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GRAVITATIONAL EQUATION WITH AND OF WITHOUT USING THE UNIVERSAL OF NEWTON Rodolfo Sergio González Castro

To cite this version:

Rodolfo Sergio González Castro. GRAVITATIONAL ACCELERATION EQUATION WITH - AND WITHOUT USING THE UNIVERSAL GRAVITATIONAL CONSTANT OF NEWTON. 2014. ￿hal-00947254v5￿

HAL Id: hal-00947254 https://hal.archives-ouvertes.fr/hal-00947254v5 Preprint submitted on 18 Mar 2014 (v5), last revised 25 Mar 2014 (v8)

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. GRAVITATIONAL ACCELERATION EQUATION WITH WAVELENGTH AND SPEED OF LIGHT WITHOUT USING THE UNIVERSAL GRAVITATIONAL CONSTANT OF NEWTON

Rodolfo Sergio González Castro

Research Institute, University of Tijuana CUT, Av. Lucrecia Toriz 1010, Col. Altamira, Tijuana, Baja California, México. CP 22150. Email: [email protected]

Abstract

In this paper I derive an equation relating the gravitational acceleration with speed of light and the gravitational wavelength corresponding to the density at a point in without using the gravitational constant of Newton (G).

OAI: hal.archives-ouvertes.fr:hal-00947254

PACS: 04.50.Kd Keywords: , particles, gravity, constant, Newton

1. INTRODUCTION For Newton, gravity is considered a exerted on a remote Instant. Moreover, when The mathematical physicist Sir the gravity force is exerted by two or more in 1687 published his book "Philosophiae bodies extremely , Newton's law has Naturalis Principia Mathematica" where he serious limitations and then must resort to the presented the law of universal gravitation General stated by Albert empirically derived to describe and calculate Einstein in 1915, who says that gravity is not a quantitatively the mutual attraction of each force exerted to distance but a contraction of particle and massive objects in the . In Space-Time produced by the presence of that document, Newton concluded that the Energy- (1). attraction together two bodies is proportional to product of their and inversely However in the final formulation of the proportional to the square of the distance that equations of the universe, to make it separates them. compatible with the law of conservation of energy and principles of , However, these must be adjusted Einstein included geometric concepts such as proportionalities by introducing a constant the Ricci Tensor and scalar, but mainly the called Universal Gravitation (G) with an Energy- tensor, but fails to approximate value of 6.67428 × 10-11 N m2 integrate into said Energy-Momentum Tensor kg-2 units in the International System. Without the constant Universal Gravitation Newton the introduction of this constant, the equation, (G), as this is finally out of tensor in the lose their rationality and is impractical to member of the equation. calculate the force of gravity without it. While Einstein equation establishes the relationship between gravity, energy and geometry distortions space-time, it does not gravitational acceleration with electrostatic define the origin of the relationship. acceleration requires first that the equation derived from the principal component of In this regard, in 1995 Jacobson achieved gravity, energy and its corresponding considerable progress in linking the laws of relationship with the invariance of Planck area, thermodynamics to the Einstein equation and and the two tension vectors emerging from this the equation of state correlates entropy with area and generate electrostatic and the area of energy flow (2). gravitational as seen in the present Erik Verlinde published on January 6, 2010, work. his work "On the Origin of Gravity and the Newton's equation establishes the relationship Laws of Newton" (3), which proposed that between mass and gravity, while the equation gravity is a reality entropic force emerging Einstein relates the Energy-Momentum Tensor space. In its formulation, includes reduced with the modification or distortion of space: Planck's constant, N as a Screen of information bits of space, adds a new constant called G, which ultimately found to be equivalent to the Universal Gravitational Constant. On that basis, Verlinde forecast to gravity as a For particles at absolute rest, Einstein's equation has only one active component ) fundamental force. of the Energy-Momentum Tensor , In March 2010, Jae-Weon Lee, Hyeong-Chan dimensionally same as is defined by the Kim and Lee Jungjai published a paper in equation: which suggest that the Einstein equation can (1.1) be derived from the principle of Landauers on the Elimination of information causal Where is the , c the speed of horizons, and conclude that gravity has its light and ƿ the energy density, so if you divide origin in quantum information (4). From then between we simply obtain the energy such work is also supported by Jacobson density. That is, in real terms the Einstein equation defines the energy density is the linking between thermodynamics and the curving space. equation Einstein, as well as on the work of Verlinde entropic force. The problem of the Einstein equation is that the energy density in order to be equivalent to Thus, today we already have a strong linkage the curvature of space, requires the tensor between energy, heat, temperature, laws of is multiplied by the constant of universal thermodynamics, general theory of relativity, gravitation Newton (G) and their perturbation of the geometry of space-time, corresponding dimensions. entropy and quantum information, but somehow linking gravity and electrostatic In view of the above, this paper begins from force has failed in all these works that the energy (E) component and distribution or eventually drift to the Einstein equation and displacement in the area occupied promptly in repeatedly use the constant of universal space. gravitation Newton (G). 2. GRAVITATIONAL ENERGY DERIVED Derive a gravity equation that eliminates the FROM PLANCK AREA universal gravitational constant and link the To derive the gravity equation, I need to the same length, and in the case of (Y) the hypothesize a priori that the particles at Vector Pressure is orthogonal and contractive absolute rest, energy its distributed in a (from to the X axis), then the two specific area of space and that this area has Tension Vectors cancel each other. In terms of two main components (x, y) from which the de Broglie wavelength to be exactly equal emerge two tension vectors on the space with in length and width but opposite to each other, different action and different time. they cancel each other, so that an external observer cannot feel the presence of the The first tension vector is considered by particle or Planck on electrostatic terms or in Einstein and in principle only generates the gravity terms. It is then a particle "Null" or electrostatic force and the related space "Empty". electromagnetic phenomena, as the second tension vector becomes the cause of the By the law of conservation of the area, an gravitational interaction. However, the or any other particle in the universe at immediate question is where did emerge the absolute rest always retains the Planck area. second Tension vector? That is, if the axis (X) corresponding to the To explain the above and get the answer, first Tension Vector (Wavelength De Broglie) consider a priori a hypothetical particle whose extends beyond the (lp), then Plank energy distribution in space is the area proportionally the axis (Y) which corresponds of a perfect square: to the Second Tension Vector is shortened ( length). But also at the time, in which the (X) and (Y) Y = = = 1,0155241E-34 m are different, they cease to cancel each other and so the electrostatic and gravitational effects are "visible" to outside observers.

