Black Hole Entropy is Noether Charge

Robert M. Wald

REVIEW

Black Hole Entropy is Noether Charge

Robert M. Wald

General Relativity and Quantum Cosmology · Submitted 29 Jul 1993 · 19 references

Pith's verdict: The Noether-charge entropy formula is derived cleanly from diffeomorphism invariance alone, and the paper is honest about the stationary, bifurcate-horizon scope it does not cover.

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arXiv IDgr-qc/9307038 (version 1)

Pith ID56AKUVZP, Pith's permanent record of this paper and its review

CitationRobert M. Wald. "Black Hole Entropy is Noether Charge." arXiv:gr-qc/9307038 (1993). https://pith.science/paper/56AKUVZP

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Editor's note

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The thing to know: Wald shows that in any theory of gravity with a diffeomorphism-invariant Lagrangian, the horizon term in the first law can always be written as (κ/2π)δS with S = 2π∫_Σ Q̃, where Q̃ is the Noether charge of the horizon Killing field normalized to unit surface gravity. No Einstein equations anywhere in the derivation. That is the result, and it is the one that stuck — it is why Lovelock and higher-derivative entropy calculations have a general home. The derivation itself is short and clean: j = dQ on shell, d(δQ) = d(ξ·Θ) on a stationary background, integrate over a Cauchy surface with horizon boundary, then use δQ = κδQ̃ on Σ. The Euclidean cross-check in eqs. (28)–(32) lands on the same S, which is a genuinely useful bridge to the Gibbons-Hawking machinery and not something you can see by staring at the two literatures side by side. Credit where due: the reduction to Bekenstein-Hawking in GR is immediate, and the paper says so without fanfare. The soft spots are mostly scope, and mostly acknowledged. The bifurcate-horizon assumption is stated in the abstract and discussed at length after eq. (1), including the admission that the zeroth law 'does not appear likely to generalize.' The non-stationary perturbation case is explicitly deferred to Iyer-Wald [8], which is unpublished here — that is the one place where a reader is asked to take something on faith, and it is worth flagging as such rather than as a flaw in what is actually proved. The Θ/Q ambiguity under change of derivative operator or adding an exact form to L is noted after eq. (4) and not fully resolved; the stress-test is right that it is benign on shell for stationary backgrounds, but the paper does not spell that out. One small overreach: the abstract says the results 'show' the second law is 'equivalent' to positivity of a total Noether flux, while the body correctly says this 'suggests a possible relationship.' The body is the honest version. Who should read it: anyone working on entropy in modified gravity, and anyone who wants the cleanest available statement of why black hole entropy is a Noether charge rather than a feature of the Einstein-Hilbert action. This deserves referee time without question.

Referee report

Significance

If correct, this is a foundational result: it identifies black hole entropy as a Noether charge in an arbitrary diffeomorphism-invariant theory, with no free parameters and no appeal to the Einstein equations, and it reproduces the Bekenstein–Hawking value in general relativity as a special case. The derivation is parameter-free and internally checkable, and the Euclidean cross-check in eq. (32) is a genuine independent consistency test rather than a restatement. The result supplies concrete, falsifiable entropy formulas for Lovelock and (1+1)-dimensional gravity (refs. [4], [5]) and a sharp link between the second law and positive-energy properties, which is exactly the kind of structural claim that later work can test. The paper is a letter, with the detailed extensions explicitly deferred to the companion Iyer–Wald work [8].

Major objections (2)

[Abstract (s2, s3) vs. text after eq. (27) and the second-law paragraph] The abstract states that the first law 'always holds for perturbations to nearby stationary black hole solutions' and that the validity of the second law 'is equivalent to the positivity of a total Noether flux.' The body is more careful: eq. (25) is derived only for perturbations to nearby stationary black holes, and the paragraph following eq. (27) states that for non-stationary perturbations eq. (24) continues to hold only in 'sufficiently low derivative' theories where Q depends on ξ^a and its first antisymmetrized derivative, with the general case 'presently under investigation [8].' Likewise, the second-law discussion says only that the Noether-charge relation 'suggests a possible relationship' and that these issues 'are under investigation [8].' Since the abstract is what will be cited, please align its wording with the body: state that the entropy formula (26) and the first law (25) are established for stationary perturbations, and that the second-law statement is a conjectured link rather than an established equivalence.
[Eq. (24) and the sentence defining Q̃ immediately preceding it] Eq. (24), δQ = κδQ̃, is the step that converts eq. (23) into the first law and is therefore load-bearing, but the normalization stated in the preceding sentence appears inconsistent as printed. If ξ^a satisfies eq. (1) with surface gravity κ, then ξ̃^a = κξ^a has surface gravity κ², not unit surface gravity; the field with unit surface gravity is ξ̃^a = ξ^a/κ. With that reading, Q̃ = Q[ξ̃] = Q[ξ]/κ and eq. (24) follows with κ held fixed at its unperturbed value, which is presumably the intent. Please correct the relation and state explicitly which quantity (the unperturbed κ) is held fixed in the variation, since the identification of the two spacetimes described in the following paragraph is delicate when the perturbation changes κ or Ω_H. The result itself is unaffected, but as written a reader checking eq. (24) will be led to a factor-of-κ² discrepancy.

