1+1D simple vacuum EFE solution

Question

Can there be any solutions for simple vacuum Einstein Field Equations in 1+1D (1 space and 1 time dimension) i.e $R_{\mu\nu} = 0$ except for flat space? I tried different combinations of random Riemann Curvature Tensor Components - it seems finding such a solution is impossible in 1+1D.

Answer 1

In 1+1D the Einstein tensor always vanishes identically, i.e., in any 2D manifold it holds that $R_{ab} = \frac{1}{2} R g_{ab}$. Hence, the Einstein equations reduce to an imposition that the stress tensor must vanish.

In a way, any solution to the Einstein equations in 1+1D is a vacuum solution, since the EFE end up reducing to the imposition that the stress tensor vanishes. As for the geometry, you see that any Lorentzian manifold will do.