Unification of Electromagnetism and Gravitation. Raymond J. Beach

Transcription

1 Preprints ( NOT PEER-REVIEWED Posted: 1 Septeber 17 doi:1.944/preprints v3 Unification of Electroagnetis and Gravitation Rayond J. Beach Lawrence Liverore National Laboratory, L-465, 7 East Avenue, Liverore, CA E-ail: beach@llnl.gov Using four field equations, a recently proposed theory that covers the phenoenology of classical physics at the level of the Maxwell and Einstein ield Equations (M&EEs) but then goes further by unifying electroagnetic and gravitational phenoena in a fundaentally new way is reviewed. Maxwell s field equations are a consequence of the new theory, as are Einstein s field equations augented by a ter that can replicate both dark atter and dark energy. To ephasize the unification brought to electroagnetic and gravitational phenoena by the new theory, specific solutions for a spherically-syetric charged particle, and both electroagnetic and gravitational waves are presented. A unique feature of the new theory is the treatent of both charge and ass density as dynaic fields, this as opposed to their treatent in the classical M&EEs where they are external paraeters. This feature enables a procedure for quantizing the ass, charge and angular oentu of particle-like solutions. inally, antiatter is naturally accoodated by the new theory and its interaction with a gravitational field is reviewed. (key words: unified field theory, electroagnetis, Maxwell s equations, gravitation, General Relativity, Einstein s field equations, gravitational radiation, hidden variable, dark atter, cosological constant) 17 by the author(s). Distributed under a Creative Coons CC BY license.

2 Preprints ( NOT PEER-REVIEWED Posted: 1 Septeber 17 doi:1.944/preprints v3 Introduction Assuing the geoetry of nature is Rieannian with four diensions, the following four equations have been proposed as an alternative description of classical physics at the level of the M&EEs, i but one that then goes further by reconciling gravity and electroagnetis κ = a R (1) κ ar = ρ u () c uu = 1 (3) uu 1 4 g ρσ ρ + ρσ = (4) Both equations (1) and () are new, as is the vector field a that appears in the and serves to couple gravitational to electroagnetic phenoena. Equation (1) couples the Maxwell tensor to the Rieann-Christoffel (R-C) tensor R κ. Equations () couples the Ricci Tensor R to the coulobic current density ρ c u. Suppleenting these first two equations are equations (3) and (4), both of which are well known. Equation (3) noralizes the four-velocity vector field u that describes the otion of both the charge density field ρ c, and the ass density field ρ, which are assued to be cooving. Equation (4) describes the conservation of energy and oentu for a specific choice of energy-oentu tensor. Much of the discussion that follows will be focused on describing solutions to these equations and deonstrating that such solutions are consistent with those of the classical M&EEs, but then go further by unifying electroagnetic and gravitational phenoena. Taken together, the four field equations are used to axioatically build up a description of nature in ters of the six dynaic fields described in Table I. Table I. Dynaic fields ield Description Nuber of coponents g Metric tensor 1 Maxwell tensor 6

3 Preprints ( NOT PEER-REVIEWED Posted: 1 Septeber 17 doi:1.944/preprints v3 u our-velocity 4 a our-vector coupling electroagnetis to 4 gravitation ρc Charge density 1 ρ Mass density 1 Total nuber of independent field coponents 6 An outline of the paper is as follows: Using the four fundaental field equations and the properties of the R-C tensor, Maxwell s equations are derived. The classical field theory based on equations (1) through (4) is then shown to be consistent with the requireents of general covariance after taking full account of dependent or constraining equations. Syetries of the theory iportant for the treatent of antiatter and how it responds to a gravitational field are then reviewed. A discussion of the Einstein field equations and how they fit into the fraework defined by the fundaental field equations (1) through (4) is then given. Next, a soliton solution to the field equations representing a spherically-syetric charged particle is reviewed. Ephasized in this particle-like solution are the source ters of the electroagnetic and gravitational fields, ρ c and ρ, respectively, which are theselves dynaic fields in the theory a developent which opens the possibility of quantizing such solutions for charge, ass and angular oentu using a set of self-consistency equations that flow fro the analysis. Two radiative solutions representing electroagnetic and gravitational waves are then developed and analyzed with an ephasis on the unification that the theory brings to electroagnetic and gravitational phenoena. A discussion of antiatter and how it is accoodated by the new theory is then given. inally, to gain insight into the nuerical solution of the fundaental field equations (1) through (4), an analysis of the Cauchy initial value proble as it applies to the is given in Appendix II. In this paper geoetric units are used throughout along with a etric tensor having signature [+,+,+,-]. Coas before tensor indices are used to denote ordinary derivatives while seicolons before tensor indices denote covariant derivatives. Spatial indices run fro 1 to 3, with 4 the tie index. or the definitions of the R-C curvature tensor and the Ricci tensor, the conventions used by Weinberg are followed. ii

