There are several useful known solutions to the EFE with relatively simple / trivial stress-energy-momentum tensor, such as the Schwarzschild solution. Despite the idealizations made therein they are enormously important and insightful.
I am curious to know
Do any such solutions exist for the case of relativistic elastic solids?
If not, what are the principal obstacles in constructing solutions, even approximate (say linearized) ones?
If I understand correctly, various competing versions of a "theory" of relativistic elasticity have been around for several decades. Is there a reason why there are multiple theories?
Also, IIUC, matters of existence and uniqueness for the Cauchy problems arising from such theories have also been dealt with. I came across this paper by Beig and Schmidt, in which the authors deal with questions of local existence and uniqueness for a theory they describe as a "relativistic version of classical elasticity". Is this a generally accepted state of affairs or are there caveats?