Minor objections (6)

[Introduction, paragraph following eq. (3)] The displayed form of the first law in the introduction is garbled as 'κ2πδS'; it should read (κ/2π)δS, matching eq. (25).
[Eq. (27) and the following sentence] The assertion that the w_ab term makes no contribution to the variation of Q, by appeal to the reflection isometry of lemma 2.3 of [13], is compressed into a single sentence. A short explicit argument — for example, that the reflection flips the sign of the normal-tangential part while leaving the integral invariant — would let the reader verify the claim without consulting [13].
[Paragraph following eq. (26), and refs. [8], [18]] The dynamical-black-hole entropy proposal and the non-stationary extension both rest on ref. [8] (Iyer and Wald, listed as 'to be published'), and the independent-support claim rests on ref. [18] (Simon and Whiting, also 'to be published'). This is acknowledged in the text and does not affect the stationary result, but the reader should be told clearly that these extensions are not established here.
[Eq. (18) and the discussion of E and J] The existence of an (n−1)-form B satisfying eq. (18) is a nontrivial condition on the theory, as the text itself notes. It would help to state in one sentence, near eqs. (19)–(20), what class of theories is known to admit such a B, so that the scope of the first-law derivation is transparent.
[References and typography] Several typographical errors should be corrected: 'the the' in the abstract, 't o', 'stat ionary', 'bifurcati on', 'l agrangian', 'e quations', and 'P ACS' in the front matter; 'Class and Quant. Grav.' in ref. [12] should be 'Class. Quantum Grav.'; and 'Centennary' in ref. [19] should be 'Centenary'.
[Eq. (26)] It would be worth stating explicitly that Q̃ in eq. (26) is evaluated on the bifurcation surface Σ and that, for a stationary horizon, the result is independent of which cross-section is chosen; this is implicit in the surrounding discussion but is the property that makes the dynamical proposal in the following paragraph well posed.

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The paper

The paper claims

Wald argues that in any gravity theory built from a diffeomorphism-invariant Lagrangian, black hole entropy is simply $2π$ times the Noether charge of the horizon Killing field, so the first law of black hole mechanics holds universally.

The reading

The paper asks whether the first law of black hole mechanics — the statement that a change in a black hole's mass and spin is balanced by a change in its entropy — is special to Einstein's equations or is instead a generic feature of any gravity theory that can be written as a Lagrangian without preferring any coordinate system. Earlier Hamiltonian derivations could show that some horizon surface term balances the changes in mass and angular momentum at infinity, but they could not show that this term is the change of a local, geometrical quantity on the horizon — which is what you need before you can call anything 'entropy'. Wald closes that gap. He shows that the horizon term is always $κ/2π$ times the change of $2π$ times the integral over the horizon cross-section of a Noether charge $(n-2)$-form, built from the horizon Killing field normalized to unit surface gravity. If this is right, every such theory has a black hole entropy defined by a local expression on the horizon, the first law follows from diffeomorphism invariance alone, and the Euclidean and Hamiltonian routes to entropy are the same calculation in different clothes.

What carries the argument

The Noether charge $(n-2)$-form $Q$. For any vector field $ξ^a$, diffeomorphism invariance makes $L_ξφ$ an infinitesimal local symmetry, so one can form the Noether current $(n-1)$-form $j = Θ(φ, L_ξφ) - ξ·L$, which is closed on solutions ($dj = -E L_ξφ$). Since $j$ is closed for every $ξ^a$, it can be written $j = dQ$ on solutions, with $Q$ locally constructed from the fields and $ξ^a$ and depending on at most $k-1$ derivatives of $ξ^a$ when the Lagrangian contains at most $k$ derivatives of the dynamical fields. The second piece of machinery is the reduction of $Q$ on the bifurcation surface: every higher derivative $∇_{a_1}…∇_{a_n}ξ_b$ of a Killing field can be rewritten in terms of $ξ^a$ and $∇_aξ_b$ with coefficients built from the curvature and its derivatives, so on $Σ$ one may set $ξ^a = 0$ and replace $∇_aξ_b$ by the binormal $ε_{ab}$, producing $Q̃$ — the Noether charge of the horizon Killing field at unit surface gravity. The third piece is the identification of the asymptotic surface terms with $δE$ and $δJ$ via $δE = ∫_∞(δQ[t] - t·Θ)$ and $δJ = -∫_∞ δQ[φ]$, which is what turns the integrated identity into a first law.

Extended reading notes

Core claim

The central claim is eq. (26): the entropy of a stationary black hole in an $n$-dimensional diffeomorphism-invariant Lagrangian theory of gravity is $S = 2π ∫_Σ Q̃$, where $Q̃$ is the Noether charge $(n-2)$-form associated with the horizon Killing field $ξ^a$ normalized so that its surface gravity is unity, integrated over the bifurcation surface $Σ$. With this $S$, the first law takes the standard form $(κ/2π) δS = δE - Ω^{(μ)}_H δJ_{(μ)}$ for perturbations to nearby stationary black hole solutions (eq. 25). The derivation starts from the identity $δj = Ω(φ, δφ, L_ξφ) + d(ξ·Θ)$, integrates it over an asymptotically flat hypersurface bounded inside by $Σ$, and identifies the horizon surface term as $δ∫_Σ Q$; the step to eq. (25) uses the fact that on $Σ$ the Killing field vanishes and its first derivative is $κ$ times the binormal, so that $δQ = κ δQ̃$. Because $Q̃$ is fixed by an algorithm whose only input is a diffeomorphism-invariant Lagrangian, it is a local geometrical quantity on the horizon — which is what licenses calling it entropy, and what lets the author propose the same integral over any cross-section of a dynamical horizon as a candidate entropy there.

Load-bearing premise

The derivation assumes the black hole has a bifurcate Killing horizon with constant surface gravity, and the paper itself notes that the constancy of surface gravity is proved in general relativity using the specific form of Einstein's equations and does not look likely to carry over to other theories of gravity.

