The Geometrization of Maxwell’s Equations and the Emergence of Gravity

Assuming the geometry of nature is Riemannian with four dimensions, the classical Maxwell equations are shown to be a derivable consequence of a single equation that couples the Maxwell tensor to the Riemann-Christoffel curvature tensor. This geometrization of the Maxwell tensor extends the interpretation of the classical Maxwell equations, for example, giving physical quantities such as charge density a geometric definition. Including a conserved energy-momentum tensor, the entirety of classical electromagnetism is shown to be a derivable consequence of the theory. The coupling of the Riemann-Christoffel curvature tensor to the Maxwell tensor also leads naturally to the emergence of gravity which is consistent with Einstein’s equation of General Relativity augmented by a term that can mimic the properties of dark matter and/or dark energy in the context of General Relativity. In summary, the proposed geometrization of the Maxwell tensor puts both electromagnetic and gravitational phenomena on an equal footing, with both being tied to the curvature of space-time. Using specific solutions to the proposed theory, the unification brought to electromagnetic and gravitational phenomena as well as the relationship of those solutions with the corresponding solutions of the classical Maxwell and Einstein field equations are examined.


INTRODUCTION
Electromagnetic and gravitational fields have long range interactions characterized by speed of light propagation; similarities that suggest these fields should be coupled together at the classical physics level.
Although this coupling or unification is a well-worn problem with many potential solutions having been proposed, it is fair to say that there is still no generally accepted classical field theory that can explain both electromagnetism and gravitation in a coupled or unified framework. [1] Today, the existence of electromagnetic and gravitational fields are generally understood to be distinct and independent with electromagnetic fields described by Maxwell's equations and gravitational fields described by Einstein's equation of General Relativity. The purpose of this manuscript is to assess a recently proposed field equation that geometricizes the Maxwell tensor and leads to a geometricized version of Maxwell's equations from which gravity then emerges.
Assuming the geometry of nature is Riemannian with four dimensions, the classical Maxwell equations will be shown to be a derivable consequence of, [2] ; F a R where F  is the Maxwell tensor, R  is the Riemann-Christoffel (R-C) curvature tensor, and a  is a four-vector related to the familiar vector potential A  of classical electromagnetism. Including the conserved energy-momentum tensor for matter and electromagnetic fields, where u  is the four-velocity, m  is the scalar mass density, and g  is the metric tensor, all the equations of classical electromagnetism will be shown to be a consequence of the equations (1) and (2). Notably, only equation (1) which couples the derivatives of the Maxwell tensor to the R-C tensor through the vector field a  is new; equation (2), the conserved energy-momentum for matter and electromagnetic fields is already a well-established foundational equation of classical physics.
Beyond the succinct framework for the classical Maxwell equations provided by equation (1), its coupling of the R-C tensor to the Maxwell tensor introduces gravitational effects into any solution of (1) and (2).
While the emerging gravitational fields due to this coupling are not identical to those predicted by Einstein's equation of General Relativity, they are, as will be shown, consistent with Einstein's equation of General Relativity augmented by a term that can mimic the properties of dark matter and/or dark energy. LLNL-JRNL-726138-Draft The goal of this manuscript is to show through an axiomatic development that the continuous field theory based on equations (1) and (2) covers classical electromagnetism and the emergence of gravitational phenomena in a unified manner with both tied to the curvature of space-time. Throughout the manuscript, geometric units will be used with a metric tensor having signature [+, +, +, -] in which spatial indices run from 1 to 3 and 4 is the time index. The notation within uses commas before tensor indices to indicate ordinary derivatives and semicolons before tensor indices to indicate covariant derivatives. For the definitions of the R-C curvature tensor and the Ricci tensor, the conventions used by Weinberg [3] are followed.

CONSEQUENCES OF THE FIELD EQUATIONS (1) AND (2)
Here I give a short derivation of the classical equations of electromagnetism in the framework of the proposed theory. The point in going through this purely formal development is to show that the classical equations of electromagnetism are derivative only to equations (1) and (2) and the algebraic properties of the R-C tensor. After developing the classical equations of electromagnetism from equations (1) and (2), I go on to describe the emergence of gravity that is forced by them.

