I cannot completely understand what is a regular method to solve Einstein's equations in GR when there are no handy hints like spherical symmetry or time-independence.
E.g. how can one derive Schwarzschild metric starting from arbitrary coordinates $x^0, x^1, x^2, x^3$? I don't even understand the stress-energy tensor form in such a case – obviously it should be proportional to $\delta(x – x_0(s))$, where $x_0(s)$ is a parametrized particle's world-line, but if the metric is unknown in advance how do I get $x_0(s)$ without any a priori assumptions?
Best Answer
First, there is no mechanical algorithm to solve a general differential equation. Einstein's equations are obviously no exception – in fact, they belong among the more complicated and less "solvable" equations among those one may learn about. Analytically writable solutions only exist in very special, simple, and/or symmetric cases (simple enough equations describing simple enough physical situations).
Second, Einstein's equations don't determine the metric uniquely. Even with well-defined initial/boundary conditions, they only determine the solution (metric tensor field) up to a general coordinate transformation (which may be determined by 4 functions $X^\mu(x^\nu)$ of the old coordinates). It means that out of the 10 components of the symmetric metric tensor, only 6 functions are really independently physical. When we impose 4 "gauge-fixing" conditions on the metric tensor field, we effectively define the "right" coordinates and we are left with 6 independent equations for the remaining 6 functions that determine the metric tensor as a function of the coordinates. Einstein's equations are superficially 10 equations but 4 of them (more precisely 4 equations constructed out of the derivatives of these equations and the equations themselves), the covariant divergence $\nabla_\mu (G^{\mu\nu} - K\cdot T_{\mu\nu})=0$, are obeyed identically so they don't constrain the metric.
Third, general relativity may also contain point masses, the point-like sources of the gravitational field that indeed add a delta-function of a sort to the metric tensor. If that's so, general relativity is a coupled system of mutually interacting Einstein's partial differential equations and ordinary differential equations for the world lines which may be parameterized e.g. by $t(x^i)$ or otherwise (e.g. using an auxiliary time parameter along the world line – which requires us to deal with a one-dimensional coordinate transformation redundancy analogous to the four-dimensional above). Alternatively, matter may be described by electromagnetic, Klein-Gordon, Dirac, and other fields. In that case, we deal with a coupled system of many partial differential equations – Einstein's equations plus Maxwell's equations plus the Dirac equation(s) and Klein-Gordon equation(s) with various source terms.