Increase of Black Hole Entropy in Higher Curvature Gravity

Gungwon Kang; Robert C. Myers; Ted Jacobson

REVIEW

Increase of Black Hole Entropy in Higher Curvature Gravity

Ted Jacobson, Gungwon Kang, Robert C. Myers

General Relativity and Quantum Cosmology · Submitted 11 Mar 1995 · 28 references

Pith's verdict: A clean, honest proof that Wald entropy obeys the Second Law in R + P(R) gravity — conditional on a positivity bound the authors defend only heuristically.

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CitationGungwon Kang, Robert C. Myers, Ted Jacobson. "Increase of Black Hole Entropy in Higher Curvature Gravity." arXiv:gr-qc/9503020 (1995). https://pith.science/paper/FK3V35KF

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Two things worth knowing. First, §2 is the part I'd hand to a student: for quasi-stationary accretion, ΔS ≥ 0 follows from the First Law plus the null energy condition on the matter alone, with no reference to the gravitational action. That is a genuinely general statement and it is proved in a page. Second, for R + P(R) the conformal map to Einstein plus a scalar does real work: the barred area pulls back to (1/4G)∫√h(1+P'(R)), and since mass, angular velocity and surface gravity agree between frames, the entropy variations must agree too. The Zeroth Law comes along for free. §5's extended Raychaudhuri equation is a reusable tool, and the physical-process derivation of the entropy functional at the end is a nice piece of work — it recovers (23) by integrating the flux, which is a genuinely independent route to the same answer rather than a restatement of Wald's Noether charge. The algebra in (13)–(23) checks out, including the branch structure in §4 where P''(φ₀)=0 gives spurious Einstein solutions; they flag that honestly instead of burying it. The paper is also candid about its own limits, in the body at least: the positivity bound, cosmic censorship, the stability requirement, and the concession in §5 that the 'direct' proof still borrows 1+P' > 0 from the conformal map. Where I'd push: the potential-barrier argument for 1+P' > 0 is a plausibility argument, not a dynamical existence proof, and it assumes the metric and matter fields stay nonsingular — which is exactly what is at stake if the bound is crossed. The abstract's 'in general for any dynamical process' [abstract:s2] does not carry that caveat; the body does. Second, the dismissal of the Iyer–Wald boost-invariant dynamical entropy in §3 is asserted from the conformal-frame identification rather than argued on its own terms. That's a real disagreement in the literature and it deserves more than a paragraph. Neither undercuts the central claim, which is conditional on assumptions the authors name. This is for people working on higher-curvature thermodynamics or on the Wald entropy program; the §2 argument is worth knowing even if you skip the rest. Send it out.

Referee report

Significance

The results are of clear value to the field. The §2 argument is short, general, and worth having stated explicitly: it isolates exactly which ingredients (First Law plus null energy condition plus stability of the final state) a quasi-stationary Second Law requires, and it makes no assumption about the form of S. The conformal construction of §§3–4 does more than reprove a known entropy formula: by pulling the barred area back through eq. (16) it selects a preferred dynamical entropy among the Noether-charge ambiguities, and it converts the Second Law into a checkable constraint on effective actions — the requirement that the local entropy density 1 + P'(R) be positive everywhere outside the horizon. That is a genuinely useful, falsifiable criterion for effective Lagrangians, and the observation that the same expression controls both the conformal map's validity and the entropy density is a nice structural point. The extended Raychaudhuri equation of §5 also yields a physical-process derivation of the First Law and of the entropy, which is a reusable technique.

Major objections (3)

[Abstract [abstract:s2]; §3, eq. (18) and following; §4, eq. (22) and following] The abstract states that the Second Law holds 'in general for any dynamical process', but the proof as constructed is conditional on two assumptions that are not theorems: cosmic censorship in the barred frame, and 1 + P'(R) > 0 everywhere outside and on the horizon. The latter is defended in §3 by the field redefinition φ = β^{-1} ln(1 + 2αφ) and the exponential potential barrier of eq. (18), and in §4 by the analogous P'' > 0 argument; both arguments explicitly assume 'the metric and matter fields do not become singular', which is precisely the condition that fails if the bound is crossed. The barrier is a local energy argument, not a global existence proof, and it does not by itself exclude a crossing driven by large gradients or by the e^{-Dβφ/(D-2)} coupling to matter during collapse. The authors are candid about this in the text, but the abstract and the first bullet of §6 do not carry the qualification. I would ask that the abstract and the §6 summary state the hypotheses (P'' > 0, 1 + P' > 0 outside the horizon, cosmic censorship in the conformally related frame) explicitly, since the result is a conditional theorem and reads as unconditional in the abstract.
[§2, eqs. (2), (6), (8); Abstract [abstract:s1]] The claim that the quasi-stationary result holds 'independent of the details of the gravitational action' is stronger than what §2 establishes. The derivation uses the First Law in the form of eq. (2), κ δS/(2π) = δM − Ω δJ, which presupposes that the surface gravity κ is constant over the horizon and that a definite entropy functional S exists for the theory at hand. The paper itself states in the Introduction and again in §6 that the Zeroth Law 'remains an important open question' for general higher curvature theories, and that the First Law fixes S only up to a constant on each connected family of stationary solutions. So the §2 argument is conditional on the Zeroth Law, not independent of the gravitational action. This does not damage the logic — the argument is correctly presented as taking the First Law as input — but the abstract's phrasing should be softened to something like 'given the First Law, independent of the form of the entropy functional', which is what eq. (8) actually shows.
[§5, eqs. (24)–(27) and the paragraph following eq. (27)] The 'direct proof' of §5 is not independent of the conformal analysis, as the authors themselves note at the end of the section ('we do not really have a fully direct proof'). The step that carries the argument is the passage from θ̃ → −∞ at finite affine parameter to a contradiction: the paper argues that either θ → −∞, which gives a conjugate point excluded by the standard area-theorem reasoning, or ∂_λρ → −∞, which is asserted to be a naked singularity. The second horn is the one that actually arises for these theories, and the identification of a divergence in ∂_λρ with a cosmic-censorship violation is stated rather than derived; for the P(R) theories it reduces to a divergence in R or in k^a∇_aR, and the claim that the equations of motion then force a curvature or stress-tensor divergence is asserted in one sentence. Since this is offered as the template for extending the Second Law to Lagrangians not amenable to the conformal trick, that step deserves a fuller argument, or an explicit statement that it is heuristic.

