is expresses as ρmv4x 4y 4z , which is the product of the equivalent magnetic charge density and the the elemental volume of the box.

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Figure 7. (a) Magnetic ﬁeld divergence (Gauss’ Law) experiment in the (x’,y’,z’,t’) reference frame, and (b) the same experiment as seen by an observer from the (x,y,z,t) reference frame.

0 0 0 0 0 0 0 0 0 0 0 0 0 (Bx3 − Bx1)∆y ∆z + (By4 − By2)∆x ∆z + (Bz6 − Bz5)∆y ∆x = ∆Qm 0 0 0 0 = ρmv4x 4y 4z . (56)

The above summation form of the Gauss’ Law may also be expressed using derivatives of the ﬂux 0 density B at the center of the box.

0 0 0 0 0 0 0 0 0 0 0 0 (Bx3 − Bx1)∆y ∆z + (By4 − By2)∆x ∆z + (Bz6 − Bz5)∆y ∆x = 0 0 0 ∂Bx 0 0 0 ∂By 0 0 0 ∂Bz 0 0 0 ∂x0 4x ∆y ∆z + ∂y0 4y ∆x ∆z + ∂z0 4z ∆y ∆x = 0 0 0 0 ρmv4x 4y 4z , 0 0 0 ∂Bx ∂By ∂Bz 0 ∂x0 + ∂y0 + ∂z0 = ρmv. (57)

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5.2. A Basic Gauss’ Law Experiment in the Primed Reference Frame. Let an observer in the primed coordinate system conduct an experiment to verify or validate the Gauss’ Law for the rectangular box, placed in a free-space medium. With reference to the Gauss’ Law expression of (56), the observer would need to measure the normal components of the flux density Bi = µ0Hi 0 for each of the six faces, i = 1, .., 6, of the rectangular box S . The magnetic field Hi or the flux density Bi are fundamentally defined in terms of the force experienced by a given current element 0 I of a given length ∆l0. This is equivalent to the definition of (48), applied in the primed frame, 0 0 where qv0 = I v0∆t = ∆l0I .

0 0 0 0 0 0 0 F = ∆l I × B = µ0∆l I × H . (58)

Accordingly, one may measure the force on a reference current element, from which a particular component of the magnetic field or flux density can be deduced. As per the equation (58), the component of the magnetic field which is aligned along the current element would not contribute to the total force. Therefore, one can not deduce the the component of the magnetic field along the current element from the force measurement. This implies, in order to measure the normal component of the magnetic field on a particular face of the elemental box, as needed in the validation experiment, one would like to place the reference current element parallel to the face. With this in mind, let us choose the reference currents on faces 1 and 3 to be along the y direction, but along the x direction for all other faces. For simplicity, let us select the magnitude of each reference current to be the same. Accordingly, a conducting wire is placed centered on each face of the rectangular box, directed along the x direction for faces 2, 4, 5 and 6, but along the y direction for faces 1 and 3. Each 0 0 0 wire is excited with a current of magnitude I . The forces F i experienced by a given section ∆l of the current wire is then measured using an appropriate technique. The actual length of the reference current, over which the force measurement is conducted, is not really critical, because we are interested in the magnetic field or flux density, which is defined in terms of the force per unit length. However, for simplicity, we have specified the reference sections of the currents over which the fore is measured on each face to be of equal length ∆l0. Now, the normal magnetic field components on each face are related to appropriate force components, as per (58).

0 0 0 0 0 0 0 0 0 0 Fz1 F 1 = ∆l I yˆ × B 1, −Fz1 = ∆l I Bx1,Bx1 = − ∆l0I0 , 0 0 0 0 0 0 0 0 0 0 Fz3 F 3 = ∆l I yˆ × B 3, −Fz3 = ∆l I Bx3,Bx3 = − ∆l0I0 , 0 0 0 0 0 0 0 0 0 0 Fz2 F 2 = ∆l I xˆ × B 2,Fz2 = ∆l I By2,By2 = ∆l0I0 , 0 0 0 0 0 0 0 0 0 0 Fz4 F 4 = ∆l I xˆ × B 4,Fz4 = ∆l I By4,By4 = ∆l0I0 , 0 0 0 0 0 0 0 0 0 0 Fy5 F 5 = ∆l I xˆ × B 5, −Fy5 = ∆l I Bz5,Bz5 = − ∆l0I0 , 0 0 0 0 0 0 0 0 0 0 Fy6 F 6 = ∆l I xˆ × B 6, −Fy6 = ∆l I Bz6,Bz6 = − ∆l0I0 . (59)

The Gauss’ Law equation (56) can be expressed in terms of the measured force components in (59).

0 0 0 0 0 0 0 0 0 0 0 0 (Bx3 − Bx1)∆y ∆z + (By4 − By2)∆x ∆z + (Bz6 − Bz5)∆y ∆x = 1 0 0 0 0 0 0 0 0 0 0 0 0 I0∆l0 [(−Fz3 + Fz1)∆y ∆z + (Fz4 − Fz2)∆x ∆z (−Fy6 + Fy5)∆x ∆y ] = 0 0 0 0 0 ∆Qm = ρmv4x 4y 4z . (60)

The above equation (60) is a form of the Gauss’ Law for the magnetic ﬁeld, expressed in terms 0 of the force components on the reference current elements. For a known charge density ρmv, this

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relationship can be experimentally verified by measuring the force components, for given values of the reference currents and the length elements, It may be noted that the above experiment is conducted in the primed coordinates, at a fixed time t0. In other words, all the above force measurements are conducted simultaneously in the primed coordinate system, with time difference ∆t0 between measurements at different faces equal to zero.

5.3. The Experiment as Observed from the Unprimed Frame. Now, let the above physical experiment, originally conducted in the primed frame, be “observed” from the unprimed coordinate system, as shown in Fig.7b. As per the general principle of relativity, any “basic” physical experiment conducted in one reference frame should lead to the same conclusion when the experiment is observed or measured from another coordinate frame, moving with an uniform velocity with respect to each other. The Gauss’ Law experiment may be considered one such basic experiment. It establishes a fundamental relationship between the measured forces experienced by reference current elements and any enclosed equivalent magnetic charges. Accordingly, the governing relationship (60) for the experiment performed in the primed frame is expected to remain invariant when the frame of observation is switched. In other words, the governing equation (60) should remain valid, when the diﬀerent parameters in (60) are properly substituted by their new values of measurement, when the same physical experiment is observed from the unprimed frame.

h i (− Fz3 + Fz1 )∆y∆z + ( Fz4 − Fz2 )∆x∆z + (− Fy6 + Fy5 )∆x∆y = ∆Q . (61) I3∆l3 I1∆l1 I4∆l4 I2∆l2 I6∆l6 I5∆l5 m

We will examine all the unprimed variables in (61), in contrast with the corresponding primed variables in (60). Fundamental and analytical implications of observing the physical experiment from the unprimed frame will have to be properly understood. The length element 4x would be different from 4x0, as per the transform relation (40). Length measurements only along the direction of relative velocity change through relativistic transformation. Whereas, those along directions orthogonal to the relative velocity remain the same. Accordingly, 4y = 4y0 and 4z = 4z0. The timing of measurement on locations with different x coordinates would be different (∆t 6= 0), whereas those with different y or z, but the same x coordinates would be simultaneous (∆t = 0). Using the relativistic transformation relations (10,24), one may relate the differential time (∆t) and location (∆x) parameters of observation in the unprimed reference frame. This would lead to the relationship (40) deduced in the last section.

