Mass identification in Schwarzschild spacetime

Asked Mar 11 '18 at 16:24

Active Mar 11 '18 at 21:42

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I am doing an exercise where I'm asked to rewrite the Schwarzschild metric using the following coordinate redefinition $$ r=r'\left(1+\frac{M}{2r'} \right)^2. $$

I know the metric, in these the new coordinates, would be the following (isotropic) one:$$ ds^2=-\left(\frac{1-M/2r}{1+M/2r} \right)^2 dt^2+(1+M/2r)^4 (dr^2+r^2d \Omega^2). $$

Then I'm asked to take the limit $r\to \infty$ and identify $M$ with the mass of the black hole. Ok, the problem is that I'm asked to justify why this isotropic change of coordinates is necessary to do the mass identification, and that doing so in the original coordinates (as I have always seen) would be incorrect.

Can anybody explain to me why this is necessary?

Thank you!

Reference: see question $2$ of problem $4$ here: http://www.sissa.it/app/phdsection/pastexams/2011/App11.pdf

edited Mar 11 '18 at 21:42

asked Mar 11 '18 at 16:24

As a side note let me give you a small tip off : You will find "mass" mentioned nowhere in the Einstein Field Equations. On crafting the Schwarzchild metric, there comes a factor that we can identify with mass. Pretty cool... – Yuzuriha Inori Mar 11 '18 at 16:35
How is this related to my question? – Mar 11 '18 at 17:11
Because on solving the equations, we get a condition when without the redefinition, the mass term is not visible (that was the historical approach) – Yuzuriha Inori Mar 11 '18 at 17:15
Btw, $r \to \infty$ does not give the mass of a black hole. It can be identified with the mass of a spherical isolated body. – Yuzuriha Inori Mar 11 '18 at 17:16
What definition of mass are you using? – AccidentalFourierTransform Mar 11 '18 at 21:52
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Related: https://physics.stackexchange.com/q/252044/2451 and links thetein. – Qmechanic Mar 11 '18 at 21:54

Mass identification in Schwarzschild spacetime

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