X = = = 1,0155241E-34 m In other words, we insist, Planck area is when all particles are in absolute rest Both the axis (X) and the axis (Y) or even to non-relativistic . corresponding to the Planck length (lp), if we multiply it by 2 then we get the Planck Pilot For example, let’s see here graphically wavelength . Then immediately notice that proportional representation of an electron the Planck energy density, has not one (1) but according to Planck area: two (2) waves associated interaction.

Y = = Both sides of the square shafts or exert 4,2504295E-57 m pressure on space, or in terms of Einstein, X = = = 2,4263178E-12 m produce a Tension Vector. In the case of Axis (X) the Vector of Tension is exactly the same as Einstein, but in the case of axis (Y), a Note: The graph of the rectangle is not hypothetical still correspond Second Tension exactly proportional to the scalar Vector. quantities for obvious reasons.

However, in the case of the hypothetical As can be observed in the electron because (X) particle Planck because either axes, or sides of is extended too much relative to the Planck length, then (Y) is shortened proportionately While the physical interpretation of tension then being a wavelength excessively small but Vector II corresponds to the component of sufficient to generate the gravitational effects normal acceleration (gravitational of the electron. acceleration) occurring Wave inward, towards the center of mass-energy. In this regard there In the case of the electron, the De Broglie are three possible interpretations: wavelength is about 2.4263170 E-12 meters, while the gravitational wave length is I) The energy density distributed in the Wave, approximately 4.2504295 E-57 meters. has a "thickness or wide" limit set by the Planck area (yarn thickness wave). In this In the case of the , being an object case, the energy density pressure inside the (particle) with a large mass, the proportion is particle itself. reversed; the De Broglie wavelength is about 3.7009 E-67 meters, while the gravitational II) A second possible interpretation is that the wave length is about 0.027866232 meters, wave to advance, the vector has a torque space according to calculations using the equations, that causes progress turning on itself elliptical. described later. Therefore the rotation or tension energy density causes a second acceleration towards In other words, the particles and the mass the center of the elliptical rotation, then this objects have a spatial configuration with acceleration corresponding to the gravitational respect to the wave packet associated vibrating acceleration. like strings or threads whose length in the X axis is much longer or shorter with respect to III) In a third interpretation, a second Tension the Planck length but a width or thickness (Y vector actually corresponds to the wave axis) is proportional to the axis X excessively amplitude (A) that is not symmetrical with longer or shorter, but enough to be the sources respect to the wavelength, then said amplitude of the gravitational effects.. wave then generates a second acceleration whose second component (normal To understand this, let's draw a conceptually acceleration) is orthogonal to with respect to X wave example: (wavelength) axis.

