A Covariant form of Maxwell's (inhomogeneous) equations (in GR)

[Kostik]

TL;DR Summary

We *prove* the covariant form of Maxwell's homogeneous equations, $$F_{\mu\nu;\sigma} + F_{\nu\sigma;\mu} + F_{\sigma\mu;\nu} = 0$$ but simply *postulate* the covariant form of the inhomogeneous equations, $${F^{\mu\nu}}_{;\nu} = 4\pi J^\mu \, .$$

Background: In flat spacetime (special relativity), Maxwell's homogeneous equations $$\text{curl} \textbf{ E}=-\frac{d\textbf{B}}{dt} \qquad\qquad \text{div} \textbf{ B} = 0$$ can be written in the single equation
$$F_{\mu\nu,\sigma} + F_{\nu\sigma,\mu} + F_{\sigma\mu,\nu} = 0 \qquad\qquad (*)$$ To pass to the case of curved spacetime (general relativity), we write this in covariant form as follows. First, recall the electromagnetic field tensor is defined in terms of the electromagnetic vector potential: $$F_{\mu\nu} = A_{\mu,\nu} - A_{\nu,\mu}$$ where $A^\mu = (\phi, \textbf{A})$, where $\phi$ is the electric potential and $\textbf{A}$ is the magnetic vector potential. Since "covariant curl equals ordinary curl", we see immediately that $$F_{\mu\nu} =A_{\mu,\nu} - A_{\nu,\mu}= A_{\mu;\nu} - A_{\nu;\mu} \, .$$ Now, we have $$F_{\mu\nu;\sigma} = F_{\mu\nu,\sigma} - \Gamma^\alpha_{\mu\sigma}F_{\alpha\nu} - \Gamma^\alpha_{\nu\sigma}F_{\mu\alpha} \, .$$ Making cyclic permutations of $\mu, \nu, \sigma$ and adding the equations, we get $$F_{\mu\nu;\sigma} + F_{\nu\sigma;\mu} + F_{\sigma\mu;\nu} = F_{\mu\nu,\sigma} + F_{\nu\sigma,\mu} + F_{\sigma\mu,\nu} = 0 \, .$$ Thus, we arrive at the covariant form of Maxwell's homogeneous equations: $$F_{\mu\nu;\sigma} + F_{\nu\sigma;\mu} + F_{\sigma\mu;\nu} = 0 \, .$$ Now, I want to understand how we obtain the covariant form of Maxwell's inhomogeneous equations. Again, in flat spacetime, the inhomogeneous equations $$\text{div} \textbf{ E} = 4\pi\rho \qquad\qquad \text{curl} \textbf{ B}=-\frac{d\textbf{E}}{dt} + 4\pi\textbf{J}$$ can be written in the single equation $${F^{\mu\nu}}_{,\nu} = 4\pi J^\mu \qquad\qquad(**)$$ where $J^\mu = (\rho, \textbf{J})$.

$\qquad$ What I am finding in the textbook literature is generally a hand-wavy statement that "In general relativity, we must replace $(**)$ with $${F^{\mu\nu}}_{;\nu} = 4\pi J^\mu \, .$$ Here's where I get a bit confused. In the case of the homogenous equations, we did not say "In general relativity, we must replace $(*)$ with $$F_{\mu\nu;\sigma} + F_{\nu\sigma;\mu} + F_{\sigma\mu;\nu} = 0 \, .$$ On the contrary, we did a fair amount of work to prove it. So why do we just "postulate" the covariant form of Maxwell's inhomogeneous equations?

 

[renormalize]

Here's where I get a bit confused. In the case of the homogenous equations, we did not say "In general relativity, we must replace $(*)$ with $$F_{\mu\nu;\sigma} + F_{\nu\sigma;\mu} + F_{\sigma\mu;\nu} = 0 \, .$$ On the contrary, we did a fair amount of work to prove it. So why do we just "postulate" the covariant form of Maxwell's inhomogeneous equations?