Maxwell's homogeneous equation and gauge invariance
To begin, I demonstrate that equation (1) forces both the antisymmetry of F  and the vanishing of its antisymmetrized derivative, i.e., [ , ] 0 F   = . The antisymmetry of F  follows from equation (1) and the algebraic property of the R-C tensor, Contracting (3) with a  and using equation (1) gives, Contracting (5) (6) is justified by the antisymmetry of F  .
Having established the antisymmetry of F  and the vanishing of its anti-symmetrized derivative, which is just a statement of Maxwell's homogeneous equation (6), the converse of Poincaré's lemma establishes that F  can itself be expressed as the anti-symmetrized derivative of a vector function, i.e., ,, where A  is the classical electromagnetic vector potential. Because F  can be expressed as the antisymmetrized derivative of the vector potential A  , its value will be unaffected by a gauge transformation in which a gradient field is added to

Maxwell's inhomogeneous equation and the definitions of charge density and four-velocity
Next, Maxwell's inhomogeneous equation and the definitions of charge density c  and four-velocity u  are derived using equation (1). Contracting the  and  indices in equation (1) gives, where R   is the Ricci tensor. To establish the connection between equation (9) and Maxwell's inhomogeneous equation, I use the following identity, which is valid for any non-null four-vector W . With the aid of (10), any W satisfying 0 WW can be recast as a current density, i.e., the product of a scalar density and a four-velocity u , where the scalar density is defined as, WW , (12) and the four-velocity as, Equation (13)  Using the definitions given in (12) and (13), and identifying W with aR   in equation (9) gives, c a R u (15) where the charge density c is defined by, and the four-velocity u by, Using these definitions for charge density and four-velocity, equations (9) and (15) (16) and (17), respectively. This geometrization of c and u hints at the emergence of gravity in the proposed theory and is reminiscent of classical General Relativity's geometric interpretation of the mass density m in terms of the curvature scalar R .

The conservation of charge
Next, I derive the conservation of charge. Returning to the antisymmetry of F  which was established in equations (3) and (4), it follows that, which is an identity for all antisymmetric tensors. Comparing (19) to Maxwell's inhomogeneous equation thus, establishing the conservation of charge, The Lorentz force law and the conservation of mass With some substitutions and rearrangements using Maxwell's homogeneous equation (6) and inhomogeneous equation (18), equation (22) can be re written as, Contracting (23) with u  , the 2 nd and 3 rd terms on the LHS are zeroed due to the normalization of u  (14) and the antisymmetry of F  (4), respectively, leaving, thus, establishing the conservation of mass equation. Using (24) to zero out the conservation of mass term in (23) then leaves the Lorentz force law, Now compare (26) to equation (1) rewritten as, where the RHS of (27) follows from the commutation property of covariant derivatives. Equating the RHS's of equations (26) and (27) establishing a connection between a  and A  in the theory of electromagnetism being proposed here.
In summary, the theory of electromagnetism based on equations (1) and (2)  (2) will be developed further using specific solutions to show that electromagnetic and gravitational phenomena are effectively described in a unified manner and put on an equal footing, with both being tied to nonzero space-time curvatures.

A theory of gravitation
The preceding discussion established that the equations of classical electromagnetism follow directly from equations (1) and (2). With the R-C tensor coupled to the Maxwell tensor as it is in equation (1), some form of gravitation can be expected to emerge. The question that naturally arises is this: Will this emergent gravity be equivalent to Einstein's General Relativity, where 1 2 is the Einstein tensor?
As will be shown using the specific example of a spherically symmetric, non-rotating, charged particle, the Reissner-Nordström metric is an exact solution of equations (1) and (2), thus establishing that the emergent gravity in the proposed theory and classical General Relativity (29) support the same gravitational metric field solutions, at least in the case of spherical symmetry. However, one must go further to determine if Einstein's field equations are a derivable consequence of (1) and (2). Here I investigate this issue starting with the conserved energy-momentum tensor given in equation (2) With this definition,   is constrained to be both symmetric, and conserved, The value of the constant  in (30) is completely arbitrary and without physical significance because   as defined can absorb any change in  such that (30) remains satisfied. Taking advantage of this arbitrariness and setting the value of the constant 8  =− then gives with a slight rearrangement of (30), which is recognized as Einstein's equation of General Relativity (29) augmented on its RHS by the term   . From the perspective of classical General Relativity,   mimics the properties of the energymomentum tensor for dark matter and/or dark energy, viz., it is a conserved and symmetric tensor field, it is a source of gravitational fields in addition to energy-momentum tensor T  for normal matter and normal energy, and it has no interaction signature beyond the gravitational fields it sources.
It is important to recognize that (33) is a trivial result with no physical significance in the proposed theory based on equations (1) and (2). This follows because any solution of the equations (1) and (2) must necessarily be a solution of (33) for some choice   . In fact, the validity of (33) rests only on the existence of a conserved energy-momentum tensor and the properties of the R-C tensor, and so will be true in any physical theory that has a conserved energy-momentum tensor. However, the interesting point in the context of the proposed theory is that the value of   can be calculated from solutions to (1) and (2) without postulating the existence of dark matter and/or energy. This feature will be investigated further in subsequent sections in which specific solutions to equations (1) and (2) will be developed.
In the view being put forth here, gravitation emerges as a manifestation of the geometricized theory of electromagnetism based on equations (1) and (2), i.e., a theory of gravitation is self-contained within equations (1) and (2). Specifically, it is the coupling of the derivatives of the Maxwell tensor to the R-C tensor in (1) that brings gravitation into the picture. Importantly, the gravitational theory that emerges does not obey the classical General Relativity field equations (29), although any solution of equations (1) and (2) must necessarily be a solution of equation (33) for some choice of   . While viewing gravitation as a manifestation of electromagnetism and vice versa is not new [4,5,6,7,8] , the specific approach being followed here with equation (1) is new.

