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Kostik
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- 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?
$$F_{\mu\nu,\sigma} + F_{\nu\sigma,\mu} + F_{\sigma\mu,\nu} = 0 \qquad\qquad (*)$$ 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?
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