Einstein tensor in 2d

Question

Is the Einstein tensor in 2D or 1+1D always zero? If so, why? I recently installed EinsteinPy and started playing wing different metrics - for the 2D cases the result turned out to be always zero.

Answer 1

Please refer here. From here you will learn that in 2D

$$R_{abcd} = \frac{R}{2}(g_{ac}g_{bd}-g_{ad}g_{cb})$$ Contracting $a$ and $c$ both sides, you will obtain $$R_{bd} = \frac{R}{2}g_{bd} \implies G_{bd} = 0$$