Pith. sign in

REVIEW 3 major objections 3 minor 49 references

Anomaly quenching and dynamical cooling of Hawking evaporation in Horndeski gravity

T0 review · 3 major / 3 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper argues that a Horndeski correction to a two-dimensional CGHS black hole makes the Hawking temperature fall as the hole loses mass, so evaporation freezes at a cold remnant and the radiation's entanglement entropy grows only…

desk verdict A solid, internally consistent first-order Horndeski-CGHS calculation; the cold remnant, infinite lifetime, and log Page curve all follow from the assumed temperature profile Eq. (10), which the paper itself concedes is an extrapolation. read the letter →

arxiv 2608.07182 v1 pith:ZHNP3NH7 submitted 2026-08-07 gr-qc cond-mat.stat-mechhep-th

Amogh Srivastav , S. Shankaranarayanan (IIT Bombay) This is my paper · ORCID
classification gr-qccond-mat.stat-mechhep-th MSC 83C5783C8081T2083C47 PACS 04.70.Dy04.62.+v04.50.Kd
keywords HawkingevaporationHorndeskigravityCGHSmodelcoldremnantentanglemententropysurfacegravitationalanomalyblackholeinformation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper studies a two-dimensional CGHS black hole with a first-order Horndeski scalar-tensor correction in the $\gamma_3 \Phi X$ sector and argues that the correction changes Hawking evaporation in a striking way: the surface gravity, and with it the Hawking temperature, decreases as the hole loses mass. Extrapolating the exact first-order temperature shift through a resummed profile $T_H(M)=(\lambda/2\pi)(1-M_r^3/M^3)$, the authors derive that evaporation slows into an infinite-time freeze-out at a macroscopic cold remnant of mass $M_r=(2\gamma_3\lambda^5/3)^{1/3}$. Because the radiation never stops completely, the fine-grained entanglement entropy of the emitted radiation departs from linear growth and asymptotes to logarithmic growth, suggesting the remnant permanently stores the interior entanglement. The key physical message is that a higher-derivative correction can halt semiclassical evaporation and reshape the Page curve without any breakdown of horizon regularity.

What carries the argument

The central object is the shifted Kruskal mapping $X^\pm=(Y^\pm)^{1+s_i}$, with $s_i=\gamma_i(\lambda/M)w_c$ the coefficient of the logarithmic divergence in the lapse. This power-law reparametrization absorbs the coordinate singularity at the apparent horizon and exponentiates the linear lapse expansion $f\propto\epsilon[1-s_i\ln\epsilon]$ into $f\propto\epsilon^{1-s_i}$, converting a coordinate artifact into a physical fractional shift $\delta\kappa/\kappa=-s_i$ of the surface gravity. From that shift the paper obtains the exact first-order temperature correction, which it then resums into the closed profile $T_H(M)=(\lambda/2\pi)(1-M_r^3/M^3)$ that drives the freeze-out.

What would settle it

If a second-order or non-perturbative calculation of the surface gravity shows the temperature does not vanish at any finite mass, then evaporation reaches $M=0$ in finite time and the logarithmic entropy growth does not occur.

Watch

Extended reading notes

Core claim

Within a spherically reduced Horndeski action restricted so that the surviving two-dimensional theory is ghost-free, the linearized field equations on a collapsing-shockwave CGHS background admit closed-form solutions whose logarithmic divergences at the classical apparent horizon are coordinate artifacts. Regularizing the geometry by a power-law Kruskal shift $X^\pm=(Y^\pm)^{1+s_i}$ absorbs the logarithms and produces a finite, shifted surface gravity; for the $\gamma_3\Phi X$ sector with $\gamma_3>0$ the first-order result is $\kappa=\lambda[1-2\gamma_3\lambda^5/(3M^3)]$, so the Hawking temperature falls as the hole loses mass. Under the resummed temperature profile $T_H(M)=(\lambda/2\pi)(1-M_r^3/M^3)$, the evaporation equation $dM/dt=-(\pi/12)T_H^2$ has a double pole at $M=M_r$, giving an infinite lifetime with late-time freeze-out $M(t)-M_r\sim 1/t$, and coupling the entropy production rate $dS/du=(N/12)\kappa(M)$ to this law yields the asymptotic logarithmic growth $S(u)\simeq S_{\rm sat}+(4\pi N M_r/3\lambda)\ln u$. The paper verifies the same suppressed flux independently through the conformal trace anomaly and the Robinson-Wilczek gravitational anomaly, and stresses that the event horizon remains strictly regular while the interior singularity is pushed beyond perturbative control.

