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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1There 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) – Eletie Mar 30 '23 at 09:17
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1Related: https://physics.stackexchange.com/q/1417/2451 , https://physics.stackexchange.com/q/200346/2451 and links therein. – Qmechanic Mar 30 '23 at 09:20
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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.
Níckolas Alves
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