So suppose you have a geodesic for a small massed particle(say a proton), and are given the initial conditions of the stress energy of the system(at t=0 for simplicity). We do not, however, know the differential form of the Tensor components(R_ab, g_ab, and T_ab). Is it possible to derive a specific solution or set of solutions for the differential geometry given this information? I've been playing around with the math and it seems very difficult.