REVIEW
Dynamic black-hole entropy
General Relativity and Quantum Cosmology · Submitted 2 Oct 1998 · 19 references
Pith's verdict: A clean Clausius observation plus a Kodama-Noether proposal whose key integral is deferred to an in-preparation companion paper.
T0 · AI review
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Referee report
The paper
The paper claims
For dynamic black holes in spherical symmetry, two independent entropy definitions both give one-quarter the area of the trapping horizon.
The reading
What carries the argument
The carrying object is the Kodama vector $k=*dr$, a divergence-free preferred time flow that exists in spherical symmetry even without a Killing vector. The paper uses it twice: to define a dynamic surface gravity $κ=*dk/2$, and to stand in for the stationary Killing vector in Wald's Noether-charge formula. The trapping horizon, where $dr$ is null and $E=r/2$, is the surface on which the extra term in the energy supply drops out and the Clausius identification becomes clean. The unified first law organizes the argument by separating work density $w dV$ from energy supply $Aψ$.
Extended reading notes
Core claim
The central claim is that two independent non-statistical definitions of entropy agree for a dynamic black hole in spherical symmetry in Einstein gravity. The unified first law reads $dE = Aψ + w dV$, where the energy-supply term can be split as $Aψ = κ dA/8π + r d(E/r)$. On a trapping horizon, defined by $dr$ null and hence $E=r/2$, the second term vanishes, leaving $Aψ = κ dA/8π$; identifying $κ/2π$ as temperature makes $dA/4$ the Clausius entropy. Separately, replacing the stationary Killing vector by the Kodama vector $k=*dr$ in Wald's Noether-charge construction gives $∮ Q = Aκ/8π$, and hence $S=A/4$. The paper's point is that both definitions land on the same quarter-area expression even when the hole is not stationary.
Load-bearing premise
The load-bearing premise is that the Kodama vector can replace the stationary Killing vector in Wald's Noether-charge construction, so that the sphere integral of the Noether charge is $Aκ/8π$; the paper proposes this and defers the calculation to an in-preparation reference.
Editorial extensions
If this is right
- In Einstein gravity, a spherically symmetric dynamic black hole has entropy $A/4$ at its trapping horizon, matching the stationary result.
- Black-hole entropy can be derived from a first law with an energy-supply term, without a statistical count of microstates.
- Wald's Noether-charge method extends to non-stationary black holes when the Kodama flow replaces the Killing vector.
- Because the equations hold off the horizon as well, the paper suggests that area entropy may be associated with surfaces more generally, not only black-hole horizons.
- The result is local and spherically symmetric; removing that restriction is left to forthcoming work.
Reading between the lines
- If the Kodama-Noether construction survives beyond spherical symmetry, it would supply a quasilocal entropy for dynamical horizons without requiring any asymptotic stationarity.
- The off-horizon validity of both definitions suggests a test: compare $A/4$ for small surfaces in flat space with entanglement-entropy calculations, where the same area law is expected.
- A useful consistency check is whether the dynamic first law plus $S=A/4$ reproduces the area-increase second law for trapping horizons; if it does, entropy and horizon monotonicity become two views of one statement.
- The agreement depends on a normalization computed in an in-preparation reference; an independent derivation of $∮ Q$ would turn the coincidence into a robust correspondence.
Original abstract
We consider two non-statistical definitions of entropy for dynamic (non-stationary) black holes in spherical symmetry. The first is analogous to the original Clausius definition of thermodynamic entropy: there is a first law containing an energy-supply term which equals surface gravity times a total differential. The second is Wald's Noether-charge method, adapted to dynamic black holes by using the Kodama flow. Both definitions give the same answer for Einstein gravity: one-quarter the area of the trapping horizon.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We consider two non-statistical definitions of entropy for dynamic (non-stationary) black holes in spherical symmetry. The first is analogous to the original Clausius definition of thermodynamic entropy: there is a first law containing an energy-supply term which equals surface gravity times a total differential. The second is Wald's Noether-charge method, adapted to dynamic black holes by using the Kodama flow. Both definitions give the same answer for Einstein gravity: one-quarter the area of the trapping horizon.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We consider two non-statistical definitions of entropy for dynamic (non-stationary) black holes in spherical symmetry. | contribution 0.93 | 0.74 | 0.05 | asserted 0.70 |
| s1 | The first is analogous to the original Clausius definition of thermodynamic entropy: there is a first law containing an energy-supply term which equals surface gravity times a total differential. | method 0.58 | 0.38 | 0.06 | asserted 1.00 |
| s2 | The second is Wald's Noether-charge method, adapted to dynamic black holes by using the Kodama flow. | method 0.74 | 0.48 | 0.05 | asserted 0.99 |
| s3 | Both definitions give the same answer for Einstein gravity: one-quarter the area of the trapping horizon. | result 1.00 | 0.79 | 0.79 | asserted 1.00 |
What was examined, and what was not
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- P abstract
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References
How later papers used it
51 later papers cite it
1 later paper by other authors is listed below: 1 used its method or tool.
