A Boundary Scaling Law for Thermal Horizons

Thermal horizons are characterized by a radius, a temperature, and an area-based horizon count. This work records a structural scaling relation connecting these quantities.

For a horizon of radius R_H, temperature T_H, and horizon count N_H, with

k_B T_H = eta_H hbar c / R_H

and

N_H = gamma_H A_H / l_P^2,

one obtains

k_B T_H N_H = alpha_H (c^4/G) R_H,

where

alpha_H = 4 pi eta_H gamma_H.

The coefficient alpha_H depends on the horizon temperature convention and on the counting convention. For a Schwarzschild horizon, using the Hawking temperature and the Planck-area count N_A = A_H / l_P^2, one obtains alpha_H = 1. For a de Sitter or Gibbons-Hawking horizon with the same area count, one obtains alpha_H = 2.

The relation is not an equation of motion and not a universal conservation law with fixed unit coefficient. It is a boundary scaling law showing that horizon temperature and horizon area-count combine into the gravitational energy scale (c^4/G) R_H.