# How can the electromagnetic stress energy tensor be restricted to flat space-time

The Wikipedia article describing the electromagnetic stress energy tensor seems to suggest that this tensor can only be defined in flat space-time. How is it possible to define an electromagnetic stress energy tensor this way since any available electromagnetic energy/momentum must render the space-time curvature nonzero?

How in practice would someone extract useful information with this stress energy tensor?

The electromagnetic stress tensor can be defined in all spacetimes:

$$\frac{\delta \mathscr{L}}{\delta g^{ab}} = F_{a}{}^{c}F_{bc} - \frac{1}{2}g_{ab}F^{cd}F_{cd}$$

Which reduces to the expression in the Wikipedia article for the case of flat spacetime. Note that it is still fine to define this in flat spacetime becase:

1) Electromagnetism is perfectly consistent in special relativity

2) There are many limits where the contribution of the electromagnetic field to spacetime curvature is very, very negligible (for instance, nearly every electromagnetic experiment ever run on earth's surface).

• Purely by chance, two hours ago I happened to browse the relevant section in Misner, Thorne and Wheeler. They have no problem defining it in flat spacetime. May 19, 2014 at 15:48

In General Relativity one typically defines the stress-energy tensor from the Lagrangian, by varying it with respect to the metric. This is by assumption the source of the gravitational field. These two tensors are not a priori identical because the stress-energy-tensor you derive from Noether's current isn't necessarily symmetric, while the latter is. So you should symmetrize the former. Alas in practice this doesn't normally cause problems.

The energy-momentum tensor, $$T_{ab} = F^{ac} F_{b}{}^c - \frac{1}{4}g_{ab} F_{de} F^{de},$$ can be defined in curved, torsionless spacetime starting from the definition in Minkowski spacetime, since the two-form $F$ is a tensor in curved, torsionless spacetime. This affects spacetime curvature, as mandated by the Einstein field equation, but extracting useful quantities from it is nontrivial. For example, for a charged, stationary, axisymetric, space-time, solving the Einstein equation yields the Kerr-Newman metric with specific vector potential, from which we obtain the electric, magnetic fields and the charge as an integral in the asymptotic region.