# “Simple” solutions to the Einstein Field equation

Looking at Wikipedia articles about solutions for the Einstein Field Equation (fluid solution, dust solution) I hoped to find a table relating simple forms of the stress-energy tensor to the metric tensor.

For example, given the simplest non-zero stress energy tensor I can think of, with $$T^{00}(x^0,x^1,x^2,x^3)=\rho,$$ $$\forall (x^0,x^1,x^2,x^3)\in\mathbf{R}^4$$ and $$T^{\mu\nu}=0$$ for $$(\mu,\nu)\neq (0,0)$$, how does the metric tensor $$g_{\mu\nu}(x^0,x^1,x^2,x^3)$$ look like (in some basis)?

• Do you want the energy density $\rho$ to be a constant, independent of $t$ and any spatial coordinates? – G. Smith Oct 27 '19 at 17:40
• @G.Smith wouldn't that just be a pressure-less dust solution? – Kyle Kanos Oct 27 '19 at 19:13
• @KyleKanos It seems so to me. Dusts have energy density but no pressure. But they can have time-varying and, I think, space-varying energy density, which is why I asked the OP about conditions on $\rho$. – G. Smith Oct 27 '19 at 19:34
• @G.Smith: yes, that's what I tried to state with the "everywhere" now replaced with a $\forall$. – Harald Oct 29 '19 at 12:38

(2) This also isn't going to work because we don't normally have a unique solution for a given stress-energy tensor. For example, the simplest stress-energy is the vacuum, and this happens to be one that we can write in a coordinate-independent way, $$T=0$$. But there are many vacuum solutions, including a flat spacetime, a flat spacetime with a nontrivial topologies, the Schwarzschild spacetime (maximally extended or not), spacetimes containing $$n$$ black holes, and all kinds of gravitational wave spacetimes.