Thermodynamic temperature of the semiclassical quantum horizon

Determine which, if any, of the surface-gravity-based temperature, horizon-radius-based temperature, and mass-based temperature should be identified with the thermodynamic temperature of the semiclassical Schwarzschild horizon by coupling the semiclassical horizon to an additional quantum field and analyzing its thermodynamic properties.

Background

The paper defines three candidate temperatures for the semiclassical quantum horizon: one from the effective surface gravity, one from the quantum horizon radius, and one from the expectation value of the mass. These definitions coincide in the classical limit but differ for finite quantum mass, because the classical relations among mass, horizon radius, and surface gravity no longer hold exactly.

The authors therefore leave unresolved which candidate has the correct thermodynamic interpretation. They adopt the surface-gravity-based temperature for definiteness, but explicitly state that a definitive identification requires coupling the semiclassical horizon to an additional quantum field and studying its thermodynamic behavior.

References

At this level, it is therefore not possible to determine unambiguously which, if any, of these quantities should be identified with the thermodynamic temperature of the semiclassical horizon.

— Radial quantization of the Schwarzschild geometry: relational observables, evaporation, and remnant state  (2609.26575 - Brizuela et al., 22 Sep 2026) in Section 6.2, “Thermodynamic properties and evaporation of the quantum horizon”