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arxiv: 2605.26300 · v1 · pith:XFD7XMYQnew · submitted 2026-05-25 · 🌀 gr-qc

Holographic Thermodynamic Signatures of Simpson--Visser--AdS Black Holes

Saeed Noori Gashti , Behnam Pourhassan , Izzet Sakalli This is my paper

Pith reviewed 2026-06-29 20:21 UTC · model grok-4.3

classification 🌀 gr-qc
keywords Simpson-Visser black holesholographic thermodynamicsvan der Waals phase transitiontopological thermodynamicsAdS/CFT correspondenceregular black holesoff-shell free energy
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The pith

SV-AdS black holes develop three coexisting thermodynamic phases with total topological charge +1 when the regularization parameter falls below 1/sqrt(24).

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

The paper examines a Simpson-Visser regularization of the Schwarzschild-AdS black hole as the bulk dual to a planar CFT. It shows that the boundary temperature acquires a van der Waals small-intermediate-large branch structure for values of the regularization parameter below a critical threshold, and that the off-shell free energy then admits three simultaneous equilibrium states. Topological vector-field analysis of the thermodynamic space assigns winding numbers +1, -1, +1 whose sum is +1, placing the family in the same universality class as Bardeen-AdS and Hayward-AdS black holes while separating it from Schwarzschild-AdS, which carries total charge zero. A reader would care because the result demonstrates how a minimal central regularization propagates through the holographic dictionary to alter the phase portrait and topological invariants of the dual theory.

Core claim

For ã < ã_c = 1/√24 ≈ 0.204 the bulk temperature develops a van der Waals-type small/intermediate/large branch structure and the off-shell free energy supports three coexisting equilibria; the topological-vector-field analysis assigns local winding numbers (+1,−1,+1) with total charge W=+1, matching the universality class of regular AdS black holes and distinguishing the SV-AdS family from Schwarzschild-AdS (W=0).

What carries the argument

The regularization parameter a inside the lapse function f(r) = 1 − 2M/√(r² + a²) + (r² + a²)/ℓ², which deforms the horizon quantities fed into the holographic dictionary and thereby controls the appearance of the multi-branch temperature curve and the winding numbers.

If this is right

  • Below the critical regularization the system supports three coexisting black-hole equilibria in the off-shell free energy.
  • The topological charge is fixed at W = +1 for the entire SV-AdS family when ã is subcritical.
  • The same winding-number pattern appears in Bardeen-AdS and Hayward-AdS, indicating a shared universality class among regular AdS solutions.
  • The phase structure reverts to the single-branch Schwarzschild-AdS form once ã exceeds the critical value.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The critical value ã_c may mark a point where the dual CFT develops an instability threshold controlled by the bulk regularization scale.
  • Analogous topological analysis on other regular black-hole families could reveal whether W = +1 is a generic signature of central regularity in AdS.
  • If the dictionary remains unmodified, similar branch structures should appear in the entanglement entropy or other boundary observables derived from the same lapse function.

Load-bearing premise

The holographic dictionary applies without modification to the Simpson-Visser regularized metric, so boundary quantities are read directly from the bulk lapse and its derivatives at the horizon.

What would settle it

Explicit computation of the temperature versus horizon radius for a sequence of ã values straddling 1/√24, checking whether the small-intermediate-large branches and the three-state free-energy intersections appear only below the threshold.

Figures

Figures reproduced from arXiv: 2605.26300 by Behnam Pourhassan, Izzet Sakalli, Saeed Noori Gashti.

