PeterDonis

Mentor
In the first article in this series, we looked at the Einstein Field Equations in a static, spherically symmetric spacetime. In this article, we are going to build on what we saw in the first article to show what Maxwell’s Equations in a static, spherically symmetric spacetime look like.
The electromagnetic field tensor is given in general by
$$F_{ab} = \partial_b A_a – \partial_a A_b$$
where $A_a$ is the electromagnetic 4-potential.
In covariant form, Maxwell’s Equations in general are:
$$\partial_c F_{ab} + \partial_b F_{ca} + \partial_a F_{bc} = 0$$
$$\nabla_a F^{ab} = \partial_a F^{ab} + \Gamma^a{}_{ac} F^{cb} = 4 \pi j^b$$
The first equation is an identity given the definition of $F_{ab}$; the second equation links the EM field to its source, the charge-current 4-vector $j^b$, and is the one we will be focusing on here...

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andresB

Just want to point out that the energy-momentum tensor discussed is not indeed the most general electromagnetic tensor possible. Non-linear correction of electrodynamics affects the form of the tensor.

You can have (at least theoretically, probably not in nature) a non-rotating, electrically charged black hole with big enough charge that non-linear corrections are in order.

PAllen

Just want to point out that the energy-momentum tensor discussed is not indeed the most general electromagnetic tensor possible. Non-linear correction of electrodynamics affects the form of the tensor.

You can have (at least theoretically, probably not in nature) a non-rotating, electrically charged black hole with big enough charge that non-linear corrections are in order.
Is any of this relevant if you assume purely classical EM? It looks like it is all QED corrections to Maxwell EM.

andresB

I just wanted to point out the possibility of generalizations. However, there are real stars with high enough magnetic fields that these corrections do need to be considered (though, in those cases spherical symmetry is lost).

PeterDonis

Mentor
I just wanted to point out the possibility of generalizations.
These are possible, but beyond the intended scope of this particular article. The article is only intended to cover the standard classical Einstein-Maxwell equations.

SWystub

Are you sure this form of the SET is correct? When trying to calculate it, I get a factor of 7/8 for the tt and rr components.

PeterDonis

Mentor
Are you sure this form of the SET is correct?
Yes. You can find it in many references, including MTW, which is where I first saw it.

When trying to calculate it, I get a factor of 7/8 for the tt and rr components.
How are you calculating it?

SWystub

Yes. You can find it in many references, including MTW, which is where I first saw it.

How are you calculating it?
I'm asking because of the Kronecker Delta - in some references (Sean Carroll. Spacetime and Geometry: An Introduction to General Relativity) it is given with an \eta_{\mu \nu} instead.

The factor of 7/8th was a simple mistake on my side.

PeterDonis

Mentor
I'm asking because of the Kronecker Delta - in some references (Sean Carroll. Spacetime and Geometry: An Introduction to General Relativity) it is given with an \eta_{\mu \nu} instead.
If you write it with both indexes up or both indexes down, the metric will appear (which in a general curved spacetime will be $g_{\mu \nu}$ or $g^{\mu \nu}$, not $\eta_{\mu \nu}$ or $\eta^{\mu \nu}$). But if you write it with one index up and one index down, as I did, then the metric becomes $\delta^\mu{}_\nu$.

"Maxwell’s Equations in a Static, Spherically Symmetric Spacetime - Comments"

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