# Kruskal-Szekeres coordinates

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## Main Question or Discussion Point

I'm currently looking at Kruskal-Szekeres coordinates in relation to a static black hole.

For the exterior region, the coordinates are-

$$R=\left(\frac{r}{2GM}-1\right)^{1/2}e^{r/4GM}cosh\left(\frac{t}{4GM}\right)$$

For the interior-

$$R=\left(1-\frac{r}{2GM}\right)^{1/2}e^{r/4GM}sinh\left(\frac{t}{4GM}\right)$$

where c = 1 and G, the gravitational constant, is kept explicit.

I've looked extensively on the web but can find very little as to explaining the equations in a bit more depth. I'd appreciate it if someone could shed some light on what e represents and while I'm certain that c^2 appears adjacent to r in the first set of brackets, does it appear anywhere else in the equations? Also, would it be correct to assume that t represents time?

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Wallace
Check out http://casa.colorado.edu/~ajsh/schwp.html" [Broken] site. It goes through the various co-ordinate representations of black holes and has some nice animations as well. I think it should answer your question.

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Gold Member
Thanks for the link Wallace. I also found a paper that covered the subject 'Kruskal Coordinates and Mass of Schwarzschild Black Holes by' Abhas Mitra-

http://arxiv.org/abs/astro-ph/9904162

Unfortunately, neither actually state what the quantities e and t are, I can only assume that e is energy and t is time but don't see how they would be incorporated into the equations. It appears to be taken for granted that e and t are understood but would appreciate confirmation as to what they are.

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Wallace
t is time, though of course it is a different time co-ordinate to that appearing in the Schwarschild metric. I'm pretty sure that the e is just http://en.wikipedia.org/wiki/E_%28mathematical_constant%29" [Broken] (i.e. the same e as in Log_e = Ln).

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George Jones
Staff Emeritus
Gold Member
Thanks for the link Wallace. I also found a paper that covered the subject 'Kruskal Coordinates and Mass of Schwarzschild Black Holes by' Abhas Mitra-

http://arxiv.org/abs/astro-ph/9904162
Careful; the main results of this infamous and unpublished paper are quite wrong.

Gold Member
Thanks for the heads up George, I was under the impression that if a paper was on the arxiv website then it had passed some seal of approval, that doesn't appear to be the case; I'm assuming that Kruskal-Szekeres coordinates are still legite though. Regarding e being a constant and probably the log of something, what exactly would it be the log of?

Gold Member
I twigged within a couple of minutes of my reply that e is a constant as stipulated in the wikipedia link; Physics Forums appears to undergo some kind of maintanence around 7.45 and 8.15 am GMT (which would be around midnight PDT) so I couldn't edit my post. Does anyone have an idea of how time would be incorporated as t?

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Careful; the main results of this infamous and unpublished paper are quite wrong.
hello, i would like to know what result in the paper is wrong? it is the transformation itself?

Wallace
I haven't read that paper in detail, but from the abstract it is drawing physical significance from the properties of a particular co-ordinate system. This is a big no no! Anything with physical meaning will be invariant (not co-ordinate dependant). So for instance, you should be able to demonstrate what that paper claims to show in the Schwarschild co-ordinate system, or any other of the many BH co-ordinate charts.

Wallace