Physical Science International Journal 7(3): 152-185, 2015, Article no.PSIJ.2015.066 ISSN: 2348-0130
SCIENCEDOMAIN international
www.sciencedomain.org
Electro-Gravity via Geometric Chronon Field
Eytan H. Suchard1*
1Geometry and Algorithms, Applied Neural Biometrics, Israel.
Author’s contribution
The sole author designed, analyzed and interpreted and prepared the manuscript.
Article Information
DOI: 10.9734/PSIJ/2015/18291 Editor(s): (1) Shi-Hai Dong, Department of Physics School of Physics and Mathematics National Polytechnic Institute, Mexico. (2) David F. Mota, Institute of Theoretical Astrophysics, University of Oslo, Norway. Reviewers: (1) Auffray Jean-Paul, New York University, New York, USA. (2) Anonymous, Sãopaulo State University, Brazil. (3) Anonymous, Neijiang Normal University, China. (4) Anonymous, Italy. (5) Anonymous, Hebei University, China. Complete Peer review History: http://sciencedomain.org/review-history/9858
Received 13th April 2015 th Original Research Article Accepted 9 June 2015 Published 19th June 2015
ABSTRACT
Aim: To develop a model of matter that will account for electro-gravity. Interacting particles with non-gravitational fields can be seen as clocks whose trajectory is not Minkowsky geodesic. A field, in which each such small enough clock is not geodesic, can be described by a scalar field of time, whose gradient has non-zero curvature. This way the scalar field adds information to space-time, which is not anticipated by the metric tensor alone. The scalar field can’t be realized as a coordinate because it can be measured from a reference sub-manifold along different curves. In a “Big Bang” manifold, the field is simply an upper limit on measurable time by interacting clocks, backwards from each event to the big bang singularity as a limit only. In De Sitter / Anti De Sitter space-time, reference sub-manifolds from which such time is measured along integral curves, are described as all events in which the scalar field is zero. The solution need not be unique but the representation of the acceleration field by an anti-symmetric matrix, is unique up to SU(2) x U(1) degrees of freedom. Matter in Einstein Grossmann equation is replaced by the action of the acceleration field, i.e. by a geometric action which is not anticipated by the metric alone. This idea leads to a new formalism of matter that replaces the conventional stress-energy- momentum-tensor. The formalism will be mainly developed for classical but also for quantum physics. The result is that a positive charge manifests small attracting gravity and a stronger but small repelling acceleration field that repels even uncharged particles that measure proper time, i.e.
______
*Corresponding author: Email: [email protected], [email protected];
Suchard; PSIJ, 7(3): 152-185, 2015; Article no.PSIJ.2015.066
have rest mass. Negative charge, manifests a repelling anti-gravity but also a stronger acceleration field that attracts even uncharged particles that measure proper time, i.e. have rest mass.
Keywords: Time; general relativity; electro-gravity; reeb class; godbillon-Vey class.
1. INTRODUCTION the De Rham Cohomology. Locally, it offers foliations which their tangent bundle is the kernel The motivation of this theory is to show matter is of a 1-Form, namely the gradient of a scalar field an acceleration field i.e. prohibition of inertial of time. motion/free-fall of every small particle that can measure proper time. By the principle of parsimony, even the maximum time requirement is not necessary if the scalar Non-geodesic motion as a result of interaction field that defines proper time along curves, is with a field, is not a geodesic motion in a emergent out of a mathematical formalism. Here gravitational field, i.e. it is not free fall. Moreover, we should introduce an equivalence relation. material fields by this interpretation prohibit geodesic motion curves of particles moving at 1.1 Avoidance of Closed Time-like Curves speeds less than the speed of light and by this, reduce the measurement of proper time! The In this paper, the curves along which maximal expression of such a field is thus by a field of proper time is measured do not contain time that is not part of the geometry of space- backward travel in time. If two events e and e time but rather adds new information to the 1 2 space-time manifold by expressing how material are on a curve such that e 2 is in the future force fields bend the curve of a small enough relative to e then no geodesic exists between clock. This bending is not anticipated by general 1 relativity but can account for a complementary e 2 and e 1 such that e 1 is in the future in field to the gravitational field and it offers a new relation to e , otherwise the curve is excluded expression of matter. In this paper, the author 2 replaces the notion of a force field by an from the calculation of maximal time. acceleration field. In fact, the covariant description of such an acceleration field is by an The equations of the presented theory may anti-symmetric matrix that is multiplied by the render this definition redundant. velocity vector in order to point to a perpendicular direction i.e. to an acceleration 4-vector. A 4- 1.2 Geodesic Equivalence between vector of acceleration is only a representative of Events such a field, which as we shall see, leaves an SU(2)xU(1) degrees of freedom in the definition Given a sub-manifold M in M . Events in a set of the matrix. An acceleration can be seen as 1 curvature of a particle's trajectory, however, we of events M 2 are equivalent, if for each event seek a formalism that will be independent of any e in M there exists an event e in M specific trajectory. This goal can be achieved by M 2 2 M 1 1 an introduction of a very special scalar field, such that the maximal proper time Max namely, a field of time. By the principles of measured along curves that connect e to General Relativity, no coordinate of time should M 2 be preferable and therefore any such scalar field events in M1 is the same for each other event in should not lead to a realizable preferable M 2 , i.e. If the maximal proper time from event coordinate of time. Also, because more than one ~ ~ curve can measure the same maximal proper eM 2 to M1 is to event eM 1 M1 then the curve time between a predefined reference sub- length in proper timer is also . The main manifold and an event, a definition of maximal Max time may not lead to unique integral curves. The interest of this paper is that for each Max , there theory that will be presented is a geometric interpretation of Sam Vaknin’s Chronon Theory exists a class M 2 such that eM 2 eM 1 is a
[1] by a Godbillon-Vey [2], [3] curvature vector / surjective map. Max is subject to acceleration Reeb vector. This vector is a part of the fields, i.e. to fields that prevent any small test Godbillon-Vey 3rd order form and it belongs to
153
Suchard; PSIJ, 7(3): 152-185, 2015; Article no.PSIJ.2015.066
particle from moving along geodesic curves. M If test particles are forced to move along non- 1 geodesic curves, i.e. experience trajectory may not be unique and all such M1 sub- curvature due to an external field, a manifolds of M should be dictated by a physical mathematical formalism of such curvature will law, i.e. a minimum action principle. have to be developed and will replace matter in Einstein-Grossmann’s field equations, as such Before we continue, we have to define the “big curvature fields become a new description of bang” as it embodies the most simple case of M matter. It is quite known that acceleration can be 1 seen as a curvature and therefore acceleration . Please note that M 2 may not be a neat field is another interpretation of a curvature geometric object as a 3D sub-manifold. It can be vector perpendicular to 4-velocity [7] though it a countable unification of such sub-manifolds. leaves SU(2)xU(1) degrees of freedom. An acceleration field that acts on any particle, can't Big Bang: The Big Bang in this paper is a be expressed as a 4-vector because a 4-vector presumed event or manifold of events, such that does depend on a specific trajectory and by Tzvi if looking backwards from any clock - “Test Scarr and Yaakov Friedman, such a field is Particle”, at an event ‘e’ - that measures the expressible by an anti-symmetric matrix maximum possible time up to ‘e’ then such clock A A such that if V is the normalized 4- must have started the measurement from the big bang as a limit. The Big Bang synchronizes all velocity and VV 1 then the 4-acceleration is possible such clocks that measure the maximal 2 time to any event from the past. The definition actually a / C AV such that C is the allows prior time even before such a manifold of speed of light. events but requires that clocks will be synchronized on the "big bang" manifold or that 2. THE CLASSICAL NON-RELATIVISTIC the synchronization will be done by measuring the limit of time backwards to a "big bang" LIMIT – MASS AT REST IN A singularity. GRAVITATIONAL FIELD
The idea of a test particle measuring time and 2.1 Definitions even transferring time is not new, thanks to Sam Vaknin's dissertation from 1982 in which he Gravity: Gravity is the phenomenon that causes introduced the Chronon field [1] in an all forms of energy to be inertial if and only if they amendment to Dirac's equation. Instead of freely fall, including a projectile that starts defining the maximal time backwards to a single upwards. All forms of energy including light event, the definition can be to a sub-manifold, but appear to accelerate towards the source of to which sub-manifold and what if small test gravity. Gravity is seen as a phenomenon that particles are not allowed to move along geodesic influences the metrics of space-time. curves due to interactions with force fields? Mass Dependent Force: A mass dependent The physical law that we reach will have to solve force is a presumed force that accelerates any that problem too. massive object that does not propagate at the
An earlier incomplete paper of the author about speed of light and the force is mass dependent. this inertial motion prohibition in material fields, The mass dependent force does not change the can be found [4]. metrics of space-time. i.e. clocks in the field will not tick slower than clocks far from the field In general, the Euler number of the gradient of unless gravity coincides. the time field may not be zero [5]. To avoid such singularities, this paper harnesses some of the An Acceleration Field / Non-Inertial Field: An results known to belong to Lars Hormander by acceleration Field is Mass Dependent Force that using Distribution Theory [6]. If our time field is is not intrinsic to an object but rather appears as a property of space-time. Unlike gravity, it does , we may consider a new scalar field P not affect photons or any particle that propagates such that is complex. PP* * avoids at the speed of light. The idea is that such a field gradient field singularities where * becomes affects measurement of time by prohibition of inertial motion of particles that can measure zero if the derivatives of are discontinuous. proper time. Non geodesic curves in Minkowsky
154
Suchard; PSIJ, 7(3): 152-185, 2015; Article no.PSIJ.2015.066
space-time measure less time between events. property of the particle and not of a field. So As was already mentioned, an acceleration field eventually, we will have to derive our curvature is expressible as a / c 2 A V and vector from the gradient of a scalar field and not from the velocity of any specific particle. Since
A A is locally similar to the Tzvi Scarr our new tensor is purely geometric, the constant 8K and Yaakov Friedman matrix [7]. will be replaced by1. To be more precise, C 4 Very important: The acceleration field need not the equation will be written as 8 or another represent an interaction of a small test particle with the matter in which the field appears. As a value e.g. 4 and property of space-time, it can represent an 8K a a other _ terms 1 ( ) R Rg interaction with the entire space – time. C 4 K 2
and in some special cases, where electric Caution: Although throughout the classical non- relativistic limit sections of this paper, the author charge is not involved i.e. Q 0 , a simple uses vectors to describe trajectory dependent a a equation will be valid, . In any representative acceleration vectors, an 4 R acceleration field can’t be described as a C 2 covariant field without the Scarr-Friedman [7] a a a 2 case this implies that can be formalism a / c A V and VV 1 . As K K an author, it is important to me to make sure that construed as energy density and hopefully the the reader fully understands that classical limit reader is not annoyed by the sloppy notation a 2 . calculations are nothing more than classical limit calculations and the author is fully aware of and Using a potential field intrinsic to an object does not mix non relativistic and covariant instead of correct covariant formalism and using formalisms! gravitational pseudo-acceleration, the following will shed some light on the general intuition as to Motivation beyond this section: For the pedant the expected relation between energy and physicist there is no point in presenting a acceleration fields although as a physical potential intrinsic to a massive object and a non- argument, it is not fully acceptable. We will now relativistic potential energy as the classical limit consider classical non-relativistic gravity and of a covariant theory. To such a reader, the classical non-relativistic acceleration as author will say that the purpose of this paper is to qualitative limits that will hint at the relationship replace the conventional energy momentum between non - inertial and non - geodesic tensor T – which is part of Einstein- acceleration fields and energy. Estimates will be Grossmann’s field equation – by a tensor with discussed in the next section. The classical non- fully geometric meaning. Recall Einstein- relativistic and non-inertial acceleration caused Grossmann’s field equations in his writing by material fields will be denoted by 8K 1 a (ax , a y , az ) . By the principle of convention as T R Rg such C 4 2 equivalence, we will calculate the integral of the square acceleration of a particle at rest which is that K is the gravity constant, C , the speed of accelerated because it is prevented from inertial light, R the Ricci tensor, g the metric motion, i.e. from falling in a gravitational field of a ball of mass. We will see how this value is tensor, the Ricci scalar. The R g R related to classical potential energy. The replacement will be with a totally geometric qualitative hint starts with integration. K is the tensor and thus will achieve a gravity equation constant of gravity, M mass, r radii of a which is geometric on both sides. To give a hollowed ball quite the way it is done in the further clue, the author will say that T will be electromagnetic theory replaced by a tensor which is the result of a r0 2 2 2 KM 2 4KM a a a dV 4 r dr K (1) representative acceleration . seems as 2 2 2 r r C C VVolume 0 0 a curvature vector of a particle’s trajectory with units of 1/length but as such, it is an intrinsic