4 Preprints ( NOT PEER-REVIEWED Posted: 1 Septeber 17 doi:1.944/preprints v3 Maxwell s equations fro =a R and ρ u κ κ c Equations (1) and () which relate the Maxwell tensor derivatives to the R-C tensor and the charge current density to the Ricci tensor, respectively, are the fundaental relationships fro which all of Maxwell s equations flow. The derivation of Maxwell s hoogenous equation begins with the algebraic property of the R-C tensor Contracting (5) with a gives, Rκ + Rκ + Rκ =. (5) ar + ar + ar =. (6) κ κ κ Using (1) to substitute κ for ar κ leads to + + =, (7) κ κ κ which is Maxwell s hoogenous equation. Maxwell s inhoogeneous equation follows fro (1) by contracting its and κ indices = a R. (8) Making the connection between the RHS of (8) and the coulobic current density u J ρc using () then gives the conventional Maxwell-Einstein version of Maxwell s inhoogeneous equation = ρcu ( J ). (9) Because is antisyetric, the identity = is forced, which in turn forces the coulobic charge to be a conserved quantity ( u c ) ( a R ) ρ = =. (1) Using equations () and (3), the coulobic charge density can be solved for in ters of a, u and the Ricci tensor,

5 Preprints ( NOT PEER-REVIEWED Posted: 1 Septeber 17 doi:1.944/preprints v3 ± σ ar ar ρc = or ar u σ. (11) In the forgoing developent, only equations (1), () and (3) are fundaental to the new theory. Maxwell s equations (7) and (9), the conservation of the charge (1), and the solution for the coulobic charge density (11), are all consequences of (1), () and (3), and the properties of the R-C curvature tensor. These additional equations all represent constraints any solution of (1), () and (3) ust also satisfy (7), (9), (1) and (11). One of the new pieces of physics in the foregoing developent is the introduction of the vector field a, a vector field that has no counterpart in the conventionally accepted developent of classical physics but here serves to both couple the Maxwell tensor to the etric tensor through (1), and the charge density to the etric tensor through (). Much of the analysis and discussion that follows will be focused on the ipact of a and how it drives the developent of a classical field theory that encopasses the physics covered by the M&EEs, but then goes further by unifying gravitational and electroagnetic phenoena. A classical field theory that unifies electroagnetis and gravitation As shown in the preceding section, equations (1), () and (3) cobined with the properties of the R-C tensor provide a basis for deriving Maxwell s hoogenous and inhoogeneous equations in curved space-tie. Taking the source ters of the gravitational and electroagnetic fields, and ρ c, respectively, as dynaic fields to be solved for, a classical field theory of gravitation and electroagnetis that is logically consistent with the requireents of general covariance is possible. or a theory to be logically consistent with the requireents of general covariance, the N dynaical field coponents of the theory ust be underdeterined by N-4 independent equations, the reaining 4 degrees of freedo representing the freedo in the choice of coordinate syste. Table I lists the six dynaic fields of the theory along with the nuber of independent coponents that coprise each field, yielding a total of 6 independent field coponents. Now consider the ρ

6 Preprints ( NOT PEER-REVIEWED Posted: 1 Septeber 17 doi:1.944/preprints v3 last of the theory s fundaental equations, equation (4), the energy and oentu conservation equation uu 1 4 g ρσ ρ + ρσ =. (4) The specific for of the energy-oentu tensor in (4) ensures that ρ is conserved and that there is a Lorentz force law. These two dependent equations are derived by first contracting (4) with u which leads to the conservation of ass ( ) u ρ =, (1) and then cobining (4) and (1), which leads to the Lorentz force law Du ρ D ρcu τ = (13) Du where u σu Dτ σ. A ore coplete outline of the derivation of (1) and (13) is given in the Appendix I. Table II collects and suarizes the four fundaental equations of the new theory, along with the nuber of coponents of each equation. Table II. undaental equations Equation Equation nuber in text Nuber of coponents κ = a R (1) 4 κ a R = ρ u ( J ) () 4 uu c = 1 (3) 1 uu 1 g ρ + = ρσ ρσ 4 (4) 4 Total nuber of equations 33

7 Preprints ( NOT PEER-REVIEWED Posted: 1 Septeber 17 doi:1.944/preprints v3 The total nuber of fundaental field equations listed in Table II is 33 which is greater than the 6 field coponents listed in Table I that need to be solved for. If all the fundaental equations listed in Table II were independent, then the field coponents listed in Table I would be overdeterined and the theory would not be copatible with the requireents of general covariance. However, not all the 33 equations listed in Table II are independent. As already noted, dependent constraint equations can be derived fro the equations listed in Table II and the properties of the R-C curvature tensor. Table III collects these dependent constraint equations along with a brief description of their derivation. Equation κ κ κ Table III. Dependent equations Equation nuber in text + + = (7) ( u c ) ( a R ) Derivation (1) and Rκ + Rκ + Rκ = Nuber of coponents ρ = = (1) (1) and () 1 4 ± σ ar ar ρc = or ar u σ (11) () and (3) 1 ( ) u ρ = (1) (4), (3) and (7) 1 Du ρ = ρcu Dτ (13) (4), (3) and (1) 4 Total nuber of equations 11 The 11 dependent constraint equations listed in Table III ean that of 33 fundaental equations listed in Table II, only 33-11= are independent. These independent equations satisfy the requireents of general covariance for deterining the 6 independent field coponents of Table I. The reaining four degrees of freedo in the solution representing the four degrees of freedo in choosing a coordinate syste. To further elucidate the content of the fundaental field

8 Preprints ( NOT PEER-REVIEWED Posted: 1 Septeber 17 doi:1.944/preprints v3 equations (1) through (4), an outline of their solution when viewed as a Cauchy initial value proble is presented in Appendix II. Syetries of the fundaental field equations Before leaving the foral description of the fundaental equations listed in Table II, three iportant syetries that these equations exhibit are noted. The first of these syetries corresponds to charge-conjugation u u a a g g ρc ρc ρ ρ, (14) the second syetry corresponds to a atter to antiatter transforation, as will be discussed and justified later u u a a g g ρc ρc ρ ρ, (15) and the third syetry is the product of the first two u u a a g g ρc ρc ρ ρ. (16)