4t = 4xV/c2, 4x0 = α4x, 4y0 = 4y, 4z0 = 4z. (62)

Accordingly, the forces on the faces 1 and 3, which differ in their x coordinates, would be measured in the unprimed frame at different times. Whereas, those on all other faces 2, 4, 5, and 6 would be measured simultaneously, because the x coordinates at the center of all these faces have the same x coordinates. The forces F i on a given face maybe in general assumed to be different in magnitude 0 from F i on the corresponding face. As we have indicated before, for simplicity of the original measurement in the primed frame, the reference currents I0 and their length elements ∆l0 were selected with equal magnitude on all faces. However, they are substituted in (61) with distinct variables Ii and ∆li, i = 1, ..., 6, for the six faces. This is because the reference currents having equal magnitude in the unprimed frame may no longer be equal when measured in the unprimed frame. If desired, these reference currents and length elements can be related to I0 and ∆l0, and consequently with each other, using space-time transform relations. The final outcome of our present derivation happens to be independent of their actual magnitudes. Therefore, we choose to keep them as distinct variables just to be technically correct, without any further analysis or simplification.

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The equivalent magnetic charge and current parameters, as seen in the two frames, can be 0 treated analogous to electric charge and current parameters. The total magnetic charge ∆Qm = 0 0 0 0 ρmv4x 4y 4z is assumed to be invariant to the frame of observation, and, therefore, should be equal to the charge ∆Qm as observed from the unprimed frame. One may like to deﬁne ∆Qm in terms of a new equivalent charge density ρmv0, as observed from the unprimed frame, such that 0 0 ∆Qm = ρmv04x4y4z. Accordingly, we can relate ρmv0 and ρmv by equating ∆Qm and ∆Qm: 0 0 0 ρmv0 = ρmv4x /4x = ρmvα. This derivation uses the transformation equation (62) to relate the 0 dimensional variables. Now, ρmv or ρmv0 may also be related to the charge density ρmv one would normally measure in the unprimed coordinates at a given time t, ∆t = 0. We may extend the derivations in the last sections (3,4), equations (29,31,41), to magnetic charges.

0 2 ρmv0 = ρmvα = ρmv + ∆ρmv = ρmv − MxV/c , (63) where the parameter M represents magnetic current density, which is analogous to J used for the electric current density. By combining (63) with (61), we get,

h i (− Fz3 + Fz1 )∆y∆z + ( Fz4 − Fz2 )∆x∆z + (− Fy6 + Fy5 )∆x∆y = I3∆l3 I1∆l1 I4∆l4 I2∆l2 I6∆l6 I5∆l5 2 ∆Qm = ρmv04x4y4z = (ρmv − MxV/c )4x4y4z. (64)

5.3.1. Charged Reference Currents. Now, let us examine the forces experienced by the electric current elements Ii∆li, and relate them to suitable field components measured in the unprimed frame. Before we can characterize the forces on the new current elements, we need to examine the nature of these current elements Ii∆li as seen in the unprimed coordinates. The reference current elements I0∆l0 used in the original experiment are assumed to be free of charge, so that the forces experienced by the current elements in the primed reference frame are contributed only due to the magnetic fields. That allowed us to directly relate the magnetic fields in the Gauss’ Law to the total forces on the current element, which can be measured. Otherwise, the basic governing equation (60) for the experiment would have been invalid. However, when these charge-free currents are observed from the unprimed frame, they would look charged if the current is directed along x, but remain charge free if directed along y or z. This is governed by the derivations in section 3. An x-directed current Ix and its observed line-charge density ρl in the unprimed frame may be related using (29,31,41).

0 2 ρvα = ρv − JxV/c , 0 2 2 ρlα = ρl − IxV/c = 0, ρl = IxV/c , 0 0 ρl = 4Aρv = 0, ρl = 4Aρv,Ix = Jx4A. (65)

0 The volume distributions ρv, ρv and Jx, in (29,31,41) are substituted in terms of the corresponding 0 line distributions ρl, ρl, and Ix, respectively, leading to the above relationship (65). ∆A is the small 0 cross-sectional area of the current element, which approaches zero for an ideal line current. ρv is assumed zero, because the current element is chosen to be charge free in the primed frame. This is 0 equivalent to having ρl also to be zero.

5.3.2. Force Transformation Using Electric and Magnetic Fields. With the understanding of the nature of the reference current elements and other relativistic considerations, discussed above, let us characterize the forces components of the Gauss’ Law equation (64) using diﬀerent ﬁeld components measured in the unprimed coordinates. Let us classify the forces in two separate groups, based on the timing of the force measurement and the nature of their reference currents used. The two groups are, (a) forces for the faces 1 and 3, and (b) the forces for the faces 2, 4, 5 and 6. We will discuss the force modeling for these two groups separately.

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For the two faces in the first group, the test current elements are directed along y, and therefore do not appear charged, as we have discussed earlier. Consequently, the forces experienced by these current elements can be expressed using magnetic field alone. This group of faces differ in their x coordinates, and therefore the forces measured on these faces in the primed coordinates at a given time (t0, ∆t0 = 0) are observed in the unprimed coordinates at different times t, ∆t 6= 0. This is in accordance with the differential space-time relationship (62), as discussed before. Now, the different components of the force and magnetic flux density, as measured in the unprimed frame, can be related using an extension of (59) as follows. The primed variables in (59) are replaced by the corresponding unprimed variables, and the space and time coordinates for different parameters in (59) are specified at the center of corresponding face (see Fig.7b). Further, the fields at the center of the faces are related to those at the center of the box using partial derivatives, and equation (62) is used to relate differential space (4x) and time (4t) variables.

F 1 = ∆l1I1yˆ × B1, −Fz1 = ∆l1I1Bx1 = ∆l1I1Bx(x − 4x/2, t − 4t/2) 4x ∂Bx(x,t) 4t ∂Bx(x,t) = ∆l1I1(Bx(x, t) − 2 ∂x − 2 ∂t ) 4x ∂Bx(x,t) 4xV ∂Bx(x,t) = ∆l1I1(Bx(x, t) − 2 ∂x − 2c2 ∂t ), F 3 = ∆l3I3yˆ × B3, −Fz3 = ∆l3I3Bx3 = ∆l3I3Bx(x + 4x/2, t + 4t/2) 4x ∂Bx(x,t) 4t ∂Bx(x,t) = ∆l3I3(Bx(x, t) + 2 ∂x + 2 ∂t ) 4x ∂Bx(x,t) 4xV ∂Bx(x,t) = ∆l3I3(Bx(x, t) + 2 ∂x + 2c2 ∂t ). (66)

On the other hand, for the second group of faces 2, 4, 5 and 6, the reference currents are directed along x, and therefore they would appear charged, as discussed earlier. Equation (65) can be used to relate the observed line charge density ρl to the corresponding line current density Ix. Consequently, the force experienced by these currents are to be expressed using both the magnetic and electric fields. The centers of measurement for the four faces have the same x coordinates. Therefore, as discussed before, the experiment in the primed coordinates at a fixed time (t0, ∆t0 = 0) are observed in the unprimed coordinates also at a fixed time t, ∆t = 0. This is in accordance with equation (62). With these above issues in mind, the different components of the force, as measured in the unprimed frame, can now be modeled by adding two contributions: the force due to the magnetic field using an extension of (59), added with the force on the charges on the current elements using appropriate electric field components.

F 2 = ∆l2I2xˆ × B2 + ∆l2ρl2E2,Fz2 = ∆l2I2By2 + ∆l2ρl2Ez2 4y ∂By V = ∆l2I2(By − 2 ∂y ) + ∆l2I2 c2 Ez2,

F 4 = ∆l4I4xˆ × B4 + ∆l4ρl4E4,Fz4 = ∆l4I4By4 + ∆l4ρl4Ez4 4y ∂By V = ∆l4I4(By + 2 ∂y ) + ∆l4I4 c2 Ez4,

F 5 = ∆l5I5xˆ × B5 + ∆l5ρl5E5, −Fy5 = ∆l5I5Bz5 − ∆l5ρl5Ey5

4z ∂Bz V = ∆l5I5(Bz − 2 ∂z ) − ∆l5I5 c2 Ey5, F 6 = ∆l6I6xˆ × B6 + ∆l6ρl6E6, −Fy6 = ∆l6I6Bz6 − ∆l6ρl6Ey6

4z ∂Bz V = ∆l6I6(Bz + 2 ∂z ) − ∆l6I6 c2 Ey6. (67)

The contribution in (67) from the magnetic ﬁeld have been obtained by replacing the primed variables in (59) by the corresponding unprimed variables. The magnetic ﬁeld components are measured at the center of particular faces (see Fig.7), which are then related to those at the center of the box using partial derivatives. The force contribution in (67) due to the charges on the current

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elements is expressed as product of the electric ﬁeld and the charge. The equation (65) is used for simpliﬁcation relating line charge density to the line current.

