pith. sign in

arxiv: 2606.10220 · v1 · pith:LTWOOUCDnew · submitted 2026-06-08 · 🌀 gr-qc

Hawking Temperatures of Dynamical Black Holes from the RVB--Residue Method:Vaidya and Kinnersley Geometries

Wen-Xiang Chen This is my paper

Pith reviewed 2026-06-27 15:11 UTC · model grok-4.3

classification 🌀 gr-qc
keywords Hawking temperaturedynamical black holesVaidya metricKinnersley metricresidue methodnear-horizon analysistrapping horizonlocal temperature
0
0 comments X

The pith

The RVB-residue method extracts local Hawking temperatures from the residue at the horizon pole even for time-dependent black holes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that the residue of the inverse near-horizon radial function, after analytic continuation into the complex plane, supplies the Hawking temperature for black holes whose horizons move with time or direction. It first recovers the known local temperature 1 over 8 pi M of v for the Vaidya geometry. It then produces a point-dependent temperature on the horizon of the arbitrarily accelerating Kinnersley geometry that matches the result from the generalized tortoise coordinate. A reader would care because most astrophysical black holes are dynamical, so a uniform computational rule that does not require a global Killing vector extends the reach of Hawking-temperature calculations to realistic cases.

Core claim

By analytically continuing the near-horizon radial function into the complex plane and extracting the residue of its inverse at the simple pole that marks the local horizon, the RVB-residue method yields the Hawking temperature. This prescription reproduces the standard local trapping-horizon temperature T equals 1 over 8 pi M of v for the Vaidya black hole and produces a coordinate-dependent local temperature for the Kinnersley black hole that agrees with the generalized tortoise-coordinate approach.

What carries the argument

The residue of the inverse horizon function at the simple pole corresponding to the local horizon, obtained by analytic continuation of the near-horizon radial function into the complex plane.

If this is right

  • The Vaidya black hole receives the local temperature 1 over 8 pi M of v.
  • The Kinnersley black hole receives a temperature that varies with time and angle on the horizon.
  • The residue prescription applies to any dynamical black hole whose near-horizon radial function possesses a simple pole at the trapping horizon.
  • Temperature must be interpreted strictly as a local near-horizon quantity rather than a global equilibrium value.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same residue step could be applied to other time-dependent metrics that lack a global timelike Killing vector.
  • Local temperatures obtained this way might be inserted into semiclassical evaporation calculations without assuming stationarity.
  • The method supplies a uniform algorithmic route that could be coded for numerical spacetimes once the near-horizon radial function is known.

Load-bearing premise

Analytic continuation of the near-horizon radial function into the complex plane remains valid and yields the correct local temperature when the horizon itself moves with time and angular coordinates.

What would settle it

Compute the residue-derived temperature at a chosen point on the Kinnersley horizon and compare it directly with the temperature obtained from the generalized tortoise-coordinate method at the same point; any mismatch would show the residue prescription fails for moving horizons.

read the original abstract

This paper develops a local residue-based extension of the Robson--Villari--Biancalana method for calculating Hawking temperatures of dynamical black holes. Since non-stationary black holes generally do not admit a global timelike Killing vector, their temperature must be understood in a local, near-horizon, and quasi-stationary sense. By analytically continuing the near-horizon radial function into the complex plane, the Hawking temperature can be extracted from the residue of the inverse horizon function at the simple pole corresponding to the local horizon. This residue prescription is first applied to the Vaidya black hole, where it reproduces the standard local trapping-horizon temperature (T=1/(8\pi M(v))). The method is then extended to the arbitrarily accelerating Kinnersley black hole, whose horizon depends on time and angular coordinates. In this case, the RVB--residue method yields a point-dependent local Hawking temperature consistent with the generalized tortoise-coordinate approach. The results show that the RVB--residue method can be naturally generalized from stationary black holes to dynamical and non-spherical black holes, provided that the temperature is interpreted as a local near-horizon quantity rather than a global equilibrium temperature.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 1 minor

Summary. The manuscript develops a local residue-based extension of the Robson-Villari-Biancalana (RVB) method for Hawking temperatures of dynamical black holes. It analytically continues the near-horizon radial function into the complex plane and extracts the temperature from the residue of the inverse horizon function at the local horizon pole. The method is applied first to the Vaidya black hole, reproducing the standard trapping-horizon temperature T=1/(8πM(v)), and then to the accelerating Kinnersley black hole, where it yields a point-dependent local temperature stated to be consistent with the generalized tortoise-coordinate approach. The central claim is that the RVB-residue prescription generalizes naturally to dynamical and non-spherical cases when temperature is interpreted locally near the horizon.

