---
title: Finite Carrollian Black-Hole Thermodynamics
url: https://www.emergentmind.com/topics/finite-carrollian-black-hole-thermodynamics
type: topic
---

# Finite Carrollian Black-Hole Thermodynamics

Searching arXiv for the main paper and closely related Carrollian black-hole thermodynamics work.
Finite Carrollian black-hole thermodynamics denotes the study of black-hole thermodynamic laws in an ultra-relativistic \(c\to 0\) regime where the relevant limit is taken not only on the metric but on the full thermodynamic phase space. In the formulation developed for Schwarzschild–AdS and related AdS black holes, the Carroll limit sends the timelike generator toward a zero-norm direction, and the ordinary first law degenerates unless the time generator and Newton’s constant are co-scaled so that the thermodynamic products remain finite [2604.27449]. In parallel, an intrinsically Carrollian line of work defines Carroll black holes as C-thermal states with a Carroll extremal surface and finite entropy, with energy, temperature, entropy, and specific heat determined directly in Carroll gravity and 2D Carroll dilaton models [2308.10947]. A complementary near-horizon viewpoint identifies black-hole horizons as Carrollian manifolds whose dynamics obey Carrollian conservation laws and whose finite horizon variables emerge from an ultra-relativistic limit of stretched-horizon data [1903.09654]. Together these approaches establish that Carrollian black-hole thermodynamics is not a single formalism but a family of related frameworks: finite first-law contractions in AdS, intrinsic Carroll black-hole thermodynamics, horizon Carrollian fluids, and near-horizon Carrollian matter probes.

## 1. Carrollian limit, extremal surfaces, and horizon geometry

The Carrollian limit is the ultra-relativistic contraction \(c\to 0\). For Schwarzschild–AdS in four dimensions, the metric can be written in a “Carroll scaling” coordinate \(t\) as
\[
ds^2 = -c^2 f(r)\,dt^2 + \frac{dr^2}{f(r)} + r^2 d\Omega_2^2, \qquad f(r)=1-\frac{r_0}{r}+\frac{r^2}{\ell^2},
\]
with the finite Lorentzian clock \(\tau=ct\) recovering the standard Schwarzschild–AdS form [2604.27449]. In this description the static Killing generator is
\[
\xi_t=\partial_t,\qquad g(\xi_t,\xi_t)=-c^2 f(r)\xrightarrow[c\to 0]{}0,
\]
so Lorentzian time translation collapses to a zero-norm Carrollian direction [2604.27449]. This geometric degeneration is the basic kinematical input behind Carrollian thermodynamic contractions.

In intrinsic Carroll gravity, black holes are defined without reference to a Lorentzian event horizon. The relevant geometric locus is the Carroll extremal surface. In 2D Carroll dilaton gravity, Carroll extremal surfaces are loci in target space at which \(\,=0\), equivalently
\[
e^\mu\partial_\mu X = 0,\quad X>0,
\]
so the dilaton is extremal with respect to the spatial direction [2308.10947]. In higher-dimensional Carrollian black-hole constructions obtained from Schwarzschild-(A)dS and Schwarzschild-Bach-(A)dS, the Carroll extremal surface coincides with the zero of \(f(r)\),
\[
f(r_0)=0,
\]
and the strict Carroll limit freezes time evolution at this surface [2408.01836].

A different but closely related geometric formulation arises directly on the horizon. In null Gaussian coordinates \((v,\rho,x^A)\), the near-horizon metric
\[
ds^2=-2\kappa \rho\, dv^2+2\,d\rho\, dv + 2\theta_A \rho\, dv\, dx^A+(\Omega_{AB}+\lambda_{AB} \rho)\,dx^A dx^B+\mathcal{O}(\rho^2)
\]
induces on the horizon \(\rho=0\) the degenerate metric \(ds^2_{\mathcal H}=\Omega_{AB}dx^A dx^B\), and the radial coordinate plays the role of a virtual speed of light via
\[
c^2=\rho,\qquad \alpha=\sqrt{2\kappa},\qquad b_A=\frac{\theta_A}{\sqrt{2\kappa}},
\]
so the near-horizon limit is an ultra-relativistic Carrollian limit [1903.09654]. This establishes that the horizon itself is naturally a Carrollian manifold.