155
Suchard; PSIJ, 7(3): 152-185, 2015; Article no.PSIJ.2015.066
Now we calculate the negative potential energy such that 0 is the permittivity of vacuum and E Eg , is the electrostatic field. (4) has a very deep M meaning which is, that acceleration of neutral Km KM 2 dm E (2) charge-less test particles should appear also g 0 r0 2r0 within an electric field. That is because acceleration of all small enough rest mass due to So from (1) and (2) the existence of a field, is assumed in this paper to be the reason of all energy densities including 1 the electric field. Prohibition on geodesic a 2 dV E (3) Minkowsky motion can be concentrated as an g 8K V Volume acceleration field in some areas around electric charge and may not be well distributed, however,
(3) qualitatively implies the following relation such argument is less appealing if we consider between energy and non-inertial acceleration the principle of parsimony. 2 where C is the energy density and is the 2 a 0 2 mass density E a 4K 0 E (6) 8K 2 a2 8K Definition: The constant that divides the non- 4 2 Energy _ Density C C (4) geodesic square acceleration is called "Electro- 1 a2 a 2 gravity constant". In this paper we choose as an 2 4K 8K educated guess 8K . In general we can write our guess as 8 so (6) becomes: In special relativity, the square norm of a normalized by C , 4-velocity of a particle is 2 a 0 2 K 0 2 i E a E (7) constant N uiu 1 and also K 2 2 d(u u i ) 2 i i such that N ,k (uiu ),k k 0 Other constants yield different theories and have dx dramatic cosmic consequences. (6) implies a i i dx very weak acceleration i.e. mass dependent u and the normalized by C , 4- force on small enough charge-less neutral test Cd 2 d 2 x i particles, about 8.61cm/sec in a field of acceleration is i which is 1/length in 1000000 volts over 1 millimeters distance. See a 2 2 C d Timir Datta et al. work as an elegant way to focus units which is the curvature of a specific particle’s field lines by metal cone and plane and to trajectory. If N 2 was not the norm of a particle’s observe the effect [8], however, in this paper we velocity, we could think of another way to shall see that there is another effect due to describe acceleration. More or less, that will be gravity and therefore the acceleration in (6) is not the subject of more advanced sections of this the only effect that has to be taken into account. paper. The acceleration in (6) exposes non-inertial, non- gravitational acceleration of particles that can 3. THE CLASSICAL NON-RELATIVISTIC measure proper time. On its own, it is not an LIMIT – THE ELECTROSTATIC FIELD interesting acceleration but it can explain the electric interaction as repulsive when the integration of the square acceleration increases The following uses the standard definitions of electric and electrostatic fields. and attractive when this integration is reduced. The author believes the acceleration of charge-
less particles in an electric field is from positive to What can we say about the density of the negative. electrostatic field? We know it is
In “Electro-gravitational thrust, Dark Matter and Energy _ Density 0 E 2 (5) Dark Energy” it will be shown that there is an 2 electro-gravitational effect opposite in direction to the acceleration of an uncharged particle in an
156
Suchard; PSIJ, 7(3): 152-185, 2015; Article no.PSIJ.2015.066
electro-static field. There is at least informal curves from a reference sub-manifold, then evidence that the elecro-gravitational effect non-zero curvature vector means shows thrust of the entire dipole towards the acceleration. The trajectory of any small positive direction [9] and the author does not enough clock that measures non-zero imply asymmetrical capacitors of 1 - 0.1 Pico- proper time will not be parallel to any Farad with 45000 Volts. Such capacitors geodesic field as it interacts with material according to the calculations in the section fields. “Electro-gravitational thrust, Dark Matter and 2) A specific 4-vector can’t account for an Dark Energy” in this paper, can’t manifest any acceleration field because we need a measurable effect of at least 1 micro Newton representation that does not depend on thrust. Here is a testimony of Hector Luis any specific velocity 4-vector. Such an Serrano, a former NASA physicist in reply to acceleration is exactly Zvi Scarr and Peter Liddicoat: “Actually by the generally Yaakov Friedman matrix [7]. A basic accepted definition of what constitutes high singular matrix will be developed. It is vacuum 10^-6 Torr is about in the middle. This singular because 2 vectors are not enough pressure is about equal to low Earth orbit. More to describe rotation and scaling in importantly at this pressure the ‘Mean Free Path’ Minkowsky space-time. There are still at of the molecules in the chamber is far too great least two more degrees of freedom and if to support Corona/Ion wind effects. We’ve tested our curvature vector is complex then there from atmosphere to 10^-7 Torr with no change in are SU(2)xU(1) degrees of freedom. performance either. However, I’m glad the results 3) We develop a non-singular acceleration have you thinking. It looks simple, but trust me matrix in which there is SU(2)xU(1) it’s not”. degrees of freedom. 4) We represent the gradient of our scalar 3.1 Serious Experimental Problem – field of time by means of the perpendicular Electron Mobility foliation and show an additional SU(3) degrees of freedom. Although The down side of the non-geodesic acceleration SU(3)xSU(2)xU(1) is the symmetry group is that it is about 10 orders of magnitude smaller of the Standard Model, it is shown in this than the accepted and known electric field paper to be a result of geometry and not of interaction. For example, negative charge any second quantization technique. suspended above the Earth will cause charge to 5) We explore more scaling degrees of move in the ground. This charge will have a freedom in the definition of the time field. much stronger effect than the interaction with the 6) We use the square norm – the second acceleration field as is, and will cause a shielding power of the Minkowsky norm – of the effect i.e. the fields will cancel out within the curvature field as an action operator that Earth. Even the almost ideal insulator, i.e. can replace the material action in Einstein- diamond crystals, have impurities such as Grossmann-Hilbert action. Nitrogen Vacancies [10] that allow charge 7) We show that the Euler Lagrange carriers to move in the lattice i.e. high electron equations lead to a new term that mobility. In the most pure diamonds the NV expresses the divergence of the curvature impurities are about 1018 nodes per cm 3 field. This divergence is attributed to 23 electric charge because an electric field is comparing to 1.77x10 carbon atoms per a form of energy for which the simplest cm 3 . The donor electrons lie deep in the band explanation is a very small acceleration of gap of 5.47ev, at about 1.7 ev. even uncharged small clocks along electro-static field lines. So electric 4. THE NON-GEODESIC ACCELERATION interaction is predicted as either increasing FIELD or decreasing the energy of the acceleration field. The author’s strategy in developing the idea of 8) The interpretation of charge is of having an acceleration field that is not gravity is as two unpredicted fields, an acceleration field follows: and an opposite weaker gravitational field. 1) The curvature vector of the gradient of a Electrons are predicted to manifest scalar field will be developed. If the attractive acceleration field and a weaker meaning of the scalar field is that it is repulsive gravity. Free intergalactic proper time, measured along different
157
Suchard; PSIJ, 7(3): 152-185, 2015; Article no.PSIJ.2015.066
electrons are thus prime candidates for P U Dark Matter. A should emerge for some 2 9) Warp drive is mentioned as a result of very P P * large charge separation. Implementation by radio photons as possibly behaving as vector U that replaces acceleration. The oscillating pairs of virtual +e and –e charge following is simply an exercise in differential is only briefly discussed though even slight geometry. Considering a scalar field P and its imbalance of such virtual charge dP gradient in covariant writing, such distribution is equivalent to large amounts P of ordinary separated charge and may dx therefore account for Roger Shawyer’s that dx are the coordinates, find the second EMDrive, Warp Drive, experiments by Dr. power of the curvature of the field of curves David Pares from Nebraska and for dP generated by . It is a problem in NASA’s approval in the late news that P EMDrive indeed involves Warp Drive dx effects. This idea also offers energy differential geometry that can be left for the production promises to developing reader as an exercise. However, if the reader countries by Dennis Sciama inertial wants to get the answer without too much effort induction. This idea will be reminded. along with some physical interpretations, he/she 10) The time field in Schwarzschild solution is should read the following. discussed. The idea is to use a scalar field of time - that 11) Ideas of how to extend this theory to represents the maximum possible time measured quantum mechanics are offered by by test particles - back to the big bang singularity stochastic calculus. as a limit or to a sub-manifold of events - and It is first required to achieve a curvature field from this non -physical observable, to generate without resorting to Tzvi Scarr and Yaakov observable local measurements. Friedman representation [7] that is required for a The square curvature of a conserving vector field general acceleration field. A A such that is defined by an arc length parameterization t if is the 4-velocity such that then along the curves it forms. V VV 1 a Caution: This t may not be the time measured the 4-acceleration is actually . In 2 AV by any physical particle because the scalar field c from which the vector field is derived may be the
1,vx / c,v y / c,vz / c result of an intersection of multiple trajectories. special relativity V However, a particle that follows the gradient 2 2 1 v / c curves will indeed measure t even if its trajectory such that x, y, z are the well known three is not geodesic. dimensional Cartesian coordinates, vx ,v y ,vz Let our time field be denoted by P and let P are three dimensional velocity coordinates, c dP the speed of light. The first coordinate is denote the derivative by coordinates P 2 2 dx 1/ 1 v / c is the speed along the time axis. or in Einstein-Grossmann’s convention P, P The vector field that this paper uses is the . Let t be the arc length measured along the gradient of the scalar field of time, curves formed by the vector field P which may dP and it will replace the velocity not be always geodesic due to intersections. By P, P dx differential geometry, we know that the second vector of specific particles. Please note that due power of curvature along these curves is simply P to intersection of different curves, is 2 d P d P P P* Curv ( ) ( )g (8) dt P k P dt P k P not a velocity, however it is a unit vector and k k
158
Suchard; PSIJ, 7(3): 152-185, 2015; Article no.PSIJ.2015.066
such that g is the metric tensor. For constant because P is NOT the 4-velocity of a k specific particle, also but not only (see convenience we will write Norm P Pk and “APPENDIX – The time field in the d Schwarzschild solution”), due to intersections of P P . For the arc length parameter t . more than one possible particle trajectory curve. dt Here it is the main trick, Norm may not be
Let W denote: d P P P k W ( ) Pk P g (9) dt k Norm Norm 3 P Pk
Obviously P P g k P P g s P P g k P P g k W P g k k s P P g k k k 0 (10) k Norm Norm 3 k Norm Norm
Thus P P g P P g s P P P P Curv 2 W W s P P g k ( ) 2 (11) Norm 2 Norm 4 k Norm 2 Norm 2
dP dx r P Following the curves formed by P P, , The term is the derivative of the dx dt Norm normalized curve or normalized “velocity”, using the upper Christoffel symbols, d P ; P P s . r dx r s r
Caution: Using normalized velocity, here has a differential geometry meaning but not a physical meaning because a physical particle will not necessarily follow the lines which are generated by the curves parallel to the gradient P unless in vacuum. P may result from an intersection of curves along which particles move but may not be parallel to any one of such curves intersecting with an event !!!
