Comments on Sen's Classical Entropy Function for Static and Rotating AdS$_4$ Black Holes

Jewel K. Ghosh; Leopoldo A. Pando Zayas

REVIEW

Comments on Sen's Classical Entropy Function for Static and Rotating AdS$_4$ Black Holes

High Energy Physics - Theory · Submitted 23 Sep 2020 · 47 references

Pith's verdict: A genuinely new rotating AdS4 entropy-function computation, but the paper's central claim overreaches: the rotating entropy is never expressed in terms of conserved charges, and the only rotating higher-derivative term is topological.

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arXiv ID2009.11147 (version 1)

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CitationJewel K. Ghosh, Leopoldo A. Pando Zayas. "Comments on Sen's Classical Entropy Function for Static and Rotating AdS$_4$ Black Holes." arXiv:2009.11147 (2020). https://pith.science/paper/MBVTKJHJ

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Short version: the rotating AdS4 entropy-function calculation is new and worth having; the static part is a clean confirmation. But the paper's headline claim that Sen's formalism is a "highly efficient approach" is stronger than what the rotating section actually demonstrates, and the authors half-say so in §4.4.3.

What is good: the static analysis works for arbitrary horizon topology and several higher-derivative terms, and reproduces the Wald entropy of Bobev, Charles, Hristov, and Reys. The matching requires the sign mapping in Eq. (3.36), which is asserted rather than proved, but it is plausibly a convention and is the kind of thing a referee can push on. The genuinely new part is §4: the modified Sen functional with the rotating ansatz, solved without inputting the full Kerr-Newman-AdS solution, gives a near-horizon geometry that matches the Bardeen-Horowitz limit. The equations are laid out explicitly enough to check, and the Gauss-Bonnet correction is clean.

Soft spots, in order of importance. First, the rotating entropy is never written as S(J,Q). Eq. (4.62) gives S as a function of r+ and a, with the inversion left as "in principle". The stress-tester is right that this matters: for comparison with microscopic counting, or with the known thermodynamics of Kerr-Newman-AdS, the entropy has to be a function of conserved charges. The authors' own §4.4.3 concedes that the near-horizon data misses the global J–Q relation, which means the formalism as applied here cannot identify which supersymmetric near-horizon geometry corresponds to a given black hole. That is a real limitation for the "swift approach" claim, not a minor technicality. Second, the rotating higher-derivative example is Gauss-Bonnet only, which is purely topological in four dimensions. That cannot demonstrate the formalism's ability to capture non-topological corrections in the rotating case. The static section does better on this score. Third, the static sign mapping: as noted above, this is minor if it is indeed a convention, but it should be pinned down.

Who this is for: readers working with Sen's entropy function, near-horizon symmetries, or AdS4 black hole thermodynamics. It is a useful computational contribution, honest about some of its limitations in §4.4.3 even if the abstract oversells. It deserves a serious referee; I would send it out rather than desk reject. The referee should ask for the sign issue to be resolved, for the rotating entropy to be expressed in terms of J and q (or the claim explicitly scoped to near-horizon data), and ideally for one non-topological higher-derivative term in the rotating case. With those changes it would be solid.

Referee report

Significance

If fully established, the paper would provide a useful near-horizon-only route to higher-derivative corrections of AdS4 black hole entropy, complementing full-solution Wald computations and microscopic localization results. The static derivation is explicit, and the rotating background match with the Bardeen-Horowitz limit is a clean and valuable consistency check. The paper is also honest about the limitations of the near-horizon approach, explicitly conceding that global charge relations are not captured. However, the demonstrated rotating result is narrower than the abstract's claim: the entropy is not expressed as a function of conserved charges, and the only higher-derivative term treated in the rotating case is the topological Gauss-Bonnet term. The static comparison with [14] also rests on an asserted sign convention that is not demonstrated. These gaps prevent the paper from fully supporting its central claim as stated, though the underlying approach appears sound and worth publishing after revision.

Major objections (4)

