A Electromagnetic Stress Energy Tensor

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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

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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

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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.
 
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.
 
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