Spin angular momentum of black hole

[spidey]

how to measure the spin angular momentum of black hole..is there any specific formula for black hole Or the usual J=mvr is used...if so,what does r stand for? Is r the radius of black hole Or the distance from the center to the event horizon?

 

[stevebd1]

As no-one has answered this question, I'll give it a go.

As you've realized, establishing angular momentum for a black hole is quite tricky. One way is to take a look at the proximity of the inner edge of the accretion disk (sometimes referred to as the marginally stable orbit) with the event horizon. For a static black hole, the MS orbit is quantified as 6Gm/c^2 (while the event horizon is 2Gm/c^2). For a rotating black hole the equation for the MS orbit is more complex but based on the spin parameter (a unitless quantity between 0 and 1, 0 being a static black hole and 1 being an extreme Kerr black hole), 0 would put the MS orbit at 6Gm/c^2 and 1 would put it at the event horizon. The mass of a black hole is relatively easy to establish, so if we have established the mass and we look at the proximity of the marginally stable orbit to the event horizon, based on where it should be if the black hole was static, we can make a reasonably accurate guess at the spin parameter (sometimes expressed as a, sometimes as j), if we say the spin parameter is 0.8, we can use the following equation to establish angular momentum-

$$j=\frac{Jc}{Gm^2}$$

which becomes

$$J=\frac{jGm^2}{c}$$

where j is the spin parameter between 0 and 1, J is the angular momentum in Nms, c is the speed of light in m/s, G is the gravitational constant and m is mass

Based on a 10 sol mass black hole with a spin parameter of 0.8, the angular momentum would be 7.0454x10^43 Nms.

It's also worth noting that the event horizon of a black hole also reduces due to rotation (though not to the same extent as the MS orbit) so it might be a process of trial and error before the correct spin parameter is established. The equation for the outer event horizon is-

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

where R₊ is the outer event horizon, M is the gravitational radius (M=Gm/c^2) and a is the spin parameter in metres (a=J/mc).

There are also be other ways of establishing spin, such as looking for frame-dragging effects and studying the redshift/blueshift of matter as it rotates around the black hole.