Whatever the correct physical interpretation of Tension Vector II, in the end, this also generates a second emission wavelength with a corresponding length.

Moreover to calculate both wavelengths: De Broglie wave length (Tension Vector I), and wave length Gravitational (Tension Vector II), The physical interpretation of the Tension then we have to set the "Energy" which is the Vector I, not a major problem, because it is the source of both. product of the tangential acceleration of energy distributed in the wave curve along the If we make this definition from the Planck wavelength (X axis) with the corresponding hypothetical particle that transforms into pressure on space. another particle conserving Planck area, then we need to be the axis (X) or Tension Vector I, which corresponds to the energy at absolute By then calculating said sum of displaced rest and which also corresponds to the quanta of Planck (Y) to the axis (X) extension or horizontal length, then we have to occupies the following equation: multiply the Energy Plank, quantified for this energy density at absolute rest. While the axis (2.4)

(Y) or Tension Vector II corresponds to the "gravitational energy", which in turn Should clarify here, first, that the boundaries corresponds to the width or orthogonal of physics established by the laws of Planck, extension, then we also divided quantified not defined in linear terms of length or Planck according to the following equation: energy, ( , Ep) because then we would have a universe whose total contained energy not

= (2.1) would be greater than Ep.

1.9560852E+09 Joules In that sense, the limits of Planck must actually correspond to the Planck Area (Ap)

4.6735316E+31 Joules 2 and about the limits of energy; the multiplication of the energy corresponding to Where Ep is Planck's energy and N is the ratio the Tension Vector I (E) for the energy of the rest energy of any particle or object corresponding to the Tension Vector II , mass and the Planck energy given by the and whose both product becomes Planck equation

energy squared as defined by equation (2.2) 2.1.

Moreover, to calculate the "gravitational Substituting N into Equation (2.1) then the energy" of an electron, a proton, a or energy distribution in an area Planck is given other particle those are located in the "absolute by: " we get a magnitude of such

"gravitational energy" which apparently (2.3) violates the limit of Planck (Ep) but not the In this case (E) corresponds to the rest energy limit in equation 2.1 or 2.3. of any known particle or massive object, while I should also clarify that Tension Vector I is the "Gravitational Energy". corresponding to the energy at absolute rest (E) has a direction from the particle to infinity, However, here I must clarify that the so that almost all the energy is contained "gravitational energy", which is excessive and within the particle itself. inexplicable in the case of the electron and the proton, does not correspond to an existing Meanwhile, the Tension Vector II energy within the particle, because if so would corresponding to the "gravitational energy" has violate the very principles of Plank, but which an orthogonal inverse direction from infinity to is the sum of quanta energies of all Plank the particle, so that the majority of the particles which are displaced towards the axis "gravitational energy" is outside of the (X) by the contraction of (Y). particle.

Thus "gravitational energy" is the sum of quanta energies of Planck that is moved from infinity to the Tension Vector I (axis X) by the Where is the factor of gravitational contraction of Tension Vector II (Y) at a rate coupling. defined by equation 2.4. For the electrostatic force, value coupling is Both the rest energy (E) as the "gravitational considered constant () about energy" (Eg) is "moving" or "distributed” 7.297352568E-03. along the corresponding axis (X, Y), and thus generate their own wave packet with pilot In the case of gravity, said coupling depends wave associated to the front. on the ratio of both wavelengths as shown in the equation 3.1. The two Pilot waves have a corresponding wavelength according to the principles of Defining the gravitational engagement is Broglie known equation given by: critical to the development of a theory of gravity on the equivalence to approach the electrostatic force and to calculate the force of gravity, but such coupling is not necessary to That in the case of "gravitational energy" or calculate the gravitational acceleration on the Tension vector II, its wavelength can be arguments presented here and as it will show obtained directly from the following equation: at the end of this paper.

