What is the most compelling evidence of General Relativity in the presence of matter and energy?

Question

The most oft-cited [triumphs](http://en.wikipedia.org/wiki/Tests_of_general_relativity) of GR are things such as the shifting perihelion of Mercury, gravitational redshift experiments, and gravitational lensing. But, as far as I know, these are only verifications of Einstein's equation in a vacuum: $$R_{\\mu\\nu}=0.$$(In fact, I think the former two are predicted in the Schwarzchild metric, the spherical solution to the vacuum equation.)But what do we have as far as compelling evidence that the Einstein equation is valid in the presence of matter and energy? i.e.$$R_{\\mu\\nu} - \\frac{1}{2} R g_{\\mu\\nu} = 8 \\pi G T_{\\mu\\nu}.$$Are we simply happy with this because it is the simplest form of an equation in the presence of matter and energy where the left hand side has the same conservation law as the right side (due to the Bianchi identity)? Or are there actual experimental/observational verifications of solutions to this equation where $T_{\\mu\\nu}$ is not trivial?EDIT: I just want to be clear. There's a difference between "_in_ the presence of matter and energy" and "_near_ the presence of matter and energy". People keep on citing verifications of the vacuum field equation $R_{\\mu\\nu}=0$ that happen to have nonzero curvature because of nearby matter (the Schwarzchild metric is an example of this). The _full_ Einstein equation comes to play when $T_{\\mu\\nu} \\ne 0$, and this describes the expression of the metric tensor at the _same_ point in the manifold where T is nonzero, not just a nearby point. I'm asking if anybody knows of any tangible verification of a solution to the Einstein equation for a _nonzero_ energy-momentum tensor, so that excludes most of the examples people gave. And if there is none, how confident are we to assume the full equation just by symmetry arguments alone? I hope this makes things clearer.

Answer 1

Think for example about the work of LIGO on finding gravitational waves of two collapsing black holes. To do this, they had to solve Einstein's equations for two moving black holes using computational methods. In this case the momentum-energy tensor is not only non-zero (at least not in the regions where the black holes are passing by), but also is a dynamic one which is changing in time.

Also another easier example is the Friedmann–Lemaître–Robertson–Walker (FLRW) metric. This one is a cosmological solution for a expanding universe filled with a uniform and isotropic fluid of matter. Again $T_{\mu\nu}$ is non-zero in this case. (It is nowhere zero!)