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REVIEW 3 major objections 3 minor 54 references

A smooth BTZ black bounce with an extremal null throat

T0 review · 3 major / 3 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A smoothed BTZ horizon is a black bounce, not a signature change

desk verdict Solid black-bounce construction with a rigorous analytic extension; the Aretakis instability headline overreaches because the linear growth rests on an unproved settling assumption. read the letter →

arxiv 2608.04461 v1 pith:KWP3KN22 submitted 2026-08-05 gr-qc

Farzad Milani This is my paper · ORCID
classification gr-qc PACS 04.70.Bw04.70.Dy04.60.Kz04.20.Jb
keywords BTZblackholebouncedegenerateKillinghorizonAdS2xS1throatWaldentropyAretakisinstabilitysignaturechangelower-dimensionalgravity
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

This paper establishes that inserting a smooth tanh transition function into the inverse radial component of the non-rotating BTZ metric, $g^{rr}=S_\delta(r)F(r)$, does not produce a Lorentzian-to-Riemannian signature change at the horizon. In coordinates $r-r_h=q^2$ with an advanced null time, the metric extends real-analytically through $q=0$ to a second isometric exterior, so the would-be horizon becomes a degenerate null throat, a regular black bounce with vanishing surface gravity. The construction matters because it provides a fully controlled example of a BTZ black bounce whose bounce surface sits exactly at the horizon, with closed-form curvature, effective source, energy-condition, and entropy data, and because it exposes a generic obstruction: smoothing $g_{tt}$ instead is singular for every finite width. The paper also proves strict positivity of every scalar mode's effective potential and derives an Aretakis-type instability on the throat, with an exactly conserved leading transverse derivative and a linearly growing subleading one.

What carries the argument

The machine that carries the argument is the square-root coordinate transformation $r-r_h=q^2$ together with the signed root $\sigma=\sqrt{S_\delta}$ and the advanced time $dv=dt+dr/(F\sigma)$. Because $\tanh(y)=y\,g(y)$ with $g$ analytic and $g(0)=1$, the root $\sigma$ is analytic and odd in $q$, and the cross term $\beta=4\sqrt{\delta}/\sqrt{g(q^2/\delta)}$ is analytic and nonzero at $q=0$; this cancels the would-be divergence of $dr^2$ exactly. The same analyticity makes the extension unique, makes all Christoffel symbols analytic near $q=0$ so that geodesics cross the throat analytically for every conserved charge, and turns the scalar wave equation into the exact form whose evaluation at $q=0$ yields the conserved Aretakis constant $H_0=\partial_q\psi|_{q=0}$ and the linear growth law for $\partial_q^2\psi|_{q=0}$.

What would settle it

Evolve Eq. (21) from a family of smooth $L=0$ initial data with $H_0\neq0$ for which $\partial_v\psi|_{q=0}$ does not decay; if the late-time slope of $\partial_q^2\psi|_{q=0}$ deviates from $-\frac{r_h}{\ell^2\sqrt{\delta}}H_0\,v$, the claimed Aretakis growth rate is falsified.

Watch

Extended reading notes

Core claim

At $r=r_h$ the original chart is singular, but the regular chart $r-r_h=q^2$, $dv=dt+dr/(F\sigma)$ with $\sigma=\sqrt{S_\delta}$, turns the metric into $ds^2=-F(r_h+q^2)\,dv^2+\beta(q^2)\,dv\,dq+(r_h+q^2)^2\,d\varphi^2$, with $\beta(0)=4\sqrt{\delta}\neq0$. This line element is real-analytic at $q=0$, so it has a unique analytic extension to $q<0$, where $r>r_h$ again. The surface $q=0$ is a null hypersurface, a degenerate Killing horizon with vanishing surface gravity, and the two exterior sheets are isometric. The areal radius has a strict minimum there, so the geometry is a member of the black-bounce family rather than a signature-changing spacetime; near the throat it is $\mathrm{AdS}_2\times S^1$, and the throat circle is a minimal surface.

