# Applying variational principles to that metric describes a black hole!

physx_420
ds$$^{2}$$ = -c$$^{2}$$(1 - $$\frac{2Gm}{c^{2}r}$$)dt$$^{2}$$ + (1 - $$\frac{2Gm}{c^{2}r}$$)$$^{-1}$$ dr$$^{2}$$ + r$$^{2}$$d$$\Omega$$$$^{2}$$

This equation was posted on a different website and the O.P said:"Applying variational principles to that metric describes a black hole!"

I was wondering if anyone could explain it a little better. Also, to anyone knows who Miguel Alcubierre is (the guy that created an equation for a hypothetical warp-drive); the above equation shows some similarities to his:

ds$$^{2}$$ = -dt$$^{2}$$ + (dx - v$$_{s}$$f(r$$_{s}$$dt)$$^{2}$$ + dy$$^{2}$$ + dz$$^{2}$$

Does this have any implications, be they big or small? Anyone have any inputs on this?

physx_420
in M. Alcubierre's equation the "s" in the superscript of "v and r" are supposed to be subscripts, I just couldn't get them to work. btw

Staff Emeritus
Gold Member
This equation was posted on a different website and the O.P said:"Applying variational principles to that metric describes a black hole!"

Unless I'm missing something, you can cut the part about "Applying variational principles to..." The correct statement would simply be: "[T]hat metric describes a black hole!" This is simply the standard form of the Schwarzschild metric, as far as I can see.

Mentz114
The first metric you show is the Scwarzschild exterior of a radially symmetric source with a singularty at r=0 and a horizon at r=2GM/c^2.

Look up 'Scwarzschild metric' on Wiki.

[Ben - snap]

yuiop
Unless I'm missing something, you can cut the part about "Applying variational principles to..." The correct statement would simply be: "[T]hat metric describes a black hole!" This is simply the standard form of the Schwarzschild metric, as far as I can see.

I think we should acknowledge that the standard Schwarzschild metric can also represent the vacuum region outside a regular non-rotating uncharged non-singular massive body that is not a black hole.

Staff Emeritus