Editorial extensions

If this is right

The first law of black hole mechanics holds, with entropy given by $S = 2π∫_Σ Q̃$, in any theory of gravity derived from a diffeomorphism-invariant Lagrangian — including higher-derivative theories — provided the theory admits stationary black holes with bifurcate Killing horizons and well-defined canonical mass and angular momentum at infinity.
Black hole entropy is a local geometrical quantity on the horizon rather than an artifact of a particular asymptotic or Hamiltonian construction, which supplies a natural candidate definition of entropy for a dynamical black hole: integrate $Q̃$ over an arbitrary cross-section of the horizon.
For a black hole that starts stationary, evolves, and settles to a stationary final state, the net change in entropy equals the total flux of Noether current through the horizon, so the second law of black hole mechanics becomes the statement that this Noether flux is positive.
The Euclidean path-integral route to entropy and the Noether-charge route agree: writing the Euclidean action as $I = -(2π/κ)[∫_C ξ·L + ∫_∞ t·B]$ reproduces $S = 2π∫_Σ Q$ exactly.
In general relativity and in $(1+1)$-dimensional theories, where $Q$ depends only on $ξ^a$ and its first antisymmetrized derivative, the first law in the form of eq. (25) continues to hold even for non-stationary perturbations.

Reading between the lines

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

Because the entropy formula depends only on the Lagrangian and the horizon geometry, the same bookkeeping should reproduce known entropy expressions in specific higher-curvature theories once the Noether charge is evaluated for their Lagrangians — a check that would test the claim without needing new physics.
If the second law really reduces to positivity of a Noether flux, then a theory that violates a positive-energy property should be a candidate for violating the second law; that pairing is a concrete, searchable link between energy conditions and horizon thermodynamics.
The proposal to define entropy on an arbitrary horizon cross-section by the same algorithm is testable in principle against time-dependent numerical black hole evolutions, where the candidate entropy should be monotonic in any theory whose Noether flux is positive.
The derivation leans on the horizon being bifurcate with constant surface gravity, so a theory whose stationary black holes have non-bifurcate Killing horizons is the natural place to look for a counterexample to the universality of the formula.

Original abstract

We consider a general, classical theory of gravity in $n$ dimensions, arising from a diffeomorphism invariant Lagrangian. In any such theory, to each vector field, $\xi^a$, on spacetime one can associate a local symmetry and, hence, a Noether current $(n-1)$-form, ${\bf j}$, and (for solutions to the field equations) a Noether charge $(n-2)$-form, ${\bf Q}$. Assuming only that the theory admits stationary black hole solutions with a bifurcate Killing horizon, and that the canonical mass and angular momentum of solutions are well defined at infinity, we show that the first law of black hole mechanics always holds for perturbations to nearby stationary black hole solutions. The quantity playing the role of black hole entropy in this formula is simply $2 \pi$ times the integral over $\Sigma$ of the Noether charge $(n-2)$-form associated with the horizon Killing field, normalized so as to have unit surface gravity. Furthermore, we show that this black hole entropy always is given by a local geometrical expression on the horizon of the black hole. We thereby obtain a natural candidate for the entropy of a dynamical black hole in a general theory of gravity. Our results show that the validity of the ``second law" of black hole mechanics in dynamical evolution from an initially stationary black hole to a final stationary state is equivalent to the positivity of a total Noether flux, and thus may be intimately related to the positive energy properties of the theory. The relationship between the derivation of our formula for black hole entropy and the derivation via ``Euclidean methods" also is explained.

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.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,565 input tokens, $0.0003 reported.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.67 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.64 · compares with prior results 0.92 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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

We consider a general, classical theory of gravity in $n$ dimensions, arising from a diffeomorphism invariant Lagrangian. In any such theory, to each vector field, $\xi^a$, on spacetime one can associate a local symmetry and, hence, a Noether current $(n-1)$-form, ${\bf j}$, and (for solutions to the field equations) a Noether charge $(n-2)$-form, ${\bf Q}$. Assuming only that the theory admits stationary black hole solutions with a bifurcate Killing horizon, and that the canonical mass and angular momentum of solutions are well defined at infinity, we show that the first law of black hole mechanics always holds for perturbations to nearby stationary black hole solutions. The quantity playing the role of black hole entropy in this formula is simply $2 \pi$ times the integral over $\Sigma$ of the Noether charge $(n-2)$-form associated with the horizon Killing field, normalized so as to have unit surface gravity. Furthermore, we show that this black hole entropy always is given by a local geometrical expression on the horizon of the black hole. We thereby obtain a natural candidate for the entropy of a dynamical black hole in a general theory of gravity. Our results show that the validity of the ``second law" of black hole mechanics in dynamical evolution from an initially stationary black hole to a final stationary state is equivalent to the positivity of a total Noether flux, and thus may be intimately related to the positive energy properties of the theory. The relationship between the derivation of our formula for black hole entropy and the derivation via ``Euclidean methods" also is explained.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We consider a general, classical theory of gravity in $n$ dimensions, arising from a diffeomorphism invariant Lagrangian.	background 0.85	0.27	0.02	asserted 0.66
s1	In any such theory, to each vector field, $\xi^a$, on spacetime one can associate a local symmetry and, hence, a Noether current $(n-1)$-form, ${\bf j}$, and (for solutions to the field equations) a Noether charge $(n-2)$-form, ${\bf Q}$.	background 0.98	0.23	0.04	asserted 0.98
s2	Assuming only that the theory admits stationary black hole solutions with a bifurcate Killing horizon, and that the canonical mass and angular momentum of solutions are well defined at infinity, we show that the first law of black hole mechanics always holds for perturbations to nearby stationary black hole solutions.	result 0.91	0.93	0.16	asserted 1.00
s3	The quantity playing the role of black hole entropy in this formula is simply $2 \pi$ times the integral over $\Sigma$ of the Noether charge $(n-2)$-form associated with the horizon Killing field, normalized so as to have unit surface gravity.	result 0.99	0.92	0.79	asserted 1.00
s4	Furthermore, we show that this black hole entropy always is given by a local geometrical expression on the horizon of the black hole.	result 1.00	0.94	0.18	asserted 1.00
s5	We thereby obtain a natural candidate for the entropy of a dynamical black hole in a general theory of gravity.	interpretation 0.36	0.90	0.13	hedged possibility 0.50
s6	Our results show that the validity of the ``second law" of black hole mechanics in dynamical evolution from an initially stationary black hole to a final stationary state is equivalent to the positivity of a total Noether flux, and thus may be intimately related to the positive energy properties of the theory.	interpretation 0.91	0.92	0.12	hedged possibility 0.55
s7	The relationship between the derivation of our formula for black hole entropy and the derivation via ``Euclidean methods" also is explained.	interpretation 0.55	0.66	0.04	asserted 0.62