d d dx r P r P ( P P s ) (P ; ) such that x r denotes the local coordinates. If P is dt dx r s r dt r Norm 1 a conserving field then P ; P ; and thus P , P r Norm 2 , and r r r 2
P P P P Curv 2 ( ) 2 Norm2 Norm2 1 Norm2 , Norm2 , g k Norm2 , P g sr ( k ( s r )2 ) (12) 4 Norm4 Norm3
We define the Curvature Vector
2 2 (P P ), (P P ), P Norm , Norm , P m m (13) U m i i 2 Pm 2 4 Pm P Pi (P Pi ) Norm Norm
159
Suchard; PSIJ, 7(3): 152-185, 2015; Article no.PSIJ.2015.066
1 a which from [7] and simple calculations, should have the meaning U B m such that a 2 m C 2 m denotes a 4-acceleration field that will accelerate every particle, that can measure proper time, by an anti-symmetric matrix, and C is the speed of light and B m is a rotation matrix, i.e. m i k B mV B iV g VkV , Vm is a vector and g is the General Relativity metric tensor. The curvature itself does not depend on any specific acceleration since it is a scalar field.
1 Curv 2 U U m (14) 4 m
Obviously U P 0 and therefore like 4-acceleration that is perpendicular to 4-velocity U is perpendicular to P . In its complex form (13) becomes
k P P* P* P dZ Z Z P * P Z N 2 and Z and Uˆ k (15) 2 dx Z Z 2
1 and by using (Uˆ Uˆ *k Uˆ * Uˆ k ) 2 k k
1 1 Curv 2 ( (Uˆ Uˆ *k Uˆ * Uˆ k )) (16) 4 2 k k
Obviously U P * 0 .
Possible sources for an acceleration field: An acceleration field can be represented by the Tzvi P * U Scarr and Yaakov Friedman [7] matrix as A such that A A k 2 Pk P
We may write this matrix explicitly by (15) and (16) and also require an additional 1/ 2 scaling and reach the following anti-symmetric singular matrix
U P PU P* U P P* PU P* A A U / 2 2 Z Z 2Z
P* (U U * ) P So A (A )* k . It easily verifiable that Det(A ) 0 . What is required, k Z 4 Z however, is that Det(A ) 0 .
So we need a modified A . On our path we will see symmetries that are usually achieved by second quantization and particle symmetries but with a very different geometric source.
5. THE STANDARD MODEL SU(3) x SU(2) x U(1) SYMMETRIES VIA THE GODBILLON- VEY CLASS
Caution: Please note, the following is NOT a quantum theory of the discussed acceleration field U . The sole purpose of the following equations, is to show the degrees of freedom in the matrix representation of the acceleration field action. We now present the curvature quite closely to the Reinhart-Wood metric formula [2],
160
Suchard; PSIJ, 7(3): 152-185, 2015; Article no.PSIJ.2015.066
k k U P PU 1 Z P Z k P * P P Z P Z k P * P P A ( ( )) 2 Z 2 Z Z Z 2 Z Z 2 (17) 1 Z P Z P P P P P ( ) ( ), ( ), ( ), ( v ), 2 Z Z Z Z Z Z Z
For some conserving field d / dx . It is easily verifiable that the exterior derivative
P U d d( )dx , can be written with dx by the Godbillon-Vey observation [2,3], as Z 2
P U P d (^)ij ( ), dx ^ dx dx ^ dx with coordinates x . Another observation Z 2 Z
P is the Godbillon-Vey class, on the foliations whose tangent bundle T (F) is perpendicular to . Z
U U , GV (F) dx ^ dx ^ dx ^ d H 3 (M ) is closed on T (F ) . More important is the 2 2 observation that [U ] is the Reeb cohomology Class [] .
From Reeb it follows that the projection of the curvature vector U on the tangent bundle T (F) of the foliations F that are perpendicular to P , have a vanishing exterior derivative. In other words,
U (F) is a closed form on F i.e. U ; (F) U ; (F) 0 . From (6) and the remark after (13) it follows that the rotor of the electric field should be zero on F , but that is possible only if even photons are pairs of charged particles, possibly oscillating with a total zero dipole moment as indeed informally predicted by physicists such as Hans W. Giertz [11]. This conclusion, as will be discussed, regards very important late findings by NASA. It is worth mentioning that the expansion of the holonomy of the foliations is dictated by the norm of the Reeb vector [12] which in the real case is U U and therefore this value is of great cosmological interest.
Following is an offered complex form of (17),
1 1 U P PU U * P * P * U * ( A A * A * A ) (( )( ) ...) 4 4 2 Z 2 Z 1 1 U * U U * U ((U * U U * U ) (U U * U U * )) ( ) (18) 16 4 2
1 which is the time field curvature action. In the real case where A A* we define F A 2 and the action (16) takes the form F F which reminds of the electro-magnetic theory but without the vector potential. As we shall see, this vector potential is indeed redundant because electric charge is a geometric phenomena. From the theory of Lie Algebras,
Det(exp(A )) exp(Trace(A )) exp(0) 1 and therefore either exp(A ) SO(4) or k exp(A ) SU (4) depending on whether A is real or not. We proved that A Ak g does
161
Suchard; PSIJ, 7(3): 152-185, 2015; Article no.PSIJ.2015.066
P* rotate and scale into U / 2 . So by the Tzvi Scarr and Yaakov Friedman representation [7], for Z every test particle with rest mass and with velocity V and the speed of light C we have,
V 1 dV A (19) C C 2 d
1 dV Just like U the term represents a curvature vector but of a specific particle related C 2 d trajectory.
We continue with the Tzvi Scarr and Yaakov Friedman acceleration representation [7] matrix and for simplicity we restrict our discussion to the real case. A is singular and we can easily define a matrix that rotates vectors in a plane perpendicular to both U and to P . That is the matrix
1 B A (20) 2
Where is the Levi-Civita symbol. It is easily verified that
(A B )(A B ) A A B B and also
1 i j 1 i j B B A A ij A A 2 ij 2
1 1 ( )(Ai j A ) (A A A A ) A A 2 i j j i 2
Therefore (A B )(A B ) A A B B 2A A j i is the Kronecker delta.
A A B (21)
(21) is the matrix we have been looking for and it also results in an immediate degree of freedom in ~ (16). In particular, rotations in SU (4) [13] that do not affect A may be applied to B . These ~ ~ ~ ~ rotations are in SU (2)xU (1) , B B and A A .
~ ~ A A B (22)
1 For the real case, where U U * , note that Curv 2 (U U k ) (A A B B ) / 4 . k k 4 k
P SU(3) degrees of freedom result from a way to represent with 3 scalar functions Z q(1),q(2), q(3) according to the Frobenius (foliation) theorem. The Lie brackets of 3 vector fields on a
162
Suchard; PSIJ, 7(3): 152-185, 2015; Article no.PSIJ.2015.066
P perpendicular foliation to , have to depend on the tangent bundle T (F) of the foliation F as Z follows,
js q(1) j q(2) s q(3) P , 0 is complex. d d d q(1) q(1),q(2) q(2),q(3) q(3). j dx j s dx s dx i j k 1) q(1)i q(2) j q(3) k 0 (23) i j k s r r 2) q(1)i ,q(2) j ,q(3) k (q(1) q(2) s ,r q(1) s ,r q(2) ) 0 i j k s r r 3) q(1)i ,q(2) j ,q(3) k (q(1) q(3) s ,r q(1) s ,r q(3) ) 0 i j k s r r 4) q(1)i ,q(2) j ,q(3) k (q(2) q(3) s ,r q(2) s ,r q(3) ) 0
Conditions 2,3,4 in (23) define the gradients of q(1) , q(2) and q(3) as Holonomic .