9 Preprints ( NOT PEER-REVIEWED Posted: 1 Septeber 17 doi:1.944/preprints v3 All three transforations (14) through (16) leave the fundaental equations (1) through (4) unchanged. Including an identity transforation with the syetries (14) through (16) fors a group, the Klein-4 group with the product of any two of the syetries (14) through (16) giving the reaining syetry. Note that aong the fundaental fields of the theory, only g and are unchanged by the syetry transforations. These syetry transforations will be useful for defining boundary conditions that lead to quantized ass, charge and angular oentu of particle-like solutions, as well as for the treatent of antiatter which will be discussed later in this anuscript. ρ How do the Einstein field equations coport with the new theory? While evident fro the preceding discussion that Maxwell s field equations and the classical physics that flows fro the are derivable fro the fundaental equations (1) through (4), at this point it is not obvious that the sae can be said of Einstein s field equations. The particle-like solution to be analyzed in the following section deonstrates that the Reissner-Nordstro etric is an exact solution of the fundaental field equations (1) through (4), thus establishing that the new theory and classical General Relativity support the sae solutions, at least in the case of spherical syetry. To go further and deterine if Einstein s field equations are in fact derivable fro the fundaental equations of the new theory, start by considering equation (4). An iediate consequence of (4) with its vanishing covariant divergence is that a nuber of trivial tensor equations can be written down relating the covariant divergence of the energy oentu tensor used in (4) to various geoetric quantities. The siplest exaple of such a tensor equation is G = α T (17) 1 where G R g R is the Einstein tensor, α is an arbitrary constant, and T is the stress energy tensor defined in (4) 1 ρσ T ρu u + g ρσ. (18) 4

10 Preprints ( NOT PEER-REVIEWED Posted: 1 Septeber 17 doi:1.944/preprints v3 While (17) is rigorously true, it s trivially satisfied because both sides are independently, the left being by the Bianchi identity and the right by fundaental equation (4). However, an iediate consequence of equation (17) on taking the anti-covariant derivative of both sides is where G = α T +Λ (19) Λ is a function of integration forced to have vanishing covariant divergence, Λ =. () In the view being put forth here, (19) is satisfied by any solution of fundaental fields that satisfy equation (4). However, (19) by itself contains no useful inforation. To see this consider that (19) is trivially satisfied for any G and T that go with any specific solution to equations (1) through (4) and any value for the constantα by taking Λ = G α T. This also deonstrates that the specific choice for the value of α is copletely arbitrary and without physical significance a change in the value of α is absorbed by a change to Λ such that (19) reains satisfied. This discussion should ake clear that (19) is of no value when attepting to find a solution to the fundaental fields. To gain a deeper understanding of how (19) fits into the new theory set α = 1, again a choice having no physical significance but one that is convenient because equation (19), except for the appearance of in this case, i.e., The presence of the function G Λ, reduces to the Einstein field equations = T +Λ. (1) Λ in equation (1) eans that (1) is not quite identical to the classical Einstein field equations. However, the required presence of Λ is interesting because it can iic exactly the properties of dark atter, viz., it is conserved with Λ =, it is a source of gravitational fields, and it has no interaction signature beyond the gravitational fields it sources. inally, note that i.e., Λ = g. Λ also includes as a special case, the possibility of a cosological constant, It is iportant to note that equation (1) with its auxiliary condition () is a consequence of fundaental field equation (4) and the properties of the R-C tensor. With questions today regarding the validity of classical General Relativity beyond the confines of our own solar

11 Preprints ( NOT PEER-REVIEWED Posted: 1 Septeber 17 doi:1.944/preprints v3 syste, iii the ost interesting aspect of Λ in (1) is that it can represent both dark atter and dark energy. Today such ters are appended to the energy-oentu tensor of Einstein s field equations in an ad hoc anner to explain both the flattening of rotational-velocity curves observed on galactic scales, and the accelerating expansion of the universe. So in several respects equation (1), which is a consequence of the new theory, is already accepted and being actively used to analyze gravitational physics probles. inally, it is iportant to reiterate that equation (1) standalone is not useful for solving for the etric field (the g s) because there is no way a priori to fix Λ. Ultiately Λ ust be found fro a solution to the full set of fundaental field equations (1) through (4), i.e., given a specific solution to the fundaental field equations both G and T can be calculated enabling the deterination of Λ = +. Λ fro G T Particle-like solution: Electric field, gravitational field, and quantization Investigated here is an exact solution of the new theory representing a charged, sphericallysyetric, particle-like soliton. This exaple is useful because an exact solution to the fundaental field equations (1) through (4) facilitates a clear coparison between the gravitational and electric fields predicted by the new theory and those predicted by the classical M&EEs. To proceed I draw on a solution for a spherically-syetric charged particle that was previously derived in reference [i]. iv Working in spherical coordinates (r, θ, φ, t), it was shown in reference [i] that the following expressions for the dynaic fields given in Table I are an exact solution to the field equations given in Table II

