Planck-scale scaling symmetry and the thermodynamic structure of black holes

Abstract

In this paper, we present a Planck-scale geometric reinterpretation of black hole thermodynamics without modifying Einstein’s field equations. The central input is a Planck-normalized scaling symmetry, formulated through a dimensionless gravitational coupling that depends quadratically on mass. Requiring geometric consistency yields a canonical mass–radius relation, from which the Schwarzschild radius is recovered exactly rather than postulated. Once this scaling structure is imposed, black hole thermodynamics is algebraically fixed. Using the Bekenstein–Hawking area law, the entropy takes the form S_QTP = 4πk_B N², with N = M_B/m_pl = R_B/l_pl. The temperature then follows uniquely from the thermodynamic definition 1/T = dS/dE, giving T_QTP = T_pl/8πN. Together with the exact relation E = 2ST, these expressions reproduce the Hawking temperature and Smarr relation. The framework implies a Planck-scale structural bound R_B ≥ l_pl, yielding finite entropy and temperature and removing the necessity of singular behavior within the thermodynamic description. This approach provides a unified geometric perspective on black hole thermodynamics fully consistent with classical general relativity while clarifying several persistent conceptual tensions.