5.4. The Faraday’s Law, Deduced from the Force Transformation. Now, using the equations (66,67) in the force equation (64)), we get a relationship between the electric and magnetic ﬁelds in the unprimed reference frame.

h ∂Bx V ∂Bx ∂By V V ( ∂x + c2 ∂t )4x∆y∆z + ( ∂y 4y + c2 Ez4 − c2 Ez2)∆x∆z + i ∂Bz V V 2 ( ∂z 4z − c2 Ey6 + c2 Ey5)∆x∆y = (ρmv − MxV/c )4x4y4z, h ∂Bx ∂By ∂Bz V ∂Bx V ( ∂x + ∂y + ∂z + c2 ∂t )4x∆y∆z + c2 (−Ey6 + Ey5)∆x∆y i 2 +(Ez4 − Ez2)∆x∆z = (ρmv − MxV/c )4x4y4z. (68)

We have established the Gauss’ Law for the magnetic field in the primed coordinates, which is expressed in the summation form (56) or a differential form (57). The fields used in these expressions are measured simultaneously in the primed frame at a given t0, ∆t0 = 0. The Gauss’ Law for the magnetic field may be independently established in the unprimed frame as well, using fields that are measured simultaneously in the unprimed frame at a given t, ∆t = 0. Let us rewrite the differential form of the Gauss’ Law in (57) for the unprimed frame. This is accomplished by simply replacing the unprimed variables in (57) by the corresponding primed variables.

∂Bx ∂By ∂Bz ∂x + ∂y + ∂z = ρmv. (69)

Using (69) in (68) we would get,

h i V ∂Bx V c2 ∂t ∆x∆y4z + c2 (−Ey6 + Ey5)∆x∆y + (Ez4 − Ez2)∆x∆z V = −Mx c2 4x4y4z, h i ∂Bx − Ey6∆y + Ey5∆y + Ez4∆z − Ez2∆z = −( ∂t + Mx)4y4z. (70)

The above relation (70) may be recognized in terms of line and surface integrals over an elemental rectangular loop ∆C and its enclosed surface ∆S, respectively.

P ∂Bx ∂B Ei · 4li = −( ∂t + Mx)4y4z = −( ∂t + M) · 4S. (71) 4C

The loop ∆C refers to a closed rectangular loop parallel to the face 3 of the rectangular box in Fig.7b, passing through the center of the box. This is separately shown in the Fig. 8. Further, the orientation of the path of the loop ∆C is such that the normal direction to the enclosed surface vector ∆S, as per the right-hand rule, is along the normal to the face 3 of the box (that is, along +ˆx). Equation (71) is the Faraday’s Law as applied to an elemental loop ∆C. The obvious implication of the above derivation is that the Faraday’s Law no longer has to be established as an independent law. It is derived from the Gauss’ Law for the magnetic field in a free-space medium, through relativistic transformation. Accordingly, the Faraday’s Law (71) may be interpreted as a necessary condition, so that the Gauss’ Law for the magnetic field is consistent through relativistic transformation across reference frames. This is a significant development. The elemental loop ∆C

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Figure 8. Deduction of the Faraday’s Law from the magnetic ﬁeld divergence (Gauss’ Law) experiment of Fig.7, as seen by an observer from the (x,y,z,t) reference frame.

in the above derivation is oriented with its normal along the x direction, which is parallel to the relative velocity between the frames. The choice of the direction of the relative velocity is arbitrary. If the analysis was performed with relative velocity in a general direction, the expression (71) for the Faraday’s Law can be shown to apply to an elemental loop with any general orientation.

P ∂B Ei · 4li = −( ∂t + M) · 4S. (72) 4C

The above equation (72) for the Faraday’s Law may also be expressed using the curl operator.

X ∂B ∂B E · ∆l = (∇ × E) · ∆S = −(M + ) · ∆S, ∇ × E = −M − . (73) i i ∂t ∂t ∆C

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6. Discussions

6.1. Application in a Material Medium. It may be noted, that we have explicitly assumed a free-space medium for the derivations of the Ampere’s Law in (51,53), as well as the Faraday’s Law in (72,73). The derivations, in principle, are theoretically complete. The results may be extended to any material medium by substituting the material with equivalent current loops and charge dipoles, that are properly arranged in the free-space as the background. These equivalent currents and dipoles would represent the internal structure of the material, that are influenced by the electric and magnetic fields. The basic forms of the Ampere’s and Faraday’s Laws in (51,53) and (72,73), respectively, can be shown to be valid for the material medium as well, by properly defining the electric and magnetic fields and flux densities inside the medium, and relating them using suitable permittivity and permeability parameters.

6.2. A New form of the Gauss’ Laws and Charge Invariance. The conventional Gauss’ Laws (equations (37,57)) are valid in the primed frame, that are verified in the respective Gauss’ Law experiments (equations (39,60)) using suitable force measurements. The conventional Gauss’ Laws would also be valid in the unprimed reference frame (equations (45,65)), that may be similarly verified using force measurements all conducted independently in its own unprimed frame. All the measurements in the conventional Gauss’ Law experiments are conducted simultaneously, at a given time, in the respective independent frames. In addition, a new form of the Gauss’ Laws are introduced, implemented in the experiments (42,61). Here, a Gauss’ Law experiment originally conducted in the primed frame at a given time t0, ∆t0 = 0, is invariant as observed from the unprimed frame with different timing (∆t 6= 0) for the individual measurements. These new Gauss’ Law experiments, definitively timed in a particular frame (primed frame), unambiguously measure the same amount of charge (electric or equivalent magnetic), assuming the charge is invariant to any relative motion. On the other hand, the conventional form of the Gauss’ Law experiments are not guaranteed to measure the same amount of charge in two frames, because a part of the charge may be moving and might escape from the measurement box during the different, independent timings in the two frames. In other words, the new form of the Gauss’ Laws is the only unambiguous way to ensure invariance of charge (electric or equivalent magnetic) across reference frames, and therefore is more fundamental. Enforcing this new fundamental form of the Gauss’ Laws naturally allows a simple “derivation” of the Ampere’s and the Faraday’s Laws (51,72), as additional required conditions for the enforcements. This is a significant discovery.

6.3. Mechanical Principles Derived from the Electromagnetic Theory. We succeeded to derive the Maxwell’s Equations (Ampere’s and Faraday’s Laws) from the Gauss’ Laws, by additionally employing only the space-time relativistic transformation equations. This does not require the transformation relations for force, mass and momentum, or their related mechanics of momentum and energy conservation. However, once the Maxwell’s Equations are rigorously established, they can always be solved for the fields and the associated forces in a given problem (two stationary electric charges in the free-space, for example), as seen by two inertial frames. The relationships between the solved forces in the two frames would in turn provide the required force-transformation formulas in the two frames. Conventionally, the force-transformation formulas in special relativity are deduced starting from the basic Newton’s Laws of force, momentum and energy, by employing the space-time relations of the special relativity [15]. This leads to transformation relations for the mass, momentum and energy as intermediate steps, leading to the force transformation relations. Now that the force transformation relations are available directly from the Gauss’ Laws through the Maxwell’s Equations, one can then retrace backwards the conventional derivations of relativistic mechanics. Accordingly, one could derive the transformation relations for the mass, momentum and energy, leading to the “derivation” of the Newton’s Laws as well as the related principles of conservation of momentum and energy. This would be a significant development, where the basic concepts of an electric and magnetic charge (as defined through their respective fields, Gauss’ Laws and

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charge invariance) would completely describe all electromagnetic as well as material phenomena of nature, making the conventional mechanics of matter (as deﬁned through mass, Newton’s Laws and momentum/energy conservation principles) theoretically redundant. What it means is that all mechanical principles and parameters may not be fundamental after all, but are somehow intrinsic to the basic deﬁnitions of electric and magnetic charge, and their invariance, and concepts of space and time.