12 Preprints ( NOT PEER-REVIEWED Posted: 1 Septeber 17 doi:1.944/preprints v3 g 1 q 1 + r r = r r Sin[ θ ] q 1 + r r 3 q q / B 3 φ B E θ r r r B Br E φ θ = = s Bθ Br E φ E 3 r Eθ Eφ q q / + 3 r r 1 u = s,,, q 1 + r r q a = s,,, q ρc = ρ = q 1 + r r 4 r q ( 1 + ) 3 4 q r r 4 r () where s =± 1 as will be explained later. Solution () is straightforward to verify by direct substitution into the equations of Table II. v The physical interpretation of this solution is that of a particle having charge ± q and ass. Of note is the etric tensor which is identical to the Reissner-Nordstro etric, establishing that the new theory predicts gravitational fields that are in agreeent with the Einstein field equations. urtherore, the electric field is radial and agrees with the coulob field of the conventional Maxwell equations to leading order in 1/r. Regarding solution (), several points are worth ephasizing. irst, the fundaental equations in Table II which look very different than do the M&EEs give the sae solutions for the gravitational field, and the electric field to leading order in 1/r as do the M&EEs, at least for the case of the spherically-syetric charged particle investigated here. This lends credence to

13 Preprints ( NOT PEER-REVIEWED Posted: 1 Septeber 17 doi:1.944/preprints v3 the clai that the new theory s predictions are consistent with those of the M&EEs. Second, solution () does not satisfy Einstein s field equations, i.e., G satisfy G G, T and T. However, () does = T +Λ with Λ = as outlined earlier. or copleteness the values of Λ that go with solution () are given here G ( + ( + )) q q r r 6 r q 6 r = q csc[ θ ] 6 r q ( + ( + )) r q r r 3 ( q qr) ( q + r( + r) ) 8 r 3 ( q qr) 8 r T = q ( q r) csc[ θ ] 8 r 6 4 3q + q r + q r( 3+ r) 4 r ( q + r( + r) ) 6 4 ( q q r q r )( q + r( + r) ) 8 r 3 q ( q qr) r r Λ = 4 q ( q q r r ) csc[ θ ] 8 r 6 4 3q + q ( 3 r) r+ q r 4 r ( q + r( + r) ) (3) inally, the new theory s predictions go further than the M&EEs by giving the spatial distribution of the ass and charge density as part of their solution, i.e., the ass and charge density are dynaic fields in the new theory. As discussed below, having the ass density and charge density as dynaic fields, when cobined with boundary conditions that ipose selfconsistency on the field solutions leads to quantization conditions on the particle s ass and charge. As previously proposed in reference [i], a ethodology for quantizing the charge of particle-like solutions such as () proceeds by iposing a boundary condition requiring the asyptotic value of the electric field be consistent with the spatially integrated charge density

14 Preprints ( NOT PEER-REVIEWED Posted: 1 Septeber 17 doi:1.944/preprints v3 q= ρcu γ d x= li r (4) 4 3 sp 14 r where q is the total charge of the particle, 14 is the radial electric field coponent of the Maxwell tensor, and γ sp is the deterinant of the spatial etric defined by vi g g g i4 j4 γ sp i j = i j (5) g44 where i and j run over the spatial diensions 1, and 3. An analogous quantizing boundary condition for the ass of the particle is arrived at by requiring the asyptotic value of its gravitational field be consistent with the spatially integrated ass density of the solution g44 = ρ u γ spd x = li r (6) r where is the total ass of the particle. The reason for the absolute value of u 4 in the ass boundary condition (6) but not in the charge boundary condition (4) is the syetry (15) exhibited by the theory s fundaental field equations and the requireent that the boundary conditions exhibit the sae syetry. The boundary conditions (4) and (6) represent selfconsistency constraints on the charge paraeter q and the ass paraeter that appear in the etric (). The proposal here is that these boundary or self-consistency conditions represent additional constraints on physically allowable solutions beyond the fundaental equations presented in Table II. or the spherically-syetric solution investigated in (), the LHS of both (4) and (6) diverge leaving no hope for satisfying those quantization boundary conditions. The upshot of this observation is that while () represents a solution that describes the gravitational and electrical fields of a point charge that forally satisfy the equations of the theory in Table II, () cannot represent a physically allowed solution. The possibility of finding solutions that satisfy both the equations of the theory in Table II and the quantized charge and ass boundary conditions reains an open question at this point. However, interesting possibilities exist beyond the specific solution investigated here. or exaple, the odified Reissner-Nordstro and odified Kerr-Newan etrics developed by S.M. Blinder, vii give finite values for the LHS of both (4) and (6). inally, when considering etrics that include nonzero angular oentu, as for exaple would be

15 Preprints ( NOT PEER-REVIEWED Posted: 1 Septeber 17 doi:1.944/preprints v3 required for particles having an intrinsic agnetic field, the sae ethodology used here to quantize the particle s ass and charge can be used to quantize its angular oentu. The particle-like solution () illustrates one restriction that the charge-conjugation syetry (14) places on etrics that contain a charge paraeter q. By (14), the charge-conjugation transforation takes g g, and ρc ρc or equivalently q q by (4). This forces the conclusion that the sign of q has no ipact on the etric, i.e., the etric can only depend on the absolute value of q since the etric is unchanged by the transforation q q. This result is in line with the known charge containing solutions of the Einstein field equations such as the Reissner-Nordstro and Kerr-Newan etrics both of which depend on One of the unique features of the classical field theory being proposed here is that it allows for the inclusion of antiatter in a very natural way. The ultiplicative factor s in the expressions for, a and u in solution () is defined by q. + 1 s = 1 for atter for antiatter (7) and accounts for the atter-antiatter syetry expressed in (15). The physical interpretation is the s = 1 solution represents a particle having the sae ass but opposite charge and fourvelocity as the s =+ 1 solution. This is equivalent to the view today that a particle s antiparticle is the particle oving backwards through tie. viii Said another way, the tie-like coponent of the four-velocity is positive for atter and negative for antiatter 4 > u < for atter for antiatter. (8) With these definitions for the four-velocity of atter and antiatter, charged ass density can annihilate siilarly charged antiass density and satisfy both local conservation of charge (1) and local conservation of ass (1). Additionally, such annihilation reactions conserve total energy by (4). Because I a endeavoring to develop a theory that flows fro the four fundaental equations in Table II axioatically, an interesting observation is that there appears to be nothing at this point

