Bounda

Bounda crams endless scrolls into a sphere. At the limit, the sphere becomes a black hole.

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.

S ≤ 2π k_B R E / (ħc), S_BH = k_B c³ A / (4Għ)

• S: entropy (J/K)
• k_B: Boltzmann constant, 1.380649 × 10⁻²³ J/K (exact)
• R: radius of a sphere enclosing the system (m)
• E: total energy, including rest energy (J)
• ħ: reduced Planck constant, 1.054571817 × 10⁻³⁴ J s
• c: speed of light, 299 792 458 m/s (exact)
• S_BH: black-hole entropy (J/K)
• A: horizon area (m²)
• G: 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻²

Bekenstein stated the bound in 1981; Hawking's 1975 radiation fixed the 1/4 in the black-hole entropy. A Planck area, ħG/c³, is about 2.61 × 10⁻⁷⁰ m².

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.

On the shirt

Front: Bounda, the pixel-art character, with the title. Back: the entropy bound, the black-hole entropy, bits per Planck area and the holography note. Printed on a unisex tee.