Einstein Field Equations and Electromagnetic Stress-Energy Tensor

Question

My question is: if we write Einstein field equations in this form:

$$R_{\mu\nu} - \dfrac{1}{2}g_{\mu\nu}R=8\pi \dfrac{G}{c^4}T_{\mu\nu}$$

Then the left hand side is one statement about the geometry of space-time and the right hand side is one statment about the distribution of momentun and energy right? My point is: what if we use the electromagnetic stress-energy tensor as the energy-momentum tensor?

My thought was: if I understood correctly, does this says that electromagnetic fields can also change the geometry of space-time making it bend as does the presence of matter?

Sorry if it makes no sense, or if it's completely nonsense, it's just a thought that came out, I'm just starting to study those things.

Answer 1

Yes. It does in fact mean that electromagnetic fields can also change the geometry of spacetime. Anything with energy and/or momentum affects the geometry of spacetime because, as you point out, the gravitational field equations exhibit a coupling of spacetime geometry to energy-momentum.

For more info in the case of electromagnetism coupling to gravity, see THIS.

Answer 2

Yes, all energy-momentum distributions have an effect on the geometry of spacetime, including the the energy-momentum due to the electromagnetic field. For example, the Reissner-Nordstrom solution (which describes a charged, non-rotating spherical mass) has extra terms in it which are dependent on charge. This is because the EM field generated by the charge also has an effect on the geometry of spacetime.

Also note that when you're working with the EM field, the Einstein Field Equations simplify a bit. The stress-energy of the EM field is traceless, which implies $R=0$. So the EFE's simplify to just:

$$R_{\mu \nu} = \frac{8 \pi G}{c^4} ~T_{\mu \nu}$$