- #1
inottoe
- 25
- 0
Hi. I can't find a source that shows how to superpose two metrics.
For example, superposing Schwarzschild metric
with de Sitter metric
yields de Sitter-Schwarzschild metric
I've tried letting
which works for the time component of the metric but not the radial. Any ideas?
For example, superposing Schwarzschild metric
[itex]ds^2=\left(1-\frac{2M}{r}\right)dt^2-\frac{dr^2}{1-\frac{2M}{r}}-r^2d\Omega^2[/itex]
with de Sitter metric
[itex]ds^2=\left(1-\frac{r^2}{\alpha^2}\right)dt^2-\frac{dr^2}{1-\frac{r^2}{\alpha^2}}-r^2d\Omega^2[/itex]
yields de Sitter-Schwarzschild metric
[itex]ds^2=\left(1-\frac{2M}{r}-\frac{r^2}{\alpha^2}\right)dt^2-\frac{dr^2}{1-\frac{2M}{r}-\frac{r^2}{\alpha^2}}-r^2d\Omega^2[/itex]
I've tried letting
[itex]g_{\mu\nu}=g_{\mu\nu}\left(Schwarzschild\right)+g_{\mu\nu}\left(de Sitter\right)-\eta_{\mu\nu}[/itex]
which works for the time component of the metric but not the radial. Any ideas?