# Why does angular momentum shorten the Schwarzschild Radius of a black hole?

Angular momentum causes the event horizon of a black hole to recede. At maximum angular momentum, $J=GM^2/c$, the Schwarzschild radius is half of what it would be if the black hole wasn't spinning.

Can someone explain why angular momentum reduces the Schwarzschild radius?

• interesting question... does the M in that expression include its rotational energy? Jun 26 '14 at 3:15
• @AlanSE No, M is just the mass of the black hole. It doesn't include any rotational energy.
– Drew
Jun 26 '14 at 3:24
• Here is a rough, qualitative and Newtonian argument that can at least make it plausible: Put yourself in a rotating reference frame following the rotation of the BH close to $r=r_s$. The BH is seen to be at rest, but in this frame you will experience a centrifugal force. The total force on you $GMm/r^2−mv^2/r=GM_{\rm eff}m/r^2$ implies the object has a smaller effective mass: $(M_{\rm eff}−M)/M \sim -\frac{J^2}{J^2_{\rm max}}$ and therefore a smaller effective $r_s$. I think this can be done properly in GR by roughly the same approach (rotating reference frame). Jun 26 '14 at 4:15
• I'm not sure I'd use "Schwarzschild radius" to describe the size of the inner horizon in a Kerr black hole. That seems misleading to me, especially since we generally write $r_{inner}$ as a function of $r_s$ and $J/mc$. Jun 26 '14 at 5:49
• Dear @AlanSE and Drew, it is problematic to say that the mass includes the rotational energy or not. The answer depends on other things in your hypothetical "decomposition" of the total mass. It's like asking whether 23 already includes 10. Well, it does if you manage to write it as 13+10, but it doesn't if it is 15+8. ;-) If two fixed objects rotate and have some relative distance, both their kinetic energy and potential energy related to the motion contributes to the total energy - and therefore mass - of the composite object. But it's not easy to "decompose" a black hole in this way. Jun 26 '14 at 7:17

I will approach this question theoretically, although I feel the intuition follows nicely. If we talk about Kerr black holes - rotating black holes described by their mass and angular momentum, with no additional parameters such as charge etc. - then you can show that the radius of the event horizon is given by

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

where $a=\frac{J}{M}$.

(This value of $r$ is found by finding where the Kerr metric blows up; hence event horizon. In fact, finding where the metric blows up involves solving a quadratic equation, so we get two values of $r$ and in Kerr black holes we therefore have two event horizons; unlike in Schwarzschild black holes.)

Regarding your first point about maximum angular momentum, if we set $G=1$ and $c=1$, the maximum angular momentum you stated is given by $a=M$ and if we plug this into our equation for $r$ above we see that we have

$r=M$.

We know that the radius of the event horizon in a Schwarzschild black hole (no rotation) is $r=2M$. So therefore we can see that at maximum angular momentum, the radius of the event horizon is half of what it would be if the black hole weren't spinning.

To this end, we can also see that at zero angular momentum, $a=0$, we have

$r=2M$

which is what we want as at zero angular momentum we of course should have the Schwarzschild radius.

Using the boxed equation for $r$ at the top, it's easy to test out different values of $a$ to see what happens to the event horizon. For example, this equation alone is sufficient to show that for $a>M$ we don't have an event horizon, in which case we have what is a called "Fast Kerr" which is just a singularity with no event horizon.

• It's not that the complex values for the horizon dont' work in general relativity, so much as that equation has no real solution. For $a > M$, you don't have a horizon. Those solutions describe naked singularities, not black holes. Jun 28 '14 at 12:15
• Yes that's right, it's commonly called "Fast Kerr" Jun 28 '14 at 12:18
• This strikes me as begging the question: sure the Kerr metric has critical points at these values, which get smaller as the angular momentum gets larger...but why does the metric have these properties in the first place, and why is it the right metric? Sep 15 '19 at 3:56

Maybe a qualitative answer motivated from thermodynamics: If you let your black hole rotate, you reduce the number of symmetries of your system, this will decrease your entropy $S$ which is proportional to the surface area. The surface area however is for sure monotonic increasing with your Schwarzschild radius, therefore, if your break symmetries, $r$ will decrease.