Load-bearing premise

The infinite lifetime, cold remnant, and logarithmic entropy all rest on a resummed temperature profile that the paper selects to vanish at a finite remnant mass; this resummation is an extrapolation, not a derived non-perturbative result.

Editorial extensions

If this is right

  • For positive $\gamma_3$, the Hawking temperature of the CGHS black hole decreases as it evaporates, so the hole's luminosity and lifetime are no longer the standard ones.
  • Evaporation freezes at $M_r$ after infinite time, leaving a macroscopic cold remnant at the boundary of perturbative control.
  • The fine-grained entanglement entropy of the radiation grows only logarithmically at late times, rather than linearly, and never reaches a finite maximum.
  • The microscopic radiation flux computed from the gravitational and conformal anomalies matches the macroscopic temperature shift, so the cooling is a real backreaction effect, not a gauge artifact.
  • A positive $\gamma_5$ sector would have the opposite effect, heating the hole instead of cooling it, so the direction of the correction depends on the sign and type of Horndeski coupling.

Reading between the lines

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

  • If the resummation is replaced by any other function with the same first Taylor coefficient, the remnant mass and the entropy rate change, but the qualitative freeze-out survives only while that function has a zero at a positive mass; this makes the zero a testable prediction rather than a theorem.
  • The same coordinate-absorption mechanism may apply to other higher-derivative or quantum-gravity corrections in two-dimensional dilaton gravity, suggesting a general principle: horizon regularity can hide genuine thermodynamic deformations.
  • The logarithmic divergence of $S$ at $M\to M_r$ implies the radiation's fine-grained entropy exceeds the remnant's finite microstate capacity in this semiclassical treatment, so a full resolution of the information paradox requires modeling the remnant's internal degrees of freedom.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The manuscript derives a two-dimensional CGHS-like dilaton gravity by spherical reduction of Horndeski theory, solves the linearized field equations for the G3 = γ3 Φ X and G5 = γ5 Φ couplings, and studies the semiclassical thermodynamics. It finds logarithmic divergences in Kruskal-frame functions that cancel in curvature invariants, obtains a first-order surface-gravity shift δT_H/T_H = −(2 γ3 λ^5)/(3 M^3), and then extrapolates this to a resummed temperature profile T_H(M) = (λ/2π)(1 − M_r^3/M^3) with M_r = (2 γ3 λ^5 / 3)^{1/3}. On this basis it claims that evaporation freezes at M_r after infinite time, that the Robinson-Wilczek anomaly flux is quenched correspondingly, and that the radiation entanglement entropy grows only logarithmically, suggesting information retention in a cold remnant.

Significance. The linear-order computation is detailed and largely credible: the master equation (7), the closed-form profiles (B29)–(B30), the cancellation of logarithms in R and (∇R)^2, and the surface-gravity shift (D14) are substantial pieces of work. The anomaly and entropy sections are internally consistent once the temperature profile is granted. However, the headline results—the cold remnant, the infinite lifetime, and the logarithmic Page-curve saturation—are not established because they rest on an unproven extrapolation. If a genuine nonperturbative derivation of Eq. (10) were supplied, the cold-remnant scenario would be significant for the information-loss problem; as it stands, the paper is a careful linear-order calculation plus a physically motivated but uncontrolled ansatz.