Symmetries of equations (1) and (2)
Listed in Table I are the continuous field variables that the theory based on equations (1) and (2) solves for.
Also included in Table I are the charge density c and the four-velocity field u that are defined for any solution in terms of g and a  as given by (16) and (17), respectively.
the second corresponds to a matter-antimatter transformation as will be discussed in section 5.4, and the third to the product of the first two, All three transformations leave equations (1) and (2) unchanged. Adding the identity transformation to these symmetries forms the Klein-4 group, with the product of any two of the symmetries (34) through (36) giving the remaining symmetry.
Note that among the fields of the theory, only g  and m  are unchanged by all the symmetry transformations, a fact that will be useful in section 5.6 for defining boundary conditions that lead to quantized mass, charge, and angular momentum of particle-like solutions as well as for the treatment of antimatter. Finally, in addition to the proposed theory's general covariance and global symmetries (34) through (36), it also exhibits the electromagnetic gauge covariance of classical electromagnetism as detailed in equations (7) and (8).

Do solutions exist to equation (1)?
Equation (1) represents a mixed system of first order partial differential equations for F  and illustrates one of the mathematical complexities of equations (1) and (2) that must be dealt with when attempting to find solutions. [9] Specifically, mixed systems of first order partial differential equations must satisfy integrability conditions if solutions are to exist. [10] Although there are several ways of stating what these integrability conditions are, perhaps the simplest is given by, which can be derived using the commutation relations for covariant derivatives. Using (1) to substitute for which can be interpreted as conditions that are automatically satisfied by any solution consisting of expressions for g  , a  and F  that satisfy (1). With (38) as integrability conditions that must be satisfied by any solution of (1), the question that naturally arises is this: Are these integrability conditions so restrictive that perhaps no solution to the proposed theory exists? Although this view could be construed as making the proposed field theory uninteresting because perhaps no solutions exist, it will be shown that solutions that are consistent with known solutions of the classical Maxwell and Einstein Field Equations (M&EFEs) can be found. Additionally, equation (38), which is linear in F  is often useful in developing solutions to equations (1) and (2), an approach that will be used in the solution to be found in section 4.1.
Finally, to further elucidate questions regarding solutions of the proposed theory, an outline showing how the field equations can be solved numerically is given in the appendix where an analysis is presented of equations (1) and (2) in terms of a Cauchy initial value problem.

SOLUTIONS TO EQUATIONS (1) AND (2)
In this section three solutions to equations (1) and (2) are presented. The first solution is spherically symmetric, representing the electric and gravitational fields of a non-rotating, charged particle. The second solution is radiative with two distinct sub solutions, one with electromagnetic radiation in the presence of gravitational radiation and the other with standalone gravitational radiation. The third solution has a maximally symmetric 3-dimensional subspace, for example, representing an isotropic and homogenous universe. The purpose in developing these solutions is twofold: First, to provide a comparison of solutions to equations (1) and (2) with those corresponding to the classical M&EFEs, and second to demonstrate that the solutions to equations (1) and (2) go further than the classical M&EFEs by uniting electromagnetic and gravitational phenomena.

