Angular Momentum of Kerr Black Hole

In summary, to find the Angular Momentum, J, of a Kerr Black Hole using given Angular Velocity, ω, you can use the equation J = I*ω, where I = mass*r^2 and r is the reduced circumference. However, to calculate the reduced circumference, you need to know the value of J, which can be found using the frame dragging rate equation \omega=\frac{2Mra}{\Sigma^2} where \Sigma^2=(r^2+a^2)^2-a^2\Delta \sin^2\theta and a=J/M. This equation can be rearranged to find the value of J, but it may involve a quadratic equation.
  • #1
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Homework Statement


How do you find the Angular Momentum, J, if you are given the Angular Velocity, ω of a Kerr Black Hole.

Homework Equations


J = I*ω
##I = mass*r^2##
Event Horizon:
##r_+ = M + (M^2 − (J/mass/c)^2)^{1/2}##
Static Limit and the Ergosphere:
##r_0 = M + (M^2 − (J/mass/c)^2*cos^2θ)^{1/2}##
where
##M=G*MassOfBlackHole/c^2##
c=Speed Of Light

The Attempt at a Solution


Unsure where the Moment of Inertial mass radius is.
The Black Hole is not visible to the naked eye.
Is it the mass and radius of the observer or is it the mass and radius of the Black Hole itself that the Event Horizon, Static Limit and all the orbiting Stars, planets,etc. are based on.
 
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  • #2
I know when calculating tangential velocity in Kerr metric, r is the reduced circumference (R)-

[tex]v_T=\omega R[/tex]

where

[tex]R=\frac{\Sigma}{\rho}\sin\theta[/tex]

and [itex]\Sigma=\sqrt((r^2+a^2)^2-a^2\Delta \sin^2\theta)[/itex], [itex]\Delta= r^{2}+a^{2}-2Mr[/itex], [itex]\rho=\sqrt(r^2+a^2 \cos^2\theta)[/itex] and [itex]a=J/M[/itex] so I'm guessing you would use the reduced circumference when considering inertia.

Though in order to calculate the reduced circumference you need [itex]a[/itex] which means you need to know J. If the only info you have is [itex]\omega[/itex] and M, then the equation for the frame dragging rate is-

[tex]\omega=\frac{2Mra}{\Sigma^2}[/tex]

where [itex]\Sigma^2=(r^2+a^2)^2-a^2\Delta \sin^2\theta[/itex], replace [itex]a[/itex] with J/M and rearrange relative to J, which might be quadratic.

Source-
'Compact Objects in Astrophysics' by Max Camenzind page 379
 
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FAQ: Angular Momentum of Kerr Black Hole

1) What is the Kerr black hole?

The Kerr black hole is a type of rotating black hole that was proposed by physicist Roy Kerr in 1963. It is a solution to Einstein's equations of general relativity and is characterized by its mass, angular momentum, and electric charge.

2) What is the significance of the angular momentum of a Kerr black hole?

The angular momentum of a Kerr black hole is important because it determines the shape and size of the event horizon, the point of no return for anything that enters the black hole. It also affects the behavior of matter and radiation near the black hole, leading to unique phenomena such as frame dragging.

3) How is the angular momentum of a Kerr black hole measured?

The angular momentum of a Kerr black hole can be calculated using the spin parameter, which is a dimensionless quantity that ranges from 0 for a non-rotating black hole to 1 for a maximally rotating black hole. Observations of surrounding matter and radiation can also provide estimates of the angular momentum.

4) Can the angular momentum of a Kerr black hole change?

Yes, the angular momentum of a Kerr black hole can change over time due to processes such as accretion (the capture of matter) and mergers with other black holes. However, the maximum possible value of the spin parameter is always limited to 1, so a Kerr black hole cannot become more rotating than it already is.

5) What are some practical applications of studying the angular momentum of Kerr black holes?

Studying the angular momentum of Kerr black holes can provide insights into the formation and evolution of these objects, as well as the behavior of matter and radiation in extreme gravitational environments. This knowledge can also help us understand and test the predictions of general relativity, which is crucial for our understanding of the fundamental laws of the universe.

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