major comments (3)
  1. [§V, Eq. (10); §VI, Eq. (F11); Appendix E] Equation (10) is the single load-bearing assumption of the paper, and it is not derived. With M_r^3 = 2 γ3 λ^5 / 3, Eq. (10) is exactly the linear-order result T_H = (λ/2π)(1 − 2 γ3 λ^5/(3 M^3)) rewritten; it contains no O(γ3^2) or higher terms and is therefore not a resummation of a perturbation series in γ3. It is one of infinitely many functions with the same first Taylor coefficient, and the zero at M = M_r is imposed by expressing the linear coefficient in terms of M_r. The infinite lifetime in Appendix E (the double pole in Eq. (E1) at M = M_r), the 1/t freeze-out (E7), and the logarithmic entropy (G5) and (14) all require T_H(M_r) = 0. The Robinson-Wilczek calculation in Sec. VI does not provide independent support: Eq. (F11) uses the same first-order κ, and the integrated anomaly identity (F15) is an identity once κ is given. The paper is transparent about this in Sec. V and Appendix E, but the abstract and conclusions present the halt of evaporation, the cold remnant, and information retention as results. These claims are not supported unless Eq. (10) is derived or independently verified beyond linear order.
  2. [§IV, Eq. (C5)] The perturbative-control statement used to justify horizon regularity fails exactly at the remnant endpoint. For the G3 sector, γ3 R^(1)_hor / R^(0)_hor = 2 γ3 λ^5 / M^3 = (4/3) (M_r/M)^3, so at M = M_r this 'small' ratio is 4/3, not ≪ 1. Thus the claim in Sec. IV that the ratio 'remains less than 1 throughout the near-horizon regime' is inconsistent with the remnant scenario itself. The linear expansion therefore does not establish a strictly regular horizon at the mass where evaporation is claimed to freeze; at best it is an uncontrolled extrapolation.
  3. [§V, Eq. (9); Appendix D2] The power-law map X^± = (Y^±)^{1+s_i} is introduced to absorb the logarithmic divergence, but the equivalence between 'exponentiating the linear log' and the physical temperature profile is assumed, not demonstrated. Since only the O(γ_i) solution is known, any lapse of the form f ∝ ϵ [1 − s ln ϵ + c_2 (γ_i ln ϵ)^2 + ...] has the same first-order logarithm; the choice that produces the zero at M_r is a choice about all higher coefficients. A derivation of Eq. (10) requires control of the coefficients of all higher powers of γ_i ln ϵ or an independent nonperturbative argument.
minor comments (3)
  1. [Abstract; §IV] The abstract and Sec. IV call the curvature invariants 'exact', but the paper computes only O(γ_i) corrections R^(0) + γ_i R^(1); the terminology should be 'exact on the static slice to first order' or similar.
  2. [§VII, Eq. (12)] The symbol U(u) is used both for the Kruskal coordinate and for the ray-tracing map; please clarify that U is the ingoing Kruskal coordinate evaluated on the outgoing ray, and state the normalization and range of the interval [u1, u2].
  3. [§VI, Fig. 8 reference] The text in Sec. VI refers to Fig. 8 in Appendix F, but the appendix as printed does not contain a figure; please renumber or include it.

Circularity Check

2 steps flagged · score 6.0 of 10

The cold remnant, infinite lifetime, and logarithmic entropy are consequences of the assumed resummed profile in Eq. (10), not of independent non-perturbative input; the linear-order shift and anomaly identity are genuine but do not force freeze-out.

  1. fitted input called prediction [Sec. V, Eq. (10); also Introduction and Appendix D.3]
    "Resumming this exact linear-order shift produces the corrected temperature profile: TH(M) = λ/2π (1 − M^3_r/M^3), (10) where M_r≡(2γ3λ^5/3)^{1/3}... Equation (10) is under strict perturbative control in the regime M≫M_r, where it matches the exact linear-order calculation. As M→M_r, the expansion parameter becomes O(1). Therefore, Eq.(10) functions as an extrapolation ... rather than an exact non-perturbative derivation."

    Equation (10) is algebraically identical to the linear-order result already derived in Eq. (D14): since M_r^3 = 2γ3λ^5/3, it reads TH = (λ/2π)(1 − 2γ3λ^5/(3M^3)) with no O(γ3^2) content. The zero at M = M_r is imposed by this rewriting, not derived. All subsequent headline results—the double-pole luminosity (E1), the infinite evaporation time (E5), the 1/t freeze-out (E7), and the logarithmic entropy (G9)–(G10)—require TH(M_r) = 0 and hence follow from the chosen profile, not from Horndeski dynamics beyond linear order. The paper acknowledges the extrapolation, but still presents the freeze-out and information retention as robust predictions, so those claims reduce by construction to the assumed functional form.

  2. other [Sec. VI; App. F, Eqs. (F11)–(F15)]
    "Restoring general covariance requires a compensating flux equal to the value of this anomaly precisely at the horizon... Φ = Nrt|rH = f′(rH)^2/192π = κ^2/48π. ... Integrating this localized anomaly exactly yields the compensating flux required to restore diffeomorphism invariance: ∫∞_rH At dr = −κ^2/48π."

    The Robinson-Wilczek flux is defined to equal κ^2/48π, and the κ used in Eq. (F11) is the same linear-order surface gravity of Eq. (D13)/(D14) that fixes Eq. (10). The integrated-anomaly identity is an immediate application of the fundamental theorem of calculus once the anomaly is written as ∂_r N^r_t. Therefore this 'microscopic proof' cannot validate the freeze-out profile: at O(γ3) it returns the same linear coefficient already inserted into Eq. (10), and it contains no information beyond that order. It confirms the linear temperature shift, but not the existence of the cold remnant or the logarithmic entropy regime.

