Pre-metric electromagnetism as a path to unification.

DAVID DELPHENICH

Independent researcher Spring Valley, OH 45370, USA feedback@neo-classical-physics.info

It is shown that the pre-metric approach to Maxwell’s equations provides an alternative to the traditional Einstein- Maxwell unification problem, namely, that electromagnetism and gravitation are unified in a different way that makes the gravitational field a consequence of the electromagnetic constitutive properties of spacetime, by way of the dispersion law for the propagation of electromagnetic waves.

Keyw ords: Pre-metric electromagnetism, Einstein-Maxwell unification problem, line geometry, electromagnetic constitutive laws

1. The Einstein-Maxwell unification problem. fundamental distinctions between them, as well. In particular, the analogy between mass and Ever since Einstein succeeded in accounting for charge was not complete, since at the time (and the presence of gravitation in the universe by to this point in time, as well), no one had ever showing how it was a natural consequence of the observed what one might call “negative” mass or curvature of the Levi-Civita connection that one “anti-gravitation.” Of course, the possibility that derived from the Lorentzian metric on the such a unification of gravitation and spacetime manifold, he naturally wondered if the electromagnetism might lead to such tantalizing other fundamental force of nature that was consequences has been an ongoing source of known at the time – namely, electromagnetism – impetus for the search for that theory. could also be explained in a similar way. Since Several attempts followed by Einstein and the best-accepted theory of electromagnetism at others (cf., e.g., [1] and part II of [2]) at the time (as well as the best-accepted “classical” achieving such a unification. They seemed to theory to this day) was Maxwell’s theory, that fall into two basic categories: Extensions of g gave rise to what one might called the Einstein- with a four-dimensional spacetime and Maxwell unification problem: Find some extensions of spacetime to something higher- intrinsic (presumably geometric) structure on dimensional. The former models included spacetime (suitably extended) that will teleparallelism 1 and the Einstein-Schrödinger decompose into (or at least lead to) the theory [2], while the latter include the Kaluza- Lorentzian metric tensor g and the Klein models [1, 2, 4, 5], and some of the electromagnetic field strength 2-form F, along attempts to extend the tangent bundle of the four- with a set of field equations for that extended dimensional spacetime manifold to an geometric structure that would imply the anholonomic (i.e., non-integrable) rank-4 sub- Einstein field equations for g and the Maxwell bundle of the tangent bundle to a five- equations for F (at least, in some dimensional manifold (cf., e.g., [6-8]). approximation). All of the attempts were regarded as failures The suspicion that such a “unified field for one reason or another. One problem with theory” might actually exist was perhaps based upon the fact that Maxwell’s theory itself represented a “unified field theory” of the 1 The author has compiled an anthology [3] of electric and magnetic fields, and that there English translations of many of the early papers existed a well-known analogy between on teleparallelism that is available as a free PDF Coulomb’s law of electrostatics and Newton’s download at his website (neo-classical- law of universal gravitation, although there were physics.info) 2 Pre-metric electromagnetism as a path to unification teleparallelism was that it included unphysical passed over that “classical” problem 1 and started solutions, such as a static distribution of with the exchange-particle concept, combined gravitating, uncharged masses. The main with the scattering approximation for the field problem with the other theories was that they dynamics. That means: Rather than speculate on implied no new consequences of that unification; what might constitute the “field equations of i.e., they were “concatenations” of the field QED” or the nature of the electromagnetic theories, not unifications. What was missing “force” that acts between elementary charges at were any sort of “gravito-electromagnetic the quantum level, quantum physics was going to inductions,” that would suggest an analogue to replace the force of interaction with the exchange the electromagnetic induction in Maxwell’s of an elementary particle that would mediate the theory. Indeed, it is important to note that the interaction; for QED, that particle would be the latter inductions had been established photon. Then, rather than posing “classical” experimentally by Faraday before Maxwell problems, such as boundary-value problems in formulated his theory, while to date, no such statics and the Cauchy problem in dynamics, couplings of electromagnetic and gravitational QED would simply pass to the approximation in fields seem to have materialized in the which the initial time was − ∞ and the final time laboratory. (“Gravitomagnetism” is a different was + ∞, which is equivalent to assuming that matter, and we shall discuss it below.) the interaction of particles takes place inside a However, as quantum physics evolved, the very small “black box” time interval in which the nature of the Einstein-Maxwell unification nonlinear nature of the interaction can be problem changed, as well. Increasingly, Einstein enclosed in such a way that the time evolution suspected that gravitation could only be unified operator that takes incoming fields to outgoing with electromagnetism when one went to ones becomes a linear operator that takes quantum electromagnetism. That possibility incoming scattering states (which are asymptotic seems more reasonable nowadays, since the free fields) to outgoing ones. That allows one to phenomenon of “gravitomagnetism” had not use the methods of Fourier analysis and discuss been observed experimentally until relatively the scattering operator in momentum space recently [9]. The essence of that phenomenon is without having to worry that the perturbation that the analogy between Coulomb’s law and series that one defines (i.e., Feynman diagrams Newton’s law goes beyond the scope of statics, or loop expansions) is unphysical. since there is, in fact, a field that is induced by Of course, it is precisely the fundamental the relative motion of a mass that is analogous to distinction between Einstein’s theory of the magnetic field that is induced by the relative gravitation as a “classical” field theory (i.e., one motion of an electric charge, and which is in which one can pose boundary-value problems commonly called the gravitomagnetic field . As a in statics and the Cauchy problem in dynamics), result, one sees that Maxwell’s equations are while QED is a “quantum” field theory (i.e., one closely analogous to the weak-field equations of that begins in the scattering approximation to gravitation, which suggests that perhaps that Cauchy problem) that is the greatest Einstein’s equations of gravitation, which are the obstruction to the unification of those theories, strong-field equations, should be somehow although that fact is rarely addressed in quantum analogous to some hitherto-unknown “strong- gravity, which takes more of a “play it where it field” equations of electromagnetism. lays” approach. The physical realm in which one would Another common critique of the Einstein- expect to find the strongest electromagnetic Maxwell unification problem is that it is fields is in the atomic to subatomic domain, currently a partial unification problem, in the where one approaches the Schwinger critical sense that since the time of Einstein’s early work field strengths at which photons resolve into on gravitation, two other “quantum” interactions electron-positron field-pairs. However, since the time of Heisenberg and Pauli [10], quantum 1 One might suspect that the quantum use of the electrodynamics has not started with a set of word “classical” in a pejorative way is probably “strong-field equations of electromagnetism” an imitation of the pure-mathematical usage of that might perhaps be analogous to Einstein’s the word “trivial,” which often represents little equations of gravitation. Rather, it has simply more than a lack of personal curiosity about the subject, combined with an acceptance of the fact that the problem in question is hard to pose and even harder to solve. David Delphenich 3 have been added to the fundamental interactions, makes perfect sense in the context of the “pre- namely, the weak and strong interactions. metric” approach to electromagnetism. That Furthermore, once Yang and Mills had revisited approach is based upon the observation that the the gauge-field approach to elementary only place in which the Lorentzian metric on interactions that Weyl, Fock, and Ivanenko had spacetime enters into Maxwell’s equations is in studied in the context of electromagnetism at the Hodge * operator. In order to see that, we about the same time that Einstein was pondering express those equations in terms of the unified field theory and the Copenhagen school Minkowski electromagnetic field-strength 2- was defining their foundations for quantum form F as 1: physics, the unification of electromagnetism with the weak interactions as gauge field theories dF = 0, δF = 4 πJ, δJ = 0, (1) defined an entirely different approach to unification that usually took the form of looking in which d represents the exterior derivative for higher-dimensional gauge groups that might operator and: contain the more elementary gauge groups as δ = ± * d* (2) subgroups. Interestingly, although gravitation was the first of the fundamental interactions to represents the codifferential operator (whose sign present a manifestly geometric character, and will be negative for 2-forms on a four- gauge fields also have a manifestly geometric dimensional Lorentzian manifold). character (as connection 1-forms), nonetheless, Now, let us express that * operator as the finding a gauge theory of gravitation that might composition # ⋅ C of two invertible linear maps. be absorbed into the other gauge field theories in 2 Namely, C: Λ → Λ2 is the map that “raises both a unified way has proved to be more problematic indices” of the 2-form F, so locally, one can than one would expect. express C in components as:

