# Do black holes have a moment of inertia?

My question is in the title: Do black holes have a moment of inertia?

I would say that it is: $$I ~\propto~ M R_S^2,$$ where $R_S$ is the Schwarzschild radius, but I cannot find anything in the literature.

• Since BH can have measurable spin, they must have moments of inertia. I doubt that it is as large as you suggest, but I'm not sure how one would calculate it.. – Lewis Miller Feb 9 '17 at 16:16

The angular velocity of a Kerr black hole with mass $M$ and angular momentum $J$ is

$$\Omega = \frac{J/M}{2M^2 + 2M \sqrt{M^2 - J^2/M^2}}$$

The moment of inertia of an object can be thought of as a map from the object's angular velocity to its angular momentum. However, here we see that the relationship between these two quantities is non-linear. If we want to think of moment of inertia in the usual sense, we should linearise the above equation. When we do so, we find the relationship

$$J = 4 M^3 \Omega \qquad (\mathrm{to\ first\ order})$$

And so the moment of inertia is

$$I = 4 M^3$$

In other words, the expression you guessed is correct, and the constant of proportion is unity. Note that since the Schwarschild radius of a black hole is merely twice its mass, and since the only two parameters that describe the black hole are its mass and angular momentum, any linear relationship between the angular velocity and angular momentum of our black hole must be of the form $J = k\, M R_S^2\, \Omega$ on dimensional grounds.

Note that $G = c = 1$ throughout.

EDIT.

As pointed out in the comments, it's not obvious how one should define the angular velocity of a black hole. At the risk of being overly technical, we can do this as follows. First consider the Killing vector field $\xi = \partial_t + \Omega \partial_\phi$ (using Boyer-Lindquist coordinates), where $\Omega$ is defined to be as above. The orbits, or integral curves, of this vector field are the lines $\phi = \Omega t + \mathrm{const.}$, which correspond to rotation at angular velocity $\Omega$ with respect to a stationary observer at infinity.

One can show that this vector field is tangent to the event horizon, and its orbits lying on the event horizon are geodesics. These geodesics hence rotate at angular velocity $\Omega$ (with respect to an observer at infinity), and hence it is natural to interpret the quantity $\Omega$ as the angular velocity of the black hole. Whether it is possible to make a more definite statement than this I do not know.

• Also, note that the moment of inertial, even differentially, will have to be treated like a tensor -- you will get different incremental amounts of angular velocity added to the hole depending on the direct of added angular momentum. – Jerry Schirmer Feb 9 '17 at 19:48
• @gj255 What do you mean by 'first order'? – Mockingbird Feb 10 '17 at 13:07
• How are you defining the angular velocity of a black hole? That seems like a particularly slippery concept to me. – Emilio Pisanty Feb 10 '17 at 13:09
• Just a comment to say that the relation $Ω(J)$ can be inverted easily : \begin{equation}J(\Omega) = \frac{4M^3}{1 + (2M \, \Omega)^2} \; \Omega.\end{equation} If $2M \, Ω \ll 1$, then you get the linear relation. It is interesting to note that the moment $I$ is dependant on $\Omega$. You don't need a linear relation. – Cham Feb 10 '17 at 17:59
• Just another comment. The non-linear relation $J(\Omega)$ given above implies that $J$ increases linearly with $\Omega$, then gets the maximal value $J_{\text{max}} = M^2$ at $\Omega = 1/2M$, then decreases when $\Omega > 1/2M$. The moment of inertia $I$ is a decreasing function of $\Omega$. – Cham Feb 10 '17 at 18:47

Moments of inertia are defined about a given axis of rotation.

Moment of inertia is the name given to rotational inertia, the rotational analog of mass for linear motion. It appears in the relationships for the dynamics of rotational motion. The moment of inertia must be specified with respect to a chosen axis of rotation

There exist rotating black holes.

A rotating black hole is a black hole that possesses angular momentum. In particular, it rotates about one of its axes of symmetry.

Thus by definition a black hole, since it is massive , must have a moment of inertia, for details see this link, or this link.