It's perfectly fine to impose the $\partial\rightarrow\nabla$ rule on the cyclic equation $F_{\mu\nu,\sigma}+F_{\nu\sigma,\mu}+F_{\sigma\mu,\nu} = 0$ to get the generally-covariant version. That's exactly what Dirac does in his General Theory of Relativity:

It's just that the Christoffel connections nicely cancel-out of the covariant equation, so the flat-space equation is already generally-covariant.

 

[Kostik]

It's perfectly fine to impose the $\partial\rightarrow\nabla$ rule on the cyclic equation $F_{\mu\nu,\sigma}+F_{\nu\sigma,\mu}+F_{\sigma\mu,\nu} = 0$ to get the generally-covariant version. That's exactly what Dirac does in his General Theory of Relativity:
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It's just that the Christoffel connections nicely cancel-out of the covariant equation, so the flat-space equation is already generally-covariant.

Actually, Dirac does not "impose" the change $\partial\rightarrow\nabla$; in fact, my derivation of the covariant statement of the homogeneous Maxwell equations came straight out of Dirac. Dirac grinds out the covariant form using the definition of covariant derivatives, and, as you say, all the Christoffel symbols happily cancel out. Thus, he obtains the covariant form with no "imposing" or "postulating" ... it is rigorously derived.

Why can't he do the same thing for the inhomogeneous Maxwell equations? See his Eq. (23.13): "This is not a valid equation in general relativity and must be replaced by the covariant equation..."

 

[Kostik]

When I try to "follow my nose", I get: $${F^{\mu\nu}}_{;\nu}={F^{\mu\nu}}_{,\nu} + \Gamma^\mu_{\alpha\nu}F^{\alpha\nu} + \Gamma^\nu_{\alpha\nu}F^{\mu\alpha} = {F^{\mu\nu}}_{,\nu} + √^{-1}√_{,\alpha}F^{\mu\alpha}$$ which gives $${F^{\mu\nu}}_{;\nu}√ = {F^{\mu\nu}}_{,\nu}√ + {F^{\mu\nu}}√_{,\nu}$$ hence $${F^{\mu\nu}}_{;\nu} = √^{-1} ({F^{\mu\nu}}√ )_{,\nu}$$ and how do I get from that to $4\pi J^\mu$?

 

[Kostik]

Dirac seems to think the $\partial \rightarrow \nabla$ rule is self-evident. He says in chapter 10 (p. 19), “The laws of physics must be valid in all systems of coordinates. They must thus be expressible as tensor equations. Whenever they involve the derivative of a field quantity, it must be a covariant derivative. The field equations of physics must all be rewritten with the ordinary derivatives replaced by covariant derivatives.”

 

[dextercioby]

Dirac was a field theorist, and as far as I remember, his 75-page GR brochure is an advert for the action principle. So all you need to do is apply the action principle to the Lagrangian action of matter coupled with electromagnetic fields in curved spacetime. Then the covariant derivative generalization of Maxwell's equations is a consequence, i.e. they are Euler-Lagrange equations for the aforementionend action.

 

[PAllen]

MTW has discussion of the subtleties and physical basis of the comma to semicolon rule, including cases where it fails. Ultimately, you have to rely on physical arguments from the equivalence principle in a local inertial frame.

 

[Kostik]

MTW has discussion of the subtleties and physical basis of the comma to semicolon rule, including cases where it fails. Ultimately, you have to rely on physical arguments from the equivalence principle in a local inertial frame.

Could you kindly give a reference within MTW?

 

[PeterDonis]

Could you kindly give a reference within MTW?

Sections 16.2 and 16.3. The latter section and the accompanying Box 16.1 specifically discuss the case of electrodynamics.

 

[PAllen]

Could you kindly give a reference within MTW?

The main discussion is in and around box 16.1, p. 390 of my edition.