How Does the Electromagnetic Field Couple to Gravity, in Particular to Metric, Nonmetricity, Torsion, and Curvature?

Abstract

The coupling of the electromagnetic field to gravity is an age-old problem. Presently, there is a resurgence of interest in it, mainly for two reasons: (i) Experimental investigations are under way with ever increasing precision, be it in the laboratory or by observing outer space. (ii) One desires to test out alternatives to Einstein's gravitational theory, in particular those of a gauge-theoretical nature, like Einstein-Cartan theory or metric-affine gravity.-A clean discussion requires a reflection on the foundations of electrodynamics. If one bases electrodynamics on the conservation laws of electric charge and magnetic flux, one finds Maxwell's equations expressed in terms of the excitation H = (D, H) and the field strength F = (E, B) without any intervention of the metric or the linear connection of spacetime. In other words, there is still no coupling to gravity. Only the constitutive law H = functional(F ) mediates such a coupling. We discuss the different ways of how metric, nonmetricity, torsion, and curvature can come into play here. Along the way, we touch on non-local laws (Mashhoon), non-linear ones (Born-Infeld, Heisenberg-Euler, Plebański), linear ones, including the Abelian axion (Ni), and find a method for deriving the metric from linear electrodynamics (Toupin, Schönberg). Finally, we discuss possible non-minimal coupling schemes.1 Of course, we could have non-minimal coupling as, e.g., in F +γ eα⌋e β ⌋R αβ ∧F = 1 ε 0 c d ⋆ J , see Sec.7.

A Unique Mathematical Derivation of the Fundamental Laws of Nature Based on a New Algebraic-Axiomatic (Matrix) Approach ‡

“…[44][45][46][47][48], in agreement with the recent astronomical data, we can directly establish a lower bound for a constant quantity which is equivalent to the constant α = (82) that is generated by a coupling torsion field of the type (79-3) of the background curved space-time.…”

“…However, similar to the system of Equation (29), in the course of obtaining the solutions (49) or (49-1) from the system of linear Equation ( that are similar to condition (47). Here also by the same approach, since the parameters m 4 , m 6 , m 7 , m 8 , m 9 have not appeared in the solution (49), it can be assumed that m 4 = m 6 = m 7 = m 8 = m 9 = 0, and the set of conditions (50) are reduced to the following system of homogeneous quadratic equations which are similar to the quadratic Equation (20) (corresponding to the system of linear Equation (28)): The conditions (50-1) are also similar to the quadratic Equation (21).…”

“…Moreover, the trace part of torsion field Z τµν in (79-3) is obtained as: [17,39,[44][45][46][47][48]. However, in Section 3.15, we show that assuming the spin-1/2 fermion fields (describing generally by the field Equation (110-9) compatible with specific gauge symmetry group (110-12), as shown in Section 3.15, Formulas (110-9)-(110-12)) and their compositions as the source currents of the (1 + 3)-dimensional cases of general covariant massive field Equation (72) (describing the spin-1 boson field), then this field equation would be invariant under two types of gauge symmetry groups, including: SU(2) L ⊗U(2) R and SU(3), corresponding with a group of seven bosons and a groups of eight bosons (as shown in Section 3.15, Formulas (114-4)-(114-9)).…”

Equivalence principle and electromagnetic field: no birefringence, no dilaton, and no axion

“…In order to understand the coupling of electromagnetism to gravity, it is most convenient and reasonable to start from the premetric formulation of the electrodynamical theory. Following the early work of Kottler, Cartan, and van Dantzig, one can indeed develop an axiomatic approach to the electromagnetic field [3,4,5] without assuming any specific geometric structure beyond the differentiable structure of the spacetime manifold, see also [6,7]. At the heart of this approach, there are the well established experimental facts (that can be formulated as fundamental axioms) of electric charge and magnetic flux conservation.…”

Generally covariant Maxwell theory for media with a local response: Progress since 2000

“…Therefore, the QHE cannot depend on an external gravitational field, see [19]. 4) Electromagnetism can couple to a possible Cartan torsion of spacetime only nonminimally: Solanki, Preuss, et al [68], [77] pointed out that a non-minimal coupling of gravity to electromagnetism, in particular to the torsion field, is possible, see also Hehl and Obukhov [16], Rubilar et al [71], and Itin et al [25], [26]. 5) Signature of the metric can be derived from electrodynamics: In [18] it has been pointed out that the signature of the metric can be derived from the Lenz rule and the positivity of the energy of the electromagnetic field.…”