[3.1, Eqs. (3.35)-(3.37)] The claimed agreement with the Wald entropy of [14] is only up to an overall sign, and the mapping in Eq. (3.36), G4 = -GN/(1+α) and a = -16πGN/(1+α)c1, does not by itself remove that sign. With c1=0, Eq. (3.37) gives exactly minus the Wald result of Eq. (3.35). The paper asserts that the sign difference is a matter of convention between Sen's entropy function and Wald's formula, but it does not demonstrate this by recomputing either quantity under a common convention. Since the static agreement is the main validation of the formalism against [14], this sign issue is load-bearing and needs to be either resolved by an explicit calculation or corrected.
[4.3, Eq. (4.62)] The rotating result does not deliver the entropy as a function of the conserved charges. Eq. (4.62) gives S = 16π²√λ + 64π²γ, with λ = (r+²+a²)²/(1-a²/ℓ²)², and the text states that the expression can be inverted 'in principle' using Eq. (4.60). That inversion is never performed, no explicit S(J,q) is given, and no comparison is made with the Kerr-Newman-AdS entropy as a function of (J,q). Moreover, the final formula contains no explicit q dependence; the charge enters only through the unsolved extremality relation (4.57). As written, the claim that the formalism 'provides a swift approach to the black hole entropy' for rotating AdS4 black holes remains incomplete.
[4.4.3] The paper explicitly concedes that the near-horizon formalism 'seems to miss important global constraints,' specifically the BPS relation (4.81) between J and Q. That relation is needed to select which near-horizon data correspond to a given supersymmetric black hole, so the entropy cannot be extracted as S(J,Q) from near-horizon data alone in the rotating BPS setting. This is not a problem with the two-derivative matching, but it substantially qualifies the abstract's claim that the formalism provides the entropy of rotating AdS4 black holes including higher-derivative corrections. Either the scope of the claim should be narrowed, or a procedure for incorporating such global constraints into the entropy function should be supplied.
[4.2-4.3] The rotating higher-derivative analysis is restricted to the four-dimensional Gauss-Bonnet term, which is topological and does not affect the equations of motion. The correction obtained, 64π²γ, is fixed by the Euler characteristic of the horizon and is the same as in the static calculation. Consequently, the rotating computation does not test the formalism's ability to capture non-topological higher-derivative corrections, such as the Weyl-squared terms that are central to the static comparison with [14]. The claim that the formalism captures higher-derivative corrections for rotating AdS4 black holes is therefore only established in the topological sector.

Minor objections (4)

[2 (heading)] The section heading contains a typo: 'formalim' should be 'formalism'.
[3 and 4] The symbol 'a' is overloaded: in Section 3 it denotes the Gauss-Bonnet coupling, while in Section 4 it denotes the rotation parameter. This makes equations such as (3.18) and (4.56) potentially confusing, especially because both appear in the same paper.
[4.4.1, Eq. (4.63)] In the Kerr-AdS metric (4.63), the term W(dr²/Δ_r + dθ²/Δ_θ)² appears with a square that should not be there; the same typo is repeated in Eq. (4.72).
[4.3] The sentence 'The entropy should, of course, be expressed as S(J,q)' directly acknowledges the main limitation of Eq. (4.62). This sentence would be better placed in the conclusions as a clear limitation rather than appearing as a parenthetical remark inside the derivation.

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The paper

The paper claims

Sen's entropy function formalism captures higher-derivative corrections to AdS4 black hole entropy from near-horizon data alone.

The reading

This paper sets out to show that Sen's classical entropy function formalism can capture higher-derivative corrections to the entropy of asymptotically AdS4 black holes without knowing the full black hole solution or its AdS4 asymptotics. In the static case, it claims the formalism reproduces the Wald entropy computed from full solutions, for higher-derivative terms motivated by conformal supergravity. In the rotating case, it claims a modified version generates a near-horizon background that matches the Bardeen-Horowitz limit of known rotating, electrically charged AdS4 black holes and gives their entropy including Gauss-Bonnet corrections. The payoff would be a shortcut to black hole entropy and its corrections using only near-horizon symmetries, with the caveat that certain relations coming from global aspects of the full solution are missed.

What carries the argument

The central object is Sen's entropy function: the Legendre transform, with respect to electric fields, of the action density integrated over the compact directions of the near-horizon geometry $AdS_2 \times \Sigma_g$. Extremizing this function with respect to the horizon radii, scalars, and electric fields yields the attractor equations, and the value at the extremum is the entropy. For rotating black holes the paper uses a modified functional built from a near-horizon ansatz with a rotation parameter $\alpha$ and $\theta$-dependent metric functions, extremized as a functional with regularity boundary conditions at the poles. The Gauss-Bonnet term plays a special role: in four dimensions it is a total derivative, so it does not affect the equations of motion, and its entropy contribution is a topological term $32\pi^2 \gamma \chi(H)/G_N$ that is the same for static and rotating cases. This mechanism is what lets the paper read off higher-derivative corrections from near-horizon symmetries without interpolating to the asymptotic region.

