I am trying to understand a derivation posted by Pervect a long time ago (I suspect Pevect is no longer active) that involves differentiation and I was hoping someone here could fill in some of the steps to make it clearer.(adsbygoogle = window.adsbygoogle || []).push({});

The original posts by Pervect are here:

https://www.physicsforums.com/newreply.php?do=newreply&p=1019121 [Broken]

https://www.physicsforums.com/showpost.php?p=1020560&postcount=5

I have taken the liberty of posting a cut down version of Pervects posts here. I have added equation numbers and comment. The comments in blue are just notes and the comments in magenta are the places where I am stuck and need some more explanation.

Paraphrased said:Let us start with the initial Schwarzschild metric:

[tex] (Eq1.1) \hskip 1 cm ds^2 = -(1-2M/r) dt^2 + 1/(1-2M/r) dr^2 + r^2 d\Omega^2 [/tex]

Now make the following variable substitutions. (These are from Wald, not that it particularly matters BTW).

(Eq1.2) (r/2M - 1) exp(r/2M) = X^2 - T^2

(Eq1.3) (t/2M) = ln [(X+T)/(X-T)]

Note that these are of the form Comment said:

It is useful to know at this point that the last two equations are easily obtained from the Kruskal coordinates which are defined as:

[tex]X=e^{(r/2M)}*\sqrt{(r/2M-1)}*cosh(t/4M) [/tex]

[tex]T=e^{(r/(2M))}*\sqrt{(r/(2M)-1)}*sinh(t/4M) [/tex]

REf: http://en.wikipedia.org/wiki/Kruskal-Szekeres_coordinates

(Eq1.4) (X+T)(X-T) = f(r)

(Eq1.5) (X+T)/(X-T) = g(t)

hence we can solve them for X+T = sqrt(f*g) and X-T = sqrt(f/g)

We can then write: Comment said:

So from Eq 1.2 and Eq1.3

[tex]X+T= \sqrt{(r/2M - 1)} e^{(r/4M)}e^{(t/4M)} [/tex]

[tex]X-T= \sqrt{(r/2M - 1)} e^{(r/4M)}e^{(-t/4M)} [/tex]

[tex] (Eq1.6) \hskip 1 cm dr = 4\, \left( 2\,X{\it dX}-2\,T{\it dT} \right) {M}^{2} \left( {e^{{

\frac {r}{2M}}}} \right) ^{-1}{r}^{-1}

[/tex]

[tex] (Eq1.7) \hskip 1 cm dt = 2\,M \left( 2\,X{\it dT}-2\,T{\it dX} \right) \left( {\frac {r}{

2M}}-1 \right) ^{-1} \left( {e^{{\frac {r}{2M}}}} \right) ^{-1}

[/tex]

Comment said:

OK, this is one part I need help with. How does Pervect obtain dr and dt in Eq1.6 and Eq1.7?

Is he using ordinary or partial differential equations?

Is his result correct?

I am not too good with differentials so do not worry about over explaining ;)

Now we can re-write the Schwarzschild metric, with r(X,T) implicictly defined by

(Eq1.8) (r/2M - 1) exp(r/2M) = X^2 - T^2

as

[tex] (Eq1.9) \hskip 1 cm \frac{32 M^3 e^{r/2M}}{r}(-dT^2 + dX^2)+r^2 d\Omega^2[/tex]

We see that the new expression is perfectly finite in the new variables X,T at r=2M (which is at X=T), removing the coordinate singularity at the event horizon (r=2M, or X=T). Comment said:

There is a typo in the Kruskal metric here. Eq1.9 should read:

[tex] (Eq1.9b) \hskip 1 cm \frac{32 M^3 e^{-r/2M}}{r}(-dT^2 + dX^2) + r^2 d \Omega^2

[/tex]

REf: http://en.wikipedia.org/wiki/Kruskal-Szekeres_coordinates

However, a singularity remains at r=0. We know we can't eliminate that because the curvature scalar diverges.

In short, a "simple" (it's simple with computer algebra, anyway) variable substitution eliminates the singularity at the event horizon.

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