Spherically symmetric solution
Here a solution representing a non-rotating, spherically symmetric charged mass is investigated. It is demonstrated that the Reissner-Nordström metric with an appropriate choice for the fields , , , c F a u    and m  satisfies equations (1) and (2). Although the presentation in this section is purely formal, it is included here for several reasons. First, if the theory could not describe the asymptotic electric and gravitational fields of a charged particle it would be of no interest on physical grounds. Second, as already discussed, equation (1) requires the solution of a mixed system of first order partial differential equations, a system that may be so restrictive that no solutions exist, and so an outline of at least one methodology to a solution is warranted.
To proceed, I draw on a solution for a spherically symmetric charged particle that was previously derived. [11] Starting with the Reissner-Nordström metric [12] , I investigate a trial solution for a  , The next step is to satisfy (1) by solving for F  . Rather than tackle this head on by directly trying to find a solution to the mixed system of first order partial differential equations that is (1) By direct substitution it is easily verified that F  as given in (44) is a solution of (1). [13] Choosing the value of the undetermined constant 1 / c q m = then gives an electric field which agrees with the Coulomb field of a point charge to leading order in 1/r . Finally, the remaining unknown field, the scalar mass density m  is found using the conserved energy-momentum tensor (2). Substituting the known fields into (2) and then solving for m  gives, 4 2 2

26
( 2 ) m q q mr r mr (45) To summarize, the following expressions for , , , , which agrees with the Coulomb field 2 / qr to leading order in 1/r does have a higher order term. This next term in the radial electric field depends on both the charge and mass of the particle. Taking an electron as an example, its electric field as given by (47) would be, 15 22 2.82 10 11 In the context of classical General Relativity (29)   This solution illustrates several ways in which the new theory departs from the classical physics view of electromagnetic radiation. Of most significance, the undulations in the electromagnetic field are due to undulations in the underlying metric field g  given in (55). This result also underscores that the existence of electromagnetic radiation is forbidden in strictly flat space-time. An interesting aspect of this solution is that while electromagnetic radiation necessitates the presence of an underlying gravitational radiation field, the underlying gravitational radiation is not completely defined by the electromagnetic radiation. The supporting gravitational radiation has 6 undetermined constants ( ) 11 and only retaining terms to first order in the h's. Doing this, the metric (55) is transformed to,  hh =− which makes the gravitational plane wave solution (58) identical to the gravitational plane wave solution of the classical Einstein field equations. [15], [16] Gravitational radiation The forgoing analysis demonstrated the necessity of having an underlying gravitational wave to support the presence of an electromagnetic wave, but the converse is not true and gravitational radiation can exist independent of electromagnetic radiation. The following analysis demonstrates this by solving for the structure of gravitational radiation in the absence of electromagnetic radiation. Following the same weak field formalism for the unknown fields h  given in (52) where k equals +1, 0 or -1 depending on whether the spatial curvature is positive, zero or negative, Rt. To derive the time dependence of () s Rt, I note the 3-dimensional spatial subspace of (62) is maximally symmetric and so any tensor fields that inhabit that subspace must also be maximally symmetric. [17] Specifically, this restricts the form of a  to be, and forces the antisymmetric Maxwell tensor to vanish, 0 F  = .
This in turn forces, LLNL-JRNL-726138-Draft which gives 0 c  = by equation (16). Substituting a  given by (63), and the FLRW metric given by (62) into (66) then leads to the following set of equations to be satisfied, or,

The classical Maxwell's field equations from F ;κ = a λ Rλκμν
As developed within, equation (1) leads directly to the classical Maxwell equations. In fact, equation (1) was empirically chosen to reproduce the classical Maxwell equations under the assumption that the geometry of nature is Riemannian with four dimensions. Going further, equation (1) extends the interpretation of the classical Maxwell equations in that both the charge density c  and the four-velocity u  are defined in terms of the Ricci tensor and the vector field a  as given by (16) and (17), respectively.
In this sense, the fields c  and u  are not fundamental, but rather are determined from the other fields listed in Table I. Finally, the vector field a  , which is not familiar to classical physics and here serves to couple the Maxwell tensor to the R-C tensor is directly relatable to the familiar vector potential A  of classical electromagnetism through equation (28).
It is an unusual circumstance that equation (1)