## 2. Covariant phase space and the extended first law

The AdS phase-space approach is built on the Iyer–Wald covariant phase-space formalism with variable cosmological constant. For four-dimensional Einstein gravity with
\[
\mathbf L = \frac{1}{16\pi G} \left(R-2\Lambda\right)\boldsymbol\epsilon,
\]
metric variations obey
\[
\delta \mathbf L = \mathbf E_g^{\mu\nu}\delta g_{\mu\nu} + d\boldsymbol\Theta(g,\delta g) + \mathbf E_\Lambda\,\delta\Lambda,\qquad \mathbf E_\Lambda = -\frac{1}{8\pi G}\boldsymbol\epsilon.
\]
For a Killing field \(\xi\), the Iyer–Wald surface form \(\boldsymbol\chi_\xi=\delta\mathbf Q_\xi-\xi\cdot\boldsymbol\Theta\) is no longer closed when \(\Lambda\) varies:
\[
d\boldsymbol\chi_\xi = \frac{\delta\Lambda}{8\pi G}\,\xi\cdot\boldsymbol\epsilon.
\]
Integrating over a spacelike slice and defining
\[
P=-\frac{\Lambda}{8\pi G},\qquad V_\xi=-\int_\Sigma^{\rm ren}\xi\cdot\boldsymbol\epsilon,
\]
gives the extended Iyer–Wald identity
\[
\delta H_\xi = T_\xi\,\delta S+V_\xi\,\delta P.
\]
The renormalized bulk term proportional to \(\delta\Lambda\) is thereby identified with the generator-normalized thermodynamic volume contribution \(V_\xi\,\delta P\) [2604.27449].

For Schwarzschild–AdS with generator \(\xi_t=\partial_t\), the thermodynamic quantities are
\[
H_t = \frac{c\,r_0}{2G} = \frac{c}{2G}\Bigl(r_h+\frac{r_h^3}{\ell^2}\Bigr),
\]
\[
T_t = \frac{c}{4\pi}\Bigl(\frac{1}{r_h}+\frac{3r_h}{\ell^2}\Bigr),\qquad S=\frac{\pi r_h^2}{G},
\]
\[
V_t=\frac{4\pi c r_h^3}{3},\qquad P=\frac{3}{8\pi G\ell^2},
\]
and they satisfy
\[
\delta H_t = T_t\,\delta S + V_t\,\delta P
\]
[2604.27449]. A central structural fact is that \(H_\xi\), \(T_\xi\), and \(V_\xi\) scale linearly with the normalization of \(\xi\). This generator dependence is crucial because the Carroll contraction rescales the thermal generator itself.

The same phase-space logic extends beyond neutral Schwarzschild–AdS. For fixed-charge Reissner–Nordström–AdS,
\[
\delta H_\tau = T_\tau\delta S + V_\tau\delta P + \Phi_\tau\delta Q,
\]
while for fixed-rotation Kerr–AdS,
\[
\delta H_\tau = T_\tau\delta S + \Omega_\tau\delta J + V_\tau\delta P.
\]
In both cases the work terms scale homogeneously with the neutral sector under Carrollian contraction [2604.27449].

## 3. Phase-space contraction and the condition for finiteness

If one keeps \(\xi_t=\partial_t\) and \(G\) fixed and simply sends \(c\to 0\), then
\[
H_t\propto c\to 0,\qquad T_t\propto c\to 0,\qquad V_t\propto c\to 0,
\]
while the entropy remains finite:
\[
S=\frac{\pi r_h^2}{G}.
\]
The first law therefore contracts to
\[
0=0.
\]
In this sense, the Carroll limit contracts the full thermodynamic phase space together with the metric, producing a degenerate sector with vanishing Hamiltonian variation, temperature, and volume [2604.27449].

To obtain a finite, nontrivial limit, the time generator and Newton’s constant are rescaled as
\[
\xi_\lambda = c^{-\alpha}\partial_t,\qquad G=c^\gamma G_C.
\]
Because the relevant charges are linear in the generator, one finds
\[
H_\lambda\sim c^{1-\alpha-\gamma},\qquad T_\lambda\sim c^{1-\alpha},\qquad S\sim c^{-\gamma},
\]
\[
V_\lambda\sim c^{1-\alpha},\qquad P\sim c^{-\gamma},
\]
so all first-law terms scale homogeneously:
\[
\delta H_\lambda\sim T_\lambda\delta S\sim V_\lambda\delta P\sim c^{1-\alpha-\gamma}.
\]
Finite phase-space contractions therefore require
\[
\boxed{\alpha+\gamma=1.}
\]
This is the central scaling result of the AdS phase-space construction [2604.27449].