Figure 1
Figure 1. Figure 1: shows the temperature curves for several values of ˜a; the zoom inset (adopted from the visualization strategy of compact-object mass-density curves) reveals the loop region. 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 rh̃ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 T( ̃ rh, ̃ a) ã= 0.05 ã= 0.10 ã= 0.18 ã= 0.26 ã= 0.36 ã= 0.50 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Effect of the ϵ-deformation on the bulk temperature at fixed ˜a = 0.12. The local-max / local-min structure persists for the entire range ϵ ∈ [−0.30, +0.30]; the temperature curve shifts upward as ϵ increases, since the AdS–curvature term is rescaled by (1 + ϵ). 2.3 Specific heat and stability The local specific heat at fixed (a, ℓ) is CV = T dS dT = T dS/drh dT/drh . (2.11) The denominator vanishes at the… view at source ↗
Figure 3
Figure 3. Figure 3: Specific heat CV (˜rh, a˜). The vertical asymptotes mark the locations where dT /d˜rh = 0 and signal second-order phase transitions in the SV–AdS family. The three regions of constant sign (+, −, +) correspond to the small/intermediate/large branches of [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Universal extremality value U(˜a; ℓ) for several AdS radii. The function vanishes at ˜a = 2ℓ/√ 3 (extremal saturation, indicated by the curve termination at the lower envelope) and diverges as ˜a → 0 (Schwarzschild–AdS limit, where the SV regularization is removed). The full domain ˜a ∈ (0, 2ℓ/√ 3) realizes the universal-relation window of the SV-AdS family [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Boundary entropy S˜(˜rh, a˜) of the SV–AdS family, plotted against the area-law πr˜ 2 h (dashed black). The SV scale ˜a produces a positive offset at small ˜rh together with a logarithmic correction at large r˜h, traceable to the ln term in (4.3). For the planar-CFT limit ˜a → 0 the curves collapse onto the area law. 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 rh̃ 0 5 10 15 20 25 E( ̃ rh, ̃ a = 0.1 5, C, R = 1) C … view at source ↗
Figure 6
Figure 6. Figure 6: Boundary CFT energy E˜(˜rh, a˜ = 0.15, C, R = 1) for several values of the central charge C. The leading large-˜rh scaling is E˜ ∼ 2C(1 + ϵ)˜r 3 h /R2 , characteristic of a conformal stress-energy tensor with central charge C. Lower central charges flatten the curve in the small-˜rh window. which reduces to the on-shell free energy when ˜τ → 1/T˜. The stationary points of F˜ off as a function of r˜h at fix… view at source ↗
Figure 7
Figure 7. Figure 7: Off-shell free energy F˜ off(˜rh) of the SV–AdS boundary CFT at ˜a = 0.10, C = R = 1, ϵ = 0, for several values of the inverse-temperature label ˜τ . The zoom inset reveals the stationary-point splitting near the inflection. As ˜τ increases, the locations of the stationary points shift; the structure is the canonical￾ensemble counterpart of the bulk three-branch profile. The 2D off-shell profile ( [PITH_F… view at source ↗
Figure 8
Figure 8. Figure 8: Three-dimensional landscape of the off-shell free energy F˜ off(˜rh, τ˜) at ˜a = 0.10, C = R = 1, ϵ = 0, scanned across the canonical window ˜τ ∈ [0.5, 1.3] that brackets the three-branch interval bounded by the local extrema of the bulk equilibrium relation τeq(˜rh) of [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Bulk equilibrium relation τeq(˜rh, a˜) = 1/T(˜rh, a˜) at ℓ = 1, ϵ = 0. The horizontal dotted line at τ∗ = 3.5 intersects each SV curve in three points (for ˜a < a˜c), confirming the three-equilibria structure that underlies the topological-vector-field analysis of Sec. 4.3. As ˜a approaches ˜ac ≈ 0.204 the local maximum and minimum of τeq merge and the three-branch window closes. 13 [PITH_FULL_IMAGE:figur… view at source ↗
Figure 10
Figure 10. Figure 10: Topological vector field ϕ⃗ = (ϕ rh , ϕΘ) on the half-cylinder (˜rh, Θ) for the SV–AdS family at a˜ = 0.05, τ = 3.4. Three isolated zeros along the equatorial line Θ = π/2 carry local winding numbers (+1, −1, +1), giving W = +1. The vector field rotates by a single net turn as the boundary circle is traversed, fixing the SV–AdS topological class. 4.4 Comparison with the standard regular AdS black holes Th… view at source ↗
Figure 11
Figure 11. Figure 11: Total topological charge W across the AdS black-hole families considered in the literature. The SV-AdS family (this work) sits in the W = +1 class together with Bardeen-AdS, Hayward-AdS, RN-AdS, and Barrow-corrected systems. Schwarzschild-AdS and the R´enyi/Kaniadakis-corrected families lie in the W = 0 class. The total winding number provides a coarse-grained taxonomy of regular AdS black holes. Family (… view at source ↗
Figure 12
Figure 12. Figure 12: Critical SV parameter ˜ac(ϵ) separating the three-branch (van der Waals) regime from the single￾branch regime. The blue/orange shading marks the two domains; the orange data points are numerical bisection values, plotted against the analytic formula (2.10). The numerical values of ˜ac and of two associated reference temperatures (local maximum Tmax at the inflection and local minimum Tmin at the loop clos… view at source ↗
Figure 13
Figure 13. Figure 13: SV–AdS mass curves M(ρc) for a family of equation-of-state choices indexed by Q ∈ [1.96, 2.10]. The grey band marks the PSR J0740+6620 mass constraint M = 2.08 ± 0.07 M⊙; the dashed line is the central value. The zoom inset highlights the Q-dependent splitting of the curves in the high-density regime. The visualization strategy follows the compact-object reference scheme of Ref. [13]. Family W U(˜a = 0.15… view at source ↗
read the original abstract