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

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (19)

[8]

V. Iyer and R.M. Wald, to be published

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[4]

V.P. Frolov, Phys. Rev. D46, 5383 (1992)

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[5]

T.A. Jacobson and R.C. Myers, Phys. Rev. Lett. 70, 3684 (1993)

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[1]

S.W. Hawking, Commun. Math. Phys. 43, 199 (1975)

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[2]

J.M. Bardeen, B. Carter, S.W. Hawking, Commun. Math. Phy s. 31, 161 (1973)

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[3]

D. Sudarsky and R.M. Wald, Phys. Rev. D 46, 1453 (1992); R.M. Wald, “ The ﬁrst law of black hole mechanics” in Directions in General Relativity , vol. 1, ed. by B.L. Hu, M. Ryan, and C.V. Vishveshwara (Cambridge Uni versity Press, Cambridge, 1993). See also J.D. Brown and J.W. York Phys. Rev . D47, 1407 and 1420 (1993)

View on Pith 1992
[6]

I wish to thank Ted Jacobson for bringing this deﬁciency t o my attention

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[7]

G.W. Gibbons and S.W. Hawking, Phys. Rev. D15, 2752 (1977)

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[9]

S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time (Cam- bridge University Press, Cambridge, 1973)

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[10]

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

View on Pith 1984
[11]

It is proven in [12] that in an arbitrary theory of gravit y, global nonsingularity of the event horizon of a stationary black hole implies that the zeroth law holds. However, there are no compelling physical grounds to impose such a global non- singularity condition, since for a black hole formed by the c ollapse of matter, the parallelly propagated curv...

View on Pith
[12]

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

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[13]

B.S. Kay and R.M. Wald, Phys. Rep. 207, 49 (1991)

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[14]

J. Lee and R.M. Wald, J. Math. Phys. 31, 725 (1990). 15

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[15]

R.M. Wald, J. Math. Phys. 31, 2378 (1990). The results of this reference also can be derived using the “free variational bicomplex”; see I.M. Anderson in Mathe- matical Aspects of Classical Field Theory , ed. by M. Gotay, J. Marsden, and V. Moncrief, Cont. Math. 132, 51 (1992)

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[16]

This quantity corresponds to the “gravitational ﬁeld s trength” discussed in W. Si- mon, Gen. Rel. and Grav. 17, 439 (1985)

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[17]

T. Regge and C. Teitelboim, Ann. Phys. 88, 286 (1974)

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[18]

A much more indirect argument for this conclusion has be en given independently by J. Simon and B. Whiting (to be published), based upon the Eucl idean approach. See also, M. Visser (to be published)

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[19]

S.W. Hawking, “The Path Integral Approach to Quantum Gr avity”, in General Relativity, an Einstein Centennary Survey , ed. by S.W. Hawking and W. Israel (Cambridge University Press, Cambridge, 1979) 16

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For two-timescale perturbations about Kerr, the coarse-grained leading-order horizon shear vanishes and the angular-velocity and inaffinity corrections are uniform on each cut, defining adiabatic rigidity; the paper a...
Conformal Renormalisation of 8D Einstein Gravity
hep-th 2026-07 accept novelty 7.0 of 10

Holographic renormalisation of 8D Einstein-AdS gravity is recovered as the Einstein sector of the unique conformal gravity admitting constant-negative-curvature Einstein solutions, with matching counterterms.
Reverse Isoperimetry from Boundary--Completed Canonical Energy
gr-qc 2026-06 unverdicted novelty 7.0 of 10

The reverse isoperimetric conjecture is proven to be the fixed-volume form of a boundary-completed Noether-charge stability theorem.
Hodge Loci and Complex Multiplication via Generalized Symmetries in Calabi-Yau sigma models
hep-th 2026-05 unverdicted novelty 7.0 of 10

Proposes a CFT analogue of Hodge loci in Calabi-Yau sigma models via non-trivial TDL categories of topological defects, with CM number field embeddings at special points for elliptic curves and K3 surfaces.
Planetary formation tracks on the Hertzsprung-Russell diagram: Visualising the processes of giant planet growth
astro-ph.EP 2026-05 unverdicted novelty 7.0 of 10

Planetary formation tracks on the HR diagram show three branches: ascending during solid accretion with L proportional to T to the 8th for in-situ planetesimals, near-horizontal during gas accretion, and descending du...
Covariant unification of holographic c-functions
hep-th 2026-05 unverdicted novelty 7.0 of 10

A new covariant c-function is defined from extrinsic curvature of codimension-two bulk slices, unifying prior foliation-based definitions and exhibiting expected monotonic behavior in conformal, confining, and mixed-g...
Black holes and neutron stars in massive Hellings-Nordtvedt theory
gr-qc 2026-05 unverdicted novelty 7.0 of 10