References [1] James Clark Maxwell. A Treatise on Electricity and Magnetism, Vol. I and II (Reprint from 1873). Dover Publications, 2007. [2] Albert Einstein and Anna Beck (English Translator). The Collected Papers of Albert Einstein, Volume 2: The Swiss Years: Writings 1900-1909 (see Documents 23 and 24). Princeton University Press, 1989. [3] Albert Einstein. Zur Elektrodynamik bewegter K¨orper (On the Electrodynamics of Moving Bodies). Annalen der Physik, 322(10):891–921, 1905. [4] D. K. Cheng. Fundamentals of Engineering Electromagnetics. Addison-Wesley Publishing, 1993. [5] Jr. D. H. Hayt. Engineering Electromagnetics. McGraw-Hill, New York, 1995. [6] N. N. Rao. Elements of Engineering Electromagnetics. Prentice Hall, New Jersey, 2000. [7] E. M. Purcell. Electricity and Magnetism (Berkeley Physics Course, Vol.2, 2 Ed.). Prentice Hall, New Jersey, 1984. [8] Richarrd P. Feynman, Robert B. Leighton, and Mathew Sands. Lectures on Physics, Vol.II, Ch.25,26. Addision Wesley, 1964. [9] J. D. Jackson. Classical Electrodynamics, 2 Ed. John Wiley and Sons, New York, 1975. [10] R. S. Elliott. Electromagnetics: History, Theory and Applications. Wiley-IEEE Press, 1999. [11] J. R. Bray. From Maxwell to Einstein: Introduction of the Time-Dialation Property of Special Relativity in Undergraduate Electromagnetics. IEEE Antennas and Propagation Magazine, 48(3):109–114, June 2006. [12] J. W. Arthur. The Fundamentals of Electromagnetic Theory Revisited. IEEE Antennas and Propagation Mag- azine, 50(1):19–65, February 2008. [13] D. H. Frisch and L. Wilets. Development of the Maxwell-Lorentz Equations from Special Relativity and Gauss’ Law. American Journal of Physics, 24(8):574–579, November 1956. [14] J. R. Tessman. Maxwell - Out of Newton, Coulomb and Einstein. American Journal of Physics, 34(11):1048–1055, November 1966. [15] Ray Skinner. Relativity for Scientists and Engineers. Dover Publications, Inc., 1982.

120 9-1

Deriving the Newton’s Laws from the Maxwell’s Equations - Basic Concepts of Charge and Space-Time Supersede All Mechanical Principles

Nirod K. Das

Department of Electrical and Computer Engineering Tandon School of Engineering, New York University, Five Metrotech Center, Brooklyn NY 11201 (Dated: August, 2016)

Abstract The basic principles of the Newton’s Laws, and related concepts of conservation of momentum and energy, are derived from the Maxwell’s Equations. The electric and magnetic ﬁelds produced by an electric charge in uniform motion, as derived from the Maxwell’s Equations, are used to ﬁnd the force it exerts on another charge, as measured in two inertial frames. These force transformation relations in the two frames are extended to apply to any general physical problem involving force. The force transformation relations are then used, together with the space-time relations of special relativity, to derive the Newton’s laws of motion applicable for velocity v much smaller than the speed of light c (v << c), as well as general expressions for the mass, momentum and energy, applicable for any velocity v ≤ c. Further, the momentum or energy as expressed in one inertial frame, can be linearly related to the momentum and the energy expressed in another inertial frame. This result, when applied to a closed system with no external interaction, prove the momentum and the energy to be conserved. Fundamental and philosophical implications of the results and derivations are discussed. The principles of invariant electric and magnetic charge, upon which all electromagnetic concepts of the Maxwell’s Equations are based, are recognized to be complete, more general and fundamental than the Newton’s Laws, making the mechanical or material principles redundant.

1. Introduction

It has been recently established in [1, 2] that the Maxwell’s Equations [3] can be derived from basic principle of invariance of the electric and magnetic charges, as fundamentally defined by the Gauss’ Laws for the electric and magnetic fields, respectively, using only the space-time relations of special relativity [4, 5]. The principle of invariance of the charges, unambiguously defined using the Gauss’ Laws applicable across reference frames, allows a simple derivation of the Maxwell’s Equations from the basic charge principle, without requiring the Newton’s Laws of motion or the principles of momentum and energy conservation. Once the Maxwell’s Equations are established, they can be independently solved for the fields and the associated forces in any given problem in two reference frames. The relationships between the solved forces in the two frames would establish the required relativistic force-transformation formulas in the two frames, without any need for the Newton’s Laws. Instead, the Newton’s Laws can now be derived from the established force-transformation formulas. Conventionally, the force-transformation formulas in special relativity are deduced starting from the basic Newton’s Laws and principles of momentum and energy conservation, by employing the space-time relations of special relativity [5]. This process derives the velocity-dependent functions for the mass, momentum and energy as intermediate steps, leading to the force transformation relations. Now that the force transformation relations are available directly from the Maxwell’s Equations, one can then essentially retrace backwards the conventional derivations of the relativistic mechanics. Accordingly, one could derive the functional forms for the mass, momentum and energy, leading to the “derivation” of the Newton’s Laws and the associated principles of momentum and energy conservation. In this paper we will follow such a derivation, starting with a simple electrical problem having simple solutions for the Maxwell’s Equations. Theoretical and philosophical significance of the different results and derivations are addressed. The fundamental nature of the electromagnetic principles, in contrast with the basic material principles of the Newton’s Laws, are discussed.

2. Force Transform Relations Derived From the Forces Between Two Charges

Consider two charges of equal magnitude Q are stationary with respect to each other. Their ﬁelds and mutual forces are measured in two reference frames, one (primed frame) where the charges are at rest with respect to the observer, and the other (unprimed frame) where the charges are moving with respect to the observer at a constant velocity V along the z direction, as shown in Fig.1. This is equivalent to having individual observers attached to the unprimed and the primed frames, who see the other observer moving with a uniform velocity V in the +z and −z directions, respectively. The origins of both the frames are aligned with the location of one of the charges at time t = t0 = 0, whereas the other charge is located at (x0, y0, z0) in the primed frame, or at (x, y, z) in the unprimed frame, timed at t = 0 in the unprimed frame. The two frames, moving with uniform velocity with each other, are assumed to be naturally “unbiased” in a uniform free-space medium. The basic invariant nature of propagation of light in a uniform free-space, and consequently the validity of the associated Maxwell’s Equations, are ensured across all such equivalent, unbiased frames. The invariant nature of light to follow a straight-line path in the uniform free-space medium, having the same magnitude of its velocity across all unbiased frames, would require the reference frames to move with a uniform velocity with respect to each other, as a necessary condition. This may be veriﬁed using the space-time relations of special relativity, which was established based

