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

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?

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

bcrowell
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.

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]

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.

bcrowell
Staff Emeritus