In massive Hellings-Nordtvedt theory, a nonzero vector vacuum asymptotically forbids both curvature-vector couplings at once, and the A²R sector yields Schwarzschild-like black holes plus neutron stars that can deviat...
Derivation of the Smarr formula from the Komar charge in Einstein-nonlinear electrodynamics theories and applications to regular black holes
gr-qc 2026-05 unverdicted novelty 7.0 of 10

Derives Smarr formula from generalized Komar charge in 4D Einstein-NLED theories by treating coupling constant as dynamical field, and analyzes thermodynamics of regular Bardeen black hole.
The BEF Symplectic Form: A Lagrangian Perspective
hep-th 2026-04 unverdicted novelty 7.0 of 10

The BEF symplectic form is derived from L∞-Lagrangians via covariant phase space methods and coincides with the Barnich-Brandt form for second-order equations of motion.
All $2D$ generalised dilaton theories from $d\geq 4$ gravities
hep-th 2026-03 conditional novelty 7.0 of 10

Generic 2D Horndeski theories arise from dimensional reduction of d≥4 gravities, yielding a Birkhoff theorem for quasi-topological gravities where static spherically symmetric solutions satisfy g_tt g_rr = -1 and are ...
Non-closed scalar charge in four-dimensional Einstein-scalar-Gauss-Bonnet black hole thermodynamics
hep-th 2025-10 unverdicted novelty 7.0 of 10

A covariant framework reveals non-closed scalar charges with bulk contributions in ESGB black holes that become closed under shift symmetry and interpret spontaneous scalarization via the Smarr formula.
Equivariant localization for $D=5$ gauged supergravity
hep-th 2025-08 unverdicted novelty 7.0 of 10

A method is given to compute the D=5 on-shell action via equivariant localization after dimensional reduction to D=4 N=2 gauged supergravity for solutions admitting both the R-symmetry Killing vector and an additional...
Multicentered black hole saddles for supersymmetric indices
hep-th 2025-07 conditional novelty 7.0 of 10

Multi-center black hole saddles for the supersymmetric index exist at finite temperature, their on-shell action is the sum of extremal entropies, and the entire moduli space of saddles vanishes at walls of marginal stability.
Covariant phase space and $L_\infty$ algebras
hep-th 2025-06 conditional novelty 7.0 of 10

A covariant phase space symplectic form is constructed for any L∞ Lagrangian field theory using a 'sigmoid' operator, and is verified in scalar, Yang-Mills, general relativity, and p-adic string examples.
Black Hole Entropy, Quantum Corrections and EFT Transitions
hep-th 2025-02 conditional novelty 7.0 of 10

Quantum corrections to BPS black hole entropy in 4d N=2 string compactifications can be resummed into a finite formula that interpolates between four- and five-dimensional EFT descriptions and matches exact 5d microst...
Supersymmetric Index for Half BPS Black Holes in N=2 Supergravity with Higher Curvature Corrections
hep-th 2024-11 conditional novelty 7.0 of 10

A gravitational index for rotating half-BPS black holes in higher-derivative N=2 supergravity is shown to equal the Wald entropy of the extremal black hole carrying the same charges.
Generalized Black Hole Entropy is von Neumann Entropy
hep-th 2023-09 conditional novelty 7.0 of 10

The algebra of gravitational observables outside any Killing horizon is Type II, so its von Neumann entropy equals the generalized entropy, including for Kerr and Schwarzschild-de Sitter black holes.
Generalized entropy for general subregions in quantum gravity
hep-th 2023-06 conditional novelty 7.0 of 10

Subregions in semiclassical quantum gravity carry type II von Neumann algebras whose renormalized entropy equals the generalized entropy, up to a state-independent constant.
Black hole microstate counting from the gravitational path integral
hep-th 2022-09 conditional novelty 7.0 of 10

The gravitational path integral for 1/8-BPS black holes in N=8 supergravity reproduces the exact integer microstate index of the D1-D5 string construction, term by term in the Rademacher expansion.
Generalized quasi-topological gravities: the whole shebang
hep-th 2022-03 conditional novelty 7.0 of 10

At order n in curvature there are exactly n-1 inequivalent generalized quasi-topological gravities in D at least 5, of which exactly one is quasi-topological.
Brown-York charges at null boundaries
hep-th 2021-09 accept novelty 7.0 of 10

A null-surface Brown-York stress tensor T^i_j = -(1/8πG)(W^i_j - W δ^i_j) is derived from a Dirichlet variational principle, is independent of the auxiliary null vector, and its conservation reproduces the null constr...
On the black hole/string transition
hep-th 2021-09 accept novelty 7.0 of 10

A supersymmetric index and K-theory argument obstructs a continuous classical connection between the Horowitz-Polchinski string solution and the small black hole in type II string theory, but not in the heterotic string.
Higher-Derivative Supergravity, AdS$_4$ Holography, and Black Holes
hep-th 2021-06 accept novelty 7.0 of 10

The four-derivative action of minimal 4d N=2 gauged supergravity is fixed by two constants, which holography and localization determine, yielding the first subleading N^{1/2} corrections to AdS4 black hole entropy and...
Quantum de Sitter horizon entropy from quasicanonical bulk, edge, sphere and topological string partition functions
hep-th 2020-09 conditional novelty 7.0 of 10

The paper derives the exact one-loop de Sitter entropy for arbitrary higher-spin field content and all-loop results for 3D higher-spin gravity via a bulk-edge character formula.
Spin-induced black hole scalarization in Einstein-scalar-Gauss-Bonnet theory
gr-qc 2020-09 conditional novelty 7.0 of 10

