(adsbygoogle = window.adsbygoogle || []).push({}); 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!

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Radius of a black hole

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

Loading...

Similar Threads for Radius black hole |
---|

B Calculating Weight and mass of entire black hole |

I Gravitational difference between a black hole and a star |

B Solar Eclipse by a Black Hole |

I Relationship between star radius and luminosity |

**Physics Forums | Science Articles, Homework Help, Discussion**