3 Vectors h(s) are Holonomic if their Lie brackets depend on them [h(i),h(k)] c j h( j) for some j1 coefficients cj . Condition 1 is a transversality condition. The Lie brackets of each two vectors must depend on the vectors that span T (F) . (23) has a deep meaning that our scalar field of time is a result of 3 scalar fields of space locally perpendicular to the gradient of the scalar field of time.
The source of these fields is possibly the Sam exact "monent" of the Big Bang. A relativistic Vaknin's Chronon field [1]. This implies a simpler QFT with Chronons as Field Quanta (excited physics and it is possible that Dirac's equation states.) The integration is achieved via the [14] and spinors are an algebraic language that quantum superpositions". was required because concurrent physics theories do not have a complete analytic theory Here are more definitions we will need. of space-time. The need for algebraic abstraction may not be related to physical reality but rather Energy Conservation: In any physical system to the way we perceive it. This possibility justifies and its interaction, the sum of kinetic (visible) and further extensive research and should not be latent (dark) energy is constant, gain of energy is dismissed. maximal and loss of energy is minimal. See E. E. Escultura [15]. 5.1 Dr. Vaknin's Theory – One of Four of his Offers Information: Information is a mathematical representation of the state of a physical system The coming definitions will reflect Dr. Sam with as few labels as possible. Labels can be Vaknin's view, that even photons by numbers or any other mathematical object. entanglement of wave functions have rest mass. 4 Not that a photon as observed on its own has C 2 rest mass. That is incorrect. Matter by Vaknin's Energy Density: We define Curv , such theory [1] is a result of interaction of a field of K time that quite reminds of Quarks in which that C is the speed of light and K is the gravity summation results in positive propagation. This constant, as the Energy Density of space-time. If paper sees Dr. Sam Vaknin's theory as a starting this value is defined by (14) then P is the point. upper limit of measurable time from an event back to near big bang event or to a sub-manifold Dr. Sam Vaknin's possible description of time: of events and therefore (14) is intrinsic to the "Time as a wave function with observer-mediated space-time manifold because it is dictated by the collapse. Entanglement of all Chronons at the equations of gravity and adds no information that
163
Suchard; PSIJ, 7(3): 152-185, 2015; Article no.PSIJ.2015.066
is not included in the manifold and in the metric tensor is equations. As we shall see, if we choose to write g 00 1, g11 1, g 22 1, g 33 1 otherwise, P such that is a complex scalar field g 0 , i.e. flat geometry on the hyper-cylinder. then if is a function of only then (16) is Since space-time can be curved, we can only reduced to (14) as if P . Consider the set of discuss small volume S3 and small proper time events for which is constant. Since is not a coordinate, we can't expect that set to be a sub- Tau in local coordinates. Instead of considering manifold but a unification of such 3 dimensional the general acceleration field 3 3 geometric objects ( ) ( ) . V 1 dV 0 0 as in (18), we need a A 2 C C d We consider as a Morse function on the representative trajectory dependent boost 4- space-time M manifold. That is that M is dV acceleration a so now we can apply locally smooth and the differential of this map is d of rank 1. In such a case the Morse – Sard (6) as an exact relation as the outcome of the theorem states that the Lebesgue measure of the Critical Points of M is zero [16]. energy density a a /(8K) i.e. for the previously discussed guess 8 . Energy: The following is equivalent to rest mass energy. The integration of a C 4 E Curv2d3 ( ) is defined as the 4K 0 0 3 K ( 0 ) (25)
Energy of the scalar 0 . This value is such that E is a 4-vector that replaces the locally conserved for small neighborhoods in electrostatic field. Please notice that here 3 m ( ) if U ; 0 as any local integration of 0 0 m E 0 . Consider a 3 dimensional surface of 4 the squared norm of a vector field is conserved if dimensional cylinder around a charge Q , i.e. its divergence is zero. So if there is a possibility m Surface S2 xI around a Cylinder B3 xI of U ;m 0 local conservation of the term Energy does not hold. Photons too, by such that both the radius of the sphere S2 and of entanglement and superposition have rest mass the ball B and the length of the interval I , is but not as an isolated electro-magnetic wave. 3 L and is small. We may consider L to be an A nice thought experiment – is the use of a atom of length so by the relation we have from covariant Gauss Law and the atom of space for (18), acceleration is merely a curvature of a particle's trajectory. If there is an atom of length, 8 . After we have these definitions in mind, say L , then the maximal curvature would we may want to extend Gauss Law to a covariant format and later use the idea that there is an appear in loops of radius L so it would be 1 atomic structure of space time that can't be Curvature and by (18) and by the relation curved by gravity. For start, a generalized Gauss L law using the exterior algebra, should look like: a A V such that V V 1, the maximal Q 2 E ds E ^dxi ^dx j ^dxk cTau (24) c acceleration is therefore S3 S3 0 such that E is the electric field. ds a surface C 2 a a (26) element and x denotes the coordinates, but here L E is a 4-vector and will have to be defined, cTau is the length of the path of a charge where C is the speed of light. By (6), this enclosed in a 3 dimensional surface S3, 0 the acceleration can be the underlying acceleration permittivity of vacuum and c is the speed of light. field in an electric field. The parameterization in We also assume that in our local coordinates, the polar coordinates of the surface S2 xI is
164
Suchard; PSIJ, 7(3): 152-185, 2015; Article no.PSIJ.2015.066
(L cos() cos(), L sin() cos(), Lsin(),l) wavelength is bigger by 1/ than the where 0 l L . It is expected wave length if we assume that L is (r cos() cos(), r sin() cos(), r sin(),0) also the wavelength of a photon that has equal where l 0 and it is energy to the field E or in its matrix form A . (r cos() cos(), r sin()cos(), r sin(), L) L e K We have assign where , such that 2 l L C 4 0 r (0, L), (0,2 ), ( / 2, / 2) 1 e 2 and the normal to the surface which for and get, 0 l L is 4 0 C n (cos()cos(),sin() cos(),sin(),0) and at l L the normal vector is n (0,0,0,1) K (29) and at l 0 the normal is n (0,0,0,1) . C 3
Which is known as the "Planck Length". An acceleration field A around a physical source is expected to cause E n to cancel out A nice though exotic description of anti-matter is on the "top" and "bottom" l L and l 0 simply to describe it as charged particles going back in time. A positron that propagates back in boundaries combined together, because time will therefore become an electron with a acceleration changes sign as the time coordinate reversed acceleration field and as we shall see, changes sign. So the integration (25) will be by not with a reversed gravity because only part of (25) and by (26), even though the metric is the gravity, the part which unexpectedly results Minkowsky, from charge, will be reversed.
C 2 QL 4L3 L It is an interesting question if photons have a small oscillating dipole moment, even if it is L 4K 0 0 (27) extremely small, because then, by the Q K understanding that negative charge can be C 2 4 particles that propagate back along the time axis 0 we can understand why photons propagate at But what about gravity which results from the the speed of light and therefore measure zero energy of the acceleration field A and what proper time. about the influence of such gravity on the metric tensor within the cylinder? 6. INVARIANCE UNDER DIFFERENT FUNCTIONS OF P Since we consider an atom of length L , no field can exist below that radius and therefore, (24) is Here we are about to explore another degree of valid even if the charge Q is not small. freedom in the action operator of the acceleration field as shown by a representative vector field dP We now assign the electron charge e to Q and that curves. i (27) becomes, dx
e K Caution: Although the calculation may have a (28) quantum meaning, it is not brought here for the L 2 C 4 0 purpose of developing a quantum field theory.
What is the wave length of a photon that can be We revisit our acceleration field, 2 2 μ emitted by the field ? We know that the N , N ,μ P * E U m P s.t. N 2 energy should be times smaller than the m N 2 N 4 m (also found energy of the field, such that is known as the (P *i P P *i P ) / 2 "Fine Structure Constant". Therefore, the i i as Z in this paper) we can sloppily omit the
165
Suchard; PSIJ, 7(3): 152-185, 2015; Article no.PSIJ.2015.066
comma for the sake of brevity the same way we df(P) df ( p) dP . Let dP f(P)i i i f p(P)Pi write instead of for and write dx dp dx Pi P,i i dx 2 2 2 μ N P P then N m N μ P * . 2 2 2 U m 2 4 Pm Nˆ f(P) f(P) N f (P) and N N p 2 2 Nˆ k N k 2 f pp ( p) Suppose that we replace P by f(P) such that f pk but also Nˆ 2 N 2 f ( p) is positive and increasing, then p
2 2 s Nˆ k Nˆ s f ( p) p f ( p) p Uˆ p p k k Nˆ 2 Nˆ 2 Nˆ 2 2 2 s N k 2 f pp ( p) N s 2 f pp ( p) f p ( p) p f p ( p) pk (30) 2 pk ( 2 ps ) 2 2 N f p ( p) N f p ( p) N f p ( p) 2 2 μ N k N μ P P U N 2 N 4 k k
Consider quantum coupling between the wave function of a particle and the time field , PP* 2 * as follows. Where does this coupling P come from? It is has some common sense if we say that the sum of wave functions that intersect/coincide with an event, influence the time measurement from near the “big bang” singularity event or from a sub-manifold of events to that specific event. A much better choice that will be discussed in the appendix "Appendix - Event Theory and Lèvy Process" is P and PP* *. In that case, * is a Lèvy measure and time itself becomes a stochastic process with events in which time can be measured. The appendix presents an offer of how to quantize (14).