Minor objections (5)

[§5, eqs. (24)–(26)] The notation is inconsistent: eq. (24) writes the entropy density as e^{ρ̃}, but eq. (25) and eq. (26) use ρ (∂_λρ, ∇_a∇_bρ) without ever stating that ρ̃ = ρ. Please unify the symbol, and define S′ in eq. (25) as the derivative with respect to the affine parameter λ along the horizon generators.
[§2, eqs. (5) and (7)] Eq. (7) writes the volume integrand as ∇_a(T^{ab}χ_b) dΣ but the flux integrals carry dΣ_b; the index structure should be made uniform. It would also help to state explicitly, near eq. (5), the sign convention for the outward normal on O and on H, since the sign of the horizon term in eq. (8) is what makes the argument work.
[§3, paragraph following eq. (17)] The statement that the alternative dynamical entropy of ref. [11] 'appears inconsistent with the Second Law' rests on the premise that at most one entropy functional can satisfy a local increase law. The authors hedge with 'unless', which is appropriate, but the uniqueness premise should be stated as an assumption rather than left implicit, since it is doing the work in that paragraph.
[§4, paragraph following eq. (20)] The second class of solutions, φ = φ_0 with P''(φ_0) = 0, is dismissed as spurious because of the effective cosmological constant, with the degenerate case Λ_eff = 0 handled in a single clause. A sentence or two on why the degenerate case is harmless for the asymptotically flat solutions of interest would close an otherwise loose end.
[§6, final paragraph] The discussion of the generalized Second Law and of nonlocal effective actions is brief and somewhat disconnected from the rest of the paper. It could either be shortened or given a clearer statement of which of the present results is expected to survive once Hawking radiation is included.

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The paper

The paper claims

The quantity that plays the role of black hole entropy keeps increasing even when the gravitational action is corrected by higher powers of curvature, because a conformal change of frame turns the corrected entropy into an ordinary horizon

The reading

Black hole thermodynamics rests on two laws: the Zeroth, that surface gravity is constant over a stationary horizon, and the Second, that entropy never decreases. In ordinary general relativity both are known, but the Second Law is proved through Hawking's area theorem, which needs a null convergence condition that higher curvature terms in the gravitational action typically violate. This paper argues that the right question is not whether the area grows but whether the particular horizon integral that appears in the First Law grows, and shows that for actions of the form $R + P(R)$ with $P$ a polynomial, the answer is yes. The trick is a conformal rescaling of the metric that turns the higher curvature theory into ordinary Einstein gravity coupled to a scalar field, chosen so that the rescaled horizon area equals $S = (1/4G)int sqrt{h},(1+P'(R))$ in the original theory; the area theorem in the rescaled frame then transfers directly. A separate, shorter argument covers quasi-stationary accretion and needs no assumption about the gravitational action at all, only that the infalling matter satisfies the null energy condition.

What carries the argument

The load-bearing device is the conformal map $bar{g}_{ab} = (1+P'(varphi))^{2/(D-2)} g_{ab}$ that converts the $R+P(R)$ action into Einstein gravity coupled to a scalar with potential $V(varphi) propto (varphi P' - P)$. The factor is chosen precisely so that the barred horizon area element $sqrt{bar{h}}$ equals $sqrt{h},(1+P'(R))$, which is why the Einstein-frame area theorem lands exactly on the higher curvature entropy rather than on some other functional. The second device is the extended Raychaudhuri equation $partial_lambda tilde{theta} = -theta^2/(D-2) - sigma^2 - k^a k^b R_{ab} + k^a k^b nabla_a nabla_b rho$, which plays the role the ordinary Raychaudhuri equation plays in Hawking's proof, with $tilde{theta}$ replacing the expansion. A third, independent device is the flux argument of Section 2: integrating $nabla_a(T^{ab} chi_b)$ over the volume between two asymptotic slices and the horizon gives $Delta S = -(2pi/kappa) int_H T_{ab} chi^a dSigma^b geq 0$, which uses only the First Law and the null energy condition for matter.