2. Pre-metric electromagnetism. κλµν κµ λν κν λµ C = 1 (g g – g g ) . (3) 2 Let us now consider another possibility, namely, that the Einstein-Maxwell unification problem is The other map is the Poincaré isomorphism 2 the wrong problem to pose. The justification for # : Λ2 → Λ , which is based upon a Riemannian that is found in the fact that when one goes back volume element on spacetime, which takes the to the chronological sequence of Einstein’s early local component form: papers on relativity, one can notice a subtlety that is easy to ignore: Einstein did not start out V = −g dx 0 ^ dx 1 ^ dx 2 ^ dx 3, (4) looking for a theory of gravitation, he started out by examining the way that electromagnetic in which g is the determinant of the component waves propagate from relatively moving bodies. It was the suggestion that the light-cones (i.e., matrix gµν of the metric tensor. characteristic manifolds for the propagation of The Poincaré isomorphism will then take the those waves) represented the relativistically- bivector field B to the 2-form # B, whose local invariant objects, when combined with the components are: insight of Minkowski that the form of that 1 − ε κλ characteristic equation for electromagnetic wave (# B)µν = 2 g κλµν B . (5) propagation suggested a non-Euclidian geometry on a four-dimensional space, that led Einstein to The 1-form J represents the electric current investigate other non-Euclidian geometries. In that serves as the source of the field F, and is particular, Marcel Grossmann told him about usually given the “convective” form σv, where σ Riemannian geometry (although light-cones are is the electric charge density, and v is the actually indicative of pseudo-Riemannian covelocity 1-form that is metric-dual to the geometry), which eventually led to Einstein’s velocity vector field for the moving source theory of gravitation. charge distribution.