Currently, we defer the quantization of (14) and we define the curvature vector of P ,
2 2 j Nˆ k Nˆ j ( ) * Uˆ ( ) (31) k 2 2 2 k Nˆ (Nˆ )
Index k means derivative by coordinate x k , Nˆ 2 ( ) (*)k , N 2 k . k k
As a special case, we replace by a wave function that depends on only
iE e s.t. i -1 (32)
E is the energy of a coupled particle, is the Barred Planck constant, so we have iE ( ) (1 ) (33) k k k k
2 E 2 2 E 2 ˆ 2 k 2 (34) N k (1 2 ) N (1 2 ) and
166
Suchard; PSIJ, 7(3): 152-185, 2015; Article no.PSIJ.2015.066
2 E 2 2 2 (1 ) 2 2 2 2 2 Nˆ s N s 2 2 E N / N s 2 E s s (35) Nˆ 2 N 2 2 E 2 2 E 2 N 2 ( 2 2 E 2 ) (1 ) (1 )N 2 2 2
2 j Nˆ j ( )* Now we want to calculate ( ) so we have (Nˆ 2 )2 k 2 j Nˆ j ( ) * ( ) k (Nˆ 2 ) 2
j iE 2 2 ( *(1 )) N j 2 j E iE ( ) (1 )) (36) N 2 ( 2 2 E 2 ) 2 E 2 k (1 )N 2 2 2 2 j 2 j 2 N j 2 E N j 2 E ( j ) k k N 2 ( 2 2 E 2 ) N 2 k (N 2 ) 2 ( 2 2 E 2 )
From (31), (35) and (36) we have the result
2 2 j Nˆ k Nˆ j ( ) * Uˆ ( ) k 2 2 2 k Nˆ (Nˆ ) 2 2 2 j 2 N k 2 E N j 2 E (( k ) ( k k )) (37) N 2 ( 2 2 E 2 ) (N 2 ) 2 ( 2 2 E 2 ) 2 2 j 2 2 j N k N j N k N j P P ( k ) ( k ) U N 2 (N 2 ) 2 N 2 (N 2 ) 2 k
7. GENERAL RELATIVITY FOR THE DETERMINISTIC LIMIT
By General Relativity, We have to add the Hilbert action to the negative sign of the square curvature of the gradient of the time field. Negative means that the curvature operator is mostly negative. We assume 8 . Z Z P k P 1 Z N 2 P P and U k and L U k U Z Z 2 4 k R Ricci curvature. (38) 1 8 Min Action Min R L. gd 2 A reader that still insists on asking on where does come from, can understand that L can be developed also for and remain invariant if is only a smooth function of . If P then 1 L (U k U * U *k U ) and an integration constraint can be 8 k k
167
Suchard; PSIJ, 7(3): 152-185, 2015; Article no.PSIJ.2015.066
* gd3 ( ) 1 (39) 3 ( )
Caution: 3 ( ) is not a sub-manifold because is not a local coordinate and thus the local submersion theorem [17], [18] does not hold. However, 3 ( ) is necessarily a countable unification of three dimensional sub-manifolds - Almost Everywhere - on which is stationary due to dimensionality considerations.
R is the Ricci curvature [19], [20] and g is the determinant of the metric tensor used for the 4- volume element as in tensor densities [21].
Important: If instead of P we choose P then U k in (37) is the same if depends only on . Moreover, instead of the constraint in (39) the following
* gd 4 1 4 ( ) (40)
4 leads to a theory in which represents an event and 4 PP * gd Event represents the ( ) time of an event, i.e. see "Appendix – Event Theory and Lèvy Process". Now we return to (38). By Euler Lagrange,
(P λ Z )2 L λ s.t. Z P P μ Z 3 μ (L g ) d (L g ) μν m μν g dx (g ,m ) λ s λ s (P Pλ ),s P m (P Pλ ),s P i m i m 2( Pμ Pν P );m 2 (Γ μ m Pi Pν P Γ υ m Pμ Pi P ) Z 3 Z 3 λ s λ s (P Pλ ),s P m (P Pλ ),s P i m i m 2( Pμ Pν );m P 2 (Γ μ m Pi Pν P Γ υ m Pμ Pi P ) g Z 3 Z 3 (P λ P ), P s ((P λ P ), P s )2 1 (P λ Z )2 2( λ s )Z P 3( λ s )P P λ g 3 μ ν 4 μ ν 3 μν Z Z 2 Z (P λ P ), P m (P λ Z )2 P P (P λ P ), P s (41) 2( λ m P k ); P P 2 λ μ ν 2( λ s )Z P Z 3 k μ ν Z 3 Z Z 3 μ ν g 1 (P λ Z )2 (P λ Z )2 P P λ λ μ ν 3 g μν 3 2 Z Z Z
168
Suchard; PSIJ, 7(3): 152-185, 2015; Article no.PSIJ.2015.066
Z λ Z L λ s.t. Z P P μ Z 2 μ (L g ) d (L g ) μυ m μυ g dx g ,m m i m i m Z P P (Γ m P P Z Γ m P P Z ) 2( μ υ ); 2 μ i ν υ μ i ) Z 2 m Z 2 (42) m i m i m (P P ); Z ) (Γ m P P Z Γ m P P Z ) 2 μ υ m 2 μ i ν υ μ i ) g Z 2 Z 2 Z Z Z Z s 1 Z Z m μ ν 2 s P P m g Z 2 Z 3 μ ν 2 (P i P )2 μν i Z m Z λ Z P P 1 Z Z k Z Z ( 2( ); P P 2 λ μ ν k g μ ν ) g Z 2 m μ ν Z 2 Z 2 Z 2 μν Z 2
Z Z P k P Z P P μ and U λ k λ and L U κU μ λ Z Z 2 k (L g ) d (L g ) μυ m μυ g dx g ,m λ m λ 2 (P Pλ ),m P k (P Z λ ) Pμ Pν 2( P );k Pμ Pν 2 Z 3 Z 3 Z (P λ P ), P s (43) λ s 2( 3 )Z μ Pν Z . g λ 2 λ 2 1 (P Z λ ) (P Z λ ) Pμ Pν g μν 2 Z 3 Z 3 Z m λ k Z Z Z Pμ Pν 1 Z Z Z μ Z ν ( 2( ); P P 2 λ k g ) Z 2 m μ ν Z 2 Z 2 Z 2 μν Z 2
1 P P (U U U U k g 2U k ; μ ν ) g μ ν 2 k μν k Z
(Z s P )2 L s s.t. Z P λ P and Z (P λ P ), Z 3 λ m λ m (L g ) d (L g ) ν Pμ dx Pμ ,ν s s (Z s P ) μ ν (Z s P ) μ i ν 4( P P );ν 4 Γ i ν P P Z 3 Z 3 s s (Z s P ) μ ν (Z s P ) μ i k 4 P ;ν P 4 Γ i k P P g Z 3 Z 3 Z P m Z μ (Z P m )2 2 m 6 m P μ 3 4 Z Z (Z P s )P ν Z P m Z μ (Z P m )2 (44) ( 4( s ); P μ 2 m 6 m P μ ) g Z 3 ν Z 3 Z 4
169
Suchard; PSIJ, 7(3): 152-185, 2015; Article no.PSIJ.2015.066
Z s Z L s s.t. Z P λ P and Z (P λ P ), Z 2 λ m λ m (L g ) d (L g ) ν Pμ dx Pμ ,ν μ ν P Z 4 μ i k k 4( 2 );ν 2 Γ i P Z Z Z (45)
4 μ ν 4 μ i k k 2 P ;ν Z 2 Γ i P Z g Z Z Z Z m 4 m P μ g Z 3 Z ν Z Z m ( 4( ); 4 m )P μ g Z 2 ν Z 3
From (38) and (45), we get the equation of gravity, assuming 8 ,
Z Z P k P 1 Z N 2 P P , U k , L U U i and Z P k P Z Z 2 4 i k m m (P P ),m P k Z 2(( P );k 2( );m )P P Z 3 Z 2 8 1 (P Z ) 2 P P Z Z P P 2 2 . 4 Z 3 Z Z 2 Z 1 U U U U k g k 2
8 1 1 P P 1 (U U U U k g 2U k ; ) R Rg (46) 4 2 k k Z 2 s.t. R R g P P P p s.t. Rk j (j k ), p (p k ), j p jk p j k
R is the Ricci tensor.
In general by (7) (46) can be written as
1 8 1 P P 1 (U U U U k g 2U k ; ) R Rg 4 2 k k Z 2
k We can also see that the ordinary local conservation laws are modified if U ;k 0 .
Important: The force represented by U may greatly differ from what conventional physics defines as mutual interaction forces. This is because such a force is a property of space-time (although not anticipated from the metric alone) and as such, it can be construed as an interaction with space-time
170
Suchard; PSIJ, 7(3): 152-185, 2015; Article no.PSIJ.2015.066
itself, i.e. all matter. Please note that U is a curvature vector which means time can't be measured along geodesic curves as these curves are prohibited in the field (17,18).
dZ Denoting Z , from the vanishing of the divergence of Einstein-Grossmann’s tensor we have: L dx L
1 P P (U U U U k g 2U k ; ); 2 k k Z U U ; U U ; 1 P P (U ; U g ks g U ; U g ks g ) (2U k ; ); 2 k s s k k Z P P U U k ; U kU ; U kU ; (2U k ; ); k k k k Z (47) k U U ;k Z Z Z P L Z P L P P U k (( ); ( k ); ( L P ); ( L P ); ) (2U k ; ); Z k Z Z 2 k Z 2 k k Z Z P L Z P L P P U U k ; U k (( L ), P ( L ); P ) (2U k ; ); k Z 2 k Z 2 k k Z Z P L P P 1 U U k ; U k ( L ), P (2U k ; ); 0 4(R k Rg k ); k Z 2 k k Z 2 k
Z P L Z k Z P L The term U k ( L ), P is due to the fact that U k L P k and U k P 0 from Z 2 k Z Z 2 k Z P L which U k (( L ); P ) 0 . The bottom line is that it is possible that Z 2 k P P P P (2U k ; ); 0 . We will later refer to - 2U k ; as “electro-gravity”. k Z k Z
We can now explore the relation between local parallel translation in loops and non-zero divergence P P U k ; 0 . Contraction of (46) by yields k Z
1 8 1 P P P P 1 P P (U U U U k g 2U k ; ) (R Rg ) 4 2 k k Z Z 2 Z 1 8 1 P P 1 ( U U k 2U k ; ) R R (48) 4 2 k k Z 2
Contraction of (46) by g yields
171
Suchard; PSIJ, 7(3): 152-185, 2015; Article no.PSIJ.2015.066
1 8 1 1 ( U U k U k ; ) R 4 2 k k 2 (49)
So from both equations we have,
1 8 P P U k ; R 4 k Z (50)
k If the divergence U ;k is related to electric charge then this result implies that even photons generate pairs of oscillating charge. By the definition of the Ricci tensor
L P P L P P R R L R R L Z Z (51) P (P L ; ; P L ; ; ) L L Z
This is because by the definition of the Riemann curvature tensor, the anti-symmetric derivation of a L L L L vector field V yields V ; j ;k V ;k ; j R j kV and so
L L k L k (V ; j ;k V ;k ; j )V R j kV V and therefore L L k L k k (V ;L ;k V ;k ;L )V R L kV V R kV V
1 8 P P U k ; R 4 k Z (52) 1 8 P U k ; (P L ; ; P L ; ; ) 4 k L L Z
Definition: We will call the latter "Charge Holonomy Equation".