The endpoint \((\alpha,\gamma)=(1,0)\) corresponds to
\[
\xi_C=c^{-1}\partial_t=\partial_\tau,
\]
with \(G=G_C\) fixed. This yields the ordinary non-degenerate Lorentzian finite-clock normalization, not a Carrollian geometry, because
\[
g(\xi_C,\xi_C)=-f(r)
\]
stays finite [2604.27449]. Genuine Carrollian finite first laws lie on the segment \(\alpha<1\), with \(\gamma=1-\alpha>0\).

A particularly simple representative is the strong-gravity Carroll point
\[
(\alpha,\gamma)=(0,1),\qquad \xi_\lambda=\partial_t,\qquad G=cG_C.
\]
There
\[
H_t=\frac{1}{2G_C}\Bigl(r_h+\frac{r_h^3}{\ell^2}\Bigr)\ \text{finite},
\]
while
\[
T_t\to0,\qquad S\sim \frac{1}{c}\to\infty,\qquad P\sim\frac{1}{c}\to\infty,\qquad V_t\sim c\to0,
\]
yet the products in the first law remain finite:
\[
T_t\delta S = \frac{1}{2G_C}\Bigl(1+\frac{3r_h^2}{\ell^2}\Bigr)\delta r_h,\qquad
V_t\delta P = -\frac{r_h^3}{G_C\ell^3}\delta\ell.
\]
This explicitly realizes finite Carrollian black-hole thermodynamics as a competition between vanishing intensive quantities and divergent extensive ones [2604.27449].

## 4. Thermodynamic regimes: zero temperature, divergent entropy, and finite products

On the finite first-law line with \(\alpha<1\), the norm of the generator behaves as
\[
g(\xi_\lambda,\xi_\lambda)=-c^{2(1-\alpha)}f(r)\to0,
\]
so the geometry is Carrollian. At the same time
\[
T_\lambda\sim c^{1-\alpha},\qquad S\sim c^{-(1-\alpha)},
\]
hence
\[
T_\lambda\to0,\qquad S\to\infty,\qquad T_\lambda\delta S=O(1).
\]
Similarly,
\[
P\sim c^{-(1-\alpha)},\qquad V_\lambda\sim c^{1-\alpha},\qquad V_\lambda\delta P=O(1),
\]
and \(\delta H_\lambda\) stays finite [2604.27449]. This regime is characterized by Carrollian geometry, strong-gravity scaling \(G\to0\), zero temperature, divergent entropy, and finite thermodynamic response.

An apparently different but conceptually related picture emerges in intrinsic Carroll black holes defined directly in Carroll gravity. There the energy is finite and the temperature is finite at the level of the Carroll theory, with entropy determined by the dilaton at the Carroll extremal surface:
\[
E=\frac{k}{2\pi}M,\qquad
T=\frac{w'(X_{\rm min})}{2\pi},\qquad
S=kX_{\rm min},\qquad
\delta E=T\,\delta S
\]
[2308.10947]. In this framework, Carroll black holes are C-thermal states with finite entropy that have a Carroll extremal surface, and the first law is derived by a Noether–Wald identity between the asymptotic boundary and the extremal surface [2308.10947].

Examples illustrate that the specific heat can be finite or divergent depending on the model. For Carroll JT,
\[
U_{\rm CJT}(X)=0,\qquad V_{\rm CJT}(X)=\frac{X}{\ell^2},\qquad
w(X)=\frac{X^2}{2\ell^2},
\]
and
\[
E = \frac{k}{2\pi}M,\qquad
T = \frac{\sqrt{2M}}{2\pi\ell},\qquad
S = k\ell\sqrt{2M},\qquad
C=S
\]
[2308.10947]. By contrast, Carroll CGHS and Carroll Witten black holes have constant temperature and infinite specific heat because \(w''(X_{\rm min})=0\) [2308.10947].

A stricter Carroll limit based on Schwarzschild-(A)dS and Schwarzschild-Bach-(A)dS yields yet another regime. There the temperature scales linearly with \(c\),
\[
T\sim c\to0,
\]
the leading entropy behaves as
\[
\mathscr{S}^{(1)} = \frac{A_H k}{4 c \hbar G_M},
\]
and therefore
\[
\mathscr{S}\sim \frac{1}{c}\to\infty,
\]
while the energy remains finite [2408.01836]. In this formulation the specific heat diverges as \(1/c\) and can be positive, negative, or zero depending on parameters such as \(\Lambda\), \(r_0\), and higher-derivative couplings [2408.01836]. The authors argue that Carroll black holes then behave as an incompressible thermodynamical system with divergent entropy when the temperature goes to zero [2408.01836].