Extended reading notes

Core claim

The central claim is that the higher-derivative corrected entropy of asymptotically AdS4 black holes is fixed by near-horizon data alone. For static black holes with arbitrary horizon topology, the entropy function yields $S = A(H)/(4G_4) + 2\pi a \chi(H)/G_4$, and after mapping $G_4 = -G_N/(1+\alpha)$ and $a = -16\pi G_N c_1/(1+\alpha)$, this agrees with the Wald formula of the full-solution analysis up to an overall sign that the paper attributes to a convention difference. For rotating Kerr-AdS and Kerr-Newman-AdS black holes, the extremization of the modified entropy functional produces a near-horizon metric and gauge field that coincide with the Bardeen-Horowitz limit of the known full solutions, and the entropy again takes the form $16\pi^2\sqrt{\lambda} + 64\pi^2\gamma$, with the Gauss-Bonnet piece proportional to the horizon Euler characteristic. The paper concludes that the formalism is sufficient for the entropy and its higher-derivative corrections, while it naturally misses global constraints such as the relation between angular momentum and electric charge in supersymmetric embeddings.

Load-bearing premise

The static agreement with the Wald formula depends on the assumption that the overall minus sign between the two entropy definitions is purely a convention; if it is a physical discrepancy, the claimed match fails.

Editorial extensions

If this is right

The static match means that for a class of AdS4 supergravity theories, Sen's near-horizon method and Wald's full-solution method give the same higher-derivative corrected entropy.
The rotating result means the Bardeen-Horowitz near-horizon limit of known Kerr-AdS and Kerr-Newman-AdS solutions can be recovered without first solving the full Einstein equations.
The Gauss-Bonnet correction to the entropy is universal across static and rotating AdS4 black holes, proportional to the Euler characteristic of the horizon.
Entropy computations that previously required the full interpolating solution can be performed with near-horizon data alone, including higher-derivative corrections.
Global information, such as the charge-to-angular-momentum relation of supersymmetric black holes, cannot be obtained from the near-horizon formalism by itself and must come from the full solution or other input.

Reading between the lines

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

Editorial inference: the same near-horizon shortcut should be testable in higher-dimensional AdS black holes, where the rotating entropy function would involve multiple rotation parameters; the paper does not perform that computation.
Editorial inference: the sign-convention assumption behind the static match can be checked by recomputing the Wald entropy under one fixed sign convention; a residual minus sign would mean the agreement is not merely conventional.
Editorial inference: feeding the global charge relation back into the near-horizon ansatz as an extra boundary condition would show whether the missing angular-momentum-charge constraint can be recovered without the full solution.

Original abstract

We consider various aspects of Sen's classical entropy function formalism for asymptotically AdS$_4$ black holes with emphasis on its efficacy to capture higher derivative corrections to the Bekenstein-Hawking entropy. The formalism has the important advantage of being based on near-horizon symmetries and does not require knowledge of the full interpolating supergravity solution, nor of its AdS$_4$ asymptotics. For the static case, we focus on applying the entropy function formalism in the presence of various higher derivative terms motivated in conformal supergravities; we find agreement with recently reported results utilizing the full black hole solutions and Wald's entropy formula. For the rotating case, we demonstrate that a modified version of the formalism generates a background that coincides precisely with the Bardeen-Horowitz limit of known rotating, electrically charged AdS$_4$ black holes and provides a swift approach to the black hole entropy, including higher derivatives corrections. We conclude that Sen's classical entropy function formalism is a viable and highly efficient approach to capturing higher-derivative corrections to the entropy of asymptotically AdS$_4$ black holes albeit naturally missing certain relations arising from global aspects of the full black hole solution.

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

We consider various aspects of Sen's classical entropy function formalism for asymptotically AdS$_4$ black holes with emphasis on its efficacy to capture higher derivative corrections to the Bekenstein-Hawking entropy. The formalism has the important advantage of being based on near-horizon symmetries and does not require knowledge of the full interpolating supergravity solution, nor of its AdS$_4$ asymptotics. For the static case, we focus on applying the entropy function formalism in the presence of various higher derivative terms motivated in conformal supergravities; we find agreement with recently reported results utilizing the full black hole solutions and Wald's entropy formula. For the rotating case, we demonstrate that a modified version of the formalism generates a background that coincides precisely with the Bardeen-Horowitz limit of known rotating, electrically charged AdS$_4$ black holes and provides a swift approach to the black hole entropy, including higher derivatives corrections. We conclude that Sen's classical entropy function formalism is a viable and highly efficient approach to capturing higher-derivative corrections to the entropy of asymptotically AdS$_4$ black holes albeit naturally missing certain relations arising from global aspects of the full black hole solution.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We consider various aspects of Sen's classical entropy function formalism for asymptotically AdS$_4$ black holes with emphasis on its efficacy to capture higher derivative corrections to the Bekenstein-Hawking entropy.	contribution 0.91	0.50	0.06	asserted 0.56
s1	The formalism has the important advantage of being based on near-horizon symmetries and does not require knowledge of the full interpolating supergravity solution, nor of its AdS$_4$ asymptotics.	background 0.74	0.09	0.03	asserted 1.00
s2	For the static case, we focus on applying the entropy function formalism in the presence of various higher derivative terms motivated in conformal supergravities; we find agreement with recently reported results utilizing the full black hole solutions and Wald's entropy formula.	result 0.85	0.92	0.04	asserted 1.00
s3	For the rotating case, we demonstrate that a modified version of the formalism generates a background that coincides precisely with the Bardeen-Horowitz limit of known rotating, electrically charged AdS$_4$ black holes and provides a swift approach to the black hole entropy, including higher derivatives corrections.	result 0.87	0.94	0.07	asserted 1.00
s4	We conclude that Sen's classical entropy function formalism is a viable and highly efficient approach to capturing higher-derivative corrections to the entropy of asymptotically AdS$_4$ black holes albeit naturally missing certain relations arising from global aspects of the full black hole solution.	interpretation 0.92	0.64	0.97	asserted 0.98