2.1 The metric form of Maxwell’s equations .

Now, this chronological progression from 1 For the basic facts of this approach to electromagnetism to light-cones to gravitation Maxwell’s equations, one might confer [11-14]. 4 Pre-metric electromagnetism as a path to unification

2.2 The pre-metric form of Maxwell’s The fact that one rarely considers the equations. electromagnetic constitutive properties of spacetime in either special or general relativity is The observation of Kottler [15], which was largely due to the fact that one usually subsequently pointed out by Cartan [16], and encounters them only in the combination: expanded upon by van Dantzig [17] was that, in a sense, the linear isomorphism C (which is 1 c = , (9) where g enters into Maxwell’s equations) plays 0 ε µ 0 0 the role of an electromagnetic constitutive law. [18, 19]. Generally, such a law associates the which is then set equal to 1. electromagnetic excitation bivector field H, One then sees that the map (3) of raising which includes the electric displacement D and both indices of a 2-form does, in fact, take the the magnetic flux density H, with the form of a special case of an electromagnetic electromagnetic field strength F, which includes constitutive law 2. the E and B fields 1. Of course, the more general If one replaces the Riemannian volume association: element with the more general one:

H = C(F) (6) V = dx 0 ^ dx 1 ^ dx 2 ^ dx 3, (10) so: 1 ε κλ does not, by any means, have to be linear on the (# B)µν = 2 κλµν B , (11) fibers of the bundles in question. Indeed, nonlinear electromagnetism seems to be an this time, then one can express Maxwell’s unavoidable aspect of the “strong-field” form of equations in their “pre-metric” form 3: Maxwell’s equations (whatever that might be). In fact, even the restriction to invertible maps on dF = 0, div H = 4 π J, div J = 0, H = C(F), (12) fibers represents the restriction to “non- dispersive” media, which would make C and algebraic operator, instead of an integral one. in which: −1 In the event that linearity is an acceptable div = # d # (13) approximation, the map C can be represented by a fourth-rank tensor field whose local is the adjoint of d (and which agrees with the components take the form Cκλµν (x), such that: usual divergence operator on vector fields), and J = σ v is the electric current vector field. µν 1 κλµν One can also absorb the map C into the H = C F κλ . (7) 2 basic equations and arrive at:

The most elementary constitutive law that dF = 0, div C(F) = 4 π J, div J = 0. (14) one can impose upon a medium is precisely the one that one implicitly uses in special relativity. Furthermore, if one chooses an If one assumes that the spacetime vacuum is electromagnetic potential 1-form A (so F = dA ) characterized by being non-dispersive, linear, then this will reduce to: isotropic, and homogeneous then one can use:

□ A = 4 π J, div J = 0, (15) Di = ε Ei, Bi = µ Hi, (8) C 0 0 in which we have introduced the generalized ε in which 0 is the classical vacuum dielectric d’Alembertian operator that is associated with µ constant and 0 is its magnetic permittivity. the map C: Note that one still needs a spatial metric in order to raise the indices on E and H. 2 The author has recently investigated the electromagnetic interpretation of some of the 1 Actually, this association becomes confused most popular classes of Lorentzian metrics in somewhat by the fact that in the electrodynamics [20]. of continuous media [19], the B field is the 3 This form of the pre-metric equations is due to response of the medium to the imposition of the the author [13]. A different, but equivalent, form H field, not the other way around. is found in Hehl and Obukhov [14]. David Delphenich 5

