Time-dependent Christoffel symbols 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?
 A: 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.
