Abstract
It is easy to assume that a rotating black hole yields energy in proportion to how fast it spins. This paper counts how the upper limit on extractable energy grows with spin, in two ways: from the irreducible-mass formula and from the horizon area. No new theorem or law is claimed.
First, the extractable limit is the mass minus the irreducible mass: the irreducible mass is $\sqrt{(1+\sqrt{1-a^2})/2}$ times the mass: $0.965926$ at $a=0.5$ and $0.707107$ at $a=1$ (Section 2).
Second, and this is the core. Even at the fastest spin, at most $0.292893$ of the mass can be drawn: at $a=1$ the extractable share is $\mathbf{0.292893}$, matching $1-1/\sqrt{2}$ within $10^{-12}$ (Section 2).
Third, for slow rotation, it starts to grow as the square of the spin: at $a=0.1$ the share is $0.001254$, a ratio of $\mathbf{1.003142}$ to $a^2/8$; the ratio rises with spin, reaching $2.343146$ at $a=1$ (Section 3).
Fourth, at half the spin, far less: at $a=0.5$ the share is $0.034074$, only $1/\mathbf{8.595754}$ of the $a=1$ amount (Section 3).
Fifth, the irreducible mass is set by the horizon area: the irreducible mass recomputed from the horizon area as $\sqrt{A/16\pi}$ agreed with the formula within $10^{-12}$ for all eight spins (Section 2).
Sixth, the separator is whether one asks about the speed of rotation or the area of the horizon. What sets the extractable limit is the horizon area, which does not decrease (Section 4).
Keywords: Penrose process, Kerr black hole, irreducible mass, horizon area, rotational energy.
Scope (what is not claimed)
No new theorem or law is claimed: the Penrose process for extracting energy from a rotating black hole, the irreducible-mass formula, and that the horizon area does not decrease in classical processes, are all known in general relativity. Units are set: the gravitational constant and the speed of light are $1$, and the spin parameter $a$ is angular momentum divided by the square of the mass. No observations are claimed: how fast real objects spin, or whether energy is actually extracted, is not treated. Nothing beyond general relativity is entered: quantum effects and corrections from other theories are not included. Single-split efficiency is not entered: the limit on efficiency when one particle splits in two is not treated; what is counted is the limit on total extractable energy. Relation to earlier papers: Paper 73 anchored the “$1/4$” in black-hole entropy via the $4\pi$ solid angle, the Euler characteristic and equipartition. This paper stands beside it. There the horizon area was read as entropy. Here we count that the same area not decreasing sets the ceiling on extractable energy. What is added is laying out the extractable share for eight spins, checking that the fastest spin gives $0.292893$, measuring the ratio to $a^2/8$ for slow spin and the ratio $8.595754$ to half the spin, and placing the separator on rotation speed versus horizon area.
1. Notation and the Question
Setting: consider a black hole of mass $M$ and spin parameter $a$ ($0\le a\le1$), with the gravitational constant and speed of light set to $1$. Let the irreducible mass $M_{\text{irr}}$ be the mass that remains after all rotational energy is extracted, and let $r_+$ and $A$ be the horizon radius and area.
$$\frac{M_{\text{irr}}}{M}=\sqrt{\frac{1+\sqrt{1-a^{2}}}{2}},\qquad r_+=M\bigl(1+\sqrt{1-a^{2}}\bigr),\qquad A=8\pi M r_+,\qquad M_{\text{irr}}=\sqrt{\frac{A}{16\pi}}$$The extractable share is $1-M_{\text{irr}}/M$. The irreducible mass was computed two ways, from the spin formula and from the horizon area.
Question: how much energy can be drawn from a rotating black hole?
2. The Extractable Share
Computation 2.1: for varying spin, the irreducible mass and extractable share were computed.
| $a$ | $M_{\text{irr}}/M$ | extractable share | ratio to $a^2/8$ |
|---|---|---|---|
| $0.1$ | $0.998746$ | $0.001254$ | $1.003142$ |
| $0.3$ | $0.988418$ | $0.011582$ | $1.029535$ |
| $0.5$ | $0.965926$ | $0.034074$ | $1.090374$ |
| $0.7$ | $0.925782$ | $0.074218$ | $1.211730$ |
| $0.9$ | $0.847316$ | $0.152684$ | $1.507987$ |
| $0.99$ | $0.755337$ | $0.244663$ | $1.997047$ |
| $0.998$ | $0.729114$ | $0.270886$ | $2.175784$ |
| $1$ | $0.707107$ | $0.292893$ | $2.343146$ |
Observation 2.2 this is the core. Even at the fastest spin, at most $0.292893$ of the mass can be drawn. At $a=1$ the irreducible mass is $0.707107$, and the extractable share $0.292893$ matched $1-1/\sqrt{2}$ within $10^{-12}$: the remaining $0.707107$ cannot be drawn from rotation by any process.