full rationale

The genuine, non-circular content is the first-order perturbative computation: the corrected surface gravity κ_G3 = λ[1 − 2γ3λ^5/(3M^3)] in Eq. (D14), the finite horizon curvature in Appendix C, and the O(γ3) anomaly/flux shift. The paper's headline results, however, require Eq. (10), TH = (λ/2π)(1 − M_r^3/M^3), whose only nontrivial coefficient is the same linear-order shift and whose zero sits at the arbitrarily chosen remnant scale M_r = (2γ3λ^5/3)^{1/3}. The paper explicitly labels Eq. (10) an extrapolation 'rather than an exact non-perturbative derivation,' yet then derives the double-pole evaporation law, infinite lifetime, 1/t freeze-out, and logarithmic entropy from it. Since all of those conclusions require TH(M_r) = 0, they reduce by construction to the assumed profile: any completion of the series with that Taylor coefficient and that zero would yield the same qualitative outcome. The Section VI anomaly calculation is similarly definitional, because flux = κ^2/48π with the same κ, so it independently checks only the linear-order shift. No load-bearing circular self-citation was identified: Ref. [17] is used mainly for a standard Brans-Dicke gauge parametrization, and Appendix A rederives the dimensional reduction; the central circularity is the unproven resummation in Eq. (10). Score 6 reflects partial circularity: the linear-order geometry and anomaly matching are real, but the remnant, halted evaporation, and logarithmic information retention are mathematical consequences of the chosen extrapolated temperature profile.

Assumptions & free parameters 2 free parameters · 6 assumptions · 1 invented entities

The ledger shows that the central claim relies on one genuine coupling constant (gamma3), one auxiliary coupling (gamma5), a standard background theory, and one substantive ad hoc assumption: the resummed temperature profile. The remnant is an invented endpoint of that assumed profile rather than an independent entity with external evidence. This is the main reason the circularity burden is nonzero.

free parameters (2)
  • gamma3 = not fitted; input coupling of G3 = gamma3 Phi X; figures use gamma3 = 0.15
    Controls the temperature correction and the remnant mass M_r = (2 gamma3 lambda^5 / 3)^(1/3); all late-time results scale with it.
  • gamma5 = not fitted; input coupling of G5 = gamma5 Phi
    Analyzed in parallel to G3; for gamma5 > 0 it heats rather than cools, so it is not used for the cold-remnant claim.
assumptions (6)
  • domain assumption The 4D Horndeski action dimensionally reduces to the 2D action (6) with G4,X = G5,X = 0 and with IBP surface remainders removed via the selection rule (5).
    Sec. II, Eq. (5). This is needed to obtain a ghost-free master equation; any surviving boundary term or G3,XX contribution would change the dynamical equations.
  • domain assumption The evaporating black hole can be described by the static slice ansatz u = v = q + gamma (A q + W), with integration constants fixed by asymptotic flatness.
    Appendix B3-B4. This is a quasi-static, adiabatic approximation that neglects time dependence of M during the solution of the field equations.
  • ad hoc to paper The near-horizon logarithm is resummed by the power-law coordinate map X^pm = (Y^pm)^(1 + s_i), leading to the temperature profile Eq. (10).
    Eq. (9), Eq. (D5). This is an extrapolation rather than a derivation; it is the load-bearing assumption for the remnant and entropy claims.
  • standard math The Robinson-Wilczek chiral anomaly flux equals kappa^2 / (48 pi), with kappa the corrected surface gravity from Eq. (D13).
    Appendix F. This is a standard method, but it assumes the ingoing near-horizon modes can be integrated out in the presence of Horndeski corrections.
  • standard math The renormalized entanglement entropy is given by the HLW/FPST formula (12) with production rate dS/du = N kappa / 12.
    Sec. VII and Appendix G. This is a standard CFT result, applied here to a slowly drifting surface gravity.
  • domain assumption The 2D radiation luminosity obeys dM/dt = -(pi/12) T_H^2 for central charge c = 1.
    Appendix E. This Stefan-Boltzmann-type law determines the lifetime integral and the freeze-out power law.
invented entities (1)
  • Macroscopic cold remnant of mass M_r = (2 gamma3 lambda^5 / 3)^(1/3)
    purpose: Endpoint of evaporation that halts the Hawking flux and permanently stores interior entanglement.
    The remnant is predicted from the assumed resummed temperature Eq. (10). Its mass depends on the input coupling gamma3 and lies at the boundary of perturbative control, so it has no falsifiable handle outside the paper's own model.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Anomaly quenching and dynamical cooling of Hawking evaporation in Horndeski gravity." pith.science (2026). https://pith.science/paper/ZHNP3NH7

@misc{pith2026260807182,
  author       = {Pith},
  title        = {Pith review of: Anomaly quenching and dynamical cooling of Hawking evaporation in Horndeski gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZHNP3NH7}},
  note         = {Machine review of arXiv:2608.07182}
}
abstract

