- #1
latentcorpse
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Given the Komar integral
[itex]J(V)=\frac{1}{16 \pi G} \oint_{\partial V} dS_{\mu \nu} D^\mu m^\nu[/itex]
where [itex]V[/itex] is the volume of the spacelike hypersurface [itex]\Sigma[/itex] with boundary [itex]\partial V[/itex] and [itex]m=\frac{\partial}{\partial \phi}[/itex] is the Killing vector field This particula Komar integral is associated with, I am asked to verify that [itex]J=Ma[/itex] for the Kerr-Newman solution with parameter [itex]a[/itex].
I have not really got any idea what to do here since the only definition in my notes of [itex]a[/itex] is [itex]a=\frac{J}{M}[/itex] and so I ended up going round in circles.
I was wondering if I am supposed to extract something from the formula for [itex]J(V)[/itex] that we can write as the mass or the ADM mass or something?
Thanks.
[itex]J(V)=\frac{1}{16 \pi G} \oint_{\partial V} dS_{\mu \nu} D^\mu m^\nu[/itex]
where [itex]V[/itex] is the volume of the spacelike hypersurface [itex]\Sigma[/itex] with boundary [itex]\partial V[/itex] and [itex]m=\frac{\partial}{\partial \phi}[/itex] is the Killing vector field This particula Komar integral is associated with, I am asked to verify that [itex]J=Ma[/itex] for the Kerr-Newman solution with parameter [itex]a[/itex].
I have not really got any idea what to do here since the only definition in my notes of [itex]a[/itex] is [itex]a=\frac{J}{M}[/itex] and so I ended up going round in circles.
I was wondering if I am supposed to extract something from the formula for [itex]J(V)[/itex] that we can write as the mass or the ADM mass or something?
Thanks.