Extended reading notes

Core claim

The central claim is that in a theory with action $I = int d^Dx sqrt{-g},[R + P(R)]/(16pi G) + L_m$, the black hole entropy is $S = (1/4G) int_H d^{D-2}x, sqrt{h},(1 + P'(R))$, and this $S$ satisfies a classical increase theorem for arbitrary dynamical processes, not merely for quasi-stationary ones. The proof proceeds by introducing an auxiliary scalar $varphi$ and a conformal factor $(1+P'(varphi))^{2/(D-2)}$ that maps the theory to Einstein gravity plus a scalar with a potential; provided $P''>0$ and $1+P'(varphi)>0$, the total stress-energy in the rescaled frame obeys the dominant energy condition, so Bardeen-Carter-Hawking gives the Zeroth Law and Hawking's area theorem gives $delta bar{S} geq 0$. Because the conformal factor goes to the identity at infinity, masses, angular momenta, angular velocities, surface gravities and horizons agree between the two frames, and the First Law then forces the entropies to agree as well. The paper also gives a direct proof in the higher curvature frame using an extended Raychaudhuri equation for $tilde{theta} = theta + partial_lambda rho$, and shows that the same equation yields a physical-process derivation of the First Law and of the entropy functional itself.

Load-bearing premise

The proof needs the conformal rescaling to remain nonsingular everywhere outside the horizon, meaning $1+P'(R)$ never reaches zero there, and the authors support this only by a plausibility argument about a scalar field being blocked by a rising potential, together with the assumption that cosmic censorship holds in the rescaled frame.

Editorial extensions

If this is right

Wald's Noether-charge entropy, which the First Law fixes only up to ambiguities for non-stationary horizons, is pinned down for Ricci-polynomial theories: the correct dynamical entropy density is $1 + P'(R)$, and the competing boost-invariant proposal of Iyer and Wald is argued to be inconsistent with the Second Law.
The positivity of the entropy density $1+P'(R)$ becomes a physical selection criterion on coupling constants, since the proof requires it everywhere outside the horizon.
The Zeroth Law extends to these theories: stationary black holes with Killing horizons have constant surface gravity, obtained by transferring the dominant-energy-condition proof through the conformal map.
The physical-process derivation recovers the entropy functional by integrating mass and angular momentum changes during quasi-stationary accretion, giving an independent route to the same $1+P'(R)$ density.
The direct Raychaudhuri-based argument is offered as a template that may extend to Lagrangians not reachable by a conformal transformation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the entropy density $1+P'(R)$ must be positive for the Second Law to hold, then the requirement that entropy have a statistical origin and the requirement that the classical increase theorem hold converge on the same inequality, which suggests the positivity condition is not an artifact of the proof technique.
The paper's own remark that a divergence in $k^a nabla_a R$ would be the mildest way to violate the Second Law points to a concrete numerical test: search for horizon evolution in which curvature stays bounded but its null derivative blows up.
Because the conformal frame exchanges higher curvature terms for a scalar with an exponentially rising potential, the stability question for these theories is really a question about whether that potential confines the scalar, which is a well-posed problem in ordinary scalar-field dynamics and could be settled independently of black hole physics.
The failure of the argument for negative coupling, where the potential falls rather than rises, suggests the Second Law may serve as a filter that discards exactly the unstable members of an effective-action family, an idea the authors raise but do not push to a general criterion.

Original abstract

We examine the Zeroth Law and the Second Law of black hole thermodynamics within the context of effective gravitational actions including higher curvature interactions. We show that entropy can never decrease for quasi-stationary processes in which a black hole accretes positive energy matter, independent of the details of the gravitational action. Within a class of higher curvature theories where the Lagrangian consists of a polynomial in the Ricci scalar, we use a conformally equivalent theory to establish that stationary black hole solutions with a Killing horizon satisfy the Zeroth Law, and that the Second Law holds in general for any dynamical process. We also introduce a new method for establishing the Second Law based on a generalization of the area theorem, which may prove useful for a wider class of Lagrangians. Finally, we show how one can infer the form of the black hole entropy, at least for the Ricci polynomial theories, by integrating the changes of mass and angular momentum in a quasistationary accretion process.