We investigate the semiclassical Hawking evaporation of a two-dimensional Callan-Giddings-Harvey-Strominger (CGHS) black hole perturbed by first-order Horndeski scalar-tensor couplings. By evaluating the exact scalar curvature invariants, we demonstrate that coordinate singularities near the apparent horizon parametrize a physical deformation of the surface gravity while preserving a strictly regular event horizon. In the linear-order $\gamma_3 \Phi X$ coupling sector, the surface gravity and correspondingly the Hawking temperature decreases dynamically as the black hole loses mass. Extrapolating this cooling effect via a resummed temperature equation suggests that evaporation halts, potentially leaving behind a stable, macroscopic cold remnant at the boundary of perturbative control. Utilizing both the conformal trace anomaly and the Robinson-Wilczek gravitational anomaly, we establish that the asymptotic radiation flux identically reflects this geometric shift. Consequently, the fine-grained entanglement entropy of the Hawking radiation departs from linear growth and transitions to an asymptotic logarithmic regime. This indicates that quantum information may be permanently retained within the remnant, altering the standard semiclassical Page curve without requiring a breakdown of horizon regularity.

Figures

Figures reproduced from arXiv: 2608.07182 by the authors.

Figure 1
Figure 1. FIG. 1. Late-time evolution of [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Hawking temperature [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Geometric, state-independent part [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Total backreacted gravitational anomaly [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Fine-grained entanglement entropy of the Hawking radiation ( [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Full evaporation history [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Semiclassical luminosity [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Off-diagonal stress-tensor component [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

49 extracted references · 29 canonical work pages

  1. [1]

    Spherical reduction of the Horndeski densities The full 4D Horndeski action(1) is assembled from four sector Lagrangian densities. WithX≡− 1 2∇µΦ∇µΦ, Gi,X≡∂G i/∂X, andG(4D) µν the 4D Einstein tensor, the explicit forms are: L2 =G 2(Φ,X), L3 =−G 3(Φ,X)2 4Φ, L4 =G 4(Φ,X)R (4D) +G 4X [ (24Φ)2−(∇µ∇νΦ)2] , L5 =G 5(Φ,X)G (4D) µν ∇µ∇νΦ − 1 6G5X [ (24Φ)3−3(2 4Φ)(...

    work page

  2. [2]

    Brans-Dicke gauge and recovery of the CGHS target space We now fix the residual freedom inL2D by adopting the parametrization (see [17]) Φ =e 2φ, ρ= Λe −2φ, G 4(Φ) = Φ,(A3) whereΛis a constant of dimension length fixing the overall normalization of the areal radius andφ is the CGHS dilaton. This choice is not arbitrary but is precisely the parametrization...

    work page

  3. [3]

    We record them here for generalG3(Φ,X Φ) and G5(Φ), using the shorthandφ±≡∂±φ, φ±±≡∂ 2 ±φ, φ+− ≡∂ +∂−φ, and likewise for ω; G3,Φ ≡∂G 3/∂Φ, G3,X≡∂G 3/∂XΦ, G′ 5≡dG 5/dΦ

    General field equations in conformal null gauge Varying the gauge-fixed action of Appendix A with respect toω, φ, and the residual conformal constraints g++,g−− produces three independent equations, denoted Eω = 0,Eφ = 0, andE++ = 0(with E−− = 0following from+ ↔− ). We record them here for generalG3(Φ,X Φ) and G5(Φ), using the shorthandφ±≡∂±φ, φ±±≡∂ 2 ±φ,...

    work page

  4. [4]

    Exact equations in Kruskal-type fields Substituting the specific choicesG3 = γ3ΦXΦ, G5 = γ5Φinto Eqs. (B1)–(B10), and changing variables from (φ,ω )to the Kruskal-type fields u≡e −2φ, v≡e −2ω via the chain ruleφ± =−u±/2u, ω± =−v±/2v (and similarly for the second derivatives), yields the exact equations. The subscript notation isu±≡∂±u, u±±≡ ∂2 ±u, u+−≡∂ +...

    work page

  5. [5]

    Static reduction and the areal coordinateq To solve Eqs.(B12)–(B16) perturbatively we specialize to the static, spherically-collapsed background relevant for an eternal (or adiabatically evaporating) BH. Introduce static Kruskal coordinatesX±, in which any static field depends only on the productχ≡X +X−; on such a slice the null derivatives collapse to or...

    work page

  6. [6]

    Write, α(q) = 1 +γiA(q) +O(γ 2 i ), v(q) =q+γ iW(q) +O(γ 2 i ),(B20) withγi∈{γ 3,γ 5}, so thatu =αv =q+γi[A(q)q+W (q)]+ O(γ2 i )