Nonlinear spinning black holes with spin-induced scalar hair exist in Einstein-scalar-Gauss-Bonnet gravity with quadratic coupling, can exceed the Kerr spin bound, and are entropically favored over Kerr black holes at...
Quantum BTZ black hole
hep-th 2020-07 conditional novelty 7.0 of 10

An exact braneworld construction of a rotating quantum BTZ black hole yields a generalized entropy that satisfies the first law of thermodynamics to all orders in backreaction and curvature corrections.
All higher-curvature gravities as Generalized quasi-topological gravities
hep-th 2019-06 conditional novelty 7.0 of 10

Any higher-curvature gravitational effective action is equivalent, up to metric redefinitions, to a Generalized quasi-topological gravity, so generic black hole thermodynamics reduces to the GQTG case.
The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole
hep-th 2019-05 conditional novelty 7.0 of 10

In a 2d evaporating black hole model, large boosts create O(1/G_N) gradients in bulk entropy that move the quantum extremal surface, causing the generalized entropy to follow unitary expectations with information disa...
Holographic studies of Einsteinian cubic gravity
hep-th 2018-01 conditional novelty 7.0 of 10

In the holographic model defined by Einsteinian cubic gravity in four bulk dimensions, the shear viscosity to entropy density ratio never falls below the KSS bound, because the parameter region that would allow a viol...
Nonlinear Gravity from Entanglement in Conformal Field Theories
hep-th 2017-05 conditional novelty 7.0 of 10

Ball entanglement entropies in sourced CFT states, expanded to second order, are reproduced by an asymptotically AdS metric that necessarily satisfies Einstein's equations to the same order, provided one relation betw...
On black holes in higher-derivative gravities
hep-th 2017-03 conditional novelty 7.0 of 10

A general proof shows that single-function black hole solutions in higher-derivative gravity require an Einstein-like linearized spectrum, plus a recipe for constructing such theories and new planar AdS5 black holes.
Generalized quasi-topological gravity
hep-th 2017-03 conditional novelty 7.0 of 10

A new cubic curvature term completes the classification of gravity theories whose static spherically symmetric vacuum solutions depend on a single metric function, with Lovelock, quasi-topological, and Einsteinian cub...
Aspects of general higher-order gravities
hep-th 2016-10 accept novelty 7.0 of 10

A systematic linearization of general curvature-based gravities classifies their particle spectra, yields new ghost-free theory constructions, and produces dimension-general Newton potentials and gravitational-wave po...
The Ryu-Takayanagi Formula from Quantum Error Correction
hep-th 2016-07 accept novelty 7.0 of 10

In any quantum error-correcting code with complementary recovery, the Ryu-Takayanagi formula, subregion duality, and relative-entropy equivalence are proven to be equivalent statements.
Observables, gravitational dressing, and obstructions to locality and subsystems
hep-th 2016-07 conditional novelty 7.0 of 10

Any gravitationally dressed operator with nonzero Poincare charge must depend on the asymptotic metric, so local commuting subalgebras and subsystems do not survive in gravity.
Local subsystems in gauge theory and gravity
hep-th 2016-01 accept novelty 7.0 of 10

By adding boundary degrees of freedom, the authors construct gauge-invariant phase spaces for regions in Yang-Mills theory and gravity, and derive a surface symmetry algebra whose generators are the conformal normal m...
Relative entropy equals bulk relative entropy
hep-th 2015-12 conditional novelty 7.0 of 10

Boundary relative entropy equals bulk relative entropy inside the entanglement wedge at leading order in Newton's constant, because the boundary modular Hamiltonian is the Ryu-Takayanagi area operator plus the bulk mo...

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How later papers used it

1,003 later papers cite it

Evidence: substantial. 1,002 of the citing papers share no author with it, and 422 of those carry a confident label from Pith's citation classifier.

Pith read 37 of the 1,790 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

13 later papers by other authors are listed below: 13 used its method or tool.

Uses its method or tool 13 of 369 papers with this label

Followed the procedure proposed in this paper and another work in its derivation.

Gravity as a constrained BF theory: Noether charges and Immirzi parameter (2011) · 1.00 · paraphrased: the licence does not allow quoting
Used effective field theoretical methods to calculate quantum gravitational corrections to the entropy of a Schwarzschild black hole using this paper's Wald entropy formula.

Quantum Gravitational Corrections to the Entropy of a Schwarzschild Black Hole (2021) · 1.00 · paraphrased: the licence does not allow quoting
Instead, we need to use the formula obtained by Wald's prescription [ref] , namely [equation] where $h$ is determinant of the induced metric on the horizon, $ϵ_ab$ is the binormal to the horizon with $ϵ_abϵ^ab=-2$ and $P_abcd$ is defined in ([ref]).

Taub-NUT Black Hole in Higher Derivative Gravity and Its Thermodynamics (2024) · Thermodynamics · 1.00
The entropy of our model is calculated using the Wald formula [ref] as follows, [equation] where, [equation] Furthermore, [equation] [equation] and [equation] Calculations give the following relation for the Ricci scalar, [equation] where, at the black hole horizon takes the form [equation] because, $f(r_h)=0$.

Exponential Modification of AdS Black Hole and Thermodynamic Behavior (2024) · Exponential Modification of the AdS Black Hole · 1.00
Constructed gravitational Noether charges following this paper for several supergravity theories.

Charges from Attractors (2007) · 1.00 · paraphrased: the licence does not allow quoting
Used a dynamical definition of entropy as proposed in this paper.

The horizon-entropy increase law for causal and quasi-local horizons and conformal field redefinitions (2011) · 1.00 · paraphrased: the licence does not allow quoting
Found the entropy from this paper's Wald formula.

Entropy Function for Non-extremal D1D5 and D2D6NS5-branes (2007) · 1.00 · paraphrased: the licence does not allow quoting
Throughout this section we will employ Wald's Noether charge formalism [ref] .

