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I recently read this Phys.SE post and, since I didn't know that black holes had a spin, a question came to my mind: how can I calculate the spin velocity of a black hole? Does mass or radius affects it? I googled it but I couldn't understand much, all I found was about Orbital Velocity of a planet...

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  • $\begingroup$ There exists an upper bound for the angular momentum and charge of a black hole depending on its mass. $\endgroup$ – auxsvr Apr 25 '14 at 20:42
  • $\begingroup$ Black holes don't have a defined "spin velocity", but do have an angular momentum. $\endgroup$ – Rob Jeffries May 22 at 8:58
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Black holes don't have a "spin velocity", since there is no surface at which to measure the rotation speed. Instead, they are characterised by an angular momentum $J$ and a specific angular momentum $J/M$.

A non-zero specific angular momentum changes the space-time metric around the black hole - one uses the Kerr metric, rather than the Schwarzschild metric. This in turn changes the dynamics of material orbiting within a few Schwarzschild radii of the event horizon. In particular it alters the radius of the innermost stable circular orbit (ISCO). The dynamics of orbiting material can be measured if it is sufficiently luminous. This is often the case in accreting black holes, where accreting matter is compressed, becomes very hot and therefore emits lots of electromagnetic radiation. Detailed fitting of the profiles of spectral features can be used to estimate $J/M$. An example of this approach can be found in Patrick et al. (2011), where X-ray emission line profiles are used to constrain $J/M$ in the accreting supermassive black holes at the centres of active galactic nuclei. A similar approach has been employed for accreting black holes in stellar binary systems (e.g. Chen et al. 2016).

A second approach, that has become apparent since this question was posted is to look at the gravitational wave signature of merging black hole binaries. The spin of the black hole components imprints a subtle signature on the gravitational wave signals. At present it appears that the data are most sensitive to the relative orientation of the spin axes of the black holes, rather than the magnitude of the individual spins (e.g. Bailes et al. 2017).

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