Compute Induced Metric on Kerr Event Horizon

[Jonsson]

Hello there,

Suppose $\Delta = r^2 + 2GMr + a^2$ and $\rho^2 = r^2 + a^2 \cos ^2 \theta$. The Kerr metric is
$$
ds^2 = - (1 - \frac{2GMr}{\rho^2})dt^2 - \frac{4GMar\sin^2 \theta}{\rho^2} d t d \phi + \frac{\rho^2}{\Delta} dr^2 + \rho^2 d \theta^2 + \frac{\sin^2 \theta}{\rho^2} \left[ (r^2 + a^2)^2 - a^2 \Delta \sin^2 \theta \right] d \phi^2
$$

I want to determine the area of the inner horizon of the black hole. But first I need to determine the induced metric on the horizon. Suppose that the radius of the innermost horizon is $r_*$. My tutor stated without proof that the induced metric was
$$
ds^2 = \rho_*^2 d \theta^2 + \frac{\sin ^2 \theta}{\rho_*^2} (r_*^2 + a^2) d \phi^2
$$
How do one compute this induced metric from the Kerr metric?

Thank you for your time.

Kind regards,
Marius

 

[martinbn]

On the horizon $t$ and $r$ are constants, so $dt=0$ and $dr=0$.

 

[PeterDonis]

On the horizon $t$ and $r$ are constants, so $dt=0$ and $dr=0$.

Also $\Delta = 0$ on the horizon.

 

[Jonsson]

That's right. Got it now. Thanks

 

[PeterDonis]

Suppose $\Delta = r^2 + 2GMr + a^2$

This isn't quite correct. It should be $\Delta = r^2 - 2GMr + a^2$.

 

[stevebd1]

You're probably working through things yourself but if you want a check, the equation for the surface area of both the inner and outer horizon in Kerr metric is-
A_{\pm}=\pm4\pi\left(r^2_{\pm}+a^2\right)
where
r_{\pm}=M\pm\sqrt{M^2-a^2}
where \pm denotes the outer (+) and inner (-) horizon.

One source-
https://redirect.viglink.com/?format=go&jsonp=vglnk_149222179481113&key=6afc78eea2339e9c047ab6748b0d37e7&libId=j1illvfl010009we000DApptnkylp&loc=https%3A%2F%2Fwww.physicsforums.com%2Fthreads%2Fsecond-law-of-thermodynamics-and-the-macroscopic-world.833987%2F%23post-5238097&v=1&out=http%3A%2F%2Fwww.researchgate.net%2Fpublication%2F230923684_Entropy_of_Kerr-Newman_Black_Hole_Continuously_Goes_to_Zero_when_the_Hole_Changes_from_Nonextreme_Case_to_Extreme_Case&ref=https%3A%2F%2Fwww.physicsforums.com%2Fsearch%2F5021084%2F%3Fq%3Dkerr%26o%3Drelevance%26c%5Buser%5D%5B0%5D%3D68677&title=Second%20law%20of%20thermodynamics%20and%20the%20macroscopic%20world%20%7C%20Physics%20Forums%20-%20The%20Fusion%20of%20Science%20and%20Community&txt=Negative%20Temperature%20of%20Inner%20Horizon%20and%20Planck%20Absolute%20Entropy%20of%20a%20Kerr-Newman%20Black%20Hole by Liu Bo Liu & Wen-Biao