Or in its complex form,
1 8 P * (U k ; U *k ; ) Re((P L ; ; P L ; ; ) ) (53) 8 k k L L Z
1 P And if 8 we have U k ; (P L ; ; P L ; ; ) in the real case and 4 k L L Z 1 P * (U k ; U *k ; ) Re((P L ; ; P L ; ; ) ) in the complex case. 8 k k L L Z
P * For (P L ; ; P L ; ; ) not to be zero, it requires that locally, parallel translation along L L Z different paths that connect two events, will yield different results. Obviously because P is a scalar
172
Suchard; PSIJ, 7(3): 152-185, 2015; Article no.PSIJ.2015.066
k k field, in flat geometry (P ;L ; P ; ;L ) 0 and in curved geometry we can have k k (P ;L ; P ; ;L ) 0 .
By Lee C. Loveridge [22], another illuminating equation which is simple if 8 is,
8 1 R(3) ( U U U k ; ) (54) 4 k
8 1 1 or in the complex case R(3) ( (U U * U * U ) U k ; U *k ; ) 2 4 k k
Where R(3) is a three dimensions scalar curvature in the space perpendicular to P P, or in the complex case, to (P P * ) / 2 . (54) can be viewed as a non linear version of Proportional to square error – Differential case of PID control which is well known to physicists who also have 1 background in engineering. U U is the square proportional term and U k ; is the differential 4 k term. The three dimensional scalar curvature R(3) is merely an output of such a control system. Note 1 that both R(3) and U U are curvature terms where U describes the curvature of the gradient 4 of the time field i.e. the curvature of P .
From (38),(44),(45) we have,
d d ( )(U U k g ) W ; g 0 dx P dx P , k
P (Z P m ) W ; 4U ; 2 m U ; 0 (55) 2 Z Z
A simpler solution to zero Euler Lagrange equations, is
d k (56) ( )(U kU g ) 0 P dx P ,
Which results in a special case, “Zero Charge” as charged particles are related to non-zero m divergences and either U U 0 or Z m P 0 .
(U ); 0 (57) and (46) becomes,
8 1 1 1 (U U U U k g ) R Rg (58) 4 2 k 2
The reader can either refer to the following calculation or skip it.
173
Suchard; PSIJ, 7(3): 152-185, 2015; Article no.PSIJ.2015.066
Z Z Z m (Z P s )P (4( ); 4 m )P 4( s ); P Z 2 Z 3 Z 3 Z P m Z (Z P m ) 2 2 m 6 m P Z 3 Z 4 Z Z Z m 4( ); P 4 m P Z 2 Z 3 (59) (Z P s )P (Z P m ) 2 4( s ); P 4 m P Z 3 Z 4 Z P m Z Z P m P 2 m ( m ) Z 2 Z Z 2 U k U kU Z P m 4(( ); k )P 2 m U 0 Z k Z Z 2
-Pμ Recall that U k P 0 , multiplication by and contraction yields, k 4
Z (Z P s )P Z Z m (Z P m ) 2 (( ); ( s ); )Z m m 0 (60) Z 2 Z 3 Z 2 Z 3
Z (Z P s )P 1 Z Z m (Z P m ) 2 ( ); ( s ); ( m m ) 0 (61) Z 2 Z 3 Z Z 2 Z 3 and as a result of (61), the following term from (46) vanishes,
P μ P ν U k P μ P ν 2(U k ); 2( ); P μ P ν 2U kU k Z Z k k Z (62) Z m (P P ), P m 1 Z Z (Z P s ) 2 2( ); ( m P k ); ( s )P μ P ν 0 2 m 3 k 2 3 Z Z Z Z Z
Z (Z P s )P which yields a simpler equation (58). Recall that U s , Z Z 2 Z And that U U U Z Z (Z P s )P 1 Z Z m (Z P m ) 2 ( ); ( s ); ( m m ) Z 2 Z 3 Z Z 2 Z 3 U 1 1 1 1 ( ); (U U m ) (U ); U Z (U U m ) (63) Z Z m Z Z 2 Z m 1 (U ); 0 Z which proves (57).
174
Suchard; PSIJ, 7(3): 152-185, 2015; Article no.PSIJ.2015.066
1 P P Question to the reader: If U U U U k g 2U k ; describes the electro-magnetic 2 k k Z energy momentum tensor, where is the torsion tensor F F that is so basic to the electro- magnetic theory?
Answer: U is not any electro-magnetic field. It is a property of space-time as P is dictated by the equations of gravity. U , however, offers a way to describe an anti-symmetric tensor which is a singular Lie Algebra matrix as an acceleration field via Tzvi Scarr and Yaakov Friedman U P * representation [7], A such that A A . See "Appendix – 2 (P * P P P * ) / 2
Acceleration field representation". The field is actually represented by A but it is not an electromagnetic field, rather, it is the underlying mechanism that results in what we call, the electric U * P * field. In the complex formalism either, (A ) * or 2 (P * P P P * ) / 2 4 U P * C 1 A . Increasing or decreasing (U * U U U * ) results 2 K 8 (P * P P P * ) / 2 in change of Energy density and in the phenomena we call Electro-Magnetism. A represents a non-inertial acceleration of every particle that can measure proper time and not of photons as single particles.
8 1 1 k Inertia Tensor: We define inertia tensor as U U U kU g this reflects what we know 4 2 as energy momentum tensor.
2 k P P Question: why is the tensor U ;k not included in the Inertia Tensor? 4 k P Pk
P P Answer: Because the term is not a material field. k P Pk
2 k P P Electro gravity tensor: We define the electro-gravity tensor as U ;k . 4 k P Pk 1 1 4K As we shall see, k is equivalent to electric charge density k or by (7) U ;k U ;k 2 2 2 0 C 1 K k if . U ;k 2 8 2 2 0 C
But the sign could be also plus. Charge conservation yields the following:
1 P P From, (U U U U k g 2U k ; )g U U k 2U k ; it follows 2 k k Z k k
175
Suchard; PSIJ, 7(3): 152-185, 2015; Article no.PSIJ.2015.066
1 1 that (U U k 2U k ; ) gd U U k gd R gd . k k k 4 4
Construction and Destruction: We define construction or destruction as local appearance and 1 disappearance of non zero U k ; neighborhoods of space-time as a function of . This definition 2 k alludes to the well known terms of construction and destruction brackets in Quantum Mechanics.
8. RESULTS - ELECTRO-GRAVITATIONAL THRUST, DARK MATTER AND DARK ENERGY
Dark Matter: Dark Matter will be defined as additional Gravity not due to the Inertia Tensor. It is meant that the cause of such gravity is not inertial mass that resists non-inertial acceleration. It also emanates from the acceleration field as expressed by the Scarr-Friedman matrix [7], A A . This field prohibits Minkowsky geodesic motion of rest mass.
Dark Energy: Dark Energy will be defined as negative Gravity not due to the Inertia Tensor. It is meant that the cause of such gravity is not inertial mass that resists non-inertial acceleration. Also in this case electro-gravity is not the only cause. The acceleration field A A that prohibits Minkowsky geodesic motion of rest mass must also be taken into account. The following will describe a technology that can take energy from space-time apparently by Sciama Inertial Induction [23] and is closely related to Alcubierre Warp Drive [24]. Electro-gravity follows from (6), (46) and (55). For several reasons we may assume the weak acceleration of uncharged particles mentioned in (6) is from positive to negative charge see also [8], consider the general relativity equation
1 1 P P 1 (U U U U k g 2U k ; ) G R Rg such that the Ricci tensor is 4 2 k k Z 2 P P P p Rk j (j k ), p (p k ), j p jk p j k
G is the Einstein-Grossmann’s tensor. From (4) in a weak gravitational background field and 8 ,
λ λ μ 1 1 (P Pλ ),m (P Pλ ),μ P a Em U m ( i i 2 Pm ) 2 4K 0 2 (64) 2 2 P Pi (P Pi ) C C
Or by (7) if 8 (68) becomes and also
a K E 0 m E k ; 0 (66) 2 2 0 C 2 C The reader is requested to notice that (70) is only C is the speed of light, a is the non-relativistic a non-relativistic and non-covariant limit! From electro-magnetism weak acceleration of an uncharged particle, 0 is the permittivity constant in vacuum, K is the gravitational constant and E is a static non- k E ;k (67) relativistic electric field in weak gravity, assuming 0 that by correct choice of coordinates, Such that is the charge density Em (E0 0,E1,E2 ,E3 ) (65)
176
Suchard; PSIJ, 7(3): 152-185, 2015; Article no.PSIJ.2015.066
8 1 1 k K (U U U kU g ) G G (70) 4 2 2 tt 00 (68) 2 0 C 0 1 1 k 8K 4 (E E Ek E g ) 2 C 2 1 K And So 2 such that 4 C Gtt G00 C 2 0 k 1 P P K E ; P P 2U k ; 0 k (69) k 2 1 4 Z 2 C Z behaves like mass density and 2 0 K k therefore we can define an electro-gravitational From the electro-magnetic theory E ;k 0 virtual mass as dependent on charge Q: such that is the charge density and so for t Schwarzschild coordinate time, 1 Q M Virtual Q (71) 2 0 K 2 0 K
Q We will calculate Virtual _ Mass for 20 Coulombs by assuming 8 , 2 0 K
1 Coulomb 9 5.802331691060358888128083379341710 Kg . 16 0 K
Multiplied by 20 we have
20 Coulombs 11 1.160466338212071777625616675868310 Kg. 16 0 K
Within 1 cubic meter the effect would be a feasible electro-gravitational field because Newton's gravitational acceleration as a rough approximation yields, K Virtual _ Mass radius 2 11 2 K 1.16046633 8212071777 6256166758 683 10 Kg /1 , Meter 7.74477595 0343958808 9631666010 105 Second 2
Consider parallel metal plates of 10 cm x 10 cm with a gap of 2 cm and with low relative permittivity thin slab of 10 grams in the middle, such that the voltage of 37 Kilovolts is applied to the plates. The net classical non-relativistic gravitational force on the slab without regarding the non-geodesic acceleration field, is less than one third of a micro-Newton. That is a dauntingly small force which is very difficult to measure.