    Linearized equations and the master equation for A(q) We now perturb around the background(B19). Write, α(q) = 1 +γiA(q) +O(γ 2 i ), v(q) =q+γ iW(q) +O(γ 2 i ),(B20) withγi∈{γ 3,γ 5}, so thatu =αv =q+γi[A(q)q+W (q)]+ O(γ2 i ). AtO(γ0 i )Eqs. (B12) and (B14) are satisfied iden- tically by Eq.(B19). At O(γi), substituting Eq.(B20) into Eqs.(B12) and (B15)/(...

    work page

  7. [7]

    Boundary conditions and closed-form profiles The homogeneous version of the master equation(B25), (λq−M )A′′ +λA′ = 0, is first order inA′ and integrates immediately: writingB≡A ′,( λq−M )B′ =−λB gives B∝(λq−M) −1, so that Ahom(q) =C 1 +C 2 ln(λq−M),(B27) for constantsC1,C 2. Since ln(λq−M)→ +∞ asq→∞ , demanding asymptotic flatness (A→ 0at spatial infinit...

    work page

  8. [8]

    The logarithm divergence of the dilaton Since u = e−2φ is a genuine spacetime scalar as it is related to the dilaton and is not an auxiliary coordinate- dependent combination, like for instance,A(q)and W (q) individually. Rewritingu =q+γiu1(q)with u1≡Aq +W, substituting the exact closed-form solutions(B29)–(B30) and expanding nearϵ≡q−M/λ→0 + gives: u1 ⏐⏐ ...

    work page

Show all 49 references
  1. [9]

    Near the horizon,v =q+γiW =M/λ+ϵ+γi(wc lnϵ +w0)+··· , where wc is the coefficient oflnϵ in Wi(q)(read off from Eq

    The shifted Kruskal frame Outside the horizon, X+ > 0, X− < 0, so χ = X+X− < 0and ϵ =−λ2χ > 0, the metric isds2 = −ΩdX+dX− with conformal factorΩ =e2ω = 1/v. Near the horizon,v =q+γiW =M/λ+ϵ+γi(wc lnϵ +w0)+··· , where wc is the coefficient oflnϵ in Wi(q)(read off from Eq. (B29...

  2. [10]

    For a function ofχ alone,(∇f)2 =−4vχ(f′)2 (with′ =d/dχ), so κ2 = lim hor (∇f)2 4f =⇒κ= lim hor |v∂χf| λ =λ ⏐⏐⏐1−lim hor χv′ v ⏐⏐⏐, (D9) which reproduces κ0 = λ (i.e

    Hawking temperature of the perturbed black hole The surface gravity is defined invariantly from the norm of the horizon Killing vectorξ =λ(X+∂+−X−∂−), whose norm-squared is|ξ|2 =−f with f =−λ2χ/v. For a function ofχ alone,(∇f)2 =−4vχ(f′)2 (with′ =d/dχ), so κ2 = lim hor (∇f)2 4...

  3. [11]

    Nature of the interior singularity The classical CGHS BH possesses a curvature singular- ity in its interior atu = e−2φ→ 0, i.e.q→ 0+, where R(0) = 4Mλ/q→ +∞. Evaluating the Horndeski curva- ture corrections in the same limit gives R(1)⏐⏐ G3 = 152M 2λ2 q5 +···, R (1)⏐⏐ G5 = 44...

  4. [12]

    Hawking evaporation and the lifetime integral A two-dimensional conformal field of central charge c = 1radiates with Stefan-Boltzmann luminosity L = (π/12)T 2 H, so the BH mass obeysdM/dt =−(π/12)T 2 H = −κ2/48π. Specializing to the G3 sector with γ3 > 0, whose temperatureTH(M...

  5. [13]

    Late-time power-law freeze-out Setting M = Mr +δ with δ→ 0+ and expanding Eq.(E1)to leading order inδ (using1−M 3 r/M3≃ 3δ/Mr) linearizes the evaporation law to the Riccati-type equation dδ dt =−Aδ 2,A= 9cλ2 (2π)2M2r = 3λ2 16πM 2r ,(E6) whose solution, obtained by separating v...

  6. [14]

    Conservation equations in null gauge In the conformal null gaugeds2 =−e2ωdx+dx−, the nonzero metric components areg+− =− 1 2e2ω, g+− = −2e−2ω,√−g = 1 2e2ω, and the only nonzero Christof- fel symbols areΓ + ++ = 2∂+ω,Γ − −− = 2∂−ω,Γ µ µλ = ∂λ ln√−g = 2∂λω. Writing out∇µTµν = 0f...