What kind of paper the classifier reports this is: position or commentary 0.62 · evidence from proof or derivation 0.98.

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References

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How the literature uses it

4 arXiv papers cite this in 5 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 · 3 sentences without a confident label

Cites it as background 2 papers

Higher curvature corrections for AdS$_4$ have also been considered in [ref] and, more recently, for AdS$_5$ black holes [ref] .

Cardy Expansion of 3d Superconformal Indices and Corrections to the Dual Black Hole Entropy · Introduction · 0.89
Other higher-derivative aspects of AdS$_4$ black holes were recently considered [ref] .

Gravitational Cardy Limit and AdS Black Hole Entropy · Discussion · 0.96

What it cites · 66 sentences without a confident label

Uses the method or tool of 5 papers

Our approach to higher-derivative corrections to the entropy of AdS$_4$ black holes is rooted in Sen's entropy function formalism introduced in [ref] .

Black Hole Entropy Function and the Attractor Mechanism in Higher Derivative Gravity · Introduction · 0.98
The boundary conditions we impose follow [ref] and imply regularity of the background: [equation]

Rotating Attractors · Rotating Black Holes in Flat Space · 0.87
Therefore, near $r=r_+$ we can expand: [equation] The Bardeen-Horowitz [ref] limit is taken by [ref] [equation] After performing these transformations and taking the $ϵ→0$ limit we obtain the near horizon geometry: [equation] where we have defined: [equation] Note that this is of the same form of our ansatz [ref] and from this we can extract the relevant functions which match with Section [ref].

The Extreme Kerr Throat Geometry: A Vacuum Analog of AdS_2 x S^2 · Kerr-AdS blackholes · 0.86

Uses data or values from 1 paper

The solutions to the extremization problem with the appropriate boundary conditions were obtained in [ref] in units where $1/16πG_4=1$ and we record it below: [equation] In this case the black hole entropy is: [equation] The first term is is the usual Bekenstein-Hawking entropy formula.

Rotating Attractors · Rotating Black Holes in Flat Space · 0.90

Compares with 2 papers

In this subsection we compare the results obtained using Sen's entropy function formalism with a recent study which discussed the influence of various higher-derivative terms, motivated by conformal supergravity, on the the Wald entropy [ref] for certain black holes in AdS$_4$.

The Unreasonable Effectiveness of Higher-Derivative Supergravity in AdS$_4$ Holography · Agreement with higher curvature corrections in supergravity · 1.00
The reason for the overall minus sign is because the definition of Wald's entropy used in [ref] has a relative minus sign with respect to the definition used by Sen in [ref] .

Black Hole Entropy Function and the Attractor Mechanism in Higher Derivative Gravity · Agreement with higher curvature corrections in supergravity · 0.93

Cites as background 25 papers

We are interested in asymptotically AdS black holes, there are two studies discussion the application of Sen's entropy function formalism to AdS black holes [ref] .

Dyonic AdS_4 black hole entropy and attractors via entropy function · Review of Sen's entropy function formalim · 0.89
We are interested in asymptotically AdS black holes, there are two studies discussion the application of Sen's entropy function formalism to AdS black holes [ref] .

Entropy function and attractors for AdS black holes · Review of Sen's entropy function formalim · 0.89
There have recently been two studies exploring aspects of higher-derivative corrections for AdS black holes; one focused on AdS$_4$ solutions [ref] and another on AdS$_5$ [ref] .

The Unreasonable Effectiveness of Higher-Derivative Supergravity in AdS$_4$ Holography · Introduction · 0.96

Discussion

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