arxiv: 2604.27449 · v2 · submitted 2026-04-30 · ✦ hep-th

Recognition: unknown

Phase-Space Contractions of Carrollian Black-Hole Thermodynamics

Yingnan Xu , Shuangshuang Chu

Authors on Pith no claims yet

Pith reviewed 2026-05-08 03:13 UTC · model grok-4.3

classification ✦ hep-th

keywords Carrollian limitblack hole thermodynamicscovariant phase spaceSchwarzschild-AdSextended first lawphase-space contractionultra-relativistic gravity

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The pith

Finite Carrollian black-hole thermodynamics requires the time-rescaling exponent alpha plus the Newton-constant exponent gamma to equal one.

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

The paper shows that the ultra-relativistic Carrollian limit of Schwarzschild-AdS thermodynamics can be taken without destroying the first law only when the time vector and Newton constant are rescaled together so that their exponents sum to one. Under this condition the phase space contracts, temperature goes to zero, entropy diverges, yet the products T delta S and the renormalized volume-pressure term stay finite. A reader cares because this supplies an explicit map from ordinary relativistic black-hole thermodynamics to a consistent Carrollian version that preserves thermodynamic relations. The construction is tested on charged and rotating cases and holds in any dimension.

Core claim

By allowing the cosmological constant to vary and using the covariant phase-space formalism, the extended Iyer-Wald identity identifies the renormalized bulk term proportional to delta Lambda with the generator-normalized thermodynamic volume V_xi delta P. Introducing the rescaled generator xi_lambda = c^{-alpha} partial_t together with G = c^gamma G_C makes the first-law variation scale as c^{1-alpha-gamma}. Finite non-degenerate contractions therefore demand alpha + gamma = 1. The Carrollian segment alpha < 1 (hence gamma = 1 - alpha > 0) produces T to 0, S to infinity, while T delta S and V_xi delta P remain finite; the opposite endpoint recovers ordinary Lorentzian normalization.

What carries the argument

The paired rescalings xi_lambda = c^{-alpha} partial_t and G = c^gamma G_C that contract the full thermodynamic phase space while keeping the first-law variation finite precisely when alpha + gamma equals one.

If this is right

The endpoint (alpha, gamma) = (1, 0) recovers the standard Lorentzian finite-clock normalization with non-vanishing temperature and volume.
All Carrollian points on the line alpha < 1 give vanishing temperature, diverging entropy, yet finite T delta S and V_xi delta P.
The same scaling relation holds for fixed-charge and fixed-rotation AdS black holes.
The construction extends to Schwarzschild-AdS black holes in arbitrary spacetime dimension.

Where Pith is reading between the lines

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

The infinite-entropy Carrollian states remain thermodynamically reachable through finite heat and work exchanges.
The contraction mechanism may supply a controlled route from AdS to flat-space or other ultra-relativistic gravitational limits.
Analog condensed-matter or holographic systems that realize c to 0 limits could be used to test whether the predicted finite T delta S survives.

Load-bearing premise

The Carrollian limit can be implemented by these specific rescalings of the time generator and Newton constant while the covariant phase-space formalism stays valid and the renormalized bulk term remains identified with the thermodynamic volume.

What would settle it

An explicit computation of the Iyer-Wald identity in the c to 0 limit with alpha + gamma not equal to one, showing that the Hamiltonian variation either diverges or vanishes identically instead of yielding a finite first law.

read the original abstract

We study Carrollian limits of Schwarzschild-AdS black-hole thermodynamics using covariant phase space. Allowing the cosmological constant to vary, we derive the extended Iyer-Wald identity and identify the renormalized bulk term proportional to $\delta\Lambda$ with the generator-normalized thermodynamic volume contribution $V_\xi\,\delta P$. We show that the Carroll limit contracts the full thermodynamic phase space together with the metric. For fixed Newton constant, the Lorentzian generator $\partial_t$ collapses to a zero-norm direction as $c\to0$, yielding a degenerate sector with vanishing Hamiltonian variation, temperature and volume. Introducing $\xi_\lambda=c^{-\alpha}\partial_t$ and $G=c^\gamma G_C$, we find that the extended first law scales as $c^{1-\alpha-\gamma}$, so finite phase-space contractions require $\alpha+\gamma=1$. The endpoint $(\alpha,\gamma)=(1,0)$, obtained by $\tau=ct$, is the ordinary non-degenerate Lorentzian finite-clock normalization. Carrollian finite first laws lie on the segment $\alpha<1$, hence $\gamma=1-\alpha>0$, and give $T\to0$, $S\to\infty$, with finite $T\,\delta S$ and $V_\xi\,\delta P$. We test the scaling principle for fixed-charge and fixed-rotation AdS black holes, and extend it to arbitrary spacetime dimension within the Schwarzschild-AdS family.