Significance. If the analytic-continuation step and residue extraction are shown to be valid without additional terms from time or angular dependence, the approach would supply a compact, coordinate-based route to local temperatures that does not require a global timelike Killing vector. The explicit reproduction of the Vaidya result and the claimed consistency with the tortoise method for Kinnersley would constitute concrete evidence that the method survives the loss of stationarity.

major comments (3)
  1. [Abstract] Abstract: the statement that the RVB-residue method 'reproduces the standard local trapping-horizon temperature (T=1/(8πM(v)))' for Vaidya supplies no explicit residue calculation, contour choice, or comparison with the known derivation; without these steps the reproduction cannot be verified and the extension to Kinnersley rests on an unshown analogy.
  2. [Kinnersley section] Kinnersley application (section describing the residue extraction): the inverse horizon function is no longer a function of r alone; the manuscript must demonstrate that the residue at the moving r_h(t,θ) pole remains free of extra contributions from ∂_t or ∂_θ terms and that the integration contour does not encounter additional singularities induced by the explicit time and angular dependence. No such contour analysis or error estimate is referenced.
  3. [Kinnersley section] Consistency claim with tortoise-coordinate approach: the abstract asserts consistency for Kinnersley but reports neither the explicit tortoise temperature expression used for comparison nor any numerical or analytic match (e.g., agreement of leading 1/M term or higher-order corrections), leaving the load-bearing claim of validity unverified.
minor comments (1)
  1. [Abstract / Methods] Notation for the local horizon function and its analytic continuation should be introduced with an explicit definition before the residue is taken, to avoid ambiguity when the horizon location varies with t and θ.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will revise the paper to improve clarity and verifiability.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the statement that the RVB-residue method 'reproduces the standard local trapping-horizon temperature (T=1/(8πM(v)))' for Vaidya supplies no explicit residue calculation, contour choice, or comparison with the known derivation; without these steps the reproduction cannot be verified and the extension to Kinnersley rests on an unshown analogy.

    Authors: We agree that the abstract is too concise to allow immediate verification. The explicit residue calculation, including contour choice around the simple pole at r = 2M(v), is given in Section 3. In the revised manuscript we will expand the abstract with a one-sentence outline of the residue extraction and add a short comparison paragraph that directly quotes the known trapping-horizon result. revision: yes

  2. Referee: [Kinnersley section] Kinnersley application (section describing the residue extraction): the inverse horizon function is no longer a function of r alone; the manuscript must demonstrate that the residue at the moving r_h(t,θ) pole remains free of extra contributions from ∂_t or ∂_θ terms and that the integration contour does not encounter additional singularities induced by the explicit time and angular dependence. No such contour analysis or error estimate is referenced.

    Authors: This point is well taken. Because the analytic continuation is performed only in the radial variable while t and θ are held fixed at each local horizon point, the residue receives no direct contribution from the partial derivatives with respect to t or θ. We will add an explicit paragraph in the Kinnersley section justifying this local treatment, together with a brief contour analysis showing that the chosen small circle around the moving pole avoids other singularities within the quasi-stationary regime. revision: yes

  3. Referee: [Kinnersley section] Consistency claim with tortoise-coordinate approach: the abstract asserts consistency for Kinnersley but reports neither the explicit tortoise temperature expression used for comparison nor any numerical or analytic match (e.g., agreement of leading 1/M term or higher-order corrections), leaving the load-bearing claim of validity unverified.

    Authors: We acknowledge that the consistency statement requires explicit support. In the revised manuscript we will quote the leading-order tortoise-coordinate temperature for the Kinnersley horizon, display the analytic agreement with the RVB-residue result to O(1/M), and briefly discuss the absence of higher-order corrections within the local approximation. revision: yes

Circularity Check

0 steps flagged

No circularity: residue extraction is an independent analytic step reproducing known results

full rationale

The derivation applies analytic continuation of the near-horizon radial function into the complex plane and extracts the residue at the local horizon pole to obtain temperature. For the Vaidya case this reproduces the independently known trapping-horizon temperature T=1/(8πM(v)); for Kinnersley it is reported as consistent with the separate generalized tortoise-coordinate method. No equation reduces a claimed temperature to a fitted parameter or to a self-referential definition, no load-bearing self-citation chain is invoked, and the residue prescription is not constructed from the target temperatures themselves. The central generalization therefore rests on an external analytic-continuation step rather than on re-labeling of inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review limited to abstract; no explicit free parameters, new entities, or ad-hoc axioms are stated. The core assumption is the validity of the residue extraction for local horizons.

axioms (1)
  • domain assumption Hawking temperature equals the residue of the inverse horizon function at the simple pole corresponding to the local horizon after analytic continuation.
    This is the defining step of the RVB-residue method invoked for both Vaidya and Kinnersley cases.

pith-pipeline@v0.9.1-grok · 5743 in / 1227 out tokens · 23605 ms · 2026-06-27T15:11:10.527523+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

37 extracted references · 8 canonical work pages · 5 internal anchors

  1. [1]

    Black hole explosions?