Dark matter and dark energy
With General Relativity as the foundation of observational gravitational physics today, dark matter and dark energy have been postulated to exist because of the many galactic and cosmological scale observations that cannot be understood using General Relativity with normal matter and normal energy alone. For example, some of the large-scale gravitational features of galaxies and galactic clusters dating back to LLNL-JRNL-726138-Draft Zwicky's observations in the 1930's have been explained using dark matter [18] , and the acceleration of the universe discovered in the 1990's explained using dark energy [19] . Another example of modifications made to the original General Relativity field equations (29) to satisfy a perceived need was the cosmological constant term g   that Einstein added to their RHS, This was done to enable a static universe solution, but then subsequently dropped after expansion was discovered. Today this term is again in vogue as a possible descriptor of dark energy.
One of the vexing problems facing dark matter and dark energy-based explanations of various observational phenomena today is an ongoing inability to directly detect them. However, equations (1) and (2) offer the prospect that dark matter and dark energy effects can be explained in terms of normal matter and normal energy alone, i.e., the   term in (33) which represents the energy-momentum tensor of dark matter and/or dark energy in the context of General Relativity is provided with a mechanism for directly calculating its structure using equations (1) and (2) with only normal matter and normal energy. The already investigated spherically symmetric particle-like solution (46) is one example that outlines such a direct calculation of   . With questions today regarding the validity of classical General Relativity beyond the confines of our own solar system [20] and the inability to directly detect dark matter and dark energy, the possible interpretation of the   term in (33) using only normal matter and normal energy is an enticing feature of equations (1) and (2) . However, it must be acknowledged that one of the challenging tasks facing the theory based on equations (1) and (2), and one well beyond the analysis presented in this manuscript, is that of finding additional solutions that could be interpreted as agreeing with the rapidly developing observational understanding of galactic and cosmological structures.

The unification of gravitational and electromagnetic radiation
One of the successes of equation (1) is the existence of solutions describing both electromagnetic and gravitational radiation, with both phenomena being unified as undulations of the underlying metric field g  . Because both gravitational and electromagnetic radiation are due to undulations of the metric field g  , their speed of propagation is predicted to be identical. This prediction has recently been refined experimentally with observations made during the binary neutron star merger in NGC 4993, 130 million light years from Earth. [21] The nearly simultaneous detection, within 2 seconds of each other, of gravity waves [22] and a burst of gamma rays [23] from this event experimentally constrain the propagation speed of electromagnetic and gravitational radiation to be the same to better than 1 part in 10 15 . LLNL-JRNL-726138-Draft

The emergence of antimatter and its behavior in electromagnetic and gravitational fields
One of the unique features of equations (1) and (2) is that the properties of antimatter emerge naturally in their solutions. Traditionally, these properties emerge in quantum mechanical treatments but here emerge in the context of a classical continuous field theory due to the global symmetry (35) of equations (1) and (2); every matter containing solution has a corresponding antimatter solution generated by the symmetry transformation (35). This is evident in the spherically symmetric, particle-like solution (46) where the multiplicative factor s in the expressions for F  , a  and u  is defined by, 1 for matter 1 for antimatter and accounts for the matter-antimatter symmetry expressed in (35). The physical interpretation of the 1 s =− solution is that it represents a particle having the same mass but opposite charge and four-velocity to that of the 1 s =+ solution. This is equivalent to the view that a particle's antiparticle is the particle moving backwards through time. [24] Said another way, the time-like component of the four-velocity is positive for matter and negative for antimatter, 4 0 for matter 0 for antimatter With these definitions for the four-velocity of matter and antimatter, charged mass density can annihilate similarly charged anti-mass density and satisfy both the local conservation of charge (20) and local conservation of mass (24). Additionally, such annihilation reactions must conserve energy by (2).
Building on the distinction between matter and antimatter, their behavior in external electromagnetic and gravitational fields in the context of equations (1) and (2) is briefly reviewed here. As already mentioned, antimatter can be viewed as matter moving backwards through time. To see this more rigorously consider the four-velocity associated with a fixed quantity of charge and mass density, ( Under the matter-antimatter transformation (35), uu  →− , or equivalently dd  →− . This motivates the following expression for the four-velocity in terms of the coordinate time in locally inertial coordinate systems, where s is the matter-antimatter parameter defined in (73), v is the ordinary 3-space velocity of the charge and mass density, and   Next, I investigate the behavior of antimatter in an external gravitational field. There is no question about the gravitational fields generated by matter and antimatter, they are identical under the matter-antimatter symmetry (35), as g  is unchanged by that transformation. To understand whether antimatter is attracted or repelled by an external gravitational field, I again go to the Lorentz force law (25) but this time assume there is no electromagnetic field present, just a gravitational field given by a Schwarzschild metric generated by a central mass 0 m  that is composed of either matter or antimatter. I explicitly call out 0 m  because I am endeavoring to develop a physical theory that flows axiomatically from (1) and (2), and at this point in the development there is nothing to preclude the existence of negative mass density 0 m   , a consideration I will return to in section 5.5. Placing a test particle having mass test m composed of either matter or antimatter a distance r from the center of the gravitational field and assuming the test particle is initially at rest, the trajectory of the test particle is that of a geodesic given by the following development, where 1 s = references whether the test particle is composed of matter or antimatter as defined by (73).
In the last line of (79), I have approximated the RHS using the initial at rest value of the test particle's four- which is independent of s , and so demonstrates that the proposed theory predicts both matter and antimatter test particles will be attracted by the source of the gravitational field, and this regardless of whether the source of the gravitational field is matter or antimatter. The result that the test particle is attracted toward the source of the gravitational field is also independent of whether the test particle's mass, test m , is positive or negative, this because the geodesic trajectory (80) is independent of test m .

