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.
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1$\begingroup$ There is no Einstein gravity in 2D. (The field equations vanish trivially, which can be seen because the action, the Ricci scalar, takes the form of a total derivative) $\endgroup$– EletieCommented Mar 30, 2023 at 9:17
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1$\begingroup$ Related: physics.stackexchange.com/q/1417/2451 , physics.stackexchange.com/q/200346/2451 and links therein. $\endgroup$– Qmechanic ♦Commented Mar 30, 2023 at 9:20
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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.
answered Mar 30, 2023 at 10:12