Should all theories of gravity have Schwarzschild solution? A consistent theory of gravity must include the Newton's classical theory of gravity as a weak field approximation. Moreover,  to satisfy the experiments in the solar system, the Schwarzschild solution is necessary. Thus, I wonder whether all reasonable theories of gravity should have the Schildschwarz solution. Certainly, I know in some modified gravity no Schwarzschild solution exist. What does this mean?
 A: From an abstract, logical point of view, the Schwarzschild solution does not need to be contained in any candidate theory of gravity. The Schwarzschild solution is unphysical in the sense that no astrophysical process would ever result in a precisely non-rotating, non-charged black hole. Even if it did, we know that through Hawking radiation the solution wouldn't be precisely static. So why should it be a problem if your theory does not contain a solution that would never be found in nature?
I think the above answer is a bit formal and unsatisfactory, because all physical theories are mathematical, and even if a certain symmetric solution never appears in nature, it's existence in the theory is clearly related to the existence of less symmetric, more realistic solutions (such as Kerr-Newman in the case of stationary black holes). So from this perspective I would say that the Schwarzschild solution does not need to be contained in the theory, but something like Schwarzschild must be. By that I mean a static, spherically symmetric solution for which the Newtonian potential goes like $\Phi \sim -GM/r$ at large distances. It should also contain solutions that we could call rotating and charged black holes, even if the precise form of the solutions differed from the Kerr-Newman solution.
