Areax: Hawking's Area Theorem
Relativity & Gravity arc, page 36 of 193. Back to The Pixelated Physics Universe
Who they are
Two black holes merge into Areax, and her horizon-crown is bigger than both of theirs put together. It always is.
Areax is a character of the Relativity & Gravity arc: drawn in pixels, ruled by one piece of real physics, and never allowed to break it.
The physics
In classical general relativity the total event-horizon area of black holes can never decrease. So when two black holes merge, the final horizon must be larger than the two original horizons combined, even though energy is radiated away as gravitational waves. For a spinning (Kerr) black hole the area depends on its mass and spin. Stephen Hawking proved the theorem in 1971; the gravitational-wave event GW250114 (2025) found the final area larger than the initial area at 4.4σ in the collaboration's reference analysis (LIGO–Virgo–KAGRA, Phys. Rev. Lett. 135, 111403), a far stronger test than GW150914 (Isi et al. 2021).
$$ \begin{gathered} A_{\text{final}} \ge A_1 + A_2 \\ A = 8\pi\left(\frac{GM}{c^{2}}\right)^{2}\left(1 + \sqrt{1-\chi^{2}}\right) \qquad (\text{Kerr black hole}) \end{gathered} $$
| Symbol | Meaning | SI unit |
|---|---|---|
| $A,\ A_1,\ A_2,\ A_{\text{final}}$ | event-horizon areas | m² |
| $G$ | gravitational constant | m³·kg⁻¹·s⁻² |
| $M$ | black-hole mass | kg |
| $c$ | speed of light, 299 792 458 m/s (exact) | m/s |
| $\chi$ | dimensionless spin (0 to 1) | 1 |
Honest caveat: It is a classical result: Hawking radiation lets black holes shrink, very slowly.
Sources
- Hawking (1971) Gravitational Radiation from Colliding Black Holes, Phys. Rev. Lett. 26, 1344
- Isi, Farr, Giesler, Scheel & Teukolsky (2021) Testing the Black-Hole Area Law with GW150914, Phys. Rev. Lett. 127, 011103
- LIGO–Virgo–KAGRA Collaboration (2025) GW250114: Testing Hawking's Area Law and the Kerr Nature of Black Holes, Phys. Rev. Lett. 135, 111403
Crossovers
Areax meets Spaghetta (Black Hole Physics / Spaghettification Stretch). Spaghetta tears infalling matter apart. Areax only cares about what happens to the horizon: in classical general relativity it never shrinks, so everything Spaghetta feeds in adds to the area.
Areax meets Bounda (Bekenstein Bound). Areax's area that never shrinks became Bekenstein's clue that horizon area measures entropy. Bounda carries the idea further: a region of given size and energy can hold only so much entropy.
Areax meets Eventide Prophet (Black Hole Information Paradox). Areax's classical theorem says the horizon area never decreases. Eventide Prophet's black hole evaporates through Hawking radiation and shrinks, which is allowed because quantum effects break the theorem's assumptions. Where the information goes is still open.
Wear the lore
- Areax – Hawking's Area Theorem Physics T-Shirt
- Related infographic poster: Three Radii of a Black Hole – Plate 033 Physics Poster
Concept keywords: Hawking's Area Theorem, Areax, Relativity & Gravity, physics lore, Pixelated Physics