Observation 2.3 the irreducible mass is set by the horizon area. Recomputed from the area as $\sqrt{A/16\pi}$, it agreed with the formula within $10^{-12}$ for all eight spins: the irreducible mass goes as the square root of the area, so if the area does not decrease, neither does the irreducible mass.
3. Spin and Extractable Energy
Computation 3.1: the extractable share was compared with $a^2/8$, and half spin with the fastest spin.
Observation 3.2 for slow rotation, it starts to grow as the square of the spin. At $a=0.1$ the ratio of the extractable share to $a^2/8$ is $1.003142$: as spin increases the ratio rises to $1.090374$, $1.507987$ and $2.343146$, growing faster than the square.
Observation 3.3 at half the spin, far less. At $a=0.5$ the extractable share is $0.034074$, $1/8.595754$ of the $0.292893$ at $a=1$: halving the spin cuts the extractable amount not in half but to $1/8.595754$.
4. The Separator
Observation 4.1 the separator is whether one asks about the speed of rotation or the area of the horizon. Asking about rotation speed, it looks as if energy comes out in proportion to spin: asking about the horizon area, the area does not decrease during extraction, so the mass cannot fall below the irreducible mass. The area is what sets the limit.
Observation 4.2 placed beside Paper 73. 73 anchored the “$1/4$” in black-hole entropy to the horizon area. There the area was read as entropy. Here the same area not decreasing is counted as the ceiling on extractable energy.
Observation 4.3 what is not said. Spins and observations of real objects, quantum effects, and single-split efficiency are not treated. What is shown is that the extractable share starts as the square of the spin and reaches at most $0.292893$.
5. Census and Closing
- irreducible-mass formula (Section 1): mass left after rotation is used up.
- irreducible mass $0.707107$ at $a=1$ (Computation 2.1): $1/\sqrt2$.
- extractable limit $\mathbf{0.292893}$ (Observation 2.2): the core.
- recomputed from area, agreeing (Observation 2.3): within $10^{-12}$.
- $0.001254$ at $a=0.1$ (Computation 2.1): almost nothing.
- ratio to $a^2/8$: $\mathbf{1.003142}$ (Observation 3.2): starts as the square.
- ratio $2.343146$ at $a=1$ (Computation 2.1): faster than the square.
- $0.034074$ at $a=0.5$ (Computation 2.1): half the spin.
- $1/\mathbf{8.595754}$ (Observation 3.3): against the fastest spin.
- rotation speed, or horizon area (Observation 4.1): the separator; Paper 73.
Closing: The energy extractable from a rotating black hole starts to grow as the square of the spin and reaches at most $0.292893$ of the mass. The irreducible mass is $\sqrt{(1+\sqrt{1-a^2})/2}$ times the mass; at $a=1$, $0.707107$ remains and $0.292893$ can be drawn, because the irreducible mass goes as the square root of the horizon area, and the area does not decrease during extraction. For slow spin the share is close to $a^2/8$ (ratio $1.003142$ at $a=0.1$), and at half the spin, $a=0.5$, only $1/8.595754$ of the $a=1$ amount can be drawn: energy does not come out in proportion to spin, but only beneath a ceiling set by the area. Placed among the earlier papers: beside the horizon area anchored in Paper 73, we placed that the area not decreasing is the ceiling on extractable energy. To be honest: all the formulas are known in general relativity, and real objects and quantum effects are not treated.
References
- Christodoulou, D. (1970). Reversible and Irreversible Transformations in Black-Hole Physics. Physical Review Letters.
- Misner, C. W., Thorne, K. S., & Wheeler, J. A. (1973). Gravitation. W. H. Freeman.
- Penrose, R., & Floyd, R. M. (1971). Extraction of Rotational Energy from a Black Hole. Nature Physical Science.
- Yamagishi, Y. (2026). The “$1/4$” in Black-Hole Entropy [Paper 73]. YOI: https://www.yamagishiyuuki.com/yoi/73; DOI: 10.5281/zenodo.21666086
Reference
Yamagishi, Y. (2026). Extractable Energy from a Rotating Black Hole Starts to Grow as the Square of the Spin and Reaches at Most 0.292893 of the Mass [Paper 887]. YOI: https://www.yamagishiyuuki.com/yoi/887
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 887. This paper is a preprint and has not been peer reviewed.