Possibility of negative mass solutions and antigravity
As already noted, there appears to be nothing in equations (1) and (2)  As just shown, equation (80) with 0 m  predicts a test particle at some distance from the origin will feel an attractive gravitational force regardless of whether the test particle is comprised of matter or antimatter and regardless of whether its mass is positive or negative. Now consider equation (80) with the central mass 0 m  . Using the same argument as in the previous section, the test particle in this case will feel a repulsive gravitational force regardless of whether it is comprised of matter or antimatter and regardless of whether its mass is positive or negative. These two situations directly contradict each other. For example, in the first case the negative mass test particle is gravitationally attracted toward the positive mass particle located at the origin, but in the second case the positive mass test particle is gravitationally repelled by the negative mass particle located at the origin. This contradiction makes equations (1) and (2) logically inconsistent if negative mass density were to exist. The only way to avoid this logical contradiction is to require that mass density be non-negative always. This condition that mass density It is interesting to note that the existence of negative mass in the context of classical General Relativity has been extensively studied [25], [26] and invoked, particularly when trying to find stable particle-like solutions using the conventional Einstein field equations. [27], [28], [29] However, in the context of the present theory the existence of negative mass density leads to a logical contradiction that can only be resolved by requiring mass density be non-negative always, i.e., 0 m   .

Conjecture for quantizing the charge and mass of particle-like solutions
Consider particle-like solutions such as (46). Because the mass density and charge density are specified as part of the solution of equations (1) and (2), a self-consistency condition exists for physically allowed solutions that provides a mechanism for quantizing the charge and mass of such solutions. For example, for solution (46) to be self-consistent, the particle's total charge q and total mass m , both parameters of the Reissner-Nordström metric, must agree with the spatially integrated charge and mass density, respectively. For the charge, this amounts to requiring the asymptotic value of the electric field be consistent with the spatially integrated charge density, where q is the total charge of the particle and given by the asymptotic value of 2 14 rF per the solution given in (46), and sp  is the determinant of the spatial metric defined by, [30] 44 44 where i and j run over the spatial dimensions 1, 2 and 3. An analogous quantizing boundary condition for the mass of the particle is arrived at by requiring the asymptotic value of its gravitational field be consistent with the spatially integrated mass density of the solution,  [31] give finite values for the RHS of both (81) and (83). Finally, when considering metrics that include nonzero angular momentum, as for example would be required for particles having an intrinsic magnetic field, the same approach used here to quantize the particle's mass and charge could be used to quantize its angular momentum. Traditionally the quantization of mass, charge and angular momentum are introduced in quantum mechanical treatments but here are conjectured within the framework of a classical continuous field-theoretic description of nature and are another example of how the proposed theory differs from the classical M&EFEs.