Uses its method or tool 1 of 5 papers with this label
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The corresponding work density for a perfect fluid is [ref] [equation] The total energy inside the apparent horizon is $E = ρV = 4π/3R^3ρ$, whose differential is [equation] Using the continuity equation ([ref]) to express $dρ= -3H(ρ+ p) dt$, we obtain [equation] Now we apply the first law of thermodynamics on the apparent horizon, $dE = T_h dS_h + W dV$.
Big-Bang Nucleosynthesis and WIMP Dark Matter Freeze-Out as Probes of Yukawa Cosmology (2026) · Derivation of Modified Friedmann Equations through Yukawa Cosmology · 0.85
How the literature uses it
50 arXiv papers cite this in 81 sentences
Papers citing it · 62 sentences without a confident label
Builds on it 1 paper
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We will now present a generalization of this approach to dimensions $d≥3$[While the sources used in this work [ref] restrict their discussion to four dimensional spacetimes, a generalization of the Kodama vector to $d$ dimensions has been discussed in [ref] .
Dynamical Black Holes in 2+1 Dimensions · The Kodama vector and dynamical entropy · 0.94
Uses its method or tool 5 papers
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The modified theories of gravity under our consideration with the field equation ([ref]) are all diffeomorphism invariant, and therefore we can obtain the Wald-Kodama dynamical entropy of the FRW apparent horizon by Wald's Noether-charge method [ref] as [equation] with $G_eff=G_eff(t)$.
Friedmann equations from nonequilibrium thermodynamics of the Universe: A unified formulation for modified gravity · Method 2: Nonequilibrium Clausius relation · 0.89
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The corresponding work density for a perfect fluid is [ref] [equation] The total energy inside the apparent horizon is $E = ρV = 4π/3R^3ρ$, whose differential is [equation] Using the continuity equation ([ref]) to express $dρ= -3H(ρ+ p) dt$, we obtain [equation] Now we apply the first law of thermodynamics on the apparent horizon, $dE = T_h dS_h + W dV$.
Big-Bang Nucleosynthesis and WIMP Dark Matter Freeze-Out as Probes of Yukawa Cosmology · Derivation of Modified Friedmann Equations through Yukawa Cosmology · 0.85
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On the other side, for a spherically symmetric space-time with the metric ([ref]), using the Misner-Sharp mass [ref] [equation] the $a-b$ components of the Einstein equation give the mass formulas [ref] [equation] where $T=T^a_a$.
Modified gravity emerging from thermodynamics and holographic principle · The Misner-Sharp Mass · 0.86
Compares with it 2 papers
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Interestingly, these first laws exhibit the same structure as Hayward's unified first law for trapping horizons [ref] .
Unified formalism for the emergence of space in non-equilibrium description · Introduction · 0.92
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Interestingly this form of the first laws found to have the same structure as the first law for the trapping horizons proposed by Hayward [ref] .
Emergence of space from non-equilibrium thermodynamics in f(R) gravity · Introduction · 0.99
Cites it as support 2 papers
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Now, both of these prescriptions give back the Bekenstein-Hawking entropy in the static case [ref] , which receives support from the more involved discussion of dynamical black hole horizons.
Area-law versus R\'enyi and Tsallis black hole entropies · Impossibility of non-area law entropy · 0.99
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This claim has been supported by many references, including [ref] .
Friedmann equations of the fractal apparent horizon · Introduction · 0.98
Cites it as background 9 papers
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Before going on, let us first give a brief review on the work of Hayward [ref] .
Generalized Vaidya Spacetime in Lovelock Gravity and Thermodynamics on Apparent Horizon · Entropy and energy of the apparent horizon · 0.80
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Previous attempts to find a quasi-local alternative to the event horizon culminated in the elegant notions of a future outer trapping horizon (FOTH) [ref] or dynamical horizon [ref] (see [ref] for reviews).
Proof of a New Area Law in General Relativity · Introduction · 0.95
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[ref] proposed a unified first law of black hole dynamics and relativistic thermodynamics in spherically symmetric general relativity, see also [ref] .
Thermodynamic Origin of the Cardassian Universe · Introduction · 0.88
Discussion
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