# Upper limit on black hole angular momentum (and spin)?

I have read this question:

Angular momentum for the Kerr solution of a rotating blackhole

Angular momentum of a rotating black hole

Does angular velocity have an upper bound?

But these do not give explanatory answers.

Angular momentum of a black hole

where Rob Jeffries says:

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.

Is there a physical upper limit on how fast a physical object can rotate?

where annav says:

Therefore the speed at which an object can rotate will be limited by the electromagnetic forces holding the object together against the mechanical centrifugal forces, and will depend on the atomic structure of the object. It will answer how fast must it rotate for the surface atoms to split off due to centrifugal forces.

Do black holes have a moment of inertia?

where gj255 says:

The angular velocity of a Kerr black hole with mass $$M$$ and angular momentum $$J$$ is $$\Omega = \frac{J/M}{2M^2 + 2M \sqrt{M^2 - J^2/M^2}}$$

Now basically they are saying that Black holes can and sometimes do spin, and basically being macro objects, the angular momentum or the angular velocity is only limited by how strong the fundamental forces hold together the the Black hole as a single object.

But in the case of a Black hole, it is not the EM force, but gravity, the extreme curvature that is keeping everything inside the event horizon, so that not even light can escape it.

Based on this, however strong the angular momentum and the angular velocity of the Black hole is, it will never be able to overcome gravity (that is holding the BH together) and it will never be able to tear the BH apart.

Question:

1. Is there an upper limit on the BH angular momentum (and spin)?

## 1 Answer

Yes such a limit exists, and can easily be found by looking at the Kerr solution for a (vacuum) spacetime with ADM mass $$M$$ and ADM angular momentum $$J$$. If $$|J| \leq M^2$$, the solution has an event horizon, and is therefore a black hole.

If $$|J| > M^2$$, the solution doesn't have any event horizons and has naked singularities. The (weak) cosmic censorship conjecture thus implies that such solutions can never actually form.

• thank you so basically the mass of the BH determines its angular momentum maximum? Does that correspond to a certain velocity (if a BH has angular velocity at all)? – Árpád Szendrei Jan 10 '20 at 16:15
• so only the mass limits it? As the mass goes up, the angular momentum can go up too to theoretically infinite? – Árpád Szendrei Jan 10 '20 at 16:16
• I think it is better to say that WCC says that it happens rarely, not that this never happens. – MBN Jan 11 '20 at 15:57
• @safesphere How is it incorrect! There are counter examples to WCC. It can only be true generically. – MBN Jan 12 '20 at 13:39
• @safesphere The conjecture is mathematical. So it has to be precises. Also, how do you know that they don't exist in reality! – MBN Jan 13 '20 at 10:25