# Can centrifugal force overcome other forces (in a singularity/Kerr metric)?

Typical rotation speeds for black holes

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

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.

No need of speed of light, already if you rotate a piece of dough the outer levels will fly away :). One would have to calculate the forces for a specific size steel ball , for example, and get a limit for that ball. It will answer how fast must it rotate for the surface atoms to split off due to centrifugal forces.

Now in case of a black hole it is not only EM forces and strong force that holds material together, but gravity becomes a force to reckon with. Gravity becomes so strong in a black hole's case that not even massless (particles with no rest-mass) particles can escape.

And then this one:

How can a singularity in a black hole rotate if it's just a point?

They rotate because they are produced by matter that has net angular momentum, and angular momentum is conserved in axially symmetric space-time. So, there's nothing unusual making them rotate that's different from any other physics.

However, you are absolutely right to object that rotation of an infinitesimally small point wouldn't make much sense. In quantum mechanics, we talk about infinitesimally small particles having intrinsic angular momentum ("spin") but this is a uniquely quantum effect and General Relativity is a classical theory. So, your question is a good one. Fortunately, it has a simple answer: the singularity of a rotating black hole in GR is not a point, it's a ring around the black hole's axis of rotation. A rotating ring - even an infinitesimally small one - is sensible because it's topologically distinct from a zero dimensional point.

An ideal black hole with non-zero angular momentum is described by the Kerr metric. The singularity of such a black hole is not a point.

So I am talking about the black hole that is more then just a singularity, and even the singularity is not a point.

So the assumption (might be wrong) should be that gravity is so strong, it might even stand the test of centrifugal force near the speed of light rotation.

Obviously, no material object can rotate at the speed of light, not even a black hole, but measurements say that they can come close to it, sometimes 0.86c.

Question:

1. Is there a certain speed limit where the black hole would rotate so fast, that centrifugal forces would overcome EM/strong and even gravitational forces holding material in the black hole together and the hole would start falling apart (might be the wrong term) or stop being a singularity? So the hole would start reversing from being a black hole, and the singularity would explode/expand (like big bang)?

2. Is there any experimental/observational documentation where I could read about that they have calculated/measured the rotational speed of a black hole (and how did they measure it) and how much faster it needed to rotate to start falling apart (if it is possible at all) or stop being a black hole?

• Theoretical limits are not set by the "material" of a singularity/Black Hole as that is an undefined term...it makes no sense to talk about what it is made of or the forces holding it together. This answer here gives a nice rundown of what spin means for BHs astronomy.stackexchange.com/q/20276, and where a limit on their spin could come from May 11, 2018 at 19:14
• Can you put this as an answer? May 11, 2018 at 19:20

A black hole is described by its mass $M$, angular momentum per unit mass $a$, electric charge $Q$ and magnetic charge $P$. However the essential features persist in the absence of charges, so a rotating black hole $(M, a)$ as described by the Kerr metric is representative of cosmological black holes.

The outer event horizon in Kerr is given by
$$r_+ ~=~ M + \sqrt{M^2 -a^2} \,,$$ where:

• natural units are used such that $c = G = 1$;

• $\left(t, r, \theta, \phi \right) =$ Boyer-Lindquist coordinates;

• $a = \frac{J}{M}$;

• $J =$ angular momentum .

A stationary observer $(r, \theta = \text{constant})$ ideally on the event horizon is co-rotating with the black hole at angular velocity $\Omega = a / \left(r_+^2 + a^2\right)$.

The event horizon exists if $a \le M$. If $a \gt M$ the Kerr solution describes a naked singularity, however it is believed that is not physical. Reason being that in a gravitational collapse of a rapidly rotating object the centrifugal forces can prevent the formation of a black hole.

At the limit $a = M$ the black hole angular velocity is
$$\Omega_{\text{max}} = \frac{1}{2M} \,.$$ Note that $\Omega$ is monotonically increasing in the interval $0 \lt a \lt M$.