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Mass tells spacetime how to curve using the Einstein field equations and spacetime tells mass how to move using the geodesic equation.

The effect of the electromagnetic field on space-time is given by the Einstein field equation. But what equation gives the effect of spacetime curvature on the electromagnetic field? Is there no replacement of the geodesic equation?

Níckolas Alves
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Ryder Rude
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1 Answers1

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Maxwell's equations (albeit in curved spacetime).

The geodesic equation is, in some sense, just the equations of motion you get for a massive point-particle coupled to the gravitational field. The action would read $$S = S_{EH} + m \int \sqrt{-\frac{\mathrm{d} x^\mu}{\mathrm{d}\tau}\frac{\mathrm{d} x^\nu}{\mathrm{d}\tau} g_{\mu\nu}} \mathrm{d}\tau,$$ and varying the action leads to the Einstein equations and to the geodesic equation (there are issues with self-field corrections, but I won't dive into that in here).

For the electromagnetic field, you'll have a similar setup, but the additional term in the action will be the action for the electromagnetic field. In the end of the day, you'll get to Maxwell's equations, which read \begin{align} \nabla_a F^{ab} &= - 4 \pi j^b, \\ \nabla_{[a}F_{bc]} &= 0, \end{align} in Gaussian units. See, e.g., Wald's General Relativity, Sec. 4.3.

For other field contents, one would replace Maxwell's equations with the equations of motion for the field in question. A scalar field has the Klein–Gordon equation, spin-1/2 fields have the Dirac equation, and so on.

Níckolas Alves
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  • May I ask you a related question? How does the action from your first equation apply to massless particles? – Valac Jun 22 '22 at 02:41
  • @LibertarianFeudalistBot It would be best to keep each post to a single question, since it makes it easier to navigate the site. Nevertheless, the action would need to be different. It is discussed in the accepted answer to this question: https://physics.stackexchange.com/q/331529/168783 – Níckolas Alves Jun 22 '22 at 02:54
  • But don't Maxwell's eqns describe the behavior of free EM fields? How can the same eqns describe the response of the EM field in interaction with spacetime curvature? – Ryder Rude Jun 22 '22 at 03:15
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    @RyderRude In the answer, the Maxwell equations use covariant derivative of curved spacetime. It is not the Maxwell equations in flat spacetime. – Valac Jun 22 '22 at 03:19
  • @RyderRude It is precisely what Libertarian Feudalist Bot said. In addition, I believe I should mention that while one can solve Maxwell's equations in a previously given curved spacetime (hence not considering the backreaction of the EM field in the spacetime itself, which is valid for weak fields), the consistent (but often difficult) way of doing it would be to solve Einstein's equations and Maxwell's equations simultaneously, as a coupled system of PDEs. – Níckolas Alves Jun 22 '22 at 04:29
  • @NíckolasAlves Whoa. solving that must involve doing some nightmare math?? How would you rate that in terms of toughness? – Ryder Rude Jun 22 '22 at 04:51
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    @RyderRude I don't really know how to answer lol. I haven't worked much with solutions to Einstein-Maxwell. The case for a charged spherical black hole, though, is fairly similar to deriving the Schwarzschild solution. I guess it depends a lot on how much symmetry you are assuming – Níckolas Alves Jun 22 '22 at 13:20