The entropy of a black hole is proportional to the area of its event horizon, not its volume. Bekenstein proposed this in 1973 from thought experiments about information; Hawking derived the precise coefficient in 1975 from quantum field theory in curved spacetime. The formula — S = A/4 in Planck units — is one of the most important results in theoretical physics, connecting thermodynamics, quantum mechanics, and gravity. But the formula was derived from the outside. What the entropy counts — the microscopic degrees of freedom that the area measures — has remained contested for fifty years.
Rojas and Arenas (arXiv 2602.22360, February 2026) derive the Bekenstein-Hawking entropy from the inside, using thermo field dynamics applied to the BTZ black hole — a solution in (2+1)-dimensional gravity with a negative cosmological constant. The BTZ black hole is simpler than its four-dimensional cousins (no curvature singularity, no Penrose diagram complexity), but it has a genuine event horizon and genuine thermodynamic properties, making it the cleanest testing ground for microscopic entropy calculations.
The calculation places a massive scalar field near the horizon and computes the entanglement entropy between the field modes inside and outside the horizon. The key object is the difference between two vacuum states: the Hartle-Hawking vacuum (thermal, regular at the horizon) and the Killing-Boulware vacuum (non-thermal, singular at the horizon). The difference encodes the thermal character of the black hole — it tells you what radiation the horizon produces.
The entanglement entropy emerges proportional to the horizon area, with a coefficient that matches the Bekenstein-Hawking formula. The area law isn't imposed — it's derived from the quantum field theory. The degrees of freedom being counted are the entanglement correlations between field modes that the horizon separates: pairs of excitations where one partner falls in and the other escapes.
The entropy is entanglement entropy. The horizon isn't a surface with its own degrees of freedom — it's a boundary that divides a quantum field into two subsystems whose correlations are measured by the entropy. The area enters because the number of independent modes that can straddle the horizon scales with the area of the boundary, not the volume of either subsystem.
What the area counts: the number of ways to be entangled across the divide.