These different regimes are not identical, but they are compatible at the level of scaling logic. The AdS phase-space construction isolates finite first-law combinations; intrinsic Carroll dilaton gravity provides finite \(E\), \(T\), and \(S\) in a fully Carrollian theory; strict Carroll limits of Lorentzian black holes can instead produce \(T\to0\) and \(S\to\infty\). A plausible implication is that “finite Carrollian thermodynamics” is framework-dependent: finiteness may attach either to individual state variables or only to the thermodynamic combinations entering the first law.

## 5. Extensions: charge, rotation, higher dimensions, and holographic interpretation

The phase-space contraction principle survives in charged, rotating, and higher-dimensional AdS black holes. For fixed-charge Reissner–Nordström–AdS, after \(\tau=ct\) and the rescaling \(\xi_\lambda=c^{-\alpha}\partial_t\), \(G=c^\gamma G_C\), one finds
\[
\delta H_\lambda \sim T_\lambda\delta S \sim V_\lambda\delta P \sim \Phi_\lambda\delta Q \sim c^{1-\alpha-\gamma},
\]
so the contracted first law
\[
\delta H_\lambda = T_\lambda\delta S + V_\lambda\delta P + \Phi_\lambda\delta Q
\]
is finite and nonzero iff \(\alpha+\gamma=1\) [2604.27449]. No new exponent appears as long as the geometric charge parameter \(q\) is held \(O(1)\) [2604.27449].

For fixed-rotation Kerr–AdS, one similarly obtains
\[
\delta H_\lambda \sim T_\lambda\delta S \sim \Omega_\lambda\delta J \sim V_\lambda\delta P \sim c^{1-\alpha-\gamma},
\]
so again
\[
\delta H_\lambda = T_\lambda\delta S+\Omega_\lambda\delta J+V_\lambda\delta P
\]
is finite only on \(\alpha+\gamma=1\) [2604.27449]. In the Carrollian regime \(\alpha<1\), the normalized angular velocity tends to zero,
\[
\Omega_\lambda\sim c^{1-\alpha}\to0,
\]
while the work term \(\Omega_\lambda\delta J\) can remain finite because \(\delta J\sim c^{-\gamma}\) compensates [2604.27449]. This aligns with the statement that stationary, axisymmetric Carroll black holes are effectively static [2604.27449].

In arbitrary spacetime dimension within the Schwarzschild–AdS family, the same scaling law persists. With
\[
\xi_\lambda=c^{-\alpha}\partial_t,\qquad G=c^\gamma G_C,
\]
the thermodynamic variables obey
\[
H_\lambda\sim c^{1-\alpha-\gamma},\qquad
T_\lambda\sim c^{1-\alpha},\qquad
S\sim c^{-\gamma},\qquad
V_\lambda\sim c^{1-\alpha},\qquad
P\sim c^{-\gamma},
\]
and hence
\[
\delta H_\lambda \sim T_\lambda\delta S \sim V_\lambda\delta P \sim c^{1-\alpha-\gamma},
\]
so the finite-first-law condition \(\alpha+\gamma=1\) is dimension-independent [2604.27449].

A boundary interpretation of this finite line has been developed as a double-scaled low-temperature, large-\(N\) ensemble. The same scaling \(G_{d+1}=c^\gamma G_C\) implies
\[
N_{\rm eff}\to\infty,\qquad T_C\sim c^\gamma\to0,\qquad T_C S_C=O(1),
\]
so the Carrollian temperature decreases while the effective number of boundary degrees of freedom grows, leaving the thermodynamic products finite [2606.26163]. In this picture the finite Brown–York energy equals the finite bulk Hamiltonian, and the finite first law is the thermal zero-mode sector of the Carrollian Ward identity [2606.26163]. The Hawking–Page locus is identified with the zero of the chemical potential conjugate to the count of degrees of freedom [2606.26163].

