Electromagnetic Stress Energy Tensor Formula (-,+,+,+)

[Jay21]

TL;DR Summary

Formula for the Electromagnetic Stress Energy Tensor

I am trying to find the correct formula for the electromagnetic stress energy tensor with the sign convention of (-, +, +, +).
Is it (from Ben Cromwell at Fullerton College):

$$T^{\mu \nu} = \frac{1}{\mu_0}(F^{\mu \alpha}F^{\nu}{}_{\alpha} - \frac{1}{4}g^{\mu\nu}F_{\alpha\beta}F^{\alpha \beta})$$

but I have also seen it with a negative sign:

$$T^{\mu \nu} = -\frac{1}{\mu_0}(F^{\mu \alpha}F^{\nu}{}_{\alpha} - \frac{1}{4}g^{\mu\nu}F_{\alpha\beta}F^{\alpha \beta})$$

Which is the correct formula? Also for flat space-time $g^{\mu\nu} = \eta^{\mu\nu}$ and for curved space-time $g^{\mu\nu}$ is whatever metric being used for the curved space-time situation one is working in, correct?

Thanks.

 

[Orodruin]

This depends not only on your sign convention for the metric but also on your sign convention for the stress-energy tensor, i.e. the sign in $G_{\mu\nu} = \pm 8\pi G T_{\mu\nu}$. MTW has a nice compilation of the different sign conventions used in many of the popular texts.

In general, there are three sign conventions of note at work in GR:

• The metric signature.
• The sign of the stress-energy tensor.
• The sign of the Ricci tensor.

Combinations of those signs show up in varying formulas.

 

[Ibix]

Ben Cromwell at Fullerton College)

Crowell, to be pedantic. He used to post here.

Also for flat space-time $g^{\mu\nu} = \eta^{\mu\nu}$ and for curved space-time $g^{\mu\nu}$ is whatever metric being used for the curved space-time situation one is working in, correct?

More or less. $g^{\mu\nu}$ is the inverse metric - the metric has lower indices. But the same information is in both. And yes, $\eta_{\mu\nu}$ is usually used as a symbol for the metric of flat spacetime. There's nothing different about it, but it's such an important special case it gets its own symbol.

 

[Jay21]

From Misner... who uses the convention of (-, +, +, +) for the metric $g^{\mu\nu}$,
with the electromagnetic stress energy tensor being(pg.141):
$$T^{\mu \nu} = \frac{1}{\mu_0}(F^{\mu \alpha}F^{\nu}{}_{\alpha} - \frac{1}{4}g^{\mu\nu}F_{\alpha\beta}F^{\alpha \beta})$$
and then a positive sign for the stress energy tensor:
$$G_{\mu\nu} = 8\pi GT_{\mu\nu}$$
I think I now understand where all of the different signs come from.

Imagine a scenario of a charged object near Earth or at Earth's surface. Its mass is negligible but its E and B fields are not negligible. Therefore, to find the electromagnetic stress energy tensor of this object near the curvature of Earth, I would use the Schwarzschild metric in the weak field approximation, with M being the mass of Earth, as the metric (after finding the inverse of course) in the formula,$g^{\mu\nu}$, for the electromagnetic stress energy tensor, correct?

Thanks.