Experimental Tests of GR for Non-Vanishing Tensor

[Geometry_dude]

My question is the following: Have there ever been any experimental tests of the Einstein equation (a.k.a. Einstein field equations) for the case of non-vanishing energy-momentum tensor?
If so, which ones?

I know there's a wikipedia article about tests of general relativity and I have already employed the search function here and of course google, but couldn't find anything specific.

 

[bcrowell]

Cosmology. Neutron stars.

 

[Geometry_dude]

Thank you for your answer. I suppose you mean the following corresponding metrics

• FLRW metric
• Reissner-Nordström/Kerr-Newman metric

Does the latter imply that Einstein-Maxwell theory should be considered a theory as opposed to an hypothesis?