4. THE EQUATION OF GRAVITY FORCE (2.5) WITHOUT CONSTANT GRAVITATION

Where (E) is the energy at rest. In short, the "gravitational energy" ( ) "moves" or "distributed" from infinity to the The product of two wave , corresponding center of the distribution of particle energy, to the wavelength squared Planck. As Planck generating a curvature or contraction of space Area, the product of both wavelengths is defined by the length "gravitational wave” invariant in all of the particles: , in turn generating component of normal acceleration towards the center of the particle (2.6) which we interpret as gravitational Where h is Planck's constant and c is the speed acceleration. of light and where the wavelength of Broglie Because of this, we then multiply the (Tension Vector I) is and is the gravitational "gravitational energy" corresponding wave length (Tension Vector II). gravitational wave length: 3. GRAVITATIONAL COUPLING

Because the two wavelengths are extremely But because of the wide differential between different in most massive particles or objects, as rest energy and "energy gravitational" or the these originate practically in a "simultaneous", the wide differential between De Broglie gravitational wave length to be coupled to the wavelength and gravitational wave length, length of wave energy , which generates a then we have to include in the equation the coupling factor for the gravitational force given by: gravitational coupling factor obtained from equation 3.1: (3.1)

Keeping 2 in the equation is to finally remind us that the ratio between two particles or And then for the gravitational attraction massive objects is not straight but curved. between two identical particles or massive objects or with the same amount of mass only Moreover, also in the case of the gravitational remains for us to divide the radius separating interaction between a massive object and a the two particles: smaller, we can eliminate the two gravitational couplings of equation 4.4 by exchanging the

= (4.1) component of the gravitational energy ( for

the energy in rest (E) of the massive object Where is the "gravitational energy" (equation while retaining the gravitational wave length : 2.4), is the length of gravitational wave (equation 2.5), is the gravitational coupling 4.6) (equation 3.1) and (d) is the radius between the two particles interacting or massive objects. Where (F) is the force of gravitational And for the case of two different particles or interaction, (E) the rest energy of the massive objects mass: object, and the length of pilot gravitational wave associated with the

gravitational energy gained by the equation = (4.2) 2.5. 5. DERIVATION OF THE EQUATION OF Then comes to be an equation for calculating GRAVITATIONAL ACCELERATION the gravitational interaction without the use of the gravitational constant of Newton and use Then we can use Equation 4.6 to derive a new the length of de Broglie wave, for particle or equation for calculating the gravitational massive objects at rest. acceleration of a massive body like the earth. According to Newton, force (F) is equal to But due to the existence of a constant in this mass (m) by acceleration (a): equation, it is possible to reduce it further. = 1.986451698E-25 joules per meter We make the Newton equation equal to (4.3) equation 4.6: Due to this constant, we can use in the

(5.1) equation 4.2, only the values of "gravitational energy" and gravitational wave length of the particle or object with higher energy (or mass) and then the equation is abbreviated : Convert (E) to the Einstein equation:

(5.2) 4)

Spent the terms:

)

(5.3) Then we exchanged the term corresponding to Wavelength Gravitational of the Earth: the mass to remove it from the equation:

Eliminates m: 0,027866232 m

(5.5) Squared radius of the Earth: 4,05896E+13 m

We returned the radius Substituting the values for the gravitational acceleration of the earth:

=

And Eureka: =

Which is the same result obtained with = (5.6) Newton's equation. 6. DERIVATION UNIVERSAL That is, the gravitational acceleration is equal to the length gravitational wave of the mass GRAVITATIONAL CONSTANT (G) object (eg earth) by the speed of light squared Conventionally the gravitational acceleration divided radius squared. is calculated with:

As we can see in equation 5.6, noticeably = disappear two components: force (F) and mass (m). The immediate interpretation of this is that the energy density and the corresponding We then said equation equivalent to equation wavelength do not exert a force on objects that 5.6: attracts, but this force is exerted actually about the same surrounding space which is = (6.1) accelerated in an orthogonal contraction towards the center of Energy density in Eliminated radius from the two equations: proportion to the length of dominant gravitational wave. = (6.2)

Ensure in the Earth: And finally solve for G:

Mass: 5.9722 E +24 Kg

(6.3)

Energy: = 5.3675E+41 kg

Gravitational energy of the Earth: We can convert the units of Planck equation and result is the same:

= 7,12853E-24 Kg

= = (6.4)

Where ( ) is the planck wave length defined 4. DERIVATION OF EQUATION OF in 2 and is equivalent to the mass planck ELECTROSTATIC ACCELERATION of 2.1764383 E-08 kg. Following the same reasoning as in 5, then we have: 7. DERIVATION OF EQUATION CALCULATING ELECTROSTATIC

