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A Metric induced on Kerr event horizon

  1. Apr 6, 2017 #1
    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
     
  2. jcsd
  3. Apr 6, 2017 #2

    martinbn

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    Science Advisor

    On the horizon ##t## and ##r## are constants, so ##dt=0## and ##dr=0##.
     
  4. Apr 6, 2017 #3

    PeterDonis

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    2016 Award

    Staff: Mentor

    Also ##\Delta = 0## on the horizon.
     
  5. Apr 6, 2017 #4
    That's right. Got it now. Thanks
     
  6. Apr 6, 2017 #5

    PeterDonis

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    2016 Award

    Staff: Mentor

    This isn't quite correct. It should be ##\Delta = r^2 - 2GMr + a^2##.
     
  7. Apr 14, 2017 #6
    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-
    [tex]A_{\pm}=\pm4\pi\left(r^2_{\pm}+a^2\right)[/tex]
    where
    [tex]r_{\pm}=M\pm\sqrt{M^2-a^2}[/tex]
    where [itex]\pm[/itex] denotes the outer ([itex]+[/itex]) and inner ([itex]-[/itex]) 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 [Broken] by Liu Bo Liu & Wen-Biao
     
    Last edited by a moderator: May 8, 2017
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