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The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We examine the Zeroth Law and the Second Law of black hole thermodynamics within the context of effective gravitational actions including higher curvature interactions. We show that entropy can never decrease for quasi-stationary processes in which a black hole accretes positive energy matter, independent of the details of the gravitational action. Within a class of higher curvature theories where the Lagrangian consists of a polynomial in the Ricci scalar, we use a conformally equivalent theory to establish that stationary black hole solutions with a Killing horizon satisfy the Zeroth Law, and that the Second Law holds in general for any dynamical process. We also introduce a new method for establishing the Second Law based on a generalization of the area theorem, which may prove useful for a wider class of Lagrangians. Finally, we show how one can infer the form of the black hole entropy, at least for the Ricci polynomial theories, by integrating the changes of mass and angular momentum in a quasistationary accretion process.

background contribution method result interpretation main claim

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We examine the Zeroth Law and the Second Law of black hole thermodynamics within the context of effective gravitational actions including higher curvature interactions.	contribution 0.66	0.81	0.07	unclear 0.65
s1	We show that entropy can never decrease for quasi-stationary processes in which a black hole accretes positive energy matter, independent of the details of the gravitational action.	result 1.00	0.94	0.52	asserted 1.00
s2	Within a class of higher curvature theories where the Lagrangian consists of a polynomial in the Ricci scalar, we use a conformally equivalent theory to establish that stationary black hole solutions with a Killing horizon satisfy the Zeroth Law, and that the Second Law holds in general for any dynamical process.	result 0.83	0.93	0.35	asserted 1.00
s3	We also introduce a new method for establishing the Second Law based on a generalization of the area theorem, which may prove useful for a wider class of Lagrangians.	contribution 0.72	0.94	0.07	hedged possibility 0.86
s4	Finally, we show how one can infer the form of the black hole entropy, at least for the Ricci polynomial theories, by integrating the changes of mass and angular momentum in a quasistationary accretion process.	result 0.50	0.95	0.11	asserted 0.81

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 1.00.

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sources read

arXiv listing abstract v1

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levels run

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References

Works cited (28)

[1]

See for example: N.D. Birrell and P.C.W. Davies, it Quant um ﬁelds in curved space, (Cambridge University Press, 1982)

View on Pith 1982
[2]

Alternatively, this identiﬁcation follows from a semic lassical evaluation of the partition function or the density of states in quantum gravity with any action, as has been shown explicitly in the case of Einstein gravity. See for exa mple: S.W. Hawking in 18 General Relativity, An Einstein Centenary Survey , eds. S.W. Hawking and W. Israel (Cambridge Un...

View on Pith Pith review arXiv 1993
[3]

J.M. Bardeen, B. Carter and S.W. Hawking, Comm. Math. Phys. 31 (1973) 161; see also ref. [4]

View on Pith 1973
[4]

R.M. Wald, General Relativity (University of Chicago Press, Chicago, 1984)

View on Pith 1984
[5]

I. R´ acz and R.M. Wald, Class. Quant. Grav. 9 (1992) 2643

View on Pith 1992
[6]

G.W. Gibbons and S.W. Hawking, Phys. Rev. D15 (1977) 2752; S.W. Hawking in General Relativity, An Einstein Centenary Survey , eds. S.W. Hawking and W. Israel (Cambridge University Press, 1979)

View on Pith 1977
[7]

C. Callan, R.C. Myers and M. Perry, Nucl. Phys. B311 (1988) 673; R.C. Myers, Nucl. Phys. B289 (1987) 701; D.L. Wiltshire, Phys. Lett. 169B (1986) 36; Phys. Rev. D38 (1988) 2445; R.C. Myers and J.Z. Simon, Phys. Rev. D38 (1988) 2434; Gen. Rel. Grav. 21 (1989) 761; B. Whitt, Phys. Rev. D38 (1988) 3000; E. Poisson, Class. Quant. Grav. 8 (1991) 639; M. Lu and...

View on Pith Pith review arXiv 1988
[8]

T. Jacobson and R.C. Myers, Phys. Rev. Lett. 70 (1993) 3684, hep-th/9305016

View on Pith Pith review arXiv 1993
[9]

R.M. Wald, Phys. Rev. D48 (1993) 3427, gr-qc/9307038

View on Pith Pith review arXiv 1993
[10]

M. Visser, Phys. Rev. D48 (1993), 5697, gr-qc/9307194

View on Pith arXiv 1993
[11]

V. Iyer and R.M. Wald, Phys. Rev. D50 (1994) 846, gr-qc/9403028

View on Pith Pith review arXiv 1994
[12]

In higher dimensions, one has the possibility of commut ing rotations in totally or- thogonal planes. Hence the black hole is characterized by th e set of angular momenta J(α) associated with rotations in the maximal set of mutually ort hogonal planes. For example see: R.C. Myers and M.J. Perry, Ann. Phys. (N.Y.) 172 (1986) 304

View on Pith 1986
[13]

S.W. Hawking, Comm. Math. Phys. 43 (1975) 199; This result applies for Einstein gravity in any number of spacetime dimensions, but then the s urface area A refers to the volume of a space-like cross-section of the horizon, whi ch is a ( D–2)-dimensional hypersurface in D dimensions

View on Pith 1975
[14]

S.W. Hawking, Comm. Math. Phys. 25 (1972) 152; S.W. Hawking and G.R.F. Ellis, The Large Scale Structure of Spacetime (Cambridge University Press, 1973)

View on Pith 1972
[15]

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

View on Pith 1994
[16]

