I Integration of energy-momentum pseudotensor for Kerr BH

[anuttarasammyak]

TL;DR

Integration of energy-momentum pseudotensor for Kerr BH

Once, I calculated the integral of the Einstein energy–momentum pseudotensor for a Schwarzschild black hole in the region outside the Schwarzschild surface and obtained the value m, corresponding to the mass of the black hole.

Performing the same calculation for a Kerr black hole would be quite laborious. May I expect to obtain the mass m and the angular momentum M in a similar manner? Is it also possible to obtain the same results using the Landau–Lifshitz pseudotensor?

 

[PeterDonis]

Once, I calculated the integral of the Einstein energy–momentum pseudotensor for a Schwarzschild black hole in the region outside the Schwarzschild surface and obtained the value m, corresponding to the mass of the black hole.

How? This is not quite as straightforward as you seem to think.

Performing the same calculation for a Kerr black hole would be quite laborious. May I expect to obtain the mass m and the angular momentum M in a similar manner?

I have no idea unless you explain how you did the Schwarzschild calculation.

Is it also possible to obtain the same results using the Landau–Lifshitz pseudotensor?

Same answer as above.

 

[anuttarasammyak]

From "general theory of relativity" by Dirac, as for Einstein pseudotensor,
(32.3)
where $\mathcal{L}=\sqrt{}\ L$ with
(26.3)
, we got the formula for Schwartzshild coordinates and metric given by
(18.6)
and integrated in volume outside the Schwartzshild surface, r > 2m as
(31.4)

I did it with a friend of mine online before websites become familiar. We were excited to see that BH has its energy all outside the Schwartzshild surface.

[EDIT] I deleted the equation images by the suggestion post #4.

 

[PeterDonis]

From "general theory of relativity" by Dirac, as for Einstein pseudotensor,
View attachment 367131
where $\mathcal{L}=\sqrt{}\ L$ with
View attachment 367132
, we got the formula for Schwartzshild coordinates and matrix given byView attachment 367134
and integrated in volume outside the Schwartzshild surface, r > 2m as
View attachment 367135
I did it with a friend of mine online before websites become familiar. We were excited to see that BH has its energy all outside the Schwartzshild surface.

Equations in images are not permitted here. Please use the PF LaTeX feature.

 

[PeterDonis]

We were excited to see that BH has its energy all outside the Schwartzshild surface.

This is not a correct claim. You need to be very, very careful interpreting anything involving pseudotensors. The Schwarzschild BH is vacuum; it has no stress-energy anywhere. The "mass" $M$ of the hole is a global, geometric property of the spacetime; it can't be located in any particular place or region.

There are invariant mass integrals that can be done for Schwarzschild spacetime (ADM mass, Bondi mass, Komar mass--though the latter involves more technicalities). But none of them involve pseudotensors.