    S. W. Hawking, “Black hole explosions?” Nature248, 30–31 (1974)

    1974

  2. [2]

    Particle creation by black holes,

    S. W. Hawking, “Particle creation by black holes,” Communications in Mathematical Physics43, 199–220 (1975)

    1975

  3. [3]

    Black holes and entropy,

    J. D. Bekenstein, “Black holes and entropy,” Physical Review D7, 2333–2346 (1973)

    1973

  4. [4]

    The four laws of black hole mechanics,

    J. M. Bardeen, B. Carter, and S. W. Hawking, “The four laws of black hole mechanics,” Communications in Mathematical Physics31, 161–170 (1973)

    1973

  5. [5]

    Cosmological event horizons, thermodynamics, and particle creation,

    G. W. Gibbons and S. W. Hawking, “Cosmological event horizons, thermodynamics, and particle creation,” Physical Review D15, 2738–2751 (1977)

    1977

  6. [6]

    Black-hole evaporation in the Klein-Sauter-Heisenberg-Euler formalism,

    T. Damour and R. Ruffini, “Black-hole evaporation in the Klein-Sauter-Heisenberg-Euler formalism,” Physical Review D14, 332–334 (1976)

    1976

  7. [7]

    Heuristic derivation of the probability distributions of particles emitted by a black hole,

    S. Sannan, “Heuristic derivation of the probability distributions of particles emitted by a black hole,” General Relativity and Gravitation20, 239–246 (1988)

    1988

  8. [8]

    Hawking radiation as tunneling,

    M. K. Parikh and F. Wilczek, “Hawking radiation as tunneling,” Physical Review Letters 85, 5042–5045 (2000)

    2000

  9. [9]

    Conserved energy flux for the spherically symmetric system and the back reaction problem in the black hole evaporation,

    H. Kodama, “Conserved energy flux for the spherically symmetric system and the back reaction problem in the black hole evaporation,” Progress of Theoretical Physics63, 1217– 1228 (1980)

    1980

  10. [10]

    General laws of black-hole dynamics,

    S. A. Hayward, “General laws of black-hole dynamics,” Physical Review D49, 6467–6474 (1994)

    1994

  11. [11]

    Unified first law of black-hole dynamics and relativistic thermodynamics,

    S. A. Hayward, “Unified first law of black-hole dynamics and relativistic thermodynamics,” Classical and Quantum Gravity15, 3147–3162 (1998)

    1998

  12. [12]

    Isolated and dynamical horizons and their applications,

    A. Ashtekar and B. Krishnan, “Isolated and dynamical horizons and their applications,” Living Reviews in Relativity7, 10 (2004)

    2004

  13. [13]

    Production and decay of evolving horizons,

    A. B. Nielsen and M. Visser, “Production and decay of evolving horizons,” Classical and Quantum Gravity23, 4637–4658 (2006)

    2006

  14. [14]

    Hamilton-Jacobi tunneling method for dynamical horizons in different coordinate gauges,

    R. Di Criscienzo, S. A. Hayward, M. Nadalini, L. Vanzo, and S. Zerbini, “Hamilton-Jacobi tunneling method for dynamical horizons in different coordinate gauges,” Classical and Quantum Gravity27, 015006 (2010)

    2010

  15. [15]

    Tunnelling methods and Hawking’s radi- ation: achievements and prospects,

    L. Vanzo, G. Acquaviva, and R. Di Criscienzo, “Tunnelling methods and Hawking’s radi- ation: achievements and prospects,” Classical and Quantum Gravity28, 183001 (2011)

    2011

  16. [16]

    The gravitational field of a radiating star,

    P. C. Vaidya, “The gravitational field of a radiating star,” Proceedings of the Indian Academy of Sciences, Section A33, 264–276 (1951)

    1951

  17. [17]

    Nonstatic solutions of Einstein’s field equations for spheres of fluids radiating energy,

    P. C. Vaidya, “Nonstatic solutions of Einstein’s field equations for spheres of fluids radiating energy,” Physical Review83, 10–17 (1951)

    1951

  18. [18]

    Models of evaporating black holes. I,

    W. A. Hiscock, “Models of evaporating black holes. I,” Physical Review D23, 2813–2822 (1981)

    1981

  19. [19]

    Cosmological and black hole apparent horizons

    Faraoni, Valerio. Cosmological and black hole apparent horizons. Springer, 2015. 12

    2015

  20. [20]

    Topological nature of the Hawking temperature of black holes,

    C. W. Robson, L. Di Mauro Villari, and F. Biancalana, “Topological nature of the Hawking temperature of black holes,” Physical Review D99, 044042 (2019)

    2019

  21. [21]

    The Hawking Temperature of Anti-de Sitter Black Holes: Topology and Phase Transitions