T. Jacobson, G. Kang and R.C. Myers, Phys. Rev. D49 (1994) 6587, gr-qc/9312023. 19

View on Pith Pith review arXiv 1994
[17]

B. Whitt, Phys. Lett. 145B (1984) 176; G. Magnano, M. Ferraris and M. Francaviglia, Gen. Rel. Grav. 19 (1987) 465; Class. Quant. Grav. 7 (1990) 557; K. Maeda, Phys. Rev. D39 (1989) 3159; A. Jakubiec and J. Kijowski, Phys. Rev. D37 (1988) 1406; J.D. Barrow and S. Cotsakis, Phys. Lett.. B214 (1988) 515; G. Magnano and L.M. Sokolowski, Phys. Rev. D50 (1994) ...

View on Pith Pith review arXiv 1984
[18]

T. Jacobson and G. Kang, Class. Quant. Grav. 10 (1993) L1, gr-qc/9307002

View on Pith Pith review arXiv 1993
[19]

We assume that the singularities in the individual term s in the stress-energy tensor of eq. (15) do not cancel to avoid a curvature singularity

View on Pith
[20]

A. Strominger, Phys. Rev. D30 (1984) 2257

View on Pith 1984
[21]

J.Z. Simon, Phys. Rev. D41 (1990) 3720; Phys. Rev. D43 (1991) 3308; L. Parker and J.Z. Simon, Phys. Rev. D47 (1993) 1339, gr-qc/9211002

View on Pith Pith review arXiv 1990
[22]

The starting point for such a perturbative analysis is a solution of Einstein’s equations. A vacuum solution with R = 0 would therefore be unaﬀected perturbatively for the theories considered in sections 3 and 4. It is only in the pres ence of matter that R ⁄= 0, and hence that the special class of higher curvature terms co nsidered there would play a rol...

View on Pith
[23]

D.A. Eliezer and R.P. Woodard, Nucl. Phys. B325 (1989) 389

View on Pith 1989
[24]

W. Zurek, Phys. Rev. Lett. 49 (1982) 1683; W.H. Zurek and K.S. Thorne, Phys. Rev. Lett. 54 (1985) 2171; K.S. Thorne, W.H. Zurek and R.H. Price, in Black Holes: The Membrane Paradigm , eds. K.S. Thorne, R.H. Price and D.A. MacDonald (Yale University Press, New Haven, 1986) p. 280; W.G. Unruh and R.M . Wald, Phys. Rev. D 25, 942 (1982); Phys. Rev. D 27, 227...

View on Pith 1982
[25]

A.O. Barvinskii, Yu.V. Gusev, G.A. Vilkoviskii and V.V . Zhytnikov, J. Math. Phys. 35 (1994) 3543, gr-qc/9404063; J. Math. Phys. 35 (1994) 3525, gr-qc/9404061; ‘The One Loop Eﬀective Action and Trace Anomaly in Four-Dimensions’ , preprint ALBERTA- THY-15-94, hep-th/9404187

View on Pith Pith review arXiv 1994
[26]

C.G. Callan, S.B. Giddings, J.A. Harvey and A. Stroming er, Phys. Rev. D45 (1992) R1005, hep-th/9111056; V. Mukhanov, A. Wipf and A. Zelnikov , Phys. Lett. B332 (1994) 283, hep-th/9403018

View on Pith Pith review arXiv 1992
[27]

R.C. Myers, Phys. Rev. D50 (1994) 6412, hep-th/9405162. 20

View on Pith Pith review arXiv 1994
[28]

A caveat is that the entropy arising from Wald’s constru ction does not appear to provide the full contribution of the long wavelength quantu m ﬁelds. (See T.M. Fi- ola, J. Preskill, A. Strominger and S.P. Trivedi, Phys. Rev. D50 (1994) 3987, hep- th/9403137.) It may be that this discrepancy arises from imp recision in specifying the Green’s function in t...

View on Pith 1994

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Gravitational Effective Field Theories and Black Hole Mechanics
gr-qc 2024-11 conditional novelty 6.0 of 10

In effective field theories of gravity with electromagnetism and scalars, surface gravity and electric potential are constant on black hole horizons up to the accuracy of the EFT, and a modified entropy satisfies the ...
Hunting for Dark Particles with Gravitational Waves
hep-ph 2016-05 conditional novelty 6.0 of 10

LIGO at design sensitivity could probe dark bosons in the 60 MeV to 3 GeV range and dark fermions in the 0.1 to 2 GeV range, if such particles form solar-mass exotic compact objects.
The First Law of Black Hole Mechanics for Fields with Internal Gauge Freedom
gr-qc 2015-11 accept novelty 6.0 of 10

By formulating gauge fields on a principal bundle, the paper derives a general first law of black hole mechanics in which horizon contributions appear as potential times perturbed charge, including a generally non-van...
The holographic entropy increases in quadratic curvature gravity
gr-qc 2015-04 conditional novelty 6.0 of 10