□ = div ⋅⋅⋅ C ⋅⋅⋅ d . (16) outside the support of J (so J = 0) then after C some tedious, but straightforward, calculations 1,

one will get a linear map L(k): Λ1 → Λ that is 2.3 The emergence of light cones . 1 quadratic in k. It is not invertible, since the first

step in the composition of maps that gives one In order to see how one gets back to a spacetime L(k) takes A to k ^ A, which will be zero for any Lorentzian structure, one first reminds oneself A that are collinear with k, so one must first that the reason that one says “light-cones” restrict L(k) to a complementary subspace to the instead of “gravity-cones” is that the two theories line that is generated by k. In fact, in order to get are not independent of each other: The light- an invertible map, one must reduce to a two- cones are characteristic hypersurfaces for the dimensional subspace of that three-dimensional propagation of electromagnetic waves, and they subspace, and if L (k) is the restriction of L(k) to also represent of the dispersion law for those 2 that two-dimensional subspace then the waves. Those light-cones also relate to the condition for the invertibility of L (k) is the non- fundamental structure that implies the presence 2 vanishing of the determinant of that linear map. of gravitation in spacetime, namely, the That determinant will, of course, depend Lorentzian metric. upon k, and the characteristic k are the one for In fact, when one goes to pre-metric which the determinant vanishes: electromagnetism, one finds that the quadratic form of the light-cone equation: ≡ D4(k) det L2(k) = 0. (21) µ ν g (v, v) = ηµν v v −2 0 2 − 1 2 − 2 2 − 3 2 The subscript 4 indicates that the function D4 is a = c (v ) (v ) ( v ) ( v ) (17) homogeneous polynomial of degree 4 in k; in 2 fact, it will generally be quadratic in k . is a degenerate case of a more general quartic Equation (21) then represents the characteristic expression. hypersurfaces for the electromagnetic waves that In order to get that quartic expression, one propagate in spacetime according to (12), (14), first needs to restrict the scope of the theory to or (15), as well as the dispersion law for such linear electromagnetic media, since otherwise, waves. one would have to expect that the dispersion law As a polynomial of degree four, the function for “wave-like” solutions of the field equations D can also be associated with a completely- (14) would also depend upon the definition of 4 symmetric, covariant, fourth-rank tensor field on “wave-like.” For linear media, it is entirely spacetime that has been called the Tamm-Rubilar sufficient to find the dispersion law for plane- tensor [14]. Its form is very closely related to wave solutions, even though they have a the study of Kummer surfaces [21], which grew distinctly unphysical character, due to their out of the branch of projective geometry that is infinite total energy and momentum. For such called line geometry [22]. fields, one can locally set: The general electromagnetic medium will

exhibit birefringence , which means that if one F = e−ik (x) f, (18) first treats the components kµ as the in which: 3* i homogeneous coordinates for a point in RP k = ω dt – ki dx (19) then the corresponding inhomogeneous (i.e., is the frequency-wave number 1-form for a wave Plücker) coordinates ni = ki / ω, i = 1, 2, 3, will and: take the form of indices of refraction in the three µ ∂ elementary directions of space. If one then x = x µ (20) represents n as n u , where u are the components ∂x i i i of a unit vector in the (spatial) direction of

propagation, then the equation D4(ni) = 0 will is the position vector field that is defined by the 2 generally have two distinct roots for n ; i.e., the choice of coordinate chart. The field f basically same direction of propagation will be associated represents the shape of the wave; in the with two different speeds of propagation. geometrical optics approximation, one effectively sets df = 0. 1 If one substitutes (18) into (15), and For more details on the calculations, one can considers only the points of spacetime that are confer the books by the author [13] and Hehl and Obukhov [14]. 6 Pre-metric electromagnetism as a path to unification

(Although it is not clear in the present context, breaks down the light cone structure into a the distinction relates to the state of polarization bimetric structure. of the wave.) The conclusion that we have been leading A first reduction in generality for D4 is to up to through all of this is that although pre- the product of quadratic functions: metric electromagnetism does not provide a solution to the Einstein-Maxwell unification ′ problem, it does nonetheless exhibit a radically D4(k) = D2(k) D( k ) , (22) 2 different approach to the unification of the two

field theories, which is that the gravitational ′ in which D2(k) and D2 ( k ) are homogeneous, field in spacetime comes about as a consequence quadratic polynomials in k, and generally of of the electromagnetic constitutive properties of Lorentzian type. One calls this possibility bi- spacetime . Hence, the field C is more metricity [23], and it basically represents a pair fundamental to spacetime structure than g; in of distinct light-cones at every point. effect, gravitation is the shadow that is cast by The final reduction that brings one back to electromagnetism. Lorentzian structures of the kind that are treated Something that was only touched upon here in general relativity is to look at only constitutive that is also a radical departure from the usual laws for which: approach to the geometry of spacetime is that line geometry is to electromagnetism what D (k) = D′ ( k ) = g(k, k), (23) metric geometry is to gravitation. Hence, the 2 2 very type of geometry that one is considering so: 2 changes, as well. (For more on that aspect of the D4(k) = g(k, k ) . (24) problem, see the author’s work [13, 22].)