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Figure 1. Two charges, that are stationary with respect to each other, but located at different positions in space. The fields produced by one charge (source charge), acting upon the second charge (test charge), as seen in two different reference frames. The primed frame is moving with a velocity V along the z axis with respect to the unprimed frame. on the expected special nature of light. This requirement of the uniform relative velocity is consistent with the two reference frames selected in the electromagnetic analysis of Fig.1. We will find the electric and magnetic fields produced by the charge at the origin, moving with velocity V along z direction in the unprimed frame. The fields seen in the primed frame is a specific case of that in the unprimed frame, when the velocity is substituted as V = 0 and the coordinates are changed from the unprimed to the respective primed variables. Using these fields due to the charge at the origin, we can find the total force applied on the second charge at the general location, for the two cases with observers in the primed and unprimed frames. Relating the components of the two forces F¯0 and F¯ would establish the required force transformation between the frames. These force transform relations, although derived for the specific simple situations of the two charges, would be applicable to any physical problem involving force, and considered fundamental relations with universal scope. The electric E¯(x, y, z) and magnetic H¯ (x, y, z) fields of the charge in the unprimed frame, located at the origin and moving in uniform velocity V along the z axis, can be solved from the Maxwell’s Equations. One simple approach is to express the moving charge as a superposition or integration of Fourier current distributions on a plane parallel to the

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charge velocity (xz-plane). The individual Fourier currents would produce uniform plane waves propagating or evanescent in the ±y directions [6, 7], the fields of which are one of the simple solutions of the Maxwell’s Equations (in the uniform free-space medium, observed in an unbiased reference frame). The total fields can then be obtained by Fourier integration of the plane-wave fields. We will provide here the final results, which are also available from physics and engineering texts [8, 9]. Now, the special case for a stationary charge (in the primed frame) consists of only the electric field given by the Coulomb’s Law, with a zero magnetic field. This may be verified from the general results with any uniform velocity V in the unprimed frame, by simply substituting V = 0 and changing the unprimed to the primed coordinate parameters.

¯ Q E(x, y, z) = 03 (xxˆ + yyˆ + zzˆ), 4πε0r α ¯0 Q 0 0 0 E (x, y, z) = 03 (x xˆ + y yˆ + z zˆ), 4πε0r ¯ QV ¯ 0 0 0 0 H(x, y, z) = 4πr03α (−yxˆ + xyˆ), H (x , y , z ) = 0, 0 2 2 02 1/2 0 0 0 z V 2 1/2 r = (x + y + z ) , x = x, y = y, z = α , α = (1 − c2 ) , (1)

¯ ¯ ¯ Q2 2 2 F (x, y, z) = QE + Qµ0(V zˆ × H) = 03 (α xxˆ + α yyˆ + zzˆ), 4πε0r α ¯0 0 0 0 ¯0 ¯ 0 Q2 0 0 0 F (x , y , z ) = QE + Qµ0(V zˆ × H ) = 03 (x xˆ + y yˆ + z zˆ), (2) 4πε0r

0 0 0 Fx = αFx,Fy = αFy,Fz = Fz. (3)

2.1. Generality of the Force Transform Relations, Extended to Any Physical System Involving Force. The above result (3), although is derived for a simple problem, may be properly interpreted and extended for a general configuration. The charge at the origin is the “source” charge, which produces all the force fields we derived that act upon the second charge, called the “test” charge. The same result (3) would apply for any arbitrary location of the source charge, as well as for any arbitrary values of the source and test charges that may not be equal to each other. By principle or superposition, the same final result (3) would be obtained as well for an arbitrary spatial distribution of the source charges, producing an arbitrary distribution of the force field F¯(x, y, z). Further, the result (3) requires the velocity of the test charge to be directed along the z axis, with a magnitude equal to zero and V as seen in the primed and unprimed frames, respectively, only at the time of observation t = 0. The same result (3) would be valid for any arbitrary path and velocity function of the test charge, with any other velocityv ¯(t) at times t 6= 0 before or after the observation. This is because, the force (2) acting upon the test charge is dependent only on the location and velocity of the test charge at the time of observation t = 0, independent of all time-derivatives of the velocity at t = 0 or of the velocity functionv ¯(t) of the test charge at other times t 6= 0. In summary, the force-transform relationship (3) would work for a general force field as well as a general path or velocity function of the test body. Considering such generality, the above relations between the forces in the two inertial frames may be declared to be valid for a physical problem involving any arbitrary force field and arbitrary motion. A force is meant to be an agent to produce change in motion of the body, as time passes. Accordingly, the force F¯ may be defined as the time-derivative of certain physical

124 N. Das, A UEG Theory of Nature, 2018 9-5 quantity associated with the moving body, called the momentump ¯. The momentum and force are vector quantities, representing the directed, vector nature of the motion and its change. Accordingly, the component of the force vector in any given direction is equal to the time derivative of the momentum component in the particular direction. For mathematical generality, the time variation of the momentum may be expressed in the form of a general momentum function dependent on the positionr ¯, velocityv ¯, accelerationa ¯ and all higher- order time-derivatives of the velocity.

¯ dp¯(t) ¯0 dp¯0(t) F (t)∆ dt , F (t)∆ dt0 , 0 0 0 0 0 0 0 p¯(t) =p ¯(¯r, v,¯ a,¯ a¯1, a¯2,...), p¯ (t ) =p ¯(¯r , v¯ , a¯ , a¯1, a¯2,...), r¯ = xxˆ + yyˆ + zz,ˆ r¯0 = x0xˆ + y0yˆ + z0z,ˆ dr¯ 0 dr¯0 0 0 0 v¯ = dt = vxxˆ + vyyˆ + vzz,ˆ v¯ = dt0 = vxxˆ + vyyˆ + vzz,ˆ dna¯ dn+1v¯ dv¯ 0 dna¯0 dn+1v¯0 0 0 dv¯0 a¯n = dtn = dtn+1 , a¯0 =a ¯ = dt , a¯n = dt0n = dt0n+1 , a¯0 =a ¯ = dt0 , ¯ ¯0 0 0 0 F = Fxxˆ + Fyyˆ + Fzz,ˆ F = Fxxˆ + Fyyˆ + Fzz,ˆ 0 0 0 0 p¯ = pxxˆ + pyyˆ + pzz,ˆ p¯ = pxxˆ + pyyˆ + pzz.ˆ (4)

3. Dependence of the Momentum of a Body on its Motion, Derived from the Force-Transformation Relations

The speciﬁc dependence of the momentum on the parameters of motion of a body can be deduced from the fundamental force-transformation relations of (3), based on the space-time transformation relations of special relativity. ¯ Consider ﬁrst a simple case with a force F = Fzzˆ in the z direction, resulting in motion with changing velocityv ¯ = vzzˆ only along the z direction. Let the body at a given point be seen by a stationary observer (unprimed frame), and another observer (primed frame) moving with the same velocity V = vz as that of the body at the time of observation. We will derive results for this simple case of linear motion, based on the general force transformation relations (3). The results for the simple motion can be extended as well for a general motion.

3.1. Zero Force on a Stationary Body, and the Principle of an Inertial Frame. For the simple case with a motion long the z direction, consider the simplest situation of a given body in the primed frame, placed at a given location (at the origin z0 = 0). Due to the unbiased natures of the reference frame and the surrounding free-space, which we assumed in the above analysis to begin with, the particular body is expected to remain stationary at the specified location, with no natural bias to move it in any one way or another. For the naturally stationary body, with no spatial motion as time progresses, there would be no time variation of any momentum of the body. Using the definition of force in (4), which is 0 the time-derivative of the momentum, the expected applied force Fz would be zero. In other words, the stationary body would naturally remain stationary at the given location (at the origin z0 = 0) in the unbiased reference frame, requiring no force to maintain its stationary position. Accordingly, an observer attached to the reference frame, would also naturally remain fixed at the origin of the frame as a stationary body, with no influence of any force. In other words, the reference frame may be considered to be naturally “free-floating” in space. In this sense, the reference frame may be called an “inertial frame,” in reference to the mechanical concept of inertia of the observer, with its natural tendency to maintain its fixed position in

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absence of any force. The unbiased nature of the frame, originally defined electromagnetically, where light is observed to propagate in straight lines with an invariant speed, is now explained to be equivalent to its free-floating, inertial nature, defined in mechanical terms. This is a fundamental understanding.