Gravitational Thermodynamics of Causal Diamonds in (A)dS (2018) · Smarr formula for causal diamonds · 1.00
Proceeded with this paper's proposal to calculate the entropy.

Information Geometry on the Space of Equilibrium States of Black Holes in Higher Derivative Theories (2018) · 1.00 · paraphrased: the licence does not allow quoting
Calculated the entropy density using this paper's Wald entropy formula.

Weak Gravity Conjecture, Central Charges and $\eta/s$ (2010) · 0.91 · paraphrased: the licence does not allow quoting
Computed the black hole entropy using this paper's Wald formula.

Proof of the Weak Gravity Conjecture from Black Hole Entropy (2018) · 0.99 · paraphrased: the licence does not allow quoting
Followed this paper by starting with the definition of the pre-symplectic (d-1)-form.

The first law and Wald entropy formula of heterotic stringy black holes at first order in alpha prime (2020) · 0.99 · paraphrased: the licence does not allow quoting
At first order in $β$ we find [equation] Therefore, the Hawking temperature is given by [equation] According to the Wald formula [ref] , the entropy is [equation] where $dΣ=r^2sinθdθdϕ$, $ϵ_μν$ is the Levi-Civita tensor in two dimensions and $ℒ$ is the Lagrangian density of the theory.

Schwarzschild-type black holes in Starobinsky-Bel-Robinson gravity (2022) · $β$-corrections to the physical properties of a Schwarzschild black hole · 0.98

How the literature uses it

888 arXiv papers cite this in 1,607 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 · 946 sentences without a confident label

Contests it 5 papers

Despite its elegance and very wide success, it has been observed that Wald's entropy formula [ref] and/or the first law does not work for a family of solutions to certain Horndeski theories, e.g. see [ref] .

On Black Hole Temperature in Horndeski Gravity · Introduction · contest 0.83
In [ref] Wald proposes an algorithm for $X_dyn^cd$ that gives simply [equation] Although this prescription satisfies the conditions (i) and (ii) (because, as we argued above, for a stationary black hole with bifurcate surface $B$, $(X^cd)_a_3⋯a_n=-E_R^abcdϵ_aba_3⋯a_n$ on $B$, and the entropy is independent of the cross-section), it fails to satisfy (iii), in fact the addition of $dμ$ to $L$ with a suitable dependence on the Riemann tensor can change $E_R^abcd$ in such a way that the change in $S_dyn$ is nonvanishing for non-stationary black holes.

The First Law of Black Hole Mechanics · Entropy of a dynamical black hole and the second law of black hole mechanics for diffeomorphism invariant theories · contest 0.84
We observe that the standard Wald formulation [ref] fails to address the issue of entropy for extremal black holes.

Entropy of Black Holes in N=2 Supergravity · Discussions · contest 0.88

Builds on it 14 papers

Elizalde and Silva [ref] extended Jacobson's proof from the simplest Einstein-Hilbert theory of gravity to more general theories which depend on the Ricci scalar, by using the Noether charge entropy [ref] rather than assuming that the entropy satisfies a fixed theory-independent proportionality relation to the area.

The Einstein equations for generalized theories of gravity and the thermodynamic relation $\delta Q = T \delta S$ are equivalent · Introduction · 0.87
In this section we will review and extend the Iyer-Wald formalism [ref] to include variations in all the couplings of the theory.

Holographic entanglement chemistry · Iyer-Wald with varying the couplings: a closer look · 0.93
The final form originally derived in [ref] [equation] generalizes the results of [ref] , and [ref] to the case of first order gravity with the Holst modification.

Immirzi parameter and Noether charges in first order gravity · Generalized Noether charge · 0.97

Uses its method or tool 321 papers

In Section [ref] we use the covariant phase space formalism [ref] to construct the conserved charges for the solution.

Thermodynamics of accelerating AdS$_4$ black holes from the covariant phase space · Introduction · 1.00
The coefficient of shear viscosity can be very easily computed using the prescription given in [ref] and the result is same as in [ref] η=r^3_02κ^2(1-24 c_1 q^2r^6_0) and the entropy density in terms of $q/r^3_0$, can be calculated using the Wald's entropy formula [ref] s=4πr^3_02κ^2[1+8 c_1-q^2r^6_0(28 c_1+24 c_2)]

Weak Gravity Conjecture, Central Charges and $\eta/s$ · shear viscosity and $η/s$ · 0.91
For a general variation of the fields $Ψ={g_μν,A_μ,ϕ}$, the Lagrangian four-form satisfies [ref] δL = E_ΨδΨ+ dΘ(Ψ,δΨ) , where $E_ψ=0$ gives the equations of motion.

First Law of Black Hole Interior Dynamics · Iyer-Wald derivation of the interior first law · 1.00

Uses its data or values 1 paper

The conserved charge $Ẽ$ for a time-like killing vector, the area $Ã_H$, and the Hawking temperature $T̃_H$ of the black string are calculated as $Ẽ=2√(M/ℓ)$, $Ã_H=∫dx̂=2π/√(ℓM),$ and $T̃_H=1/(2πℓ)$ [ref] , and the Wald entropy is given as $S̃_W=4π√(ℓM)$ [ref] .

Validity of black hole complementarity in the BTZ black hole · No-cloning theorem in the black string and the BTZ black hole · 0.89

Compares with it 31 papers

Choosing $T_m$ as Eq. ([ref]) and substituting it into the above equations, we get [equation] The central term reads [equation] from which we can read off the central charge $C$ and zero mode $Q_0$ as [equation] Using the Cardy formula, we obtain the entropy [equation] which is exactly the Wald entropy of the Gauss-Bonnet black holes [ref] .