In particular, such a medium cannot be References 1 birefringent. [1] V. P. Vizgin, Unified Field Theories , 3. Conclusion Birkhäuser, Boston (1994). [2] * A. Lichnerowicz, Théorie relativiste de la One can now see how many restricting gravitational et de l’electromagnetisme, assumptions must go into starting from the Masson and Co., Paris (1955). electromagnetic constitutive properties of the [3] D. H, Delphenich, Selected papers on spacetime manifold and the pre-metric field teleparallelism , neo-classical-physics.info equations of the electromagnetic field strength 2- (2013). form F and concluding with a Lorentzian metric [4] T. Kaluza, Sitz. d. Preuss. Akad. d. Wiss ., g. In particular, one must assume that the 966-974 (1918). medium is non-dispersive, linear, and non- [5] O. Klein, Zeit. Phys . 37 , 895 (1926). birefringent. In fact, many of the popular forms [6] J. A. Schouten, Proc. Kon. Akad. that g takes in general relativity also prove to be Amsterdam 31 , 291-299 (1928). spatially isotropic. In effect, the only room for [7] * G. Vranceanu, J. de Phys . 7 (7), 514-526 variety in the gravitational field is when the (1936). medium is not electromagnetically [8] J. L. Synge, Math. Ann. 99 , 738-751 homogeneous. (1928). It is the restriction to linear media that [9] Ciufolini, I. and Wheeler, J. : Gravitation defines the strictest limitation in the eyes of and Inertia , Princeton University Press, quantum electrodynamics, since one generally Princeton, NJ (1996). finds that the effective electromagnetic field [10] * W. Heisenberg and W. Pauli, Zeit. Phys., equations (such as Heisenberg-Euler, which are 56 , 1-61 (1926); ibid. , 59 , 168-190 (1929). one-loop effective equations) are nonlinear [11] W. Thirring, Classical Field Theory, generalizations of Maxwell’s equations that also Springer, Berlin (1978). involve nonlinear effective constitutive laws. In [12] T. Frenkel, The Geometry of Physics: an fact, they exhibit what is commonly called introduction , Cambridge University Press, vacuum birefringence ; that is, the vacuum polarization that is associated with the 1 electromagnetic field at the Schwinger point also The references that are marked with an asterisk are available in English translation at the author’s website (neo-classical-physics.info) David Delphenich 7

Cambridge (1997). [13] D. H. Delphenich, Pre-metric electromagnetism , neo-classical- physics.info (2009). [14] F. W. Hehl and Y. N. Obukhov, Foundations of Classical Electrodynamics, Birkhäuser, Boston (2003). [15]* F. Kottler, Sitz. Akad. Wien IIa (1922), 119-146 (1922). [16] E. Cartan, On manifolds with an affine connection and the theory of relativity, (English translation by A. Ashtekar of a series of French articles from 1923 to 1926), Bibliopolis, Napoli (1986). [17] D. Van Dantzig, Proc. Camb. Phil. Soc. 30 , 421-427 (1934). [18] E. J. Post, Formal Structure of Electromagnetics , Dover, NY (1997). [19] L. D. Landau, E .M. Lifschitz and L.P. Pitaevskii, Electrodynamics of Continuous Media, 2 nd ed., Pergamon, Oxford (1984). [20] D H. Delphenich, “On the electromagnetic constitutive laws that are equivalent to spacetime metrics,” arXiv:1409.5107 (2014). [21] P. Baeckler, A. Favaro, Y. Itin and F. W. Hehl, “The Kummer tensor density in electrodynamics and in gravity.” arXiv:1403.3467 (2014). [22] D. H. Delphenich, “Line geometry and electromagnetism I: basic structures,” arXiv:1309.2933 (2014).; “Line geometry and electromagnetism II: wave motion,” arXiv:1311.6766 (2014).; “Line geometry and electromagnetism III: groups of transformations,” arXiv:1404.4330 (2014).. [23] M. Visser, C. Barcelo and S Liberati, “Bi-refringence versus bi-metricity,” arXiv: gr-qc/0204017 (2002).