Time-dependent Christoffel symbols

Question

I was studying the Schwarzschild metric, and I found out that all of the Christoffel symbols aren't time-dependent. This is because the nonzero Christoffel symbols of the Schwarzschild metric are:

$$\Gamma^0_{01} =\Gamma^0_{10} =\frac{1}{2}\frac{1}{A}(\partial_rA)$$

$$\Gamma^1_{00} =\frac{1}{2}\frac{1}{B}(\partial_rA)$$

$$\Gamma^1_{11} =\frac{1}{2}\frac{1}{B}(\partial_rB)$$

$$\Gamma^1_{22} =-\frac{r}{B}$$

$$\Gamma^1_{33} =-\frac{r(\sinθ)^2}{B}$$

$$\Gamma^2_{33} =-\sin\theta\cos\theta$$

$$\Gamma^2_{12} =\Gamma^2_{21} =\Gamma^3_{13} =\Gamma^3_{31} =\frac{1}{r}$$

$$\Gamma^3_{23} =\Gamma^3_{32} =\cot\theta$$

But since $A$ and $B$ are not time-dependent, none of these Christoffel symbols should be time-dependent, therefore performing actions such as calculating a derivative of the Christoffel symbols with respect to time would always equal to zero

$$\frac{\mathrm d \Gamma^\alpha_{\beta\nu}}{\mathrm dx^0}=0$$

But I know there are plenty of other metrics besides the Schwarzschild metric (Kerr metric, Riessner-Nordstrom metric, Kerr-Newman metric). So my question is, are there any Christoffel symbols in any other metrics that are time-dependent (hence make the term $\frac{\mathrm d\Gamma^\alpha_{\beta\nu}}{\mathrm dx^0}$ nonzero)? If so, which ones?

Answer 1

Yes there are. It is well known that the Schwarschild metric can be generalized to fulfill the field equations with dark energy, called the Schwarzschild-de Sitter metric: $$\mathrm d s^2 =\left(1-\frac{r_\mathrm S}{r}-\frac{\Lambda}{3}r^2\right)c^2\mathrm dt^2 -\left(1-\frac{r_\mathrm S}{r}-\frac{\Lambda}{3}r^2\right)^{-1}\mathrm dr^2 -r^2\mathrm d\Omega^2,$$ which is a static solution and reduces back to the Schwarzschild metric for $\Lambda=0$. Amir Abbassi presents a non-static solution in his paper "Non-Static Spherically Symmetric solution of Einstein vacuum Field Equations with $\Lambda$" found here, which is given by: $$\rho=re^{t\sqrt\frac{\Lambda}{3}}$$ $$\sigma=1-\frac{r_\mathrm S}{\rho}-\frac{\Lambda}{3}\rho^2$$ $$\mathrm ds^2 =\frac{1}{2}\left(\sqrt{\sigma^2+\frac{4\Lambda}{3}\rho^2}+\sigma\right)\mathrm dt^2 -2e^{2t\sqrt\frac{\Lambda}{3}}\left(\sqrt{\sigma^2+\frac{4\Lambda}{3}\rho^2}+\sigma\right)^{-1}\mathrm dr^2 -\rho^2\mathrm d\Omega^2,$$ which includes countless time-dependend Christoffel symbols (all formulas found in the paper) and also reduces back to the Schwarzschild metric for $\Lambda=0$.

By the way, Soheila Gharanfoli and Amir Abbassi also present a generalization of this solution with an arbitary constant $\alpha$ (which can also be done with the Schwarzschild metric) in in their paper "General Non-Static Spherically Symmetric Solutions of Einstein Vacuum Field Equations with $\Lambda$" found here.