# Radius of a black hole

1. Jul 23, 2014

### Greg Bernhardt

Definition/Summary

Mass-equivalent radius: $M$, a mathematically useful distance.

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

Inner Event Horizon: $R_-$, the radius inside which there is "normal" space-time.

Outer Event Horizon: $R_+$, the radius from which no orbit may escape the black hole.

Outer Ergosphere: $R_{e+}$, 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: $R_{ph}$, 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, $R_{\text{mb}}$ where a test particle starts (as viewed from infinity) to be gravitationally bound by the black hole.

Radius of Marginally Stable Orbit: $R_{\text{ms}}$, 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:

$$M\ =\ \frac{Gm}{c^2}$$

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

$$R_-\ =\ R_+\ =\ \frac{2Gm}{c^2}\ =\ 2M$$

$$R_{ph}\ =\ \frac{3Gm}{c^2}\ =\ 3M$$

$$R_{mb}\ =\ \frac{4Gm}{c^2}\ =\ 4M$$

$$R_{ms}\ =\ \frac{6Gm}{c^2}\ =\ 6M$$

Rotating uncharged spherically symmetric body with angular momentum $aMc$ (Kerr solution):

$$R_\pm\ =\ M\ \pm\ \sqrt{M^2\ -\ a^2}$$

$$R_{e\pm}\ =\ M\ \pm\ \sqrt{M^2\ -\ a^2\ cos^2\ \theta}$$

$R_{ph}$, $R_{mb}$ & $R_{ms}$ 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)

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

$$R_{mb}\ =\ \left(\sqrt{M}\ +\ \sqrt{M \mp a}\right)^2\ =\ 2M\ \mp \ a\ +\ 2\sqrt{M(M \mp a)}$$

$$R_{ms}\ =\ M\left(3+Z_2 \mp \sqrt{(3-Z_1)(3+Z_1+2Z_2)}\right)$$

where

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

$$Z_2=\sqrt{3\frac{a^2}{M^2}+Z_1^2}$$

Extended explanation

Fast rotating black hole:

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

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