Existence of solutions to Einstein's equation

[causalset]

It is easy to see, from bianchi identities, that if energy-momentum tensor is not conserved, then Einstein's equation does not have a solution. But is there a proof that if energy momentum tensor IS conserved then Einstein's equation ALWAYS have a solution?

 

[atyy]

I wonder: how do we know the energy-momentum tensor is conserved until we have a solution, since the metric enters the covariant derivative which defines energy momentum conservation?

 

[causalset]

I wonder: how do we know the energy-momentum tensor is conserved until we have a solution, since the metric enters the covariant derivative which defines energy momentum conservation?

What I meant is this: if we have some physics (such as electrodynamics) which is conserved WITHOUT gravity, then the solution continues to exist after we add gravity to it. My question is: how can you prove that is the case?