For Vaidya-like black holes in Ricci-squared gravity, the holographic entanglement entropy formula satisfies a local linearized Second Law, unlike the Wald entropy.
Do gauge fields really contribute negatively to black hole entropy?
hep-th 2012-06 conditional novelty 6.0 of 10

The negative 'contact term' that Maxwell fields were thought to add to black hole entropy is a gauge-fixing artifact; quantized on the physical phase space in two dimensions, the gauge field's entropy is positive, fin...
Second Law Violations in Lovelock Gravity for Black Hole Mergers
gr-qc 2010-11 conditional novelty 6.0 of 10

For any Lovelock gravity theory other than General Relativity, the standard Wald entropy can decrease instantaneously during a black hole merger, violating the classical second law.
A comparison of two constructions for dynamical corrections to Wald entropy
hep-th 2026-07 conditional novelty 5.0 of 10

For small perturbations, the dynamical entropy S_dyn and Wall's entropy S_Wall satisfy S_dyn = (1-v∂_v)S_Wall, and this paper gives a local algorithm that reconstructs S_Wall from S_dyn once a bifurcation-surface cond...
Dynamical Entropy Is a Noether Charge
hep-th 2026-07 reject novelty 5.0 of 10

A Noether-charge derivation of the known HWZ dynamical entropy for generic null surfaces, whose local second-law proof is invalid as printed because Eq. (19) has the wrong sign in K.
Dynamical black hole entropy beyond general relativity from the Einstein frame
hep-th 2024-12 accept novelty 5.0 of 10

For f(R) and scalar-tensor gravity, the dynamical black hole entropy formula is rederived via conformal frames and shown to equal the Wald entropy of a generalized apparent horizon.
Ten Proofs of the Generalized Second Law
gr-qc 2009-01 accept novelty 5.0 of 10

Of ten proposed proofs of the generalized second law, only Hawking's area theorem, the hydrodynamic arguments, and Frolov and Page's S-matrix proof survive scrutiny, and the review maps the remaining gaps.
Entropy Current from Partition Function: One Example
hep-th 2014-03 conditional novelty 4.0 of 10

For a parity-even charged fluid at second order in derivatives, the paper constructs the most general entropy current with non-negative divergence, finding 17 free functions of temperature and chemical potential.
Black Holes in the Dilatonic Einstein-Gauss-Bonnet Theory in Various Dimensions I -- Asymptotically Flat Black Holes --
gr-qc 2008-06 conditional novelty 4.0 of 10

Adding a dilaton to Einstein-Gauss-Bonnet gravity yields regular asymptotically flat black holes in 4, 5, 6 and 10 dimensions whose mass-radius curves and thermodynamic behaviour differ qualitatively between five dime...
Lectures on on Black Holes, Topological Strings and Quantum Attractors (2.0)
hep-th 2006-07 conditional novelty 4.0 of 10

Radial quantization of BPS black hole attractor flows yields an exact wave function in terms of a Bessel function, and suggests a one-parameter generalization of the topological string amplitude tied to minimal repres...
Lanczos-Lovelock models of gravity
gr-qc 2013-02 accept novelty 3.0 of 10

Lanczos-Lovelock gravity shares Einstein gravity's horizon thermodynamics and emergent-gravity structure, making it a testbed for the idea that gravity is not fundamental.
A Survey of Black Hole Thermodynamics
gr-qc 2018-04 accept novelty 2.0 of 10

A structured review that states, for each law of black hole thermodynamics and each entropy formula, what has been proved, under which assumptions, and what remains conjecture.

How later papers used it

46 later papers cite it

Evidence: substantial. 46 of the citing papers share no author with it, and 9 of those carry a confident label from Pith's citation classifier.

Pith read 18 of the 84 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

1 later paper by other authors is listed below: 1 used its method or tool.

Uses its method or tool 1 of 3 papers with this label

Used the Wald formulation to compute the entropy.

Geometric Entropy (2010) · 0.95 · paraphrased: the licence does not allow quoting

How the literature uses it

43 arXiv papers cite this in 78 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 52 sentences without a confident label

Builds on it 1 paper

In this section, we will prove the physical process version of the first law [ref] for theories with arbitrary non-minimal gauge couplings generalizing the results of [ref] .

Non-minimal coupling of scalar and gauge fields with gravity: an entropy current and linearized second law · Physical process version of the first law with non-minimal interactions · 0.98

Uses its method or tool 3 papers

In [ref] , this trickier set of terms were dealt with by taking recourse to the physical process version of the first law [ref] .

An entropy current and the second law in higher derivative theories of gravity · Introduction and summary · 0.90
Now using the Wald formulation the entropy can be computed$^ [ref] $.

Geometric Entropy · Identification of Noether charge with entropy · 0.95
We now compute the entropy of a static black hole of the form ([ref]), using Wald's formula [ref] , [ref] , which is given by [equation] where $ϵ_ab$ is the binormal to the horizon cross-section and $ϵ̂$ is the volume form induced on the spatial cross section $Σ_3$ of the horizon at $r=r_+$.