3.2. Position Independence of the Momentum. The above stationary body in the primed frame would be seen in the unprimed frame with a uniform velocity V in the z direction, having no acceleration or other time-derivatives of its velocity. As per the force transformation relation (3), the force Fz seen in the unprimed frame is required to be zero, 0 given that the force Fz in the primed frame is known to be zero. In other words, the body must not need any force in order to sustain a uniform linear motion, as observed in the unprimed frame.

dpz dpz dz dpz dz Fz = dt = dz dt = dz V, dt = vz = V, 0 0 0 0 0 dpz dpz dz dz 0 Fz = dt0 = dz0 dt0 = 0, dt0 = vz = 0, 0 dpz Fz = Fz = 0, dz = 0. (5) Recall that the unprimed frame, like the primed frame, was also originally selected to be electromagnetically unbiased. Therefore, like the primed frame discussed earlier, the unprimed frame may also be considered an “inertial frame” in mechanical terms. Accordingly, the above conclusion regarding the body in uniform linear motion, speciﬁcally deduced in the unprimed frame, may be generally stated for validity in any inertial frame. That is, a body would maintain a uniform linear motion in an inertial frame, without any assistance of force. Mathematically, the above conclusion (5), derived using (3) and (4), is equivalent to having the momentum pz to be independent of the position z. This leaves the momentum pz to be a function of its remaining variables - the velocity vz, acceleration az, and other higher-order time-derivatives of the vz (see (4)).

3.3. Independence of Momentum With All Time-Derivatives of Velocity, and the Newton’s First Law. The position independence of momentum established that a stationary body or a body with uniform velocity does not require a force. We would like to know other possible motions, if any, that also may not require force. 0 Consider a linear motion along the z axis, with a non-zero acceleration az in the 0 0 primed frame, having the velocity vz and all time derivatives of the velocity vz, except the 0 0 first derivative (or acceleration az), to be zero. The force component Fz and Fz along the z direction, as seen in the two frames, defined in (4) as time-derivatives of the momentum pz 0 or pz in the respective frames, must satisfy the transform relations (3). This would require the momentum pz to be independent of all time derivatives of the vz. The space-time relations of the special relativity may be used to deduce the consequent relations for the velocity, as well as for its time-derivatives, in the unprimed frame with those in the primed frame. It may be shown that all time-derivatives of the velocity vz in the unprimed frame would be non-zero functions of vz = V 6= 0, even though only the first 0 0 time-derivative (acceleration az) of the velocity vz is non-zero in the primed frame. This is 0 due to the non-linear nature of the relativistic relation (8) between the velocities vz and vz in the two frames. Further, the time-derivatives of the velocity vz of increasingly higher order 0 can be shown to be proportional to increasing exponents of the acceleration az. The above conditions, applied with the force transformation relations (3) in the two frames, would lead to the independence of the momentum pz with all time-derivatives of the velocity vz.

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∞ F = dpz = dpz v + dpz a + P dpz a = z dt dz z dvz z danz n+1z n=0 ∞ dpz ρ (v )a0 + P dpz ρ (v )a0 n+2, dvz 0 z z danz n+1 z z n=0 n+1 d vz 0 n+1 dvz anz = dtn+1 = ρn(vz)az ; a0z = az = dt , ρn(vz = V 6= 0) 6= 0, n ≥ 0, 0 0 0 n+1 0 0 dpz dpz 0 0 0 dvz 0 d vz F = 0 = 0 a , a = a = 0 , a = 0n+1 = 0, n ≥ 1, z dt dvz z z 0z dt nz dt F 0 = F , dpz = 0, n ≥ 0. (6) z z danz

0 0 In the above derivation, the Fz is expressed proportional to the az, with no dependence 0 0 on higher exponents of the az. The Fz expression may be viewed as a power-series of the 0 0 az, with only one term involving the first-exponent of the az. On the other hand, the Fz 0 0 is expressed as a power-series of the az, involving all exponents of the az. The expressions 0 of Fz and Fz must be equated, as required by the force transformation relations (3). This 0 would require the individual terms in the power-series expressions of the Fz and Fz, with 0 different exponents of the az, to be equated. Given that the coefficients ρn can be shown to be non-zero for all n ≥ 0, as discussed in the following section, the above process leads to requiring the momentum pz to be independent of all time-derivatives of the velocity vz. Like the position independence of the momentum deduced earlier, the independence of the momentum with all time-derivatives of the velocity is also a significant deduction from the electromagnetic theory. This leaves the momentum to be dependent only upon the velocity. Using the definition of force in (4), this means that a non-zero force would be required only when the velocity of a massive body is changed. In other words, a stationary body would remain stationary, and a body in uniform motion would maintain the uniform motion, without any force, whereas an accelerating body would certainly require a non-zero force. However, any change in the acceleration would not require any additional force. This is the Newton’s first law of motion, although the first law does not specify that higher-order time derivatives of velocity beyond the first derivative (acceleration) do not require additional force. This aspect is specified only through the Newton’s second law, to be derived in the following.

3.4. Functional Dependence of the Momentum with Velocity. As explained above, the momentum function pz is left with the velocity vz as its only valid variable. The functional expression of pz with the variable vz can be deduced from the above result (6), by using the expression of ρ0 derived from the space-time relations of special relativity.

0 0 0 0 dpz 1 dpz 1 dpz 1 dpz Fz = F , = 0 = 3 0 = 3/2 0 , z dvz ρ0 dv α dv 2 dv z z (1− vz ) z c2 0 m0vz dpz dpz pz(vz) = 1/2 , m0 = 0 = , (7) v2 dvz 0 dvz (1− z ) vz=0 vz=0 c2

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0 0 z0+V t0 t0+z0V/c2 x = x , y = y , z = α , t = α , 0 0 0 vxα vyα vz+V vx = 0 2 , vy = 0 2 , vz = 0 2 , (1+V vz/c ) (1+V vz/c ) (1+V vz/c ) dt0 0 0 2 0 0 0 0 dt = α , dvx = α(dvx), dvy = α(dvy), dvz = α (dvz); vz = vx = vy = 0, 0 0 dvz dvz 3 dvz 0 3 dt = ρ0 dt0 = α dt0 ; vz = 0, ρ0 = α . (8)

Relations between the higher order time-derivatives of the velocity vz in the unprimed frame and the acceleration in the primed frame can be similarly obtained by further diﬀer- entiating the above relations between the velocities in the two frames. This would provide the expressions for all other ρn, n > 0, which are non-zero functions of vz = V as we needed in the above derivation (6).

n 0 n+1 d az dvz 0 n V n n+3 dtn = ρn(vz)( dt0 ) , ρn(vz = V, vz = 0) = (−1) (2n + 1)!!( c2 ) α , (2n + 1)!! = (2n + 1)(2n − 1)(2n − 3) ··· (1). (9)

3.5. Expressions in the Small-Velocity Limit, and the Newton’s Second Law. Note that we have “derived” the expression (7) for the momentum component pz, as a function of the velocity component vz, starting from the Maxwell’s Equations. In the limit of a small velocity, the expression of momentum in (7), and the associated force defined in (4), would take the form of the Newton’s second law. In the small-velocity limit, the momentum is shown to be proportional to velocity, with the constant of proportionality m0 recognized as the rest mass. The force, defined in (4) as the time-derivative of the momentum, is therefore equal to the rest mass times the acceleration (time-derivative of the velocity), in the small- velocity limit. This is the most basic mechanical formula constituting the Newton’s second law [10] of motion for a body of constant rest mass m0. Accordingly, we have “derived” the Newton’s second law of motion, from the electromagnetic theory based on the Maxwell’s Equations. This is a significant development.

m0vz pz(vz) = 2 1/2 , pz(vz → 0, vz << c) = m0vz, (1− vz ) c2

dpz d(m0vz) dvz Fz = dt = dt = m0 dt = m0az; vz << c, m0vz m0 pz(vz) = m(vz)vz = 2 1/2 , m(vz) = 2 1/2 . (10) (1− vz ) (1− vz ) c2 c2

3.6. Relativistic Mass. In consistency with the momentum expression in the small-velocity limit, which we deduced above to be the product of the velocity vz and the rest mass m0, the general expression of the momentum pz(vz) in (7) may also be expressed as a product of the velocity vz and a general mass term m(vz). This new mass term, as shown in (10), is a function m(vz) of the velocity vz of motion, unlike a ﬁxed mass m0 assumed in the Newton’s Law. Further, this velocity-dependent mass, referred to as the relativistic mass, would increase indeﬁnitely as the velocity vz increases approaching the speed of light c. We have succeeded to derive the required velocity function of the relativistic mass, directly from the Maxwell’s Equations.