Surface term, Virasoro algebra, and Wald entropy of black holes in higher-curvature gravity · Entropy of a Gauss-Bonnet black hole · 0.89
The semiclassical black hole entropy is then given by $S=(1-β∂_β)I$, which reduces to [equation] and agrees with the result found in [ref] by means of Wald's formula [ref] .

Field theories with anisotropic scaling in 2D, solitons and the microscopic entropy of asymptotically Lifshitz black holes · Asymptotically Lifshitz black hole · 0.98
So the covariant phase space methods [ref] and the cohomological methods [ref] give the same results even in five dimensions.

Central Charges in Extreme Black Hole/CFT Correspondence · Central charge · 0.80

Cites it as support 6 papers

The derivation of the gravitational first law follows from diffeomorphism invariance and the gravitational field equation, and the matter entropy does not enter in the derivation [ref] .

Gravitational Thermodynamics of Causal Diamonds in (A)dS · Conformal matter · 0.97
These charges as shown in [ref] are integrable.

Gauge Invariant Derivation of Zeroth and First Laws of Black Hole Thermodynamics · Gauge invariant black hole charges · 0.98
All variations along the symmetries therefore lead to conserved charges in the bulk of spacetime as a consequence of the generalized Noether theorem [ref] .

Asymptotically flat spacetimes with BMS$_3$ symmetry · Symplectic symmetries · 0.96

Cites it as background 160 papers

In differential covariant theories of gravity, Wald showed that the entropy of a black hole is a kind of Noether charge [ref] .

On the Entropy Function and the Attractor Mechanism for Spherically Symmetric Extremal Black Holes · The Definition of Wald · 0.95
While for an arbitrary theory of gravity obtained from a diffeomorphism covariant Lagrangian, the entropy of a stationary black hole at the bifurcation surface $ℬ$ is given by the Wald entropy, which is defined as the Noether charge and for the $f(Riemann)$ theories of gravity it is written as [ref] [equation] where $v$ is the future-directed affine parameter of the null generator of the future horizon, $u$ denotes the affine null distance away from the horizon and is also future-directed.

Dynamical black hole entropy beyond general relativity from the Einstein frame · Introduction · 0.89
The covariant phase space [ref] has yet to be applied to accelerating AlAdS black holes, although analysing their charges and associated thermodynamics using different techniques have been the study of a slew of recent work [ref] which we will follow closely (see also [ref] for related works).

Thermodynamics of accelerating AdS$_4$ black holes from the covariant phase space · Introduction · 0.81

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] [8]

V. Iyer and R.M. Wald, to be published

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[2] [4]

V.P. Frolov, Phys. Rev. D46, 5383 (1992)

View on Pith 1992

[3] [5]

T.A. Jacobson and R.C. Myers, Phys. Rev. Lett. 70, 3684 (1993)

View on Pith 1993

[4] [1]

S.W. Hawking, Commun. Math. Phys. 43, 199 (1975)

View on Pith 1975

[5] [2]

J.M. Bardeen, B. Carter, S.W. Hawking, Commun. Math. Phy s. 31, 161 (1973)

View on Pith 1973

[6] [3]

D. Sudarsky and R.M. Wald, Phys. Rev. D 46, 1453 (1992); R.M. Wald, “ The ﬁrst law of black hole mechanics” in Directions in General Relativity , vol. 1, ed. by B.L. Hu, M. Ryan, and C.V. Vishveshwara (Cambridge Uni versity Press, Cambridge, 1993). See also J.D. Brown and J.W. York Phys. Rev . D47, 1407 and 1420 (1993)

View on Pith 1992

[7] [6]

I wish to thank Ted Jacobson for bringing this deﬁciency t o my attention

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[8] [7]

G.W. Gibbons and S.W. Hawking, Phys. Rev. D15, 2752 (1977)

View on Pith 1977

[9] [9]

S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time (Cam- bridge University Press, Cambridge, 1973)

View on Pith 1973

[10] [10]

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

View on Pith 1984

[11] [11]

It is proven in [12] that in an arbitrary theory of gravit y, global nonsingularity of the event horizon of a stationary black hole implies that the zeroth law holds. However, there are no compelling physical grounds to impose such a global non- singularity condition, since for a black hole formed by the c ollapse of matter, the parallelly propagated curv...

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[12] [12]

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

View on Pith 1992

[13] [13]

B.S. Kay and R.M. Wald, Phys. Rep. 207, 49 (1991)

View on Pith 1991

[14] [14]

J. Lee and R.M. Wald, J. Math. Phys. 31, 725 (1990). 15

View on Pith 1990

[15] [15]

R.M. Wald, J. Math. Phys. 31, 2378 (1990). The results of this reference also can be derived using the “free variational bicomplex”; see I.M. Anderson in Mathe- matical Aspects of Classical Field Theory , ed. by M. Gotay, J. Marsden, and V. Moncrief, Cont. Math. 132, 51 (1992)

View on Pith 1990

[16] [16]

This quantity corresponds to the “gravitational ﬁeld s trength” discussed in W. Si- mon, Gen. Rel. and Grav. 17, 439 (1985)

View on Pith 1985

[17] [17]

T. Regge and C. Teitelboim, Ann. Phys. 88, 286 (1974)

View on Pith 1974

[18] [18]

A much more indirect argument for this conclusion has be en given independently by J. Simon and B. Whiting (to be published), based upon the Eucl idean approach. See also, M. Visser (to be published)

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[19] [19]

S.W. Hawking, “The Path Integral Approach to Quantum Gr avity”, in General Relativity, an Einstein Centennary Survey , ed. by S.W. Hawking and W. Israel (Cambridge University Press, Cambridge, 1979) 16

View on Pith 1979