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Radius of a black hole

  1. Jul 23, 2014 #1
    Definition/Summary

    Mass-equivalent radius: [itex]M[/itex], a mathematically useful distance.

    Inner Ergosphere: [itex]R_{e-}[/itex], the boundary where time-like intervals return to the azimuthal plane for a rotating black hole.

    Inner Event Horizon: [itex]R_-[/itex], the radius inside which there is "normal" space-time.

    Outer Event Horizon: [itex]R_+[/itex], the radius from which no orbit may escape the black hole.

    Outer Ergosphere: [itex]R_{e+}[/itex], the boundary where space begins to rotate faster than the speed of light for a rotating black hole. Objects within the ergoregion cannot remain static in relation to the rest of the universe.

    Radius of Photon Sphere: [itex]R_{ph}[/itex], the radius where a photon may be in circular orbit around the black hole. The orbit is unstable and the photon either falls into the black hole or escapes.

    Radius of Marginally Bound Orbit: Apparently the radius, [itex]R_{\text{mb}}[/itex] where a test particle starts (as viewed from infinity) to be gravitationally bound by the black hole.

    Radius of Marginally Stable Orbit: [itex]R_{\text{ms}}[/itex], radius of smallest circular orbit for material, usually the radius of the inner edge of the accretion disk.

    Equations

    Mass-equivalent radius for body of mass m:

    [tex]M\ =\ \frac{Gm}{c^2}[/tex]

    Non-rotating uncharged spherically symmetric body (Schwarzschild solution):

    [tex]R_-\ =\ R_+\ =\ \frac{2Gm}{c^2}\ =\ 2M[/tex]

    [tex]R_{ph}\ =\ \frac{3Gm}{c^2}\ =\ 3M[/tex]

    [tex]R_{mb}\ =\ \frac{4Gm}{c^2}\ =\ 4M[/tex]

    [tex]R_{ms}\ =\ \frac{6Gm}{c^2}\ =\ 6M[/tex]

    Rotating uncharged spherically symmetric body with angular momentum [itex]aMc[/itex] (Kerr solution):

    [tex]R_\pm\ =\ M\ \pm\ \sqrt{M^2\ -\ a^2}[/tex]

    [tex]R_{e\pm}\ =\ M\ \pm\ \sqrt{M^2\ -\ a^2\ cos^2\ \theta}[/tex]

    [itex]R_{ph}[/itex], [itex]R_{mb}[/itex] & [itex]R_{ms}[/itex] have prograde and retrograde orbits around a rotating black hole. The upper sign characterizes the prograde orbit (corotating with the black hole) and the lower sign holds for the retrograde orbit (counterrotating against the black hole)

    [tex]R_{ph}\ =\ 2M\left[1\ +\ cos\left(\frac{2}{3}cos^{-1}\mp \frac{a}{M}\right)\right][/tex]

    [tex]R_{mb}\ =\ \left(\sqrt{M}\ +\ \sqrt{M \mp a}\right)^2\ =\ 2M\ \mp \ a\ +\ 2\sqrt{M(M \mp a)}[/tex]

    [tex]R_{ms}\ =\ M\left(3+Z_2 \mp \sqrt{(3-Z_1)(3+Z_1+2Z_2)}\right)[/tex]

    where

    [tex]Z_1=1+\left(1-\frac{a^2}{M^2}\right)^{1/3}\left(\left(1+\frac{a}{M}\right)^{1/3}+\left(1-\frac{a}{M}\right)^{1/3}\right)[/tex]

    [tex]Z_2=\sqrt{3\frac{a^2}{M^2}+Z_1^2}[/tex]

    Extended explanation

    Fast rotating black hole:

    For a "fast rotating black hole", with angular momentum [itex]aMc[/itex] greater than [itex]M^2c[/itex], [itex]R_-[/itex] [itex]R_+[/itex] and [itex]R_{ms}[/itex] do not exist, and so there is no event horizon, and no minimum circular orbit, and the black hole has a "naked" singularity.

    * This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
     
  2. jcsd
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