(8.1) INTERACTION

Over the original arguments, calculating the

electrostatic interaction between two particles (8.2)

is even simpler because the constant of

electrostatic coupling or fine structure. In this (8.3) case leads us to a broad general constant that

shall term Universal Electrostatic Constant: (8.4)

1.4495849660E-27 joules (8.5) per meter (7.1)

No matter how much rest energy of the

particle, the electrostatic force is always the same and only vary depending on the distance. Y Eureka: In this case, this energy corresponds to the

Tension vector I (X axis of Planck area) = (8.6) That is, the calculation of the electrostatic interaction between two identical or different That is, the electrostatic acceleration equals particles is given by the general equation: the de Broglie wavelength of the particle by

the speed of light between the squared = = (7.2) distances between the interacting particles.

Where E is the energy at rest; the Since equations can be observed 8.6 and 5.6,

wavelength of de Broglie; coupling are virtually identical, with the slight constant or Fine structure; d is the distance difference that in the case of the electrostatic between the two particles. This equation is acceleration have to stop for the time the fine equivalent to: structure constant.

Despite the existence of the fine structure = constant in equation 8.6, we can also say that Coulomb's law for electrostatic interaction the electrostatic force is not exerted on the between only two particles (electron-electron; mass of the second particle interaction but on proton-electron; proton-proton). we have the its surrounding space which is accelerated so following equivalence: repulsive or contractionary depending of the

signs of the charge of the particles in = (9.4) interaction.

4. GRAVITATIONAL ACCELERATION Then the equation of the gravitational EQUIVALENT TO ELECTROSTATIC acceleration on the 5.6, also transform to the ACCELERATION. framework of the Special Theory of Relativity including the Lorentz contraction factor for The reason why the equations 5.6 and 8.6 are gravitational wave length: not fully equivalent has a very simple reason;

5.6 is a classic equation, while the second (8.6) = (9.5) is an equation within the framework of the Special Theory of Relativity. They can be also: In this case is the Lorentz factor for the

speed of the electron in the first Bohr orbit of = (9.6)

the . Here (g), is the traditional symbol of the According to Sommerfeld calculations, gravitational acceleration. calculating the velocity of the electron in the first orbit is given by the equation: And finally we have the equivalence:

= (9.1) = =

Where (h) is Planck's constant. Substituting the Where ( ) is the electrostatic acceleration; ( ) values : is the gravitational acceleration and ( ) is the Lorentz contraction factor, we find that the = only noticeable difference between the two We now apply the Lorentz factor in summary, equations is the wavelength. because here we have obtained a speed within 7. CONCLUSION the framework of the theory of : Having derived clean and natural equation

= (9.2) gravity and shape of the same equation gravitational acceleration without using the gravitational constant of Newton, with a = different way to Newton and Einstein, using as That becomes the Lorentz contraction factor for components only terms of energy and the wavelength of an electron the first Bohr wavelength; or speed of light and wavelength, orbit, and is constant value for successive orbits the way for a new interpretation of gravity or energy levels of in the atom. So in opens: this case: 1. That to derive the equations of gravity in terms electrostatic necessarily need to include = (9.3) the concept of "Gravitational Energy" and the In basing the above arguments then modify concept of Second Tension Vector derived equation 8.6 for electrostatic acceleration: Planck Area and the limit defined by the 5. Energy density and its corresponding Planck Energy Square. wavelength does not exert a gravitational force on objects that attracts, but this force is exerted 2. That it is possible to derive equations for the on it actually surrounding space is so rapid electrostatic acceleration and gravitational orthogonal contraction towards the center of acceleration on the basis of the same density of energy in proportion overriding the principles. gravitational wave length.

3. That in the case of the force equation 6. The electrostatic force is not exerted on the Gravity, because the length of wave pilot of mass of the second particle in interaction but resting energy and "gravitational energy" wave surrounding space there on which is pilot length, are vastly different and also accelerated in a repulsive or contraction originate nearly so "simultaneous”; depending on the signs of the charge of the "Gravitational” wave length should fit variable particles interact. length electrostatic wave form, which had hitherto hindered its calculation 7. That the final equation in 8.6 here electrostatically. developed for the estimation of the gravitational acceleration, is actually an 4. However, to derive the equation of electrostatic acceleration equation where the gravitational acceleration (5.6), gravitational fine structure constant is the Lorentz coupling factor disappears, generating a contraction factor for the wavelength in the surprising equivalence of gravitational wave context of the special theory Relativity of length and gravitational acceleration. Einstein.

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