16 Preprints ( NOT PEER-REVIEWED Posted: 1 Septeber 17 doi:1.944/preprints v3 in the developent that precludes the existence of negative ass density, ρ <, and negative ass paraeter, <. Indeed, the existence of negative ass in the context of classical General Relativity has been proposed, studied, ix, x and invoked when trying to find stable particle-like solutions using the conventional Einstein field equations. xi, xii, xiii However, in the context of the present theory, the existence of negative ass density leads to a logical contradiction that can only be resolved by requiring the ass density be non-negative, i.e., ρ always. I ll coe back to this point and develop this logical inconsistency when investigating the behavior of atter and antiatter in electric and gravitational fields. Electroagnetic and gravitational wave solutions Working in the weak field liit, derived here are expressions for a propagating electroagnetic plane wave in ters of the vector field a and the etric tensor g. xiv This exaple is useful as it akes clear the relationship between electroagnetic and gravitational radiation iposed by the fundaental equations in Table II, and predicts that an electroagnetic wave cannot exist without an underlying gravitational wave. To begin, consider an electroagnetic plane wave having frequency ω, propagating in the +z-direction and polarized in the x-direction. The Maxwell tensor for this field is given by By Ex = e By Ex iω ( t z) (9) where Ex and By are the constant field aplitudes. Assuing a near Minkowski weak field etric g = η + h e iω ( t z) h 1 (3) where the h are coplex constants, and a constant vector field a, (,,, ) a = a a a a, (31)

17 Preprints ( NOT PEER-REVIEWED Posted: 1 Septeber 17 doi:1.944/preprints v3 proceed by substituting for, g and a into (1) and only retain ters to first order in the fields h and, which are both assued to be sall and of the sae order. Doing this leads to a set of 8 independent linear equations for the 16 unknown constants: h, a, E x and B y. Iposing these constraining equations, then solve for the field coponents E, B, g and a in ters of 8 free constants x y ( h + h ) Ex = iω a h B y = E x 11, (3) g h11 h1 h14 h14 h1 h11 h4 h4 = η 1 + h h h h + h 1 h14 h4 ( h33 + h44 ) h44 ( ) e iω( t z), (33) and 1 1 h1 4 4 a = a, a, a, a h11. (34) This solution illustrates several ways in which the new theory departs fro the traditional view of electroagnetic radiation. In the approach being put forth here the undulations in the electroagnetic field are due to undulations in the etrical field (33) via the coupling defined in (1). This result also underlines that the existence of electroagnetic radiation is forbidden in strictly flat space-tie. An interesting aspect of this solution is that while electroagnetic radiation necessitates the presence of an underlying gravitational radiation field, the gravitational radiation is not copletely defined by the electroagnetic radiation. The supporting gravitational radiation has 6 undeterined constants (,,,,, ) h h h h h h, with the only restriction being they satisfy h 1 and h11 as required by (3). urther insight into the content of the etric (33) is evident after aking the infinitesial coordinate transforation fro x x ' given by

18 Preprints ( NOT PEER-REVIEWED Posted: 1 Septeber 17 doi:1.944/preprints v3 i x+ h14 ω x x' i y h4 y y' + ω = z z' i z h 33 t t' ω i t h44 ω (35) and only retaining ters to first order in the h s. Doing this, the etric (33) is transfored to g' h11 h1 h h 1 11 i ( t z) e ω = η +, (36) while ' E x and B ' y, the transfored electric and agnetic field aplitudes, respectively, are identical to Ex and By given in (3). Note that only the h11and h 1 coponents of the etric (36) have an absolute physical significance and h = h11, which akes the plane wave solution (36) identical to the plane wave solution of the classical Einstein field equations. xv, xvi Because the underlying gravitational wave couples to both charged and uncharged atter, one consequence of the solution here is that there will be an uncertainty when describing the interaction of electroagnetic radiation with atter if the gravitational wave coponent of the proble is ignored. However, for a nonrelativistic atter, this gravitational interaction (36) vanishes to first order in the h s. To see this, consider the following expansion of the Lorentz force law Du ρ = ρcu Dτ du ρ = ρuuγ + ρcu dτ u. (37) The first ter on the RHS in the line above represents the gravitational interaction. This gravitational interaction ter vanishes for nonrelativistic atter with u (,,,1) because for the etric (36) all the Γ 44 vanish to first order in the h s.

