I'd like to get the metric tensor that describes a non-rotating, uncharged, perfectly spherical black hole whose radius grows with time (it gains mass). For now, I have:
$g_{\mu\nu}=\begin{pmatrix} -c^2\left(1-\dfrac{2GMa}{c^2r}\right) & A & B & C \\ A & a^2\left(1-\dfrac{2GMa}{c^2r}\right)^{-1} & 0 & 0 \\ B & 0 & a^2r^2 & 0 \\ C & 0 & 0 & a^2r^2\sin^2{\theta} \end{pmatrix}$
$a$ is the scale factor of the radius of the black hole. $A$, $B$ and $C$ are the unkown functions I'm trying to find. It must resemble Schwarzschild solution when $a=1$ and FLRW metric at infinity, so all three functions would go to zero when $a=1$ and when $r\,\to\,\infty$. Therefore, all three are $t$ and $r$ dependent and $\propto \left(a-1\right)/r$. Of course, that's because we are scaling black hole's radius by the FLRW scale factor $a$.
I'm doing the calculations by hand to get the Ricci tensor, but it is a lot. Is there a Python library or some tool that lets you compute/simplify these formulas? One that would basically operate and rearrange variables and parameters in such a way that you can rapidly find the three $A$, $B$ and $C$.
metric_to_Christoffel,metric_to_Riemann_components,metric_to_Ricci_componenst. $\endgroup$