A new cubic theory of gravity in five dimensions: Black hole, Birkhoff's theorem and C-function · Entropy function · 1.00

Compares with it 2 papers

S^v = 1+k_1 ℓ^2(2R_rv-KK̅) + 2 k_2 ℓ^2 R S^A = k_1 ℓ^2 (2D_B K^AB-D^A K+μ^AB∂_v β_B) The $k_2$ term in $S^v$ agrees with the JKM entropy [ref] .

The second law of black hole mechanics in effective field theory · Example: Ricci squared gravity · 0.99
Note that $Δ̃$ vanishes for $f(R)$ gravity [ref] , as in this case $F^tr_tr=F^tx_tx=F(R)/2$.

A holographic c-theorem for higher derivative gravity · DIscussion · 0.95

Cites it as support 2 papers

Since the conformal transformation preserves the causal structure of the spacetime, and $g^E_ab$ is asymptotically flat as long as $g_ab$ is asymptotically flat, the conformal transformation ([ref]) of an asymptotically flat black hole spacetime remains an asymptotically flat black hole spacetime with the same event horizon, at least the conformal factor is regular on the horizon [ref] .

Dynamical black hole entropy beyond general relativity from the Einstein frame · The conformal transformation and the Einstein frame · 0.96
For example, in $f(R)$ theories the entropy density depends only on the spacetime Ricci scalar [ref] and so the entropy is well-defined on a non-smooth horizon.

Creases, corners and caustics: properties of non-smooth structures on black hole horizons · Discussion · 0.97

Cites it as background 14 papers

* The Wald entropy can be constructed for arbitrary $α$ with $=0$. * In [ref] the authors construct entropy functions for theories where the higher derivative Lagrangian is written solely in terms of the Ricci scalar (the so-called $f(R)$ theories), for a finite range of $α$, and arbitrary $$ and $ω$. * In recent years, [ref] first constructed an entropy functional that was valid for higher derivative interactions of the Lovelock form (see below) for small amplitude departures from equilibrium $≪1$.

Towards a second law for Lovelock theories · Introduction · 0.83
[ref] ] horizon $ℋ$ with Killing vector $ξ$.

Gravitational focusing and horizon entropy for higher-spin fields · Focusing theorem and Bekenstein-Hawking entropy in GR · 0.92
Various attempts have been made to extend the equilibrium definition of Wald entropy to configurations involving non-stationary dynamics such that the second law is satisfied in such higher derivative theories of gravity, see [ref] .

Non-minimal coupling of scalar and gauge fields with gravity: an entropy current and linearized second law · Introduction · 0.94

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

See for example: N.D. Birrell and P.C.W. Davies, it Quant um ﬁelds in curved space, (Cambridge University Press, 1982)

View on Pith 1982

[2] [2]

Alternatively, this identiﬁcation follows from a semic lassical evaluation of the partition function or the density of states in quantum gravity with any action, as has been shown explicitly in the case of Einstein gravity. See for exa mple: S.W. Hawking in 18 General Relativity, An Einstein Centenary Survey , eds. S.W. Hawking and W. Israel (Cambridge Un...

View on Pith Pith review arXiv 1993

[3] [3]

J.M. Bardeen, B. Carter and S.W. Hawking, Comm. Math. Phys. 31 (1973) 161; see also ref. [4]

View on Pith 1973

[4] [4]

R.M. Wald, General Relativity (University of Chicago Press, Chicago, 1984)

View on Pith 1984

[5] [5]

I. R´ acz and R.M. Wald, Class. Quant. Grav. 9 (1992) 2643

View on Pith 1992

[6] [6]

G.W. Gibbons and S.W. Hawking, Phys. Rev. D15 (1977) 2752; S.W. Hawking in General Relativity, An Einstein Centenary Survey , eds. S.W. Hawking and W. Israel (Cambridge University Press, 1979)

View on Pith 1977

[7] [7]

C. Callan, R.C. Myers and M. Perry, Nucl. Phys. B311 (1988) 673; R.C. Myers, Nucl. Phys. B289 (1987) 701; D.L. Wiltshire, Phys. Lett. 169B (1986) 36; Phys. Rev. D38 (1988) 2445; R.C. Myers and J.Z. Simon, Phys. Rev. D38 (1988) 2434; Gen. Rel. Grav. 21 (1989) 761; B. Whitt, Phys. Rev. D38 (1988) 3000; E. Poisson, Class. Quant. Grav. 8 (1991) 639; M. Lu and...