19 Preprints ( NOT PEER-REVIEWED Posted: 1 Septeber 17 doi:1.944/preprints v3 The forgoing analysis deonstrates the necessity of having an underlying gravitational wave to support the presence of an electroagnetic wave, but the converse is not true, and gravitational radiation can exist independent of any electroagnetic radiation. The following analysis deonstrates this by solving for the structure of gravitational radiation in the absence of electroagnetic radiation. ollowing the sae weak field foralis for the unknown fields h given in (3), but this tie zeroing out g and a E x and B y in (9), leads to the following solutions for g h11 h1 h14 h14 h1 h1 h4 h 4 h 11 = η + 1 e h14 h4 h33 ( h33 + h44 ) 1 h14 h4 ( h33 + h44 ) h44 iω( t z) (38) and 1 1 h a = a, a, a, a h1. (39) Both g given by (38) and a given by (39) are odified fro their solutions in the presence of an electroagnetic wave as given by (33) and (34), respectively. Perforing a transforation to the sae pried coordinate syste as given in (35), here gives the etric field g' h h1 = η + h 11 h h e iω( t z) (4) again, illustrating that only the h11and h1 coponents have an absolute physical significance. The interaction of nonrelativistic atter with the gravitational wave (4) vanishes for the sae reason that it vanished for the gravitational wave (36) that accopanies electroagnetic radiation. Of

20 Preprints ( NOT PEER-REVIEWED Posted: 1 Septeber 17 doi:1.944/preprints v3 particular note is the change in the value of the h coponent depending on whether the gravitational wave supports an electroagnetic wave as in (36) or is standalone as in (4). It sees rearkable that the fundaental equations (1) through (4) that lead to Maxwell s equations and electroagnetic radiation can also lead to gravitational waves, unifying both phenoena as undulations of the etric field. On the other hand, equation (1) with = is a syste of second order partial differential equations, ar =, in the etric field coponents κ g just as the Einstein field equations are, so the fact that both sets of field equations give siilar solutions for gravitational waves is not to be copletely unexpected. Cobining the solutions in this section for gravitational and electroagnetic radiation with those of the last section for the static electric and gravitational fields of a particle-like solution lends additional credence to the clai that the new theory s solutions are consistent with those of the M&EEs. Antiatter and its behavior in electroagnetic and gravitational fields The distinction between atter and antiatter is naturally accoodated in the new theory, with antiatter solutions generated fro their corresponding atter solutions using transforation (15). As already entioned, antiatter can be viewed as atter oving backwards through tie. To see this ore rigorously, consider the four-velocity associated with a fixed quantity of charge and ass density, Under the atter-antiatter transforation (15), u u dx =. (41) dτ u, or equivalently dτ dτ. This otivates the following expression for the four-velocity in ters of the coordinate tie dx dx v u = = sγ = sγ dτ dt 1 (4) where s is the atter-antiatter paraeter defined in (7), = ( vx, vy, vz) velocity of the charge or ass density, and γ = 1/ 1 v is the ordinary 3-space v. Equation (4) establishes that the

21 Preprints ( NOT PEER-REVIEWED Posted: 1 Septeber 17 doi:1.944/preprints v3 corresponding atter-antiatter solutions travel in opposite tie directions relative to each other. One of the unusual aspects of the atter-antiatter transforation (15) is that ρc does not change sign under the transforation. To see that this is consistent with the usual view in which antiparticles have the opposite charge of their corresponding particles, I ll use (4) to illustrate the behavior of a charged atter and antiatter density in an electroagnetic field. Consider a region with an externally defined electroagnetic field Bz By Ex Bz Bx E y = By Bx Ez Ex Ey Ez (43) and with no, or at least a very weak gravitational field so that g η and Γ. Starting with the Lorentz force law (13) and expanding Du ρp = ρcu Dτ du ρp sγ = ρc u dt Bz By Exsγ vx d sγ v Bz Bx E y sγ v y ρp sγ = ρ c dt sγ By Bx E z sγ vz Ex Ey Ez sγ d γ v E+ v B ρp = s ρc dt γ v E (44) which on the last line above ends up at the conventional for of the Lorentz force law except for the extra factor of s on the RHS. This factor of s in (44) gives the product sρ c the appearance that antiatter charge density has the opposite sign to that of atter charge density. The definition of q given in (4) is also equivalent to this point of view because aking the atter-antiatter transforation (15) changes the sign of u but not ρc in (4), thus changing the sign of q.

22 Preprints ( NOT PEER-REVIEWED Posted: 1 Septeber 17 doi:1.944/preprints v3 Next, I investigate antiatter in a gravitational field. There is no question about the gravitational fields generated by atter and antiatter, they are identical under the atter-antiatter syetry (15), as g is unchanged by that transforation. To understand whether antiatter is attracted or repelled by a gravitational field, I go again to the Lorentz force law (13), but this tie assue there is no electroagnetic field present, just a gravitational field given by a Schwarzschild etric generated by a central ass > corresponding to either atter or antiatter. Placing a test particle a distance r fro the center of the gravitational field and assuing it to be initially at rest, the equation of otion for the test particle, a geodesic trajectory, is given by the following developent du sγ dt Du = Dτ = Γ u u ρ ρ r 1 r r d d θ ρ sγ sγ = Γ ρu u Γ 44 s = s dt dt φ t (45) where in the last line above I have approxiated the RHS using the initial at rest value of u, u = (,,, s) and additionally used the fact that the only nonzero Γ in a Schwarzschild etric 44 1 is Γ 44 = 1. Siplifying the LHS of the last line in (45) by noting that initially 1 r r γ =, gives dr dt (46) r independent of s, and so deonstrating that the proposed theory predicts both atter and antiatter will be attracted by a gravitational field because they follow the sae geodesic trajectory, and this regardless of whether atter or antiatter generated the gravitational field.