The calculations rule out any measurable vacuum thrust of Pico-Farad or less, asymmetrical capacitors even with 50000 volts supply, simply because the net effect depends on the total amount of separated charge which is far from sufficient in standard Biefeld Brown capacitors [25].
177
Suchard; PSIJ, 7(3): 152-185, 2015; Article no.PSIJ.2015.066
Hypothetical use of sub-luminal photons: It is 1024 photons is equivalent to over 1600 yet needed to be demonstrated that sub-luminal separated Coulombs and therefore if Hans Giertz photons do exist and that if very close to the assumption [11] holds, using photons for warp speed of light can interact with electric currents drive effects is the most feasible technological and/or fields as if they are high density electric solution. charge distributions. After (17) there is a discussion on the vanishing of the Reeb class K Virtual _ Mass based on [2][3] from which the restriction of U The shows a gravitational radius 2 to the foliation F must have a zero rotor i.e. acceleration of over 126 gees between 1600 U ; (F) U ; (F) 0 . This results in coulombs separated by a gap of 2 meters. In electric field which is always a result of electric such a case, if a photon behaves like a dynamic charge, i.e. non vanishing divergence. The idea oscillating or rotating dipole, these dipoles will be is that even photons can appear as a pair of of different dipole moments aligned or anti- oscillating negative and positive charge. Slight aligned with relativistic electric fields. The result imbalance in charge distribution in photons can is equivalent to true charge separation and can result in alleged effects such as EMDrive [26]. A have a great technological potential for the slight 1% imbalance in +e and –e charge of say development of a feasible warp drive thrust.
Use of plasma: Another idea is to use ionized plasma. Let us see what we can do with one gram of ionized hydrogen. The number of atoms by Avogadro's number is n 6.02214129 1023 . The charge of the electron is e 1.60217656510-19 Coloumbs so Q 9.64853364 595686885 10 4 Coloumbs K 6.67384 10-11 m 3 kg -1sec-1 and -12 0 8.85418781 76.. 10 F/m so 1 gram of hydrogen reaches a virtual mass of Virtual_Ma ss 5.59839925 4619768891 0422084444 814 1014 Kg . That is far less than the 24 mass of the Earth M Earth 5.97219 × 10 Kg but the distance between two clouds of positive and negative ionized hydrogen can be much less than the average Earth radius and therefore a field that overcomes the Earth gravitational field is feasible.
Dark Matter and Dark Energy follow immediately from negatively ionized gas in the galaxy and a a positively ionized gas outside or on the outskirts if we assume energy density which means our 8K electro-gravity constant is Const 8K . In other theories, e.g. Const K , the center of the galaxy should be positive. Negative charge as in the first case Const 8K , will only behave as Dark Matter more at distance because close to stars it is expected to cause induced dipoles within the star just as a small negatively charged ball above the Earth will polarize the ground. We now consider the classical non-covariant limit of the summation of two effects, the non-inertial acceleration and electro-gravity. Let Qbe the charge of a ball at radius r , then by (7) the observed non-relativistic classical acceleration a of an uncharged particle without any induced diploes is KQ K Q K Q 1 1 a 0 ( ) g a 2 2 2 Electro _ gravity r 2K 0 / 2 4 0 r 2 0 r 4
K Q 1 1 If then 8 a 2 ( ) g Electro _ gravity a 2 0 r 8 4
A friend, Mr. Yossi Avni suggested that two marble balls will be suspended on a balance above charged balls, left minus and right plus, and that by the checking if the balanced is tipped or not, we can decide 4 for unbiased balance and different value else, e.g. 8 . There is a problem, however, that charge in the ground below the balls will polarize the ground.
178
Suchard; PSIJ, 7(3): 152-185, 2015; Article no.PSIJ.2015.066
Fig. 1. Avni suggestion, the left lower ball is negatively charged and the right is positively charged. The balls above are marble or even of lower dielectric constant
This theory means that a positive charge manifests attracting gravity but has a repelling acceleration field that acts even on uncharged particles that can measure proper time, i.e. have rest mass. The k k curvature U U k 0 and for positive charge U ;k 0 .
Negative charge manifests a repelling anti-gravity but has an acceleration field that attracts even uncharged particles and acts on particles that can measure time, i.e. have rest mass.
The following table describes the relation between and the Dark Matter and Dark Energy.
Table 1. The relation between constants and Dark Matter
and classical non- Cause for Cause for Dark matter causes relativistic acceleration dark matter dark energy 4 Positive charge Negative charge Caused by gravity K Q 1 1 2 ( ) 0 2 0 r 4 4 Negative Positive charge Caused only by an Alcubierre Charge Warp Drive [24] by induced K Q 1 1 ( ) 0 dipoles. Positive side faces 2 the negative galaxy center. 2 0 r 4 4 Negative Charge Positive charge Caused by the acceleration field on particles with rest K Q 1 1 mass 2 ( ) 0 2 0 r 4
9. CONCLUSION sense. Since more than one curve on which such time can be virtually measured, intersects the An upper limit on measurable time from each same event - as is the case in material fields event backwards to the "big bang" singularity as which prohibit inertial motion, i.e. prohibit free fall a limit or from a manifold of events as in de Sitter - such time can't be realized as a coordinate. or anti - de Sitter, may exist only as a limit and is Nevertheless using such time as a scalar field, not a practical physical observable in the usual enables to describe matter as acceleration fields
179
Suchard; PSIJ, 7(3): 152-185, 2015; Article no.PSIJ.2015.066
and it allows new physics to emerge as a Also a historical justice with the philosopher replacement of the stress-energy-momentum Rabbi Joseph Albo, Circa 1380-1444 must be tensor. The punch line is electro-gravity as a neat made. In his book of principles, essay 18 explanation of the Dark Matter effect and the appears to be the first known historical account advent of Sciama's Inertial Induction, which of what Measurable Time – In Hebrew "Zman becomes realizable by separation of high electric Meshoar" and Immeasurable Time "Zman Bilti charge. This paper totally rules out any Meshoar” are. His Idea of the immeasurable time measurable Biefeld Brown effect in vacuum on as a limit [27], is the very reason for an 11 years Pico-Farad or less, Ionocrafts due to insufficient of research and for this paper. amount of electric charge [25]. The electro- gravitational effect is due to field divergence and COMPETING INTERESTS not directly due to intensity or gradient of the square norm. Inertial motion prohibition by Author has declared that no competing interests material fields, e.g. intense electrostatic field, can exist. be measured as a very small mass dependent force on neutral particles that have rest mass REFERENCES and thus can measure proper time. Such acceleration should be measured in very low 1. Vaknin S. Time Asymmetry Revisited capacitance capacitors in order to avoid electro- [microform]; 1982. Library of Congress gravitational effect. The non-gravitational Doctorate Dissertation, Microfilm 85/871 acceleration should be from the positive to the (Q). negative charge. The electro-gravitational effect 2. Reinhart B, Wood J. A Metric formula for is opposite in direction, requires large amounts of the Godbillon–Vey invariant for foliations. separated charge carriers and acts on the entire Proceedings of the American Mathematical negative to positive dipole. Society. 1973;38:427-430,. 3. Hurder S, Langevin R. Dynamics and the ACKNOWLEDGEMENTS Godbillon-Vey class of C1 foliations. preprint, September; 2000. First, I should thank Dr. Sam Vaknin for his full 4. Suchard EH. Electro-gravitational support and for his ground breaking dissertation technology via Chronon field. Physical from 1982. Also thanks to professor Larry Science International Journal. pp 1158- (Laurence) Horowitz from Tel Aviv university for 1190, June/2014, 10.9734/PSIJ/2014/9676 his guidelines on an Early work. His letter from Follow ups with minor constants update Wed, Jul 23, 2008, encouraged me to continue are: my research which started in 2003 and led to this Follow up 1 with constants update: paper. Special thanks to professor David Available:http://www.scribd.com/doc/21390 Lovelock for his online help in understanding the 3929/Electrogravity-Engine-Theory-and- calculus of variations and in fixing an error Practice several years ago. Thanks to electrical Follow up 2 with constants update: engineers, Elad Dayan, Ran Timar and Benny Available:http://vixra.org/pdf/1407.0114v6. Versano for their help in understanding how pdf profoundly difficult it is to separate charge as Follow up 3 with constants update: required by the idea of electro-gravity. Thanks to Available:http://freepdfhosting.com/f7a881 other involved engineers, Arye Aldema, Erez b129.pdf Magen and to Dr. Lior Haviv from the Weizmann In these follow ups, Institute of Science. Thanks to Zeev Jabotinsky 10.9734/PSIJ/2014/9676 is correctly cited. and to Erez Ohana for his initiative of building a 5. John W Milnor. Topology from the miniature warp drive engine with almost zero Differentiable Viewpoint, pages 32-41, funds. ISBN 0-691-04833-9. 6. Hormander Lars. The anaysis of linear Thanks to Mr. Yossi Avni who approved my partial differential operators I. 1983, research in 2003 into minimum cost Grundl. Math. Wissenschaft. 256, Springer, diffeomorphism as part of a pattern matching ISBN 3-540-12104-8 Chapter I. Test algorithm. This early research helped in acquiring Functions pages 5-31, II 2.3 Distributions special knowledge in geometry that was later with compact support, page 44, Chapter used in the presented theory. VI. Composition with Smooth Maps, 6.3. Distributions on a Manifold.
