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Hawking radiation as a flux of Noether charge

Zhi-Wei Wang1,2,* and Samuel L. Braunstein2,†

  • *Contact author: zhiweiwang.phy@gmail.com
  • †Contact author: sam.braunstein@york.ac.uk
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Phys. Rev. D 114, 044065 – Published 21 August, 2026

DOI: https://doi.org/10.1103/t75q-7bp9

Abstract

We present a geometric framework linking classical spacetime geometry to black hole thermodynamics beyond the stationary regime. By evaluating the Komar conserved current on an arbitrary timelike worldtube, with the off-boundary extension fixed by a Gaussian normal foliation, we obtain a local surface density κNoether that requires no Killing vector, no spatial symmetry, and no slow-evolution assumption. It reduces to the Killing surface gravity for stationary non-degenerate horizons, and, applied to a uniformly accelerating worldtube, recovers the Unruh temperature. Adapting the Parikh-Wilczek tunneling method to horizons lacking radial and temporal symmetry, we show the local tunneling temperature is T=κNoether/2π at leading semiclassical order. For a black hole that is genuinely evolving, the single stationary notion of surface gravity separates, at first order in the evaporation rate, into a family of inequivalent quantities; at the event horizon, two of these carry distinct roles: κNoether governs the local tracking kinematics and the proper temperature at the horizon, while the optical peeling rate κpeeling governs the asymptotic Bogoliubov spectrum. Evaluated on the exact Vaidya solution, this yields a closed-form surface gravity for the event horizon and an exact gap equation valid for arbitrary evaporation profiles. The separation has a definite causal character: the tracking rate is fixed by the instantaneous geometry, whereas the peeling rate is a functional of the horizon’s entire future, so local dynamics couples only to the former. Supplying the heat flux independently of the surface gravity, which the Vaidya geometry permits, shows further that the quasilocal first law on the apparent horizon is balanced not by the tracking rate but by the Kodama-Hayward value, with which it coincides only when the horizon’s acceleration vanishes. Four surface gravities on three surfaces therefore answer four distinct questions. These results give a local interpretation of the dynamical Komar charge, identifying the thermal emission of an evolving horizon with a flux of gravitational Noether charge.

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Phys. Rev. D 114, 044065– Published 21 August, 2026
  • Received 28 June 2026
  • Accepted 6 August 2026
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