Load-bearing premise

The headline instability rate rests on an unproved settling condition: the paper shows $\partial_q\psi|_{q=0}$ is exactly conserved, but converting Eq. (24) into linear growth of $\partial_q^2\psi|_{q=0}$ assumes that $\partial_v\psi$ tends to zero on the throat at late advanced time, a settling that is demonstrated only for one numerical initial-data family.

Editorial extensions

If this is right

  • If the central claim is right, the smoothing ansatz $g^{rr}=S_\delta F$ does not connect the Lorentzian sector to a Riemannian interior; the spacetime consists of two isometric exteriors joined at a degenerate null throat.
  • At the throat all curvature invariants are finite, with $R=-r_h/(\ell^2\delta)$ and $K=r_h^2/(\ell^4\delta^2)$, so the family is a one-parameter regularisation whose curvature grows as $\delta\to0$ and which never converges smoothly to BTZ.
  • The throat entropy $\pi r_h/2G$ is obtained independently from the minimal-surface length, the Wald–Noether charge, and a Cardy estimate using the Brown–York mass, although no first law exists because $\kappa=0$.
  • The scalar effective potential is strictly positive for every angular mode, so under the Dirichlet (Friedrichs) extension there are no exponentially growing exterior test-scalar modes; the instability that does exist is the transverse-derivative, Aretakis type at the throat.
  • The alternative ansatz that smooths $g_{tt}$ instead of $g^{rr}$ produces a curvature singularity at the horizon for every finite smoothing width.

Reading between the lines

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

  • If the settling assumption in Sec. 7.2 fails for generic smooth data, the instability reduces to exact conservation of $H_0$ without the linear growth rate; this is a direct numerical test across a wider space of initial data.
  • The rotating or charged generalisation, where a Kaluza–Klein gauge field makes Sen's entropy function nondegenerate, may be where a first law for this class of extremal throats can actually be formulated.
  • The same $r-r_h=q^2$ analytic-extension mechanism likely applies to any metric with $g^{rr}=S(r)F(r)$ where $S$ vanishes linearly at the zero of $F$, so the no-signature-change conclusion may hold for a whole class of smooth transitions, not only tanh.
  • If late-time waves probe the throat, the linearly growing $\partial_q^2\psi|_0$ may feed nonlinear or backreaction effects, and the exponentially decaying tails of the energy-condition violation make the transition shell effectively infinite; both are testable in time-domain evolutions beyond linear order.
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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. This paper studies the static, circularly symmetric BTZ deformation (2), in which a tanh transition function multiplies the inverse radial metric component, g^{rr}=tanh((r-r_h)/δ)F(r). The central claim is that the naive Lorentzian-to-Riemannian signature-change interpretation is incorrect: in coordinates r-r_h=q^2 and with an advanced time v, the metric extends real-analytically across q=0 as a regular, Lorentzian, degenerate Killing horizon, beyond which lies a second isometric exterior; the areal radius bounces at the throat. The paper also gives the curvature and effective source in closed form, analyzes energy conditions, proves analytic regularity of the geodesic crossing, identifies the near-throat AdS_2×S^1 geometry, computes the throat entropy by three routes, proves (or claims to prove) strict positivity of the scalar effective potential, and derives an Aretakis-type instability with a conserved first transverse derivative and a linearly growing second derivative. It also records a negative result: smoothing g_{tt} instead is singular at the horizon for every finite width. The paper is explicit about several limitations, including the absence of a first law, the kinematic character of the model, and the deferred construction of the maximal extension.

Significance. The analytic-extension construction is the paper's strongest contribution: the coordinate transformation and signed-root argument in Sec. 2 and Appendix A are explicit, the q-chart metric is non-degenerate and Lorentzian at q=0, and the existence of a second isometric exterior is established by real analyticity. This part is convincing and likely to be useful to the black-bounce and signature-change community. The closed-form effective source, the energy-condition accounting, and the entropy concordance (minimal-surface length, Wald--Noether charge, and a Cardy estimate explicitly labeled a consistency check) are also valuable and are presented with unusual honesty. The scalar-sector claims are less robust: the derivation of the effective potential in Appendix C contains an algebraic inconsistency, and the Aretakis linear-growth statement in Sec. 7.2 is conditional on an unproved settling assumption. These issues affect load-bearing parts of the mode-stability and instability headlines, even though the core geometric results are not in question.