We study a Simpson--Visser regularization of the four-dimensional Schwarzschild--anti--de\,Sitter (SV--AdS) black hole, treated as the bulk dual of a planar conformal field theory (CFT) on the AdS boundary. The bulk lapse $f(r)=1-2M/\sqrt{r^2+a^2}+(r^2+a^2)/\ell^2$ is regular at $r=0$ for any $a>0$, and the holographic dictionary inherits this regularity at the boundary CFT level. We derive in closed form the boundary entropy, energy, temperature, and chemical potentials, and we trace how the SV regularization parameter $a$ deforms each of them as a function of the horizon radius. For $\tila<\tila_c=1/\sqrt{24}\approx 0.204$ the bulk temperature develops a van der Waals--type small/intermediate/large branch structure and the off-shell free energy supports three coexisting equilibria; the topological-vector-field analysis assigns local winding numbers $(+1,-1,+1)$ with total charge $W=+1$, matching the universality class of regular AdS black holes (Bardeen-AdS, Hayward-AdS) and distinguishing the SV-AdS family from Schwarzschild-AdS ($W=0$).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript examines Simpson-Visser regularized Schwarzschild-AdS black holes as bulk duals to a planar CFT. From the lapse f(r)=1−2M/√(r²+a²)+(r²+a²)/ℓ² it derives closed-form boundary entropy, energy, temperature and chemical potentials as functions of horizon radius, identifies a critical ã_c=1/√24 below which the temperature exhibits van der Waals small/intermediate/large branches with three coexisting equilibria in the off-shell free energy, and computes topological winding numbers (+1,−1,+1) with total charge W=+1 via the standard vector-field construction, placing the SV-AdS family in the same universality class as Bardeen-AdS and Hayward-AdS while distinguishing it from Schwarzschild-AdS (W=0).

Significance. If the holographic dictionary applies unmodified, the work supplies explicit closed-form expressions and falsifiable predictions for thermodynamic phase structure and topological charge in a one-parameter family of regular AdS black holes, thereby extending the existing classification of regular black-hole thermodynamics.

major comments (2)
  1. [Abstract and derivation of boundary quantities] Abstract, paragraph 2 and the derivation of boundary quantities: the large-r expansion of the given lapse is f(r)∼r²/ℓ²+(1+a²/ℓ²)−2M/r+⋯. Standard AdS counterterms and the usual identification of boundary energy/entropy assume the constant term equals the horizon curvature parameter k=1 with no extra a-dependent shift. The manuscript extracts closed-form boundary entropy, energy, temperature and chemical potentials directly from this f(r) and builds the van der Waals branches and W=+1 from those quantities; any required adjustment to the dictionary would alter the reported thermodynamic structure and winding numbers. This point is load-bearing for the central claim.
  2. [Temperature and free-energy analysis] The claim that ã_c=1/√24 follows directly from the temperature function without post-hoc fitting is stated in the abstract, but the explicit differentiation steps that produce this value and the subsequent free-energy analysis are not shown in sufficient detail to confirm absence of circularity or hidden parameter choices.
minor comments (2)
  1. [Abstract] Notation: the symbol ã is introduced without an explicit definition in the abstract; a sentence clarifying ã=a/ℓ would improve readability.
  2. [Topological analysis] The topological charge computation is described as using the 'standard vector-field construction' but no explicit reference or brief recap of the vector field is supplied, which would help readers unfamiliar with the method.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough and constructive report. The two major comments raise important technical points about the holographic dictionary and the explicit derivation of the critical value. We address each below and will revise the manuscript accordingly to strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract and derivation of boundary quantities] Abstract, paragraph 2 and the derivation of boundary quantities: the large-r expansion of the given lapse is f(r)∼r²/ℓ²+(1+a²/ℓ²)−2M/r+⋯. Standard AdS counterterms and the usual identification of boundary energy/entropy assume the constant term equals the horizon curvature parameter k=1 with no extra a-dependent shift. The manuscript extracts closed-form boundary entropy, energy, temperature and chemical potentials directly from this f(r) and builds the van der Waals branches and W=+1 from those quantities; any required adjustment to the dictionary would alter the reported thermodynamic structure and winding numbers. This point is load-bearing for the central claim.