The same dependence of the relativistic mass, as a function of the velocity vz for the linear motion along z, is extended in the following to apply as well for a general motion

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along any arbitrary path, where vz may be substituted with the magnitude v of the general velocity vectorv ¯.

3.7. Generalization to Motion in the Three Dimensions. The above derivations assumed a simple linear motion along the z axis. The direction of motion along the z axis for the simple motion is an arbitrary choice. Similar results would work as well for a motion along any general direction. Accordingly, the result in (7) may be used to relate the magnitude of a general momentum functionp ¯(¯v) to the magnitude of the velocity vectorv ¯. The small-velocity limit for the general case would be an extension of the equivalent limit (10) for the simple case. Further, consistent with the small-velocity limit, the general momentum vector is also directed along the velocityv ¯. The momentum vector in the general direction can then be decomposed into its individual components px, py and pz in the x, y and z directions, respectively.

p¯(¯v) = m(v)¯v, p(v) = m(v)v = m0v , m(v) = m0 , (1− v )1/2 (1− v )1/2 c2 c2 p¯(¯v) = mv¯ = m(v xˆ + v yˆ + v zˆ) = m0(vxxˆ+vyyˆ+vzzˆ) , x y z (1− v )1/2 c2 m0vx m0vy m0vz px = (1−v2/c2)1/2 , py = (1−v2/c2)1/2 , pz = (1−v2/c2)1/2 , 2 2 2 2 v = vx + vy + vz , (11)

pz(vz) = m0vz, px(vx) = m0vx, py(vy) = m0vy; vx, vy, vz << c,

p¯(¯v) = pxxˆ + pyyˆ + pzzˆ = m0(vxxˆ + vyyˆ + vzzˆ) = m0v¯; v << c,

dpx dpy dpz dp m0 = dv = dv = dv = dv . (12) x v=0 y v=0 z v=0 v=0

We derived the above general expressions (11) for the momentum components, starting with a simple motion along the z direction. This derivation explicitly satisfied the required 0 transform relationship (3) only between the force components Fz and Fz, for the simple case, which led to relating the momentum pz to the velocity vz in (7), also for the simple case. The results were then generalized to (11) for velocity along an arbitrary direction by reorienting the velocity axis, from which the expressions for the individual momentum components for the general case were decomposed. The required transform relationships (3) between all three force components, for the general case, were expected to be implicitly satisfied by the final expressions of the momentum components of (11), through the followed process of generalization and coordinate reorientation. This is a theoretically simple, valid approach. However, the expressions for the individual momentum components in (11) for the general case may also be explicitly verified to satisfy the required transform relationships (3) for all three force components. This is possible by first differentiating the momentum components in (11), expressed in the two frames, with respect to the individual velocity components. The results are then used in steps similar to (6) to relate respective force and momentum components using the space-time relationships (8), leading to verification of the force transform relations (3).

3.8. Energy Expression Derived From the Force and Momentum. Now, let us derive the expression for the energy of a moving body, adopting the conventional deﬁnition of energy used in the Newtonian mechanics. The derivation would make use of the momentum

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expression we established above. At this point, we do not question any reasoning behind the choice of the definition of energy. The definition is likewise introduced in the Newtonian mechanics, without any justification for its special form. It is simply expected without “proof” that the conventional energy definition would provide a useful conserved quantity, which is one of the foundational principles in the Newtonian mechanics. The validity of definition of the energy used, and the proof of its conserved nature, will be addressed in the section 4.2.

¯ dp¯ dW = F · ds = dt · ds = dp¯ · v¯ = vxdpx + vydpy + vzdpz (13) 2 = m0(vxdvx+vydvy+vzdvz) = (m0/2)d(v ) , (1−v2/c2)3/2 (1−v2/c2)3/2 2 2 W = R (m0/2)d(v ) = m0c = mc2. (14) (1−v2/c2)3/2 (1−v2/c2)1/2

We have now established the basic mass-energy relationship W = mc2, derived directly from the Maxwell’s Equations. Accordingly, all forms of energy and mass may be treated in equivalent terms using (14). This would allow mechanical treatment of general systems which may include conventional massive bodies as well as electromagnetic radiation. Any exchange of energy and momentum between the conventional bodies and the radiation may be implemented using concepts of electromagnetic ﬁeld-mass/energy and ﬁeld-momentum [11].

4. Energy and Momentum Conservation in a Closed System

4.1. Momentum-Energy Transformation Relations in the Two Frames. Let us express the momentums (11) in two inertial frames, moving with velocity V with respect to each other along the z direction. This is possible using the space-time relations (8) of special relativity. The momentums in the unprimed frame can now be linearly related with those in the primed frame in terms of the energy expression of (14) in the primed frame. Using symmetry of results between the two frames, similar relationship between the momentums in the two frames and the energy in the unprimed frame can be obtained by interchanging the primed and unprimed variables, and replacing V with −V .

0 p¯ = m0v¯ , p¯0 = m0v¯ , 2 1/2 02 1/2 (1− v ) (1− v ) c2 c2 0 0 0 0 2 2 0 0 pz+m V pz+W V/c 0 pz−mV pz−W V/c px = px, py = py, pz = α = α , pz = α = α , (15)

0 0 0 vxα vyα vz+V vx = 0 2 , vy = 0 2 , vz = 0 2 , 1+vzV/c 1+vzV/c 1+vzV/c 2 2 2 0 2 0 2 0 2 v2 vx+vy+vz α2 vx +vy +vz (1 − 2 ) = (1 − 2 ) = 0 2 2 (1 − 2 ) c c (1+vzV/c ) c α2 v02 2 2 1/2 = 0 2 2 (1 − 2 ), α = (1 − V /c ) . (16) (1+vzV/c ) c

The energy expression of (14) in the unprimed frame can also be similarly related to that in the primed frame in terms of the z directed momentum in the primed frame. Similar relationship in terms of the momentum component in the unprimed frame can also be obtained by interchanging primed and unprimed variables and replacing V by −V . This is by symmetry of results between the two frames.

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2 2 W = mc2 = m0c ,W 0 = m0c2 = m0c , 2 1/2 02 1/2 (1− v ) (1− v ) c2 c2 0 0 2 vzV 0 2 vzV m0c (1+ ) m c (1+ ) 0 0 W = c2 = c2 = (W +pzV ) , 02 1/2 α α α(1− v ) c2 0 (W −pzV ) W = α . (17)

4.2. Concept of Energy as a Conserved Parameter. Consider a system with all its individual forces added to zero, as seen by an inertial frame (primed frame). In order to be relativistically consistent with the force transformation relation (3), the system would also be seen with zero total force in the unprimed inertial frame, moving with a uniform velocity V with respect to the primed frame. Although the total force is zero, the system is free to undergo any change in its internal state, produced due to individual forces that may be produced due to any interactions between the constituent bodies in the system. This would be characterized by change of velocity, momentum and energy of the constituent bodies. Based on the basic deﬁnition of force (4), having the total force zero would mean that the total change of momentum ∆¯p and ∆¯p0 of all constituent parts of the system, over any time interval, would also be zero. Given the required momentum expressions of (11) and their relationships (15) in the two frames, it would additionally require the energy of (14) to be conserved (∆W 0 = ∆W = 0).