  7. [15]

    Explicit components on the Horndeski-corrected background On the static slice, the geometric (state-independent) content of Eq.(F4) and the trace piece(F3) both reduce to functions ofqalone, Ξ(q)≡(∂ χω)2−∂ 2 χω=λ 4[ (∂qω)2−∂ 2 qω ] ,(F5) ω+−(q) =λ 2 [ −∂qω+ (M λ −q ) ∂2 qω ] ,...

  8. [16]

    The Robinson-Wilczek construction The Robinson-Wilczek derivation [ 22] obtains the same asymptotic flux from an entirely independent argu- ment, that is, thegravitational, rather than conformal, anomaly of the effective chiral theory obtained by in- tegrating out the ingoing ...

  9. [17]

    (F9) has only a timelike component, At =∂ rNr t = 1 192π∂r ( f′2 +f′′f ) =− 1 192π ( 3f′R+f∂ rR ) ,(F13) usingf′′ =−R

    The anomalous divergence and the background anomaly In the gaugeds2 =−fdt 2 +dr2/f (where√−g = 1), the anomaly of Eq. (F9) has only a timelike component, At =∂ rNr t = 1 192π∂r ( f′2 +f′′f ) =− 1 192π ( 3f′R+f∂ rR ) ,(F13) usingf′′ =−R. Evaluating Eq.(F13)on the unperturbed ba...

  10. [18]

    The HLW/FPST renormalized entropy For a(1 +1)-dimensional conformal field theory in a collapse geometry, the renormalized entropy of the right- moving (outgoing) radiation collected by a distant ob- server over a retarded-time interval[u1,u 2]is fixed entirely by the ray-traci...

  11. [19]

    Entropy production rate and the closed-form S(M) For a slowly evaporating (adiabatic) BH the surface gravity drifts with retarded time through the mass, κ = κ(M(u)), and the ray-tracing relation generalizes to d(lnU′(u))/du = −κ(M(u)). With the early end- point u1 held fixed a...

  12. [20]

    Late-time logarithmic saturation Near the remnant, writeM = Mr +ϵ; thenκ(M)≃ 3λϵ/Mr and the mass equation of Eq.(G3) becomes the same Riccati-type equation as Eq.(E6) of Appendix E (now in retarded timeu rather than coordinate timet), with solution ϵ(u) = 16πM 2 r 3λ2(u+C) u→∞...

  13. [21]

    5(a) eventually overtaking the frozen CGHS final value

    Resolving the entropy/mass-budget tension At first sight it seems paradoxical that the Horndeski BH, which radiateslesstotal mass than the CGHS BH (it retains the finite remnant massMr rather than radiating all the way toM = 0), can nonetheless producemoretotal entanglement en...

  14. [22]

    S. W. Hawking, Particle Creation by Black Holes, Com- mun. Math. Phys.43, 199 (1975), [Erratum: Com- mun.Math.Phys. 46, 206 (1976)]

    1975

  15. [23]

    S. W. Hawking, Breakdown of Predictability in Gravita- tional Collapse, Phys. Rev. D14, 2460 (1976)

    1976

  16. [24]

    Calmet and S

    X. Calmet and S. D. H. Hsu, A brief history of Hawk- ing’s information paradox, EPL139, 49001 (2022), arXiv:2207.08671 [hep-th]

    2022 arXiv

  17. [25]

    Shankaranarayanan and J

    S. Shankaranarayanan and J. P. Johnson, Modified theo- ries of gravity: Why, how and what?, Gen. Rel. Grav.54, 44 (2022), arXiv:2204.06533 [gr-qc]

    2022 arXiv

  18. [26]

    Mandal and S

    S. Mandal and S. Shankaranarayanan, Modified theories of gravity at different curvature scales, Encyclopedia of Astrophysics5, 73 (2026), arXiv:2502.07437 [gr-qc]

    2026 arXiv

  19. [27]

    Ostrogradsky, Mémoires sur les équations différen- tielles, relatives au problème des isopérimètres, Mem

    M. Ostrogradsky, Mémoires sur les équations différen- tielles, relatives au problème des isopérimètres, Mem. Acad. St. Petersbourg6, 385 (1850)

  20. [28]

    D. N. Page, Particle Emission Rates from a Black Hole: Massless Particles from an Uncharged, Nonrotating Hole, Phys. Rev. D13, 198 (1976)

    1976

  21. [29]

    N. G. Sanchez, Absorption and Emission Spectra of a Schwarzschild Black Hole, Phys. Rev. D18, 1030 (1978)

    1978

  22. [30]

