Generalized entropy and evaporating black holes

Write dimensionless entropy s=S/kB and restore the Planck area ℓP²=ħG/c³. In Einstein gravity at leading semiclassical order, the gravitational contribution is A/(4ℓP²). The exterior term is the von Neumann entropy of the chosen outside quantum fields. Entanglement across a sharp boundary is ultraviolet divergent in continuum field theory, so neither a bare area coefficient nor an unregulated outside entropy should be treated as a separately measurable finite number.

A common regulator and renormalization prescription shifts the gravitational couplings and local boundary terms along with the field entropy. The physical combination is the meaningful object within that effective description. Higher-curvature interactions replace the simple area expression with the appropriate gravitational entropy terms; quantum logarithms and finite corrections depend on fields, state, geometry, and ensemble. There is no universal correction coefficient that can be inserted into every black hole independently of its theory.

Proofs of a generalized second law use definite hypotheses about the horizon algebra, quantum state, and gravitational expansion. For example, relative-entropy methods can establish a causal-horizon result for specified classes of semiclassical fields without assuming that the exterior evolves quasistatically at every moment. This is stronger than a purely reversible thermodynamic argument, but it is not a theorem about every proposed quantum-gravity theory or every arbitrary surface chosen in a numerical plot.

The entropy of exterior fields also depends on what region and observables are included. The entropy of a finite radiation subsystem can decrease as correlations become accessible in its complement without violating a causal-horizon generalized second law. Conversely, the coarse thermodynamic entropy inferred from approximately thermal quanta need not equal the exact fine-grained radiation entropy. One must identify the subsystem before comparing curves or interpreting a negative change as a physical violation.

For a large, slowly evaporating Schwarzschild hole, the leading temperature increases as mass decreases. Combining a characteristic area with the thermal scaling gives an inverse-square power law, with a coefficient dependent on particle species and greybody transmission. Integrating that idealized law gives a cubic mass-time relation. Thresholds opening new emitted species, corrections to the spectrum, and backreaction change the coefficient or the model's validity before arbitrary extrapolation is justified.

The formal zero of the cubic is therefore not a controlled prediction of a complete evaporation endpoint. Once curvature and quantum-gravity effects become large, neither a fixed-background Hawking calculation nor the leading area entropy gives a full microscopic evolution. Remnants, final correlations, or other endpoint proposals require additional dynamics. The useful large-mass calculation still predicts parametric heating and a shrinking classical area over its valid interval, while generalized entropy provides the broader accounting appropriate to semiclassical causal horizons.