Bounda: Bekenstein Bound
Relativity & Gravity arc, page 37 of 193. Back to The Pixelated Physics Universe
Who they are
Bounda crams endless scrolls into a sphere. At the limit, the sphere becomes a black hole.
Bounda 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
Jacob Bekenstein proposed that a system of energy E inside a sphere of radius R can hold only so much entropy, and so only so much information. Black holes sit right at the edge: for a Schwarzschild hole the bound equals the Bekenstein–Hawking entropy exactly. Counted in bits, that is one bit per 4 ln 2 Planck areas of horizon.
$$ \begin{gathered} S \le \frac{2\pi k_B R E}{\hbar c} \\ S_{BH} = \frac{k_B c^{3} A}{4 G \hbar} \end{gathered} $$
| Symbol | Meaning | SI unit |
|---|---|---|
| $S$ | entropy of the system | J/K |
| $k_B$ | Boltzmann constant, 1.380 649 × 10⁻²³ J/K (exact) | J/K |
| $R$ | radius of a sphere enclosing the system | m |
| $E$ | total energy, including rest energy | J |
| $\hbar$ | reduced Planck constant, 1.054 571 817 × 10⁻³⁴ J·s | J·s |
| $c$ | speed of light, 299 792 458 m/s (exact) | m/s |
| $S_{BH}$ | black-hole entropy | J/K |
| $A$ | horizon area | m² |
| $G$ | gravitational constant, 6.674 30 × 10⁻¹¹ m³/(kg·s²) | m³/(kg·s²) |
Honest caveat: It was a key clue toward holography. A precise form was proved for quantum fields by H. Casini in 2008, but the bound's fully general meaning is still debated.
Status: part of this topic is unsolved, unconfirmed or debated. The text above says which part; treat it as an open question, not settled fact.
Sources
- Bekenstein (1981) Universal upper bound on the entropy-to-energy ratio for bounded systems, Phys. Rev. D 23, 287
- Bekenstein (1973) Black Holes and Entropy, Phys. Rev. D 7, 2333
- Casini (2008) Relative entropy and the Bekenstein bound, Class. Quantum Grav. 25, 205021
Crossovers
Bounda meets Areax (Hawking's Area Theorem). 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.
Bounda meets Erasa (Forgetting Costs Heat). Erasa says erasing a bit costs heat. Bounda says a region can only store a limited number of bits. Together they tie information to physics at both ends: storage and deletion.
Wear the lore
Concept keywords: Bekenstein Bound, Bounda, Relativity & Gravity, physics lore, Pixelated Physics