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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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PunkZebra
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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 $J/M$ (in units where $G=1$, $c=1$).

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).

ProfRob
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  • I heard that the speed of things that moves around a black hole is roughly around 1/3 the speed of light, which should make sense with how Black Holes distort spacetime. Source for reference: https://www.space.com/41923-black-hole-material-one-third-light-speed.html – C. Jordan Jan 10 '20 at 14:59
  • People have different learning styles. Mine is that I seem to need to see a numeric example to grasp the concept. Say you have a dense neutron star of mass 1e36 kg, r=1.5e9, $\omega$=0.1 rad/s. I put on my blinders and solve $a=J/M = I\omega /M = .4MR^2 \omega /M = .4R^2 \omega$ but now I've lost the mass! I would appreciate any help. – Ralph Berger Aug 03 '21 at 00:11
  • @RalphBerger the specific angular momentum wouldn't involve the mass. – ProfRob Aug 03 '21 at 07:46
  • Okay, say I have a spinning object of radius 10 km rotating at 1 radian/second. Do I say that $a = .410^21 = 40$? – Ralph Berger Aug 03 '21 at 16:20
  • @RalphBerger ah, yes you need to get the geometrised units right. – ProfRob Aug 03 '21 at 19:47
  • I'd greatly appreciate a numeric example. As I say, different people have different learning styes. If I saw an example of how the value of "a" is generated, then I could plot the equations and replicate the paths of objects spiraling into a black hole. Aside: I am a retired guy with PhD in ME playing around with physics. I started from first principles, and have developed an alternate mathematical model for frame draggging that works well for what little quantitative stuff I can find (like gravity probe B precession). I'd like to see how my model compares in other cases. – Ralph Berger Aug 04 '21 at 17:38
  • @RaphBerger it's just $J/Mc$ in SI units. So for your earlier example $a=1.33\times 10^{-7}$. – ProfRob Aug 04 '21 at 18:32