180
Suchard; PSIJ, 7(3): 152-185, 2015; Article no.PSIJ.2015.066
7. Friedman Y, Scarr T. Making the relativistic 18. Victor Guillemin, Alan Pollack. Differential dynamics equation covariant: explicit Topology, Submersions, Page 21, solutions for motion under a constant Preimage Theorem, ISBN 0-13-212605-2 force. 2012;86(6). Physica Scripta 065008. 19. Lovelock D, Rund H. Tensors, Differential
DOI: 10.1088/0031-8949/86/06/065008 Forms and Variational Principles, Dover 8. Datta T, Yin M, Dimofte A, Bleiweiss MC, Publications Inc. Mineola, N.Y., ISBN 0- Cai Z. Experimental indications of Electro- 486-65840-6, p. 262, 3.27 Gravity; 2005. 20. Lovelock D, Rund H. Tensors, Differential arXiv:physics/0509068[physics.gen-ph] Forms and Variational Principles, Dover 9. Hector Luis Serrano. (President of Gravitec Publications Inc. Mineola, N.Y., ISBN 0- Inc.) capacitor at 1.72 x 10^-6 Torr, 486-65840-6, p. 261, 3.26 Experiment by NASA and Gravitec, Video 21. Lovelock D, Rund H. The numerical from July 3; 2003. relative tensors. Tensors, Differential 10. Doi Y, Makino T, Kato H, Takeuchi D, et al. Forms and Variational Principles, 4.2, Deterministic Electrical charge-state Dover Publications Inc. Mineola, N.Y. , initialization of single Nitrogen-Vacancy ISBN 0-486-65840-6, p. 113, 2.18, p. 114, center in Diamond. Published 31 March; 2.30 2014, Physical Review X 4, 011057, 22. Lee C Loveridge. Physical and geometric DOI:10.1103/PhysRevX.4.011057, interpretations of the Riemann Tensor, Publisher: American Physical Society. Ricci Tensor, and Scalar Curvature" DOI: 11. Informal paper: Hans W Giertz. The photon 10.1103/PhysRevD.69.124009, Submitted consists of a positive and a negative on 23 Jan 2004, 7. The Einstein Tensor, charge, measuring gravity waves reveals (9), arxiv.org/abs/gr-qc/0401099 the nature of photons. 23. Sciama DW. On the Origin of Inertia. Available:http://vixra.org/pdf/1302.0127v1. Monthly Notices of the Royal Astronomical pdf Society. 1952;113:34. 12. Reeb G. Sur certaines propri´t´s 24. Alcubierre M. The warp drive: hyper-fast topologiques des vari´et´es feuillet´ees. travel within general relativity. Class. Actualit´e Sci. Indust. 1183, Hermann, Quant. Grav. 1994;11:L73-L77, Paris; 1952. DOI: 10.1088/0264-9381/11/5/001 13. Walter Pfeifer. The Lie Algebra su(N), an 25. Thomas B Bahder, Chris Fazi. Force on an Introduction. Chapter 5, The Lie Algebra Asymmetric Capacitor. Army Research SU(4), 2003;87-106. (revised 2008), ISBN Laboratory ARL; 2003. 3-7643-2418-X Available:http://arxiv.org/ftp/physics/papers 14. Elie Cartan. The Theory of Spinors, Dover /0211/0211001.pdf Publications Inc. Mineola, N.Y. ISBN 0- 26. Matt Porter at IGN News; 2015. 486-64070-1, p. 149 (14) Available:http://www.ign.com/articles/2015/ 15. Escultura EE, Chaos. Turbulence and 04/28/nasa-may-have-invented-a-warp- fractal: Theory and applications. drive International Journal of Modern Nonlinear 27. Albo J. Book of Principles (Sefer Ha- Theory (SCIRP), pp. 176 – 185. ikarim), Immeasurable time - Maamar 18 Available:http://www.scirp.org/journal/Pape measurable time by movement. (Circa rInformation.aspx?PaperID=36849; 1380-1444, unknown), The Jewish DOI: 10.4236/ijmnta.2013.23025. publication Society of America; 1946. 16. Victor Guillemin, Alan Pollack. Differential ASIN: B001EBBSIC, Chapter 2, Chapter Topology, Prentice-Hall, Englewood Cliffs, 13. NJ; 1974:39-48. Sard's Theorem and 28. Jean Bertoin. Lèvy Process Cambridge Morse Functions, ISBN 0-13-212605-2 University Press, 1998 October 26, first 17. Victor Guillemin, Alan Pollack. Differential publication; 1996. ISBN 0521646324, Topology, Submersions, Page 20, ISBN 0- Page 11. 13-212605-2
181
Suchard; PSIJ, 7(3): 152-185, 2015; Article no.PSIJ.2015.066
APPENDIX
APPENDIX – The time field in the Schwarzschild solution
Motivation: To make the reader familiar with the idea of maximal proper time and to calculate the background scalar time field of the Schwarzschild solution.
(P λ P ), (P s P ), g mk ((P λ P ), P m )2 We would like to calculate λ m s k λ m in Schwarzschild coordinates i 2 i 3 (P Pi ) (P Pi ) for a freely falling particle. This theory predicts that where there is no matter, the result must be zero. The result also must be zero along any geodesic curve but in the middle of a hollowed ball of mass the gradient of the absolute maximum proper time from "Big Bang" event or from a sub-manifold of events, derivatives by space must be zero due to symmetry which means the curves come from different directions to the same event at the center. Close to the edges, gravitational lenses due to granularity of matter become crucial. The speed U of a falling particle as measured by an observer in the gravitational field is
U 2 R 2GM V 2 (72) C 2 r rC 2
Where R is the Schwarzschild radius. If speed V is normalized in relation to the speed of light then U V . For a far observer, the deltas are denoted by dt',dr and, C
dr R r 2 ( ) 2 V 2 (1 ) (73) dt r because dr dr/ 1 R / r and dt dt 1 R / r .
R R 2 t t (1 ) t R (dr / dt) 2 R R P (1 ) dt (1 ) r r dt (1 ) 2 dt r R r R r 0 (1 ) 0 (1 ) 0 r r t R (1 )dt 0 r
Which results in,
dP R P (1 ) (74) t dt r
Please note, here t is not a tensor index and it denotes derivative by t !!! On the other hand
182
Suchard; PSIJ, 7(3): 152-185, 2015; Article no.PSIJ.2015.066
R r r R r r (1 ) r R 1 1 1 P (1 ) dr r R dr R dr 2 r r R R 2 R r R (1 ) (1 ) (1 ) r r r r r r dr R
Which results in
dP r P r dr R (75)
Please note, here r is not a tensor index and it denotes derivative by r !!! For the square norms of derivatives we use the inverse of the metric tensor,
R 1 1 R So we have (1 ) and (1 ) r R R r (1 ) (1 ) r r So we can write R 1 R r r R N 2 P P (1 )P 2 P 2 (1 )( 1) 2 r r R t r R R r 1 r r R N 2 2 (76) R r
2 2 dN N And we can calculate dx
R 2 1 R 2 2 2 (1 ) ( ) N N 2 r R r (77) 2 2 (N ) r R 2 ( 2) R r
We continue to calculate
2 2 R 2 1 R R N t Pt R 1 R R N t P (1 ) ( ) and (1 )( ) (78) t r R 2 r R r R r 2 r r (1 ) r
Please note, here t is not a tensor index and it denotes derivative by t !!!
R 2 R 1 R r (1 )N r P (1 )( ) (79) r r r R r 2 R
Please note, here r is not a tensor index and it denotes derivative by r !!!
183
Suchard; PSIJ, 7(3): 152-185, 2015; Article no.PSIJ.2015.066
2 R 1 R r R N P (1 )( )( ) And r R r 2 R r
2 2 R 2 1 R 2 r R (N P ) (1 ) ( ) ( 2) (80) r R r 2 R r
So
R 2 1 R 2 2 2 (1 ) ( ) (N P ) 2 r R r (81) 2 3 (N ) r R 2 ( 2) R r
And finally, from (77) and (81) we have,
(P P ), (P s P ), g mk ((P P ), P m ) 2 m s k m i 2 i 3 (P Pi ) (P Pi ) 2 2 2 2 N N (N P ) (82) (N 2 ) 2 (N 2 )3
R 2 1 R 2 R 2 1 R 2 (1 ) ( 2 ) (1 ) ( 2 ) r R r r R r 0 r R r R ( 2) 2 ( 2) 2 R r R r which shows that indeed the gradient of time measured, by a falling particle until it hits an event in the gravitational field, has zero curvature as expected.
APPENDIX – Event Theory and Lèvy Process
Motivation: To present one of several possible quantization offers of (38) without using Stochastic Calculus. To provide an alternative integration constraint that leads to a new possible theory – a particle is a non-zero curvature vector U and a family of event functions, . An interesting alternative to (39) is that is not a particle wave function but an event function, i.e. a collision with a particle in a 4 dimensional space-time,
* gd 4 1 (83) 4 ( )
And instead of P we choose P . The reader can verify that the same curvature vector in (37) is reached by the assignment P if depends only on .
Then from (40) we get a new random variable Event
4 4 Event 4 * g d 4 PP * g d (84) ( ) ( ) and adding this constraint to the complex form of (38) we have the following variations system,
184
Suchard; PSIJ, 7(3): 152-185, 2015; Article no.PSIJ.2015.066
P P P * P * P Z Z P *k P Z N 2 and U k and 2 Z Z 2 1 1 L ( (U k U * U *k U )) 4 2 k k R Ricci curvature, (85) 1 Min Action Min R L PP *. gd 2 and PP *. gd Event Const.
(85) is worthy of further research. To prove that (85) is consistent with quantum mechanics, e.g. Quantum Field Theory and that no other idea of how to quantize (38) is required, a family of solutions to (85) with complex functions should exist such that Event will be increasing and will describe detection events of the same particle. Because we introduce a scalar field of time , and use a probability function * , if the time s is a Lèvy process [28] then we also need the following to hold,
s t ( (s) *(s) (t) *(t)) gd (s t) *(s t) gd (86) s.t. s t . So actually a particle is a family of parameterized Lèvy measures (s) *(s) such that s itself denotes time and such that denotes the scalar field of time and such that the curvature vector k Z, j Z,k P * Pj dP field U is not zero, s.t. P and P and x j are local j Z Z 2 j dx j coordinates. This is one of the ways of the ways to try to quantize the theory. Showing that this offer works and agrees with Quantum Field Theory should be a subject of international research.
The philosophy here is that particles should emerge from geometry, stochastic processes and from stochastic calculus and not from algebra and is therefore against the concurrent physics mainstream. ______© 2015 Suchard; This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Peer-review history: The peer review history for this paper can be accessed here: http://sciencedomain.org/review-history/9858
185