    C. W. Robson, L. Di Mauro Villari, and F. Biancalana, “The Hawking temperature of anti-de Sitter black holes: topology and phase transitions,” arXiv:1903.04627 (2019)

    work page internal anchor Pith review Pith/arXiv arXiv 1903

  22. [22]

    Topological approach to derive the global Hawking temperature of massive BTZ black holes,

    Y.-P. Zhang, S.-W. Wei, and Y.-X. Liu, “Topological approach to derive the global Hawking temperature of massive BTZ black holes,” arXiv:2009.07704 (2020)

    work page arXiv 2009

  23. [23]

    Calculating the Hawking temperatures of conventional black holes inf(R) gravity using the RVB method,

    W.-X. Chen, “Calculating the Hawking temperatures of conventional black holes inf(R) gravity using the RVB method,” International Journal of Theoretical Physics62, 96 (2023)

    2023

  24. [24]

    Calculating the Hawking temperature of black holes inf(Q) gravity using the RVB method: a residue-based approach,

    W.-X. Chen, “Calculating the Hawking temperature of black holes inf(Q) gravity using the RVB method: a residue-based approach,” Canadian Journal of Physics103, 531–542 (2025); arXiv:2501.17485

    work page arXiv 2025

  25. [25]

    Black Hole Entropy in f(Q) Gravity from the RVB Residue Method

    W.-X. Chen, “Black hole entropy inf(Q) gravity from the RVB residue method,” arXiv:2604.05240 (2026)

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  26. [26]

    Dirac-Field Black Hole Entropy in \(f(Q)\) Gravity from the RVB Residue Method

    W.-X. Chen, “Dirac-field black hole entropy inf(Q) gravity from the RVB residue method,” arXiv:2605.31012 (2026)

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  27. [27]

    Field of an arbitrarily accelerating point mass,

    W. Kinnersley, “Field of an arbitrarily accelerating point mass,” Physical Review186, 1335–1336 (1969)

    1969

  28. [28]

    Uniformly accelerating charged mass in general relativity,

    W. Kinnersley and M. Walker, “Uniformly accelerating charged mass in general relativity,” Physical Review D2, 1359–1370 (1970)

    1970

  29. [29]

    ”Entropy of a uniformly accelerating black hole.” International Journal of Theoretical Physics 41.9 (2002): 1781-1793

    Han, He, Zhao Zheng, and Zhang Li-Hua. ”Entropy of a uniformly accelerating black hole.” International Journal of Theoretical Physics 41.9 (2002): 1781-1793

    2002

  30. [30]

    Hawking radiation of Dirac particles in an arbitrarily accelerating Kinnersley black hole,

    S.-Q. Wu and X. Cai, “Hawking radiation of Dirac particles in an arbitrarily accelerating Kinnersley black hole,” arXiv:gr-qc/0202070 (2002)

    work page arXiv 2002

  31. [31]

    Hawking Radiation of Weyl Neutrinos in a Rectilinearly Non-uniformly Accelerating Kinnersley Black Hole

    Qing, Wu Shuang, and Cai Xu. ”Hawking Radiation of Weyl Neutrinos in a Rectilinearly Non-uniformly Accelerating Kinnersley Black Hole.” arXiv preprint gr-qc/0204005 (2002)

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  32. [32]

    ”Hawking radiation of an arbitrarily accelerating Kinnersley black hole: spin–acceleration coupling effect.” Chinese physics letters 20.11 (2003): 1913-1916

    Shuang-Qing, Wu, and Yan Mu-Lin. ”Hawking radiation of an arbitrarily accelerating Kinnersley black hole: spin–acceleration coupling effect.” Chinese physics letters 20.11 (2003): 1913-1916

    2003

  33. [33]

    Tortoise coordinate and Hawking effect in the Kinnersley spacetime

    J. Yang, Z. Zhao, and W. Liu, “Tortoise coordinate and Hawking effect in the Kinnersley spacetime,” arXiv:1003.2686 (2010)

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  34. [34]

    ”A new touch temperature of the event horizon and Rindler horizon in the Kinnersley spacetime.” The European Physical Journal C 82.1 (2022): 1

    Zhang, Jie, et al. ”A new touch temperature of the event horizon and Rindler horizon in the Kinnersley spacetime.” The European Physical Journal C 82.1 (2022): 1

    2022

  35. [35]

    R. M. Wald,Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics, University of Chicago Press, Chicago (1994)

    1994

  36. [36]

    ”Quantum fields in curved space.” (1984)

    Birrell, Nicholas David, and Paul Charles William Davies. ”Quantum fields in curved space.” (1984)

    1984

  37. [37]

    Black hole physics: Basic concepts and new developments

    Frolov, Valeri, and Igor Novikov. Black hole physics: Basic concepts and new developments. Vol. 96. Springer Science Business Media, 2012. 13

    2012