P F¯0 = 0, ∆¯p0 = 0; P F¯ = 0, ∆¯p = 0, 0 0 0 ∆px = ∆px = 0, ∆py = ∆py = 0, ∆pz = ∆pz = 0, 0 0 2 2 ∆pz (∆W )V/c 0 0 ∆pz (∆W )V/c ∆pz = α + α = 0, ∆W = 0; ∆pz = α − α = 0, ∆W = 0. (18)

This is a significant result, which proves that the energy, as conventionally defined in the Newtonian mechanics using the incremental form (13), is in fact conserved in a system with zero total force. The conservation of energy for a zero-force system no longer needs to be accepted as a foundational mechanical principle, without proof, simply based on theoretical and observational success of the principle. Conversely, if we are looking for a useful scalar parameter to be conserved in a system with zero total force, then the conventional definition of energy (13) (written in incremental form) is now theoretically proved to be one such conserved quantity. Other possible expressions of the energy one might think of may not succeed to maintain the desired energy conservation, consistent with the force transform relations (3) and special relativity.

4.3. Momentum Conservation in a Closed System, and the Newton’s Third Law. Now consider a system physically contained inside a definite volume of space, identified with an entirely closed surface boundary, with no interaction with the external free across the boundary surface. And, this is the case as seen by any inertial observer (primed and unprimed frames). The non-interaction condition across the closed boundary may be characterized in terms of no flow of energy, or its mass equivalent as per (14), across any part of the boundary. Accordingly, the total energy or equivalent mass would remain constant inside the system (∆W = ∆W 0 = 0). This assumes that no energy or mass can spontaneously appear anywhere inside the closed system, without a definite trace of flow of the energy occurring across the closed boundary surface.

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Under the above condition, it may be shown from (17) that the total momentum pz, 0 0 pz inside the closed system in each frame would remain unchanged (∆pz = 0 = ∆pz). The choice of the z direction is arbitrary in the above discussion of the energy conservation in the closed system. Therefore, component of the momentum along any direction, or equivalently the total momentum vector, would remain unchanged (∆¯p = 0 = ∆¯p0). This is the principle of momentum conservation in a closed system. Further, because the total momentum would remain unchanged, the total of all forces in the closed system would be zero, as per the deﬁnition of force in (4). Equivalently, every force in the closed system would be balanced by a counter reaction force that is equal in magnitude but oppositely directed. This is the Newton’s third law of motion. We have now proved the Newton’s third law from the electromagnetic theory and special relativity.

0 0 0 ∆W V ∆pz 0 ∆W = ∆W = 0, ∆W = α + α = 0, ∆pz = 0, 0 ∆W V ∆pz 0 ∆W = α − α = 0, ∆pz = 0, ∆¯p = ∆¯p = 0, ∆¯p0 = 0, P F¯0 = 0; ∆¯p = 0, P F¯ = 0. (19)

It may be noted, that the two results (18) and (19) are mutually complementary to each other. That is, the condition of zero total force, or equivalently the conservation of total momentum, would require the total energy to be conserved. And conversely, the conservation of the total energy would require the total momentum to be conserved, as well as the total force to be zero.

4.4. Conservation of Total Energy and Momentum in the Universe. Consider the entire universe, which in principle contains all physical space there is, and therefore does not have any other external space across which any energy or mass can be exchanged with. Accordingly, the entire universe is in principle a closed system. Therefore, as per the above deductions, the total momentum as well the energy in the entire universe must be conserved, with every possible force in the universe balanced by an opposing force of equal magnitude, at all times. This is the universal principle of conservation of energy and momentum.

5. Discussion: Basic Concepts of Electro-Magnetic Charge and Space-Time Supersede the Newton’s Laws

We have succeeded to derive all basic mechanical principles of the Newton’s Laws from the Maxwell’s Equations. Further, we known that the Maxwell’s Equation can be established [1, 2] directly from the basic principles of electric and magnetic charge and their invariance, using only the space-time concepts of the special relativity. Accordingly, the principles of the electric and magnetic charge and the space-time relativistic transformation relations, constitute a complete set of basic rules or laws to govern the electrical as well as mechanical characteristics of the nature. In other words, we have established that the basic concepts of electro-magnetic charge and space-time are complete, which “supersede” all mechanical principles making them redundant. This interpretation may at first seem counter-intuitive. This is because we come to be educated about the mechanical principles first, which are more instinctively experienced as we come in contact with our physical world on a daily basis. Based on the mechanical principles, we are then gradually educated about more advanced principles of the electrical or magnetic forces, and their associated fields. This learning process leads to a common impression that the mechanical principles that are academically established first must be independent of, and therefore fundamentally supersede, the more advanced electromagnetic principles we learn later on. As we now understand, this impression is misleading.

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The mechanical principles are introduced based only on our common-sense faith in the Newton’s Laws without any objective “proof”, by essentially relying on our everyday experiences and experimental observations. Although the electromagnetic principles, in the form of the Maxwell’s Equations, are established later based upon these mechanical principles, with deeper insights we come to understand that the electromagnetic principles could be more fundamental. The governing basic principles of invariant electric and magnetic charges are recognized to be complete and minimal, and the underlying mechanical concepts will now have to be constrained in order to be consistent with the fundamental electromagnetic concepts. These constraints provide the desired “proof” or explanation for the Newton’s Laws and the associated momentum and energy conservation, which no longer have to be accepted only on faith in their agreement with experimental observations and common-sense experiences. This is a signiﬁcant, new scientiﬁc view.

References [1] N. Das. Introducing Maxwell’s Equations as Derived from Simple Relativity Transformation Principles. Manuscript for Publication, May 2018. [2] N. Das. A New Approach to Teaching Maxwell’s Equations, as Derived from Simple Relativistic Trans- formation Principles: A Tutorial. Paper #8, pp.88-120, in “A Uniﬁed Electro-Gravity (UEG) Theory of Nature”, 2018. [3] James Clark Maxwell. A Treatise on Electricity and Magnetism, Vol. I and II (Reprint from 1873). Dover Publications, 2007. [4] Albert Einstein and Anna Beck (English Translator). The Collected Papers of Albert Einstein, Volume 2: The Swiss Years: Writings 1900-1909 (see Documents 23 and 24). Princeton University Press, 1989. [5] Ray Skinner. Relativity for Scientists and Engineers. Dover Publications, Inc., 1982. [6] N. N. Rao. Elements of Engineering Electromagnetics, Ch.4,8. Prentice Hall, New Jersey, 2000. [7] R. F. Harrington. Time Harmonic Electromagnetic Fields, Ch.2,3. John Wiley and Sons; IEEE Press, 2001. [8] L. B. Felsen and N. Marcuvitz. Radiation and Scattering of Waves (Ch.4). Prentice Hall, 1973. [9] Richarrd P. Feynman, Robert B. Leighton, and Mathew Sands. Lectures on Physics, Vol.II, Ch.25,26. Addision Wesley, 1964. [10] Sir Isaac Newton. Principia: Mathematical Principles of Natural Philosophy. I. B. Cohen, A. Whitman and J. Budenz, English Translators from 1726 Original. University of California Press, 1999. [11] Richarrd P. Feynman, Robert B. Leighton, and Mathew Sands. Lectures on Physics, Vol.II, Ch.27,28. Addision Wesley, 1964.

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