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This will look like a physics question but it doesn't have anything to do with reality so its a vague math question if anything.

I recently learned about gravitoelectromagnetism which describes an approximation of General relativity that separates the gravitational field into an "electric-like" component and "magnetic-like" component. In particular we have a description of the gravitational field that looks identical to Maxwell's equations. So we can say we have the following hierarchy of theories of gravity:

$$ \text{Newtonian Gravity} \underbrace{\rightarrow }_{\text{approximates}} \text{ Gravitoelectromagnetism} \underbrace{\rightarrow }_{\text{approximates}} \text{General Relativity} $$

Now consider the same diagram but instead for electromagnetism:

$$ \text{Coulomb's Law} \underbrace{\rightarrow }_{\text{approximates}} \text{ Maxwell's Equations} \underbrace{\rightarrow }_{\text{approximates}} \text{???} $$

It's clear both Coulomb's law and Newton's law take on the same $\frac{\text{product of charges}}{r^2}$ form. Then Maxwell's Equations and Gravitoelectromagnetism both take an identical form as a system of PDEs of dot and cross product (they really look nearly identical up to a choice of constants).

The mathematical question: Is there some natural mathematical object that can fit into "???" above and is in some "natural" sense the "electromagnetic dual of general relativity"?

To be clear: I'm looking for a mathematical description of space time that probably gets curved by the presence of moving charges, and in turn causes those charges to move. I do not want this description to be EQUIVALENT to maxwell's equations, I want Maxwell's equations to appear as a low-energy "approximation" of this thing in the same way that gravitoelectromagnetism is a low energy approximation to GR. In particular I expect that this description should result in the same number of PDEs to describe a spacetime metric from initial conditions as General Relativity.

I believe this is not the same thing as Kaluza Klein theory because it appears KK-theory is EQUIVALENT to Maxwell's equations and we are looking for a system of field equations that Maxwell's equations approximate.

I also do not believe this is described by string theory, since that attempts to generalize the problem by merging with quantum field theory which is not the goal here.

Has such a system of PDEs been described/Is there a natural such system?

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    $\begingroup$ from a strictly mathematical point of view, you can just enter the Einstein field equations, $G_{\mu\nu}+\Lambda g_{\mu\nu}=\kappa T_{\mu\nu}$ in the question marks; the math does not know the difference between Coulomb's law or Newton's law, or between Maxwell's equations or the GEM equations, so the same mathematical operation that brings you from the Einstein equations to the GEM equations will bring you to the Maxwell equations. $\endgroup$
    – Carlo Beenakker
    Commented Jun 10 at 6:20
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    $\begingroup$ Possible Yang–Mills equations: en.wikipedia.org/wiki/Yang%E2%80%93Mills_equations $\endgroup$
    – The Tiler
    Commented Jun 10 at 8:21
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    $\begingroup$ Maxwell's equations are the Yang-Mills equations for $U(1)$ gauge field. Nothing is needed to fill the ???s above since Maxwell's equations are consistent and Lorentz invariant, unlike the "gravitoelectromagnetic" equations. $\endgroup$
    – Aaron Bergman
    Commented Jun 10 at 13:57
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    $\begingroup$ A possibility from a point of view of induction (and history) and not of deduction. the two theories are gauge theories (harmonic gauge,...). $\endgroup$
    – The Tiler
    Commented Jun 10 at 19:58

2 Answers 2

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There is a profound conceptual shift between Gravitoelectromagnetism and General Relativity that cannot analogously occur between Maxwell's equations and any "???".

In General Relativity, there is nothing (no "field") that "causes" objects "to move", as you put it. Instead, all objects are in free fall, and therefore, in particular, no intrinsic properties of the objects (such as any sort of "charge") influence their motion.

Nothing analogous can occur for electromagnetism, since the intrinsic property "charge" of objects does influence their motion - opposite charges attract, like charges repel each other. Thus, at the very least, there would have to be separate co-existing metrics for objects of differing charge, whatever that may mean.

So, "???" cannot simply reference a 4-dimensional geometry analogous to the one in General Relativity. Perhaps something larger. In that case, however, it's unclear how one would recover a description with the same number of PDEs as for the 4-dimensional geometries of General Relativity.

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  • $\begingroup$ Couldn't one incorporate electric charge in a separate dimension, yet again geometrising the mechanics as one does for general relativity? So particles follow geodesics in "spacetime + charge space". $\endgroup$
    – Jannik Pitt
    Commented Jun 12 at 10:18
  • $\begingroup$ @JannikPitt - indeed, that's what I meant by "something larger". Explorations of this idea started already very early on with the Kaluza-Klein theory mentioned in the OP, and such ideas continue to attract interest. Serious searches for experimental signatures of the associated extra dimensions are performed. However, such constructions will clearly generate more PDEs than 4-dimensional General Relativity. $\endgroup$
    – Michael Engelhardt
    Commented Jun 12 at 13:30
  • $\begingroup$ Ok i think I see what the issue is here. All objects fall down at the same rate independent of mass. But when predicting how a charged object shall move we need to not only know its charge but also its mass too. So an E&M theory can't forget the "Stress-energy" tensor and just have some kind of "E&M tensor". It needs to keep track of both of these things and will therefore necessarily be more complex than GR. $\endgroup$
    – Sidharth Ghoshal
    Commented Jun 12 at 19:12
  • $\begingroup$ @SidharthGhoshal In the second appendix to his book The Meaning of Relativity, Einstein discusses general properties of systems of PDEs in terms of how strongly they determine the fields they describe. (Unacknowledged use of E. Cartan's ideas, refer their correspondence in Letters on Absolute Parallelism Princeton 1979). It's also worth noting that in GR, movement of bodies is not only determined by mass but also by rotation (Lense-Thirring effect). $\endgroup$
    – Phil Harmsworth
    Commented Jun 13 at 2:38
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    $\begingroup$ @SidharthGhoshal - I agree with your statement, but I think that even in a hypothetical scenario in which all test objects can have only one and the same mass, "???" still has the problem that an object's motion will depend on its charge. So, even then, there can't be one unique 4-dimensional geometry in which all charges fall, but the geometry has to have an additional attribute (or dimension) that allows different charges to follow different geodesics. In contradistinction to the gravity case, some charges will fall downwards off the tower of Pisa, some upwards. $\endgroup$
    – Michael Engelhardt
    Commented Jun 13 at 5:43
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A couple of suggestions:

  1. Born-Infeld electrodynamics (Recent review article - https://arxiv.org/abs/2111.08657). The theory reduces to Maxwell's electrodynamics in the low energy limit. But it does not involve a metric and does not yield the same number of equations as general relativity.
  2. Kurşunoğlu's version of Einstein's final unified field theory ("Einstein's Unified Field Theory", Proc. Phys. Soc. Section A 65(2) (1952) 81-83. (Despite the title, the paper actually discusses a variant of Einstein's unified theory designed to give the correct equations of motion for charged particles.)
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