Re: Density of a Black Hole?
From: Tom Roberts (tjroberts_at_lucent.com)
Date: 06/11/04
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Date: Fri, 11 Jun 2004 04:35:40 GMT
Dave Snead wrote:
> The spatial part of the metric for a Schwarzschild black hole is
> ds^2 = 1/(1 - 2 m / r) dr^2 + r^2 dtheta + r^2 sin(theta)^2 dphi
That is NOT the spatial part in 0<r<2M!!!!
Look at the coefficient of dr^2 and see that in that region it is
negative -- in there r is TIMELIKE.
There is no sensible way to compute the "volume" inside the event
horizon of a black hole. The ordinary notion of "volume" really only
makes sense in a static manifold (i.e. one with a timelike Killing
vector and spatial hypersurfaces orthogonal to it) -- only in that case
is there a clear and obvious meaning to "volume" that coincides with our
ordinary notions of what it should mean. Inside the event horizon the
Schwarzschild manifold is not static.
Another way to see that the "density of a black hole" is meaningless is
to realize that a black hole really has no mass at all. To a distant
observer it _ACTS_ as if it has a mass, but the manifold inside a
Schwarzschild black hole is pure vacuum. Its "mass" is "outside the
manifold"....
And I doubt your answer for the integral is correct -- when I ask
Mathematica to do it (http://www.integrals.com ), it is quite
complicated and is singular at r=2M. As it should be....
I ignored the abs(.) you stuck in there with no justification
I can see. But just take either sign: they're both singular.
Tom Roberts tjroberts@lucent.com
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