23 Preprints ( NOT PEER-REVIEWED Posted: 1 Septeber 17 doi:1.944/preprints v3 As already noted, there appears to be nothing in the fundaental equations of Table II that preclude the possibility of negative ass density, ρ <. However, there are inconstancies that are introduced if negative ass density were to exist. As just shown, equation (46) with > predicts a test particle at soe distance fro the origin will feel an attractive gravitational force regardless of whether it is coprised of atter or antiatter. But this attraction is also independent of whether the test particle is coprised of positive or negative ass because the test particle s ass does not enter the calculation all test particles, regardless of their coposition, follow the sae geodesic trajectory. Now consider equation (46) with <. In this case a test particle at soe distance fro the origin will feel a repulsive gravitational force regardless of whether the test particle is atter or antiatter and regardless of whether the test particle has positive or negative ass. These two situations directly contradict each other, aking the new theory logically inconsistent if negative ass density exists. Thus, the only way to avoid this logical contradiction is to require ass density be non-negative always. The condition that ρ be non-negative always is also consistent with the syetry transforations (14) through (16) where it was noted that the field ρ does not change sign under any of the transforations. Discussion and suary Equations (1) and () are the ost iportant concepts being proposed in this anuscript, fundaentally tying the Maxwell tensor and charge density to the R-C curvature tensor and thereby unifying electroagnetis with gravitation. The cost of this unification is the introduction of a new field a, a field that has no counterpart in classical physics but in the new theory serves to couple electroagnetic and gravitational phenoena. The unification that ensues puts electroagnetic phenoena on par with gravitational phenoena, with both intrinsically tied to nonzero curvatures. On the surface, this central role for nonzero curvature in all electroagnetic phenoena ight be construed as probleatic due to the prevalent view today that electroagnetic phenoena can exist in flat space-tie. However, looking a little deeper it is not hard to see the connection between the new theory that requires a nonzero curvature for all electroagnetic phenoena and the classical Maxwell theory that works in flat space-tie. If one were not aware of the vector field a, but recognized the existence of the coulobic current

24 Preprints ( NOT PEER-REVIEWED Posted: 1 Septeber 17 doi:1.944/preprints v3 density as is the view today, then that part of the new theory not directly connected to a is exactly the classical Maxell theory. To see this, consider the equations in Tables II and III that result fro replacing a R with the classical coulobic current density, which is just a stateent of fundaental equation (). The before and after equations with this substitution are shown in Tables VI and VII. None of the equations updated with the substitution a R J reference a or have an explicit connection to the R-C curvature tensor, and are exactly the classical Maxwell field equations and their consequences. Table VI. undaental equations fro Table II with a R J Original equation Updated with a R J Coent a κ = Rκ ar ρ = J inhoogeneous Maxwell s equation ρ u = cu J = c Definition of classical current density uu = 1 uu = 1 No change uu 1 4 g ρσ ρ + ρσ = uu 1 4 g ρσ ρ + ρσ = No change Table VII. Dependent equations fro Table III with a R J Original equation Updated with a R J Coent + + = + + = No change κ κ κ κ κ κ ( u c ) ( a R ) ρ = = J = Conservation of charge

25 Preprints ( NOT PEER-REVIEWED Posted: 1 Septeber 17 doi:1.944/preprints v3 ± σ ar ar ρc = or ar u σ ρ = c J u Definition of classical current density ( ρ ) u = ( ) u Du ρ D ρcu τ = ρ ρc τ ρ = No change Du D = u No change In considering the new theory, one ight view classical physics at the level of the Maxwell field equations as they are accepted today as incoplete, there being a hidden field a that has gone unrecognized. The value of the new theory with its introduction of the vector field a is the reductionis it brings to classical physics at the level of the M&EEs. In addition to the electroagnetic coverage of the new theory, it also contains solutions replicating those of the Einstein field equations. In fact, the Reissner-Nordstro etric (and the Schwarzschild etric as a liiting case) are exact solutions of the fundaental equations (1) through (4) deonstrating that the new theory replicates gravitational physics at the level of the Einstein field equations, at least in this spherically-syetric case. Another reason that the particle-like solution () is so iportant is that it establishes that exact solutions to the theory do exist. This is not at all evident fro equation (1), which represents a ixed syste of first order partial differential equations for and so carries with it specific integrability conditions that ust be satisfied for solutions to exist. xvii, xviii The existence of the particle-like solution () allays this concern by direct deonstration. Equation (1), Einstein s field equations with the addition of a function of integration Λ is a consequence of the fundaental field equation (4) and the properties of the R-C tensor. Because the covariant divergence of Λ is constrained to vanish, Λ has built into it all the properties of dark atter and dark energy. Specifically, particular cases of Λ can represent dark atter halos useful for explaining the flattened rotational-velocity curves observed on galactic scales, or the cosological constant ter g useful for describing dark energy and explaining the accelerating expansion of the universe. The results of applying (1) are all very uch in line with current