major comments (3)
  1. [Appendix C, Eqs. (C1)–(C2) and Eq. (20)] The reduction from Eq. (C1) to Eq. (C2) is not correct as written. With A=√(SF), dz=dr/A, substituting into (C1) does not produce (1/r)∂_z(r∂_z R). For the representative case S=1, F=r^2/ℓ^2, the exact transformed equation is (r^2/ℓ^2) R_zz + (4r^2/ℓ^3) R_z + (ω^2ℓ^2/r^2) R=0, whereas Eq. (C2) gives (r^2/ℓ^2) R_zz + (3r^2/ℓ^3) R_z + (ω^2ℓ^2/r^2) R=0; the coefficient of R_z differs. In addition, the appendix defines A=√(SF) but then uses A^2=SF^2 when computing AA′, so the printed computation of ∂_z^2√r does not correspond to the stated tortoise coordinate. Consequently Eq. (20) is not the actual effective potential for the scalar field in the stated coordinates, and the proof that V_L>0 for every mode is not valid as written. This is load-bearing for the mode-stability claim in Sec. 7.1 and must be corrected, or the claim must be withdrawn or qualified.
  2. [Sec. 7.1, endpoint classification] The statement that the AdS endpoint is limit-circle appears incorrect. For the potential (20) as given, V_L∼3r^2/(4ℓ^4) as r→∞; in the standard tortoise coordinate for BTZ, x≈−ℓ^2/r, this is V_L∼3/(4x^2), which is the borderline limit-point case (one independent solution behaves as x^{-1/2} and is not square-integrable near x=0), not the limit-circle case. If the effective potential is corrected, the divergence at the AdS boundary remains, and the classification should be re-examined. The discussion of which self-adjoint extension must be chosen, and the claim that the Dirichlet (Friedrichs) extension is forced by normalizability, therefore need to be revisited.
  3. [Sec. 7.2, Eqs. (24)–(25), and Fig. 6] The linear-growth formula (25) is derived from Eq. (24) under the explicitly unproved assumption that ∂_vψ|_0→0 on the throat at late advanced time. The exact conservation of H_0=∂_qψ|_0 is rigorous, but the linear growth of ∂_q^2ψ|_0 is conditional on this settling assumption. The abstract and conclusions present the linear growth as an unconditional property of the throat. The numerical confirmation uses a single initial-data family, ψ(0,q)=q exp[−q^2/(2(0.3)^2)], on the finite interval q∈[0,1.1], with no statement of the outer boundary condition or a convergence study. The authors should either prove the settling for a suitable class of admissible data or explicitly state the result as conditional, describing the numerical run as an illustrative example rather than generic evidence.
minor comments (3)
  1. [Appendix A, after Eq. (A4)] At δ=0.1 the expansion is quoted as σ=√10 q − (50√10/3) q^5 + O(q^7), but the series (A4) has no q^7 term; the next nonzero term is O(q^9). This is a harmless typo but should be fixed.
  2. [Sec. 7.2 and Fig. 6] Please specify the outer boundary condition used in the finite-difference evolution of Eq. (21) and report a convergence test, since boundary reflections could affect the late-time settling of ∂_vψ|_0 on a finite q-interval.
  3. [Abstract and Sec. 6(iii)] The abstract says the entropy is 'reproduced independently' by the Wald–Noether charge and by a Cardy estimate, but Sec. 6(iii) and Sec. 8 explicitly label the Cardy route as a consistency check rather than a derivation. The abstract should carry the same caveat to avoid overstating the microscopic status of the result.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the analytic extension, potential positivity, and Aretakis identities are derived from the metric and wave equation; the Cardy entropy route is explicitly a consistency check.

full rationale

The derivation chain is self-contained. In Sec. 2 the regular chart is constructed by the explicit coordinate change r - r_h = q^2 with the signed analytic root sigma(q); the metric (6) has analytic coefficients and determinant -beta^2/4 = -4 delta at q = 0, so the Lorentzian extension and its uniqueness follow from real-analyticity, not from the target claim. The throat's null, degenerate, minimal-surface, and AdS2 x S1 properties are computed from the metric and curvature, with no fitted input. The scalar potential positivity in Sec. 7.1 is proven term-by-term from F > 0, S > 0, S' >= 0; no sampled or fitted quantity enters the proof. The Aretakis equations (21)-(25) are derived exactly from the wave operator in the regular chart; Eq. (25) is explicitly conditional on the stated settling assumption, and the numerical evolution is an independent check, not a fit. The entropy determinations are presented honestly: the Wald charge and minimal-surface length are direct geometric computations, and the Cardy route is explicitly labeled 'a consistency check, not a derivation' (Sec. 6), using Brown-York mass computed from the metric rather than inferred from the entropy. The self-citation [11] is historical, described as superseded, and carries no load-bearing argument. No prediction reduces by construction to its input, and no self-citation chain forces the conclusions.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

The only free parameter is the smoothing width delta, which is introduced by hand and left unexplained. The axioms are standard mathematical theorems and standard holographic input, plus one explicit dynamical assumption in the Aretakis section. No new particles, fields, or forces are postulated.

free parameters (1)
  • delta (transition width) = not fitted; free scale
    Entered through S_delta = tanh((r-r_h)/delta). Curvature at the throat scales as 1/delta and 1/delta^2, Aretakis growth as 1/sqrt(delta), and the paper explicitly states 'The scale delta is unexplained' in Sec. 8.
assumptions (3)
  • standard math Cauchy-Kovalevskaya theorem for analytic ODE systems applies to the geodesic equations in the (v,q) chart
    Used in Sec. 5 to establish analytic geodesic crossing through the throat for all conserved charges; requires the Christoffel symbols to be analytic in a neighbourhood of q=0, which is shown in Appendix A.
  • domain assumption The Brown-Henneaux central charge c=3*ell/(2G) and the Cardy formula apply to the asymptotically BTZ exterior
    Used in Sec. 6.1(iii) for the Cardy entropy estimate; the paper calls this a consistency check, not a derivation, because the dual state is not identified.
  • domain assumption The scalar field is a test field on the fixed background and the throat settles to a constant at late advanced time for the linear growth result
    Sec. 7.2 explicitly conditions the linear growth of d^2_q psi on d_v psi|0 -> 0; the paper proves conservation of H0 exactly but not the settling.

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Pith. "Pith review of A smooth BTZ black bounce with an extremal null throat." pith.science (2026). https://pith.science/paper/KWP3KN22

@misc{pith2026260804461,
  author       = {Pith},
  title        = {Pith review of: A smooth BTZ black bounce with an extremal null throat},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KWP3KN22}},
  note         = {Machine review of arXiv:2608.04461}
}
abstract

We study a static, circularly symmetric deformation of the non-rotating BTZ black hole obtained by inserting a smooth transition function into the \emph{inverse} radial metric component, $g^{rr}=S_\delta(r)F(r)$ with $S_\delta=\tanh[(r-r_h)/\delta]$, leaving $g_{tt}=-F$ untouched. This was motivated by the proposal that such a construction realizes a Lorentzian-to-Riemannian signature change at the horizon; we show that it does not. In coordinates $r-r_h=q^2$ with an advanced time, the metric extends real-analytically across $r=r_h$, and the extension is Lorentzian: $q=0$ is a regular null hypersurface, a degenerate Killing horizon with vanishing surface gravity, beyond which lies a second, isometric copy of the exterior. The areal radius has a minimum there, so the geometry is a black bounce; the would-be Riemannian branch is a separate geometry the Lorentzian sector never reaches. We give the effective source in closed form, an invariant account of the energy conditions, and identify the near-throat geometry as AdS$_2\times S^1$. The scalar effective potential is proven strictly positive for every mode, and the throat circle is a minimal surface whose length gives an entropy $\pi r_h/2G$, reproduced independently by the Wald--Noether charge and by a Cardy estimate from the computed Brown--York mass -- concordant results for which no first law is available since $\kappa=0$. The throat carries an Aretakis-type instability, with a conserved leading transverse derivative and a linearly growing subleading one. We also record a negative result: smoothing $g_{tt}$ instead, as in the Lorentzian-Euclidean Schwarzschild proposal, is singular at the horizon for any finite smoothing width. We state explicitly what the construction does not establish.

Figures

Figures reproduced from arXiv: 2608.04461 by the authors.

Figure 1
Figure 1. FIG. 1: Exterior structure at [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: A radial timelike geodesic integrated as a full syste [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Ricci and Kretschmann scalars on the Lorentzian sect [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Left: the effective source (12) on the Lorentzian secto [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: The effective potential (20) in the tortoise coordinat [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Numerical evolution of (21) for [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]

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Works this paper leans on

54 extracted references · 31 canonical work pages

  1. [1]

    Lorentzian–Euclidean

    INTRODUCTION Signature-changing metrics have been studied in quantum cosmolog y and in classical rel- ativity for several decades [1–6]. Capozziello, De Bianchi and Battis ta [7] brought the idea to black holes, proposing a “Lorentzian–Euclidean” Schwarzschild g eometry in which the discontinuous sign function ε(r) multiplies gtt, so that the region insid...

    work page 1989

  2. [2]

    The chart (2) is singular at the horizon Near rh, F ≃F ′(rh)x and Sδ ≃x/δ, so grr =SδF has a double zero and grr ≃ δ F ′(rh)x2 → ∞, detg = −r2 Sδ → ∞

    THE EXTENSION, THE THROA T, AND THE SECOND SHEET 2.1. The chart (2) is singular at the horizon Near rh, F ≃F ′(rh)x and Sδ ≃x/δ, so grr =SδF has a double zero and grr ≃ δ F ′(rh)x2 → ∞, detg = −r2 Sδ → ∞. (4) Consequently the proper radial distance to the horizon diverges lo garithmically, ∫ dr/√SδF ≃ √ δ/F ′(rh) | lnx|, from both sides, and the Killing t...

    work page

  3. [3]

    2: A radial timelike geodesic integrated as a full syste m in the regular chart (6)

    CUR V A TURE AND THE EFFECTIVE SOURCE On the Lorentzian sector the mixed Ricci components have finite limit s at Σ for every δ >0, Rt t(Σ) = Rr r(Σ) = −F ′(rh)S′ δ(rh) 4 , R φ φ(Σ) = 0, (9) 5 0.0 0.5 1.0 1.5 2.0 τ −0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 q sheet I sheet II 0.0 0.5 1.0 1.5 2.0 τ 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 r / ℓ FIG. 2: A radial timelike geo...

    work page

  4. [4]

    (13) The angular null energy condition is satisfied everywhere

    ENERGY CONDITIONS On the Lorentzian sector, with ρ = −T tt, pr =T rr, pφ =T φφ, 8π(ρ +pr) = −FS ′ 2r , 8π(ρ +pφ) = MS ′ 2r . (13) The angular null energy condition is satisfied everywhere. The radia l one is violated wherever F > 0 and S′ > 0—that is, throughout the exterior, with magnitude peaking at ≈ 0.40 near x ≈ δ and decaying exponentially. We emphas...

    work page

  5. [5]

    Taken at face value in that chart this suggests a turning point, but the chart is singular there and the inf erence is not legitimate; the question must be settled in (6)

    GEODESICS IN THE EXTENSION In the chart (2) the radial equation is ˙r2 =Sδ(E2 −F (κ+L2/r2)), with κ = 1 for timelike and 0 for null geodesics; it has a simple zero at rh. Taken at face value in that chart this suggests a turning point, but the chart is singular there and the inf erence is not legitimate; the question must be settled in (6). In the regular...

    work page

  6. [6]

    The (t,z ) sector has constant curvature −λ2/2: the near-throat geometry is AdS2 ×S1, ℓ AdS2 = √ 2δ/F ′(rh), (15) the standard near-horizon geometry of a degenerate horizon

    THE THROA T: NEAR-HORIZON GEOMETR Y, ENTROPY FUNCTION, AND THERMODYNAMIC ST A TUS Introducing the proper radial distance z on the exterior side, the near-throat metric takes the form ds2 ≃ −Ae−λzdt2 +dz2 +r2 hdφ2 withλ = √ F ′(rh)/δ. The (t,z ) sector has constant curvature −λ2/2: the near-throat geometry is AdS2 ×S1, ℓ AdS2 = √ 2δ/F ′(rh), (15) the stand...

    work page

  7. [7]

    SCALAR PROBE: MODE ST ABILITY AND THE ARET AKIS INST ABILITY 7.1. Positivity of the potential for all modes For a massless scalar ψ =R(r)e−iωt+iLφ on (2), with tortoise coordinate dz =dr/(√ SδF ) and u = √rR, the radial problem takes the Schr¨ odinger form u′′ + (ω2 −VL)u = 0 with (Appendix C) VL = FL 2 r2 + F [ rFS ′ +S ( 3r2/ℓ2 +M )] 4r2 . (20) Since F ...

    work page

  8. [8]

    14 Not a signature-changing spacetime

    SCOPE AND LIMIT A TIONS We state plainly what this paper does not do. 14 Not a signature-changing spacetime. The construction was motivated by signature change and does not achieve it. The Lorentzian sector extends analytically to a second Lorentzian sheet; the Riemannian branch is a separate geometry, singular at it s centre for the profile used here. Whe...

    work page

Show all 54 references
  1. [9]

    CONCLUSIONS Inserting a smooth transition function into grr of the BTZ metric does not produce a signature-changing spacetime. It produces a black bounce whose throat sits exactly at the would-be horizon and is therefore degenerate: a regular null hype rsurface with vanishing ...

  2. [10]

    Analyticity of σ(q) Write tanhy =yg (y), where g(y) = tanhy y = 1 −y2 3 + 2y4 15 − · · · (A1) is even, analytic on |y|<π/ 2 and g(0) = 1. With r −rh =q2 and y =q2/δ, Sδ = tanh q2 δ = q2 δ g ( q2 δ ) , (A2) 16 so the signed square root is σ(q) = q √ g(q2/δ) δ , (A3) which is ma...

  3. [11]

    The extension metric Under (5), dt =dv −dr/(Fσ ) and dr = 2qdq , so −Fdt 2 = −F dv2 + 2 σdvdr − dr2 Fσ 2, dr2 SδF = dr2 σ2F. (A5) The dr2 terms cancel identically—this is the point of the construction—leavin g ds2 = −F dv2 + 2 σdvdr +r2dφ2 = −Fdv 2 +βdvdq +r2dφ2, (A6) with β =...

  4. [12]

    Hence det g = −β2r2/4 is finite and negative there: the metric is Lorentzian and non-degenerate

    Determinant and the null character of Σ In the ordering ( v,q,φ ), gµν =      −F β/2 0 β/2 0 0 0 0 r2     , (A8) 17 whose (v,q ) block has determinant (−F )(0) − ( β 2 ) 2 = −β2 4 , (A9) independent of F and equal to −4δ < 0 at q = 0. Hence det g = −β2r2/4 is finite a...

  5. [13]

    (A11) Every symbol is a rational expression in F,β,F ′,β ′,q,r whose only denominators are powers of β and r

    Christoffel symbols and geodesic regularity With F =F (q), β =β(q) and r =rh +q2, the nonvanishing symbols are Γv vv = F ′ β , Γv φφ = −4qr β , Γq vv = 2FF ′ β2 , Γq vq = −F ′ β , Γq qq = β′ β, Γq φφ = −8qrF β2 , Γφ qφ = 2q r . (A11) Every symbol is a rational expression in F,β...

  6. [14]

    dv = 0 (ingoing) , dq dv = F β (outgoing)

    The throat is a minimal surface Null directions orthogonal to a circle q = const in the chart (6) satisfy −F dv2 +βdvdq = 0, i.e. dv = 0 (ingoing) , dq dv = F β (outgoing). (D1) The expansion of the circle along either congruence is θ = r−1dr/dλ with r = rh +q2, so dr = 2qdq a...