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.

Desk Editor's Note private letter to a colleague

The paper organizes Carrollian limits of extended black-hole thermodynamics by a scaling condition α + γ = 1 that keeps the first law finite.

read the letter

The main takeaway is that finite Carrollian contractions of the extended first law for Schwarzschild-AdS black holes require α + γ = 1 when you rescale the time generator as ξ_λ = c^{-α} ∂_t and Newton's constant as G = c^γ G_C. Carrollian cases (α < 1) then drive T to zero and S to infinity while T δS and V_ξ δP stay finite. The endpoint α = 1, γ = 0 recovers the usual Lorentzian normalization with finite clock rate. They derive this from the extended Iyer-Wald identity in covariant phase space, identify the renormalized bulk term proportional to δΛ with the thermodynamic volume contribution, and verify the scaling holds for fixed-charge and fixed-rotation cases plus arbitrary dimension within the Schwarzschild-AdS family. That scaling family is new in the literature they cite and gives a clean way to organize the contractions. The calculation is direct and they spell out how the phase space contracts together with the metric. The potential soft spot is whether the symplectic current, Noether charge, and renormalization procedure really commute with the limit once the metric becomes degenerate for α < 1. The abstract states that the bulk-term identification survives, but any extra boundary or degeneracy-induced pieces that only appear in the Carrollian regime could shift the scaling or the volume identification. They appear to have checked this by explicit rescaling, so the claim is not obviously circular, but it is the step that would need the closest look in review. This is niche work aimed at people already working on Carrollian gravity and flat-space holography. It does not resolve a long-standing puzzle but supplies a useful organizing principle with concrete checks. I would send it to peer review because the derivation is self-contained and the result is falsifiable within the formalism they use.

Referee Report

2 major / 2 minor

Summary. The paper uses covariant phase-space methods to derive an extended Iyer-Wald identity for Schwarzschild-AdS black holes with varying cosmological constant, identifying the renormalized bulk term ∝ δΛ with the generator-normalized thermodynamic volume V_ξ δP. It then implements Carrollian contractions via rescalings ξ_λ = c^{-α} ∂_t and G = c^γ G_C, showing that the extended first law scales as c^{1-α-γ} and remains finite only for α + γ = 1. For Carrollian cases (α < 1, γ = 1 - α > 0) this yields T → 0, S → ∞ with finite T δS and V_ξ δP; the scaling is tested for fixed-charge and fixed-rotation cases and extended to arbitrary dimensions.

Significance. If the extended Iyer-Wald identity and bulk-term identification survive the metric degeneracy of the Carrollian limit without extra contributions, the work supplies a controlled, scaling-based route to finite Carrollian black-hole thermodynamics from the Lorentzian theory. This could be relevant for Carrollian gravity and limits of AdS/CFT. The explicit scaling relation c^{1-α-γ} and the parameter choice α + γ = 1 constitute a clear, falsifiable organizing principle for the contractions.

major comments (2)

[Derivation of extended Iyer-Wald identity and Carrollian limit] The central claim that the renormalized bulk term ∝ δΛ continues to be identified with V_ξ δP after the Carrollian contraction (abstract and the derivation of the extended first law) rests on the assumption that the symplectic current, Noether charge, and renormalization procedure commute with the limit. For α < 1 the rescaled generator ξ_λ has vanishing norm and the volume form degenerates; it is not shown that no additional boundary or degeneracy-induced terms appear that would alter the scaling or the volume identification.
[Phase-space contractions and scaling analysis] The scaling of the extended first law as c^{1-α-γ} is obtained from the rescalings of ξ_λ and G. However, the paper must explicitly verify that the Hamiltonian variation remains non-vanishing and that the identification with V_ξ δP holds without additional renormalization adjustments when the Killing vector becomes null in the Carrollian sector.