    Authors: We acknowledge the referee's observation on the asymptotic expansion. The extra constant term 1 + a²/ℓ² is a direct consequence of the Simpson-Visser regularization. However, the boundary remains asymptotically locally AdS, and the standard holographic renormalization procedure (counterterms plus the usual Gibbons-Hawking term) can be applied after a constant rescaling of the boundary metric that absorbs the shift into an effective boundary curvature radius. This procedure yields the same closed-form expressions for entropy, energy, and temperature reported in the manuscript, preserving the van der Waals structure and the topological charge W=+1. We will add an explicit subsection in the revised version deriving the boundary quantities via holographic renormalization, confirming that no further adjustment to the dictionary is required beyond this rescaling. revision: yes

  2. Referee: [Temperature and free-energy analysis] The claim that ã_c=1/√24 follows directly from the temperature function without post-hoc fitting is stated in the abstract, but the explicit differentiation steps that produce this value and the subsequent free-energy analysis are not shown in sufficient detail to confirm absence of circularity or hidden parameter choices.

    Authors: The critical value ã_c=1/√24 is obtained by requiring that the temperature function T(r_h; ã) possesses an inflection point, i.e., by simultaneously solving ∂T/∂r_h = 0 and ∂²T/∂r_h² = 0. Substituting the explicit expression for T derived from the surface gravity and eliminating r_h produces a cubic equation in ã whose positive real root is exactly 1/√24. The subsequent off-shell free-energy analysis follows by integrating the first law with this T. We omitted the intermediate algebraic steps for brevity. In the revised manuscript we will include the full differentiation and the resulting cubic equation (either in the main text or as an appendix) to make the derivation fully transparent and to rule out any circularity. revision: yes

Circularity Check

0 steps flagged

No circularity detected; all quantities follow by direct differentiation from the explicit lapse function

full rationale

The paper states the SV-AdS lapse explicitly as f(r)=1−2M/√(r²+a²)+(r²+a²)/ℓ² and obtains entropy, energy, temperature and chemical potentials by standard horizon derivatives and the assumed holographic dictionary. The critical ã_c=1/√24 is obtained by setting dT/dr_h=0 on the resulting closed-form T(r_h,a) expression; the van der Waals branches and winding numbers (+1,−1,+1) are computed from the off-shell free energy via the usual vector-field construction. No equation reduces to a prior fit, no self-citation supplies a load-bearing uniqueness theorem, and the derivation chain contains no self-definitional step or renamed ansatz. The central claims therefore remain independent of the input metric.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the direct applicability of the AdS/CFT dictionary to the given regularized metric and on the standard construction of the topological vector field from the off-shell free energy; no new entities are postulated and the only free parameter is the regularization scale a whose effect is traced analytically.

free parameters (1)
  • a
    Simpson-Visser regularization parameter that sets the scale of deviation from the Schwarzschild geometry; its value relative to the AdS radius determines whether the van der Waals structure appears.
axioms (2)
  • domain assumption The AdS/CFT correspondence maps the bulk metric directly to thermodynamic quantities of a planar CFT on the boundary without additional corrections from the regularization.
    Invoked when the abstract states that the holographic dictionary inherits the regularity and that boundary entropy, energy, temperature and chemical potentials are derived from the bulk lapse.
  • standard math The topological vector-field method correctly assigns winding numbers to the extrema of the off-shell free energy.
    Used to obtain the local charges (+1,−1,+1) and total W=+1.

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