Possibility of superluminal transport if a λ Rλ  is space-like
Having chosen the form of equations (1) and (2), all subsequent results presented in this manuscript have been mathematically derivative to them. As an example, after the definitions of the charge density c  and the four-velocity u  were developed in equations (16) and (17), respectively, Maxwell's inhomogeneous equation (18) was shown to follow from equation (1). Noteworthy in the definition for c  is that in addition to its motion being described in terms of subluminal transport, the development naturally includes the case of superluminal transport. Because I am attempting to develop the theory that flows axiomatically from equations (1) and (2), and because there is nothing a priori that precludes the possibility of superluminal transport, I have carried it as a possibility, although one that must be regarded as speculative at this point because the specific solutions investigated within have not exhibited it. Although not pursued here further, the possibility of superluminal transport in the context of a classical field theory may be an interesting and timely avenue of investigation as recent research has suggested the possible existence of nonlocal correlations stronger than those predicted by quantum theory. [32] 6. CONCLUSION The choice of the equations (1) and (2) was empirically driven by the desire to preserve as much as possible the physics embodied in the classical theory of electromagnetism, while providing that theory with a geometric foundation under the assumption that nature is Riemannian with four dimensions. Using the fourvector field a  that is related to the familiar vector potential A  of classical electromagnetism, equation LLNL-JRNL-726138-Draft (1) which couples the Maxwell tensor to the Riemann-Christoffel curvature tensor was shown to reproduce the classical Maxwell equations in their entirety. Next, the interpretation of the Maxwell equations based on equation (1) was shown to go further than the classical interpretation of them in that the charge density c  and the four-velocity u  were given a geometric underpinning with both dependent on the Rici Tensor.
It is this geometric underpinning that ties electromagnetism to gravitation. Although the gravity emerging form (1) and (2) is different than that described by General Relativity, it is consistent with Einstein's field equations of General Relativity augmented by a symmetric and conserved tensor field, i.e., a field exhibiting the properties of an energy-momentum tensor for dark matter and/or dark energy. However, in the context of equations (1) and (2), and in contrast to that in General Relativity, this augmenting field is determined by conventional matter and energy.
Using specific solutions to the theory based on equations (1) and (2), the unification brought to electromagnetic and gravitational phenomena as well as the relation of these solutions to those of the classical M&EFEs was emphasized throughout. Also discussed were unique features/interpretations of the theory based on equations (1) and (2) that set it apart from the classical M&EFEs. These distinguishing features include the emergence of antimatter and its behavior in electromagnetic and gravitational fields, the emergence of dark matter and dark energy mimicking terms in the context of General Relativity, an underlying relationship between electromagnetic and gravitational radiation, and the impossibility of negative mass solutions that would generate repulsive gravitational fields or antigravity.
Although not yet based on specific solutions to the proposed theory, a method for quantizing the charge, mass, and angular momentum of particle-like solutions, as well as the possibility of superluminal transport when aR   is space-like were conjectured.
The genesis of the work presented here was reported in a preliminary form in references [2]. The same coupling between the Maxwell tensor and the R-C tensor given in equation (1) was first reported there, although in a somewhat modified form. The discussion of systems of first order partial differential equations and the existence of solutions to such systems was also given in reference [2] but is included here to keep the mathematical description of the proposed theory self-contained. New to this manuscript is the discussion of the global symmetries of equations (1) and (2), and based on those global symmetries the interpretation of the particle-like solution has been advanced, as has the discussion of boundary conditions. The discussion of Einstein's equation of General Relativity augmented by a term that can mimic the properties of dark matter and/or dark energy is also new to this manuscript, as is the discussion of the solution based on the FLRW metric. The present manuscript also corrects an error in the weak field analysis of reference [2], leading to an expanded discussion of electromagnetic radiation and its underlying gravitational radiation. The discussion of the impossibility of both negative mass solutions and antigravity is new. The LLNL-JRNL-726138-Draft speculation on superluminal transport if aR   is space-like is also new. Finally, the appendix containing the analysis of the Cauchy initial value problem as it relates to the theory's equations (1) and (2) is new and included to replace an incorrect discussion of the logical consistency of the fundamental field equations that was given in reference [2].

ACKNOWLEDGEMENTS
For his penetrating insights and suggestions offered during the early phases of this work [33] I would like to   t . To see that this is so I examine the general form of the R-C tensor in a locally inertial coordinate system where all first derivatives of g  vanish, i.e., 22 22  (85) Note, having at most a single time index on the RHS of (85) means that the R-C tensor is made up entirely of terms from    can be determined from the four coordinate conditions that are fixed by the choice of coordinate system. [34] Recapping, at 0 t the following quantities are now known: