Can two metrics be superposed?

[inottoe]

Hi. I can't find a source that shows how to superpose two metrics.

For example, superposing Schwarzschild 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-Schwarzschild 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

$g_{\mu\nu}=g_{\mu\nu}\left(Schwarzschild\right)+g_{\mu\nu}\left(de Sitter\right)-\eta_{\mu\nu}$​

which works for the time component of the metric but not the radial. Any ideas?

 

[WannabeNewton]

Can you even add them like that? The solutions aren't linear.

 

[inottoe]

Can you even add them like that? The solutions aren't linear.

I'm pretty sure you can - you're just adding matrices.

 

[WannabeNewton]

No what I mean is that linear superpositions of two metrics won't result in another solution to the EFEs so why do you want to add them like that.

 

[inottoe]

No what I mean is that linear superpositions of two metrics won't result in another solution to the EFEs so why do you want to add them like that.

I didn't know that. I'm basically stuck as to how to superpose the two metric spaces. Maybe the answer is obvious and I'm being thick.

 

[WannabeNewton]

It isn't easy to properly superpose two metrics. Unlike electric fields under maxwell's equations, which linearly superpose, in GR two interacting gravitational fields present a much, much more complicated interaction (release of gravitational waves etc.) because the gravitational field is coupled to itself. You can simply add the two matrices, sure, but that won't give you anything physically relevant; combining two metrics is non - trivial. There is a linearized form of the EFEs under which linear superposition does result in another solution but the linearized EFEs are only valid for weak fields (like low amplitude gravitational waves in vacuum).

 

[inottoe]

OK, thanks very much for that.