inottoe
Aug26-11, 12:09 PM
Hi. I can't find a source that shows how to superpose two metrics.
For example, superposing Schwarzchild metric
ds^2=\left(1-\frac{2M}{r}\right)dt^2-\frac{dr^2}{1-\frac{2M}{r}}-r^2d\Omega^2
with de Sitter metric
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
yields de Sitter-Schwarzchild metric
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
I've tried letting
g_{\mu\nu}=g_{\mu\nu}\left(Schwarzchild\right)+g_{ \mu\nu}\left(de Sitter\right)-\eta_{\mu\nu}
which works for the time component of the metric but not the radial. Any ideas?
For example, superposing Schwarzchild metric
ds^2=\left(1-\frac{2M}{r}\right)dt^2-\frac{dr^2}{1-\frac{2M}{r}}-r^2d\Omega^2
with de Sitter metric
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
yields de Sitter-Schwarzchild metric
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
I've tried letting
g_{\mu\nu}=g_{\mu\nu}\left(Schwarzchild\right)+g_{ \mu\nu}\left(de Sitter\right)-\eta_{\mu\nu}
which works for the time component of the metric but not the radial. Any ideas?