minor comments (2)

Notation for the rescaled generator (ξ_λ) and the Carrollian Newton constant (G_C) should be introduced with a clear table or explicit definitions early in the text to avoid ambiguity when comparing Lorentzian and Carrollian cases.
The abstract states that the scaling principle is tested for fixed-charge and fixed-rotation AdS black holes; the corresponding explicit expressions for the first law in those cases would benefit from a short summary table.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The major comments correctly identify that the Carrollian limit requires explicit checks to confirm the absence of additional terms from degeneracy. We respond point by point below and will incorporate the requested verifications in a revised version.

read point-by-point responses

Referee: The central claim that the renormalized bulk term ∝ δΛ continues to be identified with V_ξ δP after the Carrollian contraction (abstract and the derivation of the extended first law) rests on the assumption that the symplectic current, Noether charge, and renormalization procedure commute with the limit. For α < 1 the rescaled generator ξ_λ has vanishing norm and the volume form degenerates; it is not shown that no additional boundary or degeneracy-induced terms appear that would alter the scaling or the volume identification.

Authors: We agree that the commutation of the limit with the phase-space constructions is assumed rather than explicitly verified in the current manuscript, and this constitutes a genuine gap given the metric degeneracy for α < 1. In the revision we will add an appendix that computes the symplectic current and Noether charge explicitly after the rescalings ξ_λ = c^{-α} ∂_t and G = c^γ G_C, demonstrating term-by-term that the degeneracy is compensated by the G rescaling and that no extra boundary or degeneracy-induced contributions appear in the extended Iyer-Wald identity. This will confirm that the bulk-term identification with V_ξ δP survives unchanged. revision: yes
Referee: The scaling of the extended first law as c^{1-α-γ} is obtained from the rescalings of ξ_λ and G. However, the paper must explicitly verify that the Hamiltonian variation remains non-vanishing and that the identification with V_ξ δP holds without additional renormalization adjustments when the Killing vector becomes null in the Carrollian sector.

Authors: We accept that the manuscript derives the overall scaling but does not supply an explicit check that δH stays non-vanishing and equals V_ξ δP once the generator is null. In the revised manuscript we will include direct evaluations of the Hamiltonian variation for the rescaled ξ_λ in the Carrollian regime (α < 1), both for the fixed-charge and fixed-rotation cases, showing that δH remains finite, matches V_ξ δP, and requires no further renormalization adjustments beyond the already-stated G rescaling. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is a scaling analysis under explicit rescalings

full rationale

The paper introduces explicit rescaling parameters α and γ for the Carrollian limit, applies the covariant phase-space formalism to derive the scaling of the extended first law as c^{1-α-γ}, and concludes that finiteness requires α+γ=1. This is a direct computation from the chosen ansatz rather than a reduction of the output to the input by construction. The identification of the renormalized bulk term ∝ δΛ with V_ξ δP follows from the standard extended Iyer-Wald identity in the presence of varying Λ and is not shown to be imposed rather than derived within the paper's equations. No self-citations are invoked as load-bearing premises, and the central result does not rename a known empirical pattern or fit a parameter to then predict a related quantity. The derivation chain remains self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The derivation rests on the validity of the extended Iyer-Wald identity when Lambda varies, the identification of the bulk term with V delta P after renormalization, and the assumption that the Carrollian limit is taken by power-law rescalings of the generator and G. No free parameters are fitted to data; the exponents alpha and gamma are chosen to keep the first law finite.

axioms (2)

domain assumption The extended Iyer-Wald identity holds when the cosmological constant is allowed to vary.
Invoked to derive the thermodynamic volume term.
ad hoc to paper The Carrollian limit is implemented by the rescalings xi_lambda = c^{-alpha} partial_t and G = c^gamma G_C.
Introduced to obtain finite contractions; the specific power-law form is chosen by the authors.

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