26 Preprints ( NOT PEER-REVIEWED Posted: 1 Septeber 17 doi:1.944/preprints v3 investigations of gravitational interactions that use Einstein s field equations with the ad hoc addition of ters to their energy-oentu tensor to account for the effects of dark atter and dark energy. However, in the view of the new theory equation (1) is not required when solving for the fundaental fields listed in Table I. Rather (1) is consequence of the new theory, so any solution of the fundaental field equations (1) through (4) is also a solution of (1) for soe choice of Λ. The weak field solution for electroagnetic radiation requires that it be supported by an underlying gravitational radiation. Because of this a test particle in the path of an electroagnetic wave would in addition to feeling the effects of an undulating electroagnetic field, also feel the effects of the underlying gravitational wave. This prediction of the new theory, which does not agree with the classical physics predictions of the M&EEs suggests one direction for investigations that could yield epirical results either supporting or refuting the predictions of the new theory. or exaple, if the new theory is the ore correct description of nature, then taking only electroagnetic effects into account for relativistic particles interacting with electroagnetic radiation would introduce an error in the calculated trajectory of particles due to the neglect of the interaction with the underlying gravitational wave. A detraction of the new theory, at least at its present level of developent, is the soewhat ad hoc inclusion of the conserved energy-oentu tensor (4) as one of its four fundaental equations this rather than a ore elegant and integrated variational derivation of the field equations fro an appropriate action. The strongest arguent for the inclusion of the conserved energy-oentu tensor (4) is the logical copleteness it brings to the theory, i.e., the nuber of independent field equations is four less than the nuber of independent field variables leaving four degrees of freedo in the deterination of the independent field variables a situation that is consistent with the requireents of general covariance and the freedo in choice of coordinate syste. Internal consistency with the requireents of general covariance is a very strong arguent for the theory when copared to the situation with the classical M&EEs, where, for exaple, attepts to find particle-like solutions have noted that there is generally one ore unknown than there are equations for deterining the unknowns. xi An attractive feature of the new theory is the way in which antiatter is naturally accoodated. Using the atter-antiatter transforation (15), a solution corresponding to antiatter can be

27 Preprints ( NOT PEER-REVIEWED Posted: 1 Septeber 17 doi:1.944/preprints v3 generated fro its syetric atter solution by changing the sign of a, u and, and leaving ρ, ρ and g unchanged. This leads to the sae picture as is accepted today in which c antiatter density is viewed as atter density oving backwards through tie. The change in sign of u between atter and antiatter ensures that atter density and siilarly charged antiatter density can annihilate each other and conserve: charge (1), ass (1), and overall energy (4). By direct calculation, the behavior of atter and antiatter in a gravitational field is deonstrated to be attractive always. Additionally, the ass density field ρ ust be nonnegative always to eliinate the possibility of a logical contradiction within the theory. The ability of the fundaental field equations (1) through (4) to incorporate a logical description of antiatter is both surprising and unique when one considers that the new theory is a classical field theory and not a quantu field theory. This along with the new theory s introduction of a vector field a which has no counterpart in the accepted description of classical physics today, and in fact can be considered a hidden variable in that description raises interesting questions regarding how the new theory could potentially be bridged to the quantu echanical world. As proposed here, the new theory is a theory of everything at the level of classical physics. This clai rests on the fact that both charge density and ass density are treated as dynaic fields in the theory, leaving no external paraeters to be introduced. This of course highlights one shortcoing of the particle-like solution (). As already noted that solution cannot satisfy the charge and ass boundary conditions (4) and (6), respectively, because the spatial integrals in both of those equations diverge due to singularities at the origin. This is a technical proble due to the etric solution in (), the Reissner-Nordstro etric with its singularity at the origin. One way to get around this difficulty ight be to investigate other choices of etric such as, for exaple, the Blinder-Reissner-Nordstro etric [vii] which is well behaved at the origin. Still other possibilities include relaxing the spherical syetry of the solutions investigated within to that of cylindrical syetry, thus allowing for angular oentu about an axis and solutions capable of odeling particles having a agnetic field but this goes well beyond the level of analysis presented within.

28 Preprints ( NOT PEER-REVIEWED Posted: 1 Septeber 17 doi:1.944/preprints v3 Conclusion The proposed classical field theory of electroagnetis and gravitation developed here encopasses classical physics at the level of the M&EEs using the four fundaental field equations as detailed in Table II, but then goes further by unifying electroagnetic and gravitational phenoena in a fundaentally new and atheatically coplete way. Maxwell s field equations, and the Einstein field equations with the addition of a ter that can iic dark atter and dark energy are consequences of the four fundaental field equations and the properties of the R-C tensor. The coupling between electroagnetic and gravitational physics is accoplished through the introduction of a vector field a that has no counterpart in the presently accepted description of nature based on the classical M&EEs. In deriving Maxwell s field equations fro the new theory it is clear that a can be viewed as a hidden variable in the classical Maxwell equation description of electroagnetis. This observation explains the apparent contradiction between the new theory s requireent that all electroagnetic phenoena require a nonzero curvature, and the classical Maxwell equation description in which electroagnetic phenoenon can occur is flat space tie. Considering gravitational physics in the new theory, both dark atter and dark energy effects are subsued by the Λ ter in (1), a ter which is ultiately deterined by a solution of the full set of field equations (1) through (4). The unification brought to electroagnetic and gravitational phenoena by the new theory is deonstrated through several specific exaples including the electric and gravitational fields of a spherically-syetric particle-like solution, and radiative solutions representing both electroagnetic and gravitational waves. One of the strengths of the new theory s field equations, in fact a guiding principle in their developent, is the requireent that the full set of fundaental field equations be logically consistent with the requireents of general covariance. Another strength of the new theory is the reductionis brought to electroagnetic and gravitational phenoena by treating the sources of these fields as dynaic variables rather than external paraeters that need to be added into the theory a developent which potentially explains the quantization of the ass, charge and angular oentu of particles in the context of a classical field theory. inally, an outline for nuerically solving the fundaental field equations is given in the for of an analysis of the Cauchy initial value proble applied to the fundaental field equations.