    E. J. Martinec, Two-dimensional models of black-hole evaporation, Classical and Quantum Gravity13, 1 (1996)

    1996

  23. [31]

    C. G. Callan, Jr., S. B. Giddings, J. A. Harvey, and A. Strominger, Evanescent black holes, Phys. Rev. D45, R1005 (1992), arXiv:hep-th/9111056

    1992 arXiv

  24. [32]

    J. G. Russo, L. Susskind, and L. Thorlacius, The Endpoint of Hawking radiation, Phys. Rev. D46, 3444 (1992), arXiv:hep-th/9206070

    1992 arXiv

  25. [33]

    J. G. Russo, L. Susskind, and L. Thorlacius, Black hole evaporation in (1+1)-dimensions, Phys. Lett. B292, 13 (1992), arXiv:hep-th/9201074

    1992 arXiv

  26. [34]

    175–223, arXiv:hep-th/9209055

    J.A.HarveyandA.Strominger,Quantumaspectsofblack holes, inSpring School on String Theory and Quantum Gravity (To be followed by Workshop on String Theory 8-10 Apr)(1993) pp. 175–223, arXiv:hep-th/9209055

    1993 arXiv

  27. [35]

    A. Strominger, Two-dimensional black holes and the in- formation puzzle, in1994 DPF Summer Study on High- energy Physics: Particle and Nuclear Astrophysics and Cosmology in the Next Millenium (Snowmass 94)(1995) pp. 433–435

    1995

  28. [36]

    D. N. Page, Average entropy of a subsystem, Phys. Rev. Lett.71, 1291 (1993), arXiv:gr-qc/9305007

    1993 arXiv

  29. [37]

    D. N. Page, Information in black hole radiation, Phys. Rev. Lett.71, 3743 (1993), arXiv:hep-th/9306083

    1993 arXiv

  30. [38]

    Mandal, T

    S. Mandal, T. Parvez, and S. Shankaranarayanan, From the Horndeski action to the Callan-Giddings-Harvey- Stromingermodelandbeyond,Phys.Rev.D109,L041502 (2024), arXiv:2311.07921 [gr-qc]

    2024 arXiv

  31. [39]

    G. W. Horndeski, Second-order scalar-tensor field equa- tions in a four-dimensional space, Int. J. Theor. Phys.10, 363 (1974)

    1974

  32. [40]

    Kobayashi, Horndeski theory and beyond: a review, Rept

    T. Kobayashi, Horndeski theory and beyond: a review, Rept. Prog. Phys.82, 086901 (2019), arXiv:1901.07183 [gr-qc]

    2019 arXiv

  33. [41]

    M. J. Duff, Observations on Conformal Anomalies, Nucl. Phys. B125, 334 (1977)

    1977

  34. [42]

    S. M. Christensen and S. A. Fulling, Trace Anomalies and the Hawking Effect, Phys. Rev. D15, 2088 (1977)

    1977

  35. [43]

    S. P. Robinson and F. Wilczek, A Relationship between Hawking radiation and gravitational anomalies, Phys. Rev. Lett.95, 011303 (2005), arXiv:gr-qc/0502074

    2005 arXiv

  36. [44]

    S. S. Kumar and S. Shankaranarayanan, Quantum entan- glement and Hawking temperature, Eur. Phys. J. C76, 400 (2016), arXiv:1504.00501 [quant-ph]

    2016 arXiv

  37. [45]

    S. M. Chandran and S. Shankaranarayanan, One-to-one correspondence between entanglement mechanics and black hole thermodynamics, Phys. Rev. D102, 125025 (2020), arXiv:2010.03418 [gr-qc]

    2020 arXiv

  38. [46]

    S. M. Chandran, K. Rajeev, and S. Shankaranarayanan, Real-space quantum-to-classical transition of time depen- dent background fluctuations, Phys. Rev. D109, 023503 (2024), arXiv:2307.13611 [gr-qc]

    2024 arXiv

  39. [47]

    Holzhey, F

    C. Holzhey, F. Larsen, and F. Wilczek, Geometric and renormalizedentropyinconformalfieldtheory,Nucl.Phys. B424, 443 (1994), arXiv:hep-th/9403108

    1994 arXiv

  40. [48]

    T. M. Fiola, J. Preskill, A. Strominger, and S. P. Trivedi, Black hole thermodynamics and information loss in two- dimensions, Phys. Rev. D50, 3987 (1994), arXiv:hep- th/9403137

    1994

  41. [49]

    Calabrese and J

    P. Calabrese and J. L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech.0406, P06002 (2004), arXiv:hep-th/0405152

    2004 arXiv

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.