  7. [15]

    W ald–Noether charge The Wald entropy [20, 21] is SW ald = −2π ∮ ∂L ∂Rµνρσ ǫµνǫρσdA, (D4) with ǫµν the binormal normalized by ǫµνǫµν = −2. For L = (R − 2Λ)/16πG, ∂L ∂Rµνρσ = 1 32πG ( gµρgνσ −gµσgνρ) , (D5) whose contraction with ǫµνǫρσ gives −2/32πG, so the integrand is 1 /4G ...

  8. [16]

    Brown–Y ork quasilocal mass For ds2 = −N 2dt2 +dr2/f +r2dφ2 with N 2 = F and f = SδF , the boundary circle at r = R on a static slice has outward unit normal n = √ f∂ r and extrinsic curvature k = √f/R . The Brown–York energy [23] relative to a background with k0 = 1/ℓ (pure A...

  9. [17]

    As noted in the main text this is a consistency check rather than a derivation: it uses only the asymptotic charges, and the dua l state is not determined here

    Cardy consistency check With Brown–Henneaux [22] c = 3ℓ/2G and, for a non-rotating configuration of mass MADM =M/8G, L0 = ¯L0 =MADMℓ/2 = Mℓ/ 16G, the Cardy formula [24, 25] gives SCardy = 4π √ cL 0 6 = 4π √ 1 6 · 3ℓ 2G · Mℓ 16G = π √ Mℓ 2G = πrh 2G, (D9) 22 coinciding with Smin...

  10. [18]

    The master equation Using the inverse metric (A10) and √ −g = βr/2, the massless wave operator in the chart (6) is □ψ = 2 βr [ ∂v ( βr 2gvq∂qψ ) +∂q ( βr 2 (gqv∂vψ +gqq∂qψ) ) ] +gφφ∂2 φψ. (E1) With gvq = 2/β and gqq = 4F/β 2 the bracket becomes r∂ v∂qψ +∂q ( r∂vψ ) +∂q ( 2rF β...

  11. [19]

    Then W = 2rF β = 2rhf1 β0 q2 +O(q4), (E4) so W (0) = 0, W ′(0) = 0, W ′′(0) = 4rhf1 β0

    The conserved transverse derivative Nearq = 0 write F =f1q2 +f2q4 +O(q6) with f1 =F ′(rh)> 0 — the double zero being exactly the statement that the horizon is degenerate — and β = β0 +β2q2 +O(q4) with β0 = 4 √ δ. Then W = 2rF β = 2rhf1 β0 q2 +O(q4), (E4) so W (0) = 0, W ′(0) =...

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    (E7) Atq = 0, using r′ = 0, r′′ = 2 and (E5), this reduces to 2rh∂v ( ∂2 qψ ) ⏐ ⏐ 0 + 2∂vψ ⏐ ⏐ 0 + 4rhf1 β0 H0 = 0, (E8) i.e

    Linear growth of the second transverse derivative Differentiating (21) (with L = 0) once with respect to q, 2r′∂v∂qψ + 2r∂ v∂2 qψ +r′′∂vψ +r′∂v∂qψ +W ′′∂qψ + 2W ′∂2 qψ +W∂ 3 qψ = 0. (E7) Atq = 0, using r′ = 0, r′′ = 2 and (E5), this reduces to 2rh∂v ( ∂2 qψ ) ⏐ ⏐ 0 + 2∂vψ ⏐ ⏐ 0...

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    Writing the equation as ∂q(2√ ru ) = −r−1/2∂q(W∂ qψ) for u = ∂vψ and integrating inward from the outer boundary avoids differentiating the flux twice and is numerically stable

    Numerical confirmation Equation (21) with L = 0 was integrated on q ∈ [0, 1.1] with 1600 points and dv = 1 .2 × 10−4 to v = 7 ℓ, at M = ℓ = rh = 1, δ = 0 .1. Writing the equation as ∂q(2√ ru ) = −r−1/2∂q(W∂ qψ) for u = ∂vψ and integrating inward from the outer boundary avoids d...

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