View on Pith Pith review arXiv 1988

[8] [8]

T. Jacobson and R.C. Myers, Phys. Rev. Lett. 70 (1993) 3684, hep-th/9305016

View on Pith Pith review arXiv 1993

[9] [9]

R.M. Wald, Phys. Rev. D48 (1993) 3427, gr-qc/9307038

View on Pith Pith review arXiv 1993

[10] [10]

M. Visser, Phys. Rev. D48 (1993), 5697, gr-qc/9307194

View on Pith arXiv 1993

[11] [11]

V. Iyer and R.M. Wald, Phys. Rev. D50 (1994) 846, gr-qc/9403028

View on Pith Pith review arXiv 1994

[12] [12]

In higher dimensions, one has the possibility of commut ing rotations in totally or- thogonal planes. Hence the black hole is characterized by th e set of angular momenta J(α) associated with rotations in the maximal set of mutually ort hogonal planes. For example see: R.C. Myers and M.J. Perry, Ann. Phys. (N.Y.) 172 (1986) 304

View on Pith 1986

[13] [13]

S.W. Hawking, Comm. Math. Phys. 43 (1975) 199; This result applies for Einstein gravity in any number of spacetime dimensions, but then the s urface area A refers to the volume of a space-like cross-section of the horizon, whi ch is a ( D–2)-dimensional hypersurface in D dimensions

View on Pith 1975

[14] [14]

S.W. Hawking, Comm. Math. Phys. 25 (1972) 152; S.W. Hawking and G.R.F. Ellis, The Large Scale Structure of Spacetime (Cambridge University Press, 1973)

View on Pith 1972

[15] [15]

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

View on Pith 1994

[16] [16]

T. Jacobson, G. Kang and R.C. Myers, Phys. Rev. D49 (1994) 6587, gr-qc/9312023. 19

View on Pith Pith review arXiv 1994

[17] [17]

B. Whitt, Phys. Lett. 145B (1984) 176; G. Magnano, M. Ferraris and M. Francaviglia, Gen. Rel. Grav. 19 (1987) 465; Class. Quant. Grav. 7 (1990) 557; K. Maeda, Phys. Rev. D39 (1989) 3159; A. Jakubiec and J. Kijowski, Phys. Rev. D37 (1988) 1406; J.D. Barrow and S. Cotsakis, Phys. Lett.. B214 (1988) 515; G. Magnano and L.M. Sokolowski, Phys. Rev. D50 (1994) ...

View on Pith Pith review arXiv 1984

[18] [18]

T. Jacobson and G. Kang, Class. Quant. Grav. 10 (1993) L1, gr-qc/9307002

View on Pith Pith review arXiv 1993

[19] [19]

We assume that the singularities in the individual term s in the stress-energy tensor of eq. (15) do not cancel to avoid a curvature singularity

View on Pith

[20] [20]

A. Strominger, Phys. Rev. D30 (1984) 2257

View on Pith 1984

[21] [21]

J.Z. Simon, Phys. Rev. D41 (1990) 3720; Phys. Rev. D43 (1991) 3308; L. Parker and J.Z. Simon, Phys. Rev. D47 (1993) 1339, gr-qc/9211002

View on Pith Pith review arXiv 1990

[22] [22]

The starting point for such a perturbative analysis is a solution of Einstein’s equations. A vacuum solution with R = 0 would therefore be unaﬀected perturbatively for the theories considered in sections 3 and 4. It is only in the pres ence of matter that R ⁄= 0, and hence that the special class of higher curvature terms co nsidered there would play a rol...

View on Pith

[23] [23]

D.A. Eliezer and R.P. Woodard, Nucl. Phys. B325 (1989) 389

View on Pith 1989

[24] [24]

W. Zurek, Phys. Rev. Lett. 49 (1982) 1683; W.H. Zurek and K.S. Thorne, Phys. Rev. Lett. 54 (1985) 2171; K.S. Thorne, W.H. Zurek and R.H. Price, in Black Holes: The Membrane Paradigm , eds. K.S. Thorne, R.H. Price and D.A. MacDonald (Yale University Press, New Haven, 1986) p. 280; W.G. Unruh and R.M . Wald, Phys. Rev. D 25, 942 (1982); Phys. Rev. D 27, 227...

View on Pith 1982

[25] [25]

A.O. Barvinskii, Yu.V. Gusev, G.A. Vilkoviskii and V.V . Zhytnikov, J. Math. Phys. 35 (1994) 3543, gr-qc/9404063; J. Math. Phys. 35 (1994) 3525, gr-qc/9404061; ‘The One Loop Eﬀective Action and Trace Anomaly in Four-Dimensions’ , preprint ALBERTA- THY-15-94, hep-th/9404187

View on Pith Pith review arXiv 1994

[26] [26]

C.G. Callan, S.B. Giddings, J.A. Harvey and A. Stroming er, Phys. Rev. D45 (1992) R1005, hep-th/9111056; V. Mukhanov, A. Wipf and A. Zelnikov , Phys. Lett. B332 (1994) 283, hep-th/9403018

View on Pith Pith review arXiv 1992

[27] [27]

R.C. Myers, Phys. Rev. D50 (1994) 6412, hep-th/9405162. 20

View on Pith Pith review arXiv 1994

[28] [28]

A caveat is that the entropy arising from Wald’s constru ction does not appear to provide the full contribution of the long wavelength quantu m ﬁelds. (See T.M. Fi- ola, J. Preskill, A. Strominger and S.P. Trivedi, Phys. Rev. D50 (1994) 3987, hep- th/9403137.) It may be that this discrepancy arises from imp recision in specifying the Green’s function in t...

View on Pith 1994