## 6. Horizon fluids, Carrollian Hawking effect, and open problems

At the horizon, Einstein’s equations themselves can be rewritten as Carrollian fluid equations. For a generic null surface \(\mathbf H\), the horizon variables satisfy the Raychaudhuri and Damour–Navier–Stokes equations
\[
\dot{\theta}^{(l)} - \kappa\theta^{(l)}  + \tfrac12(\theta^{(l)})^2 + N^{(l)}_{AB} N^{AB\,(l)} = 0,
\]
\[
\dot{\mathcal H}_A + \theta^{(l)}\mathcal H_A - \nabla_A\kappa - \tfrac12\nabla_A\theta^{(l)} + \nabla^B N^{(l)}_{AB} = 0,
\]
which match Carrollian hydrodynamic equations through the dictionary
\[
e=\theta^{(l)},\qquad p=-\kappa,\qquad
\Pi_{AB}=2\eta N^{(l)}_{AB}+\zeta\theta^{(l)}\Omega_{AB},
\]
with
\[
\eta=\tfrac12,\qquad \zeta=-\tfrac12
\]
[2212.06175]. In equilibrium the horizon fluid has vanishing energy density and constant negative pressure [2212.06175]. In dynamical settings, the Carrollian fluid describes finite-time relaxation, mode couplings, and teleological equilibration of the horizon [2212.06175]. This suggests that finite Carrollian black-hole thermodynamics is not limited to stationary first laws but extends to local, time-dependent horizon thermodynamics.

Semi-classical matter on Carroll black-hole backgrounds exhibits a Hawking-like effect. In the 2D Carroll–Schwarzschild background
\[
\tau=d\tilde t,\qquad e=dr,\qquad v=\partial_{\tilde t},\qquad f=1-\frac{r_s}{r},
\]
the Carroll extremal surface is at \(r=r_s\), and the Carroll temperature satisfies
\[
T^{-1}=4\pi r_s
\]
[2403.00073]. The asymptotic energy density of a conformal scalar in the unique regular Carroll Hartle–Hawking state is
\[
\lim_{r\to\infty}\langle \mathcal E\rangle = \frac{\pi}{6}T^2,
\]
precisely compatible with the 2D Stefan–Boltzmann law [2403.00073]. However, the Carroll Ward identities enforce
\[
\langle T_{++}\rangle=\langle T_{--}\rangle,
\]
so there is no net flux and no evaporation [2403.00073]. This corrects a common misconception: a Carrollian Hawking effect need not imply radiative mass loss. In the Carrollian setting one obtains a thermal energy density without an Unruh-like flux.

String probes near non-extremal horizons supply another perspective. Near-horizon Schwarzschild, Reissner–Nordström, and Kerr geometries admit string Carroll expansions, and the solution space of relativistic strings bifurcates into magnetic and electric Carroll sectors [2407.12911]. Magnetic Carroll strings shrink to a point on the two-sphere and either follow null geodesics or form folded strings in the 2D Rindler spacetime, while electric Carroll strings wrap the two-sphere and follow a massive geodesic in the Rindler space [2407.12911]. This suggests that Carrollian horizon thermodynamics may admit a microscopic description in terms of Carrollian string sectors, although such a derivation is not yet available.

Several issues remain open. Intrinsic Carroll black-hole thermodynamics and phase-space-contracted AdS thermodynamics do not yet form a single unified formalism. The relation between finite entropies in 2D Carroll dilaton gravity and divergent entropies in strict Carroll limits of higher-dimensional black holes is unresolved. A microscopic derivation of the Carroll Hawking effect, backreaction in Carrollian semiclassical gravity, rotating higher-dimensional Carroll black holes, and a full entropy-current formulation for Carrollian horizon fluids are all explicitly identified as open directions [2403.00073] [2212.06175] [2308.10947].

Finite Carrollian black-hole thermodynamics therefore refers most precisely to the existence of controlled, nontrivial thermodynamic structures in Carrollian or Carroll-contracted black-hole systems. In the AdS phase-space framework, finiteness means that the first-law combinations remain \(O(1)\) under the correlated scaling \(\alpha+\gamma=1\) [2604.27449]. In intrinsic Carroll gravity, finiteness means that \(E\), \(T\), and \(S\) can be defined directly from Carrollian geometry and Noether charges [2308.10947]. At the horizon, finiteness means that the divergent stretched-horizon stress tensor reorganizes into finite Carrollian momenta and charges obeying Carrollian conservation laws [1903.09654]. These formulations differ in detail, but all support the same general conclusion: Carrollian black-hole thermodynamics is a genuine thermodynamic arena rather than a trivial zero-temperature collapse.

Source: https://www.emergentmind.com/topics/finite-carrollian-black-hole-thermodynamics