Why is it so that the maximum rotational kinetic energy that a rotating black hole can have is 29% of its rest mass?

Question

I am currently in high school and I have very basic understanding of relativity and rotational mechanics. Is there a reason why a rotating black hole has maximally only 29% of its rest mass energy as the rotational kinetic energy or is there a valid proof for this.

Answer 1

1. The surface area $A$ of the outer event horizon of a Kerr-Newman black hole with mass $M$, angular momentum $J$, and charge $Q$, is in geometrical-Gaussian units $$ A~=~4\pi\left(2M^2-Q^2+2M\sqrt{M^2-\frac{J^2}{M^2}-Q^2}\right), \tag{*}$$ cf. e.g. this Phys.SE post.

2. In particular, a Schwarzschild black hole has $$ A_S~\stackrel{(*)}{=}~16\pi M_S^2, \tag{6}$$ while an extremal Kerr black hole has $$ A_K~\stackrel{(*)}{=}~8\pi M_K^2. \tag{8}$$

3. From Stephen Hawking's area theorem, $$ A_K~\leq~ A_S,\tag{9} $$ cf. the second law of thermodynamics.

4. So $$ M_K~\stackrel{(6)+(8)+(9)}{\leq}~\sqrt{2} M_S .\tag{12} $$

5. The extracted rotational energy is $$ E_{\rm rot}~=~M_K-M_S .\tag{13}$$

6. So $$ \frac{E_{\rm rot}}{M_K}~\stackrel{(12)+(13)}{\leq}~1-\frac{1}{\sqrt{2}}~\approx~ 29\%.\tag{15}$$

References:

1. J. Pinochet, Rotating black holes: The most fantastic source of energy in the universe, arXiv:2502.15784; eq. (15).

Answer 2

The faster a black hole rotates the higher is its rotational energy, which is kind of obvious. What is less obvious is that the rotational energy contributes to the total mass of the black hole in accordance with Einstein's famous equation $E = mc^2$.

Anyhow, there is an upper limit to how fast a black hole can spin because above this limit the event horizon disappears and the black hole becomes a naked singularity. Theoretically naked singularities are troublesome, but fortunately it is believed to be impossible for any physical process ever to spin a black hole that fast. Astronomers have observed black holes spinning just below this limit, but never above it.

The 29% figure you've read applies to an extremal black hole i.e. a black hole that has a spin right on the limit of becoming extremal. Real black holes would have a smaller fraction of their mass as rotational energy.

In a comment Qmechanic refers to the paper Rotating black holes: The most fantastic source of energy in the universe and this gives a derivation of that 29% figure. The maths involved in the derivation is likely to be impenetrable to anyone still at high school but the final result (given by equation 15 in the paper) is fairly simple:

$$ \varepsilon = \frac{E_{rot}}{M_ic^2} \le 1 - \frac{M_f}{\sqrt2 M_f} = 1 - \frac{1}{\sqrt2} \approx 0.29 \tag{15} $$