Finite Carrollian Black-Hole Thermodynamics

• The paper demonstrates that correlating the time generator and Newton’s constant scaling yields finite first-law thermodynamic products in Carrollian black-hole systems.
• It introduces multiple frameworks—such as Carrollian geometry at horizons, intrinsic 2D dilaton gravity, and AdS phase-space contractions—to capture the finite thermodynamics.
• It establishes the scaling condition (α + γ = 1) as central to preserving nontrivial thermodynamic behavior across diverse black-hole scenarios including charged, rotating, and higher-dimensional cases.

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 (Xu et al., 30 Apr 2026). 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 (Ecker et al., 2023). 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 (Donnay et al., 2019). 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 (Xu et al., 30 Apr 2026). 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 (Xu et al., 30 Apr 2026). 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 (Ecker et al., 2023). 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 (Tadros et al., 2024).

A different but closely related geometric formulation arises directly on the horizon. In null Gaussian coordinates $c\to 0$0, the near-horizon metric

$c\to 0$1

induces on the horizon $c\to 0$2 the degenerate metric $c\to 0$3, and the radial coordinate plays the role of a virtual speed of light via

$c\to 0$4

so the near-horizon limit is an ultra-relativistic Carrollian limit (Donnay et al., 2019). 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

$c\to 0$5

metric variations obey

$c\to 0$6

For a Killing field $c\to 0$7, the Iyer–Wald surface form $c\to 0$8 is no longer closed when $c\to 0$9 varies: $t$0 Integrating over a spacelike slice and defining

$t$1

gives the extended Iyer–Wald identity

$t$2

The renormalized bulk term proportional to $t$3 is thereby identified with the generator-normalized thermodynamic volume contribution $t$4 (Xu et al., 30 Apr 2026).

For Schwarzschild–AdS with generator $t$5, the thermodynamic quantities are

$t$6

$t$7

$t$8

and they satisfy

$t$9

(Xu et al., 30 Apr 2026). A central structural fact is that $$ds^2 = -c^2 f(r)\,dt^2 + \frac{dr^2}{f(r)} + r^2 d\Omega_2^2, \qquad f(r)=1-\frac{r_0}{r}+\frac{r^2}{\ell^2},$$0, $$ds^2 = -c^2 f(r)\,dt^2 + \frac{dr^2}{f(r)} + r^2 d\Omega_2^2, \qquad f(r)=1-\frac{r_0}{r}+\frac{r^2}{\ell^2},$$1, and $$ds^2 = -c^2 f(r)\,dt^2 + \frac{dr^2}{f(r)} + r^2 d\Omega_2^2, \qquad f(r)=1-\frac{r_0}{r}+\frac{r^2}{\ell^2},$$2 scale linearly with the normalization of $$ds^2 = -c^2 f(r)\,dt^2 + \frac{dr^2}{f(r)} + r^2 d\Omega_2^2, \qquad f(r)=1-\frac{r_0}{r}+\frac{r^2}{\ell^2},$$3. 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,

$$ds^2 = -c^2 f(r)\,dt^2 + \frac{dr^2}{f(r)} + r^2 d\Omega_2^2, \qquad f(r)=1-\frac{r_0}{r}+\frac{r^2}{\ell^2},$$4

while for fixed-rotation Kerr–AdS,

$$ds^2 = -c^2 f(r)\,dt^2 + \frac{dr^2}{f(r)} + r^2 d\Omega_2^2, \qquad f(r)=1-\frac{r_0}{r}+\frac{r^2}{\ell^2},$$5

In both cases the work terms scale homogeneously with the neutral sector under Carrollian contraction (Xu et al., 30 Apr 2026).

3. Phase-space contraction and the condition for finiteness

If one keeps $$ds^2 = -c^2 f(r)\,dt^2 + \frac{dr^2}{f(r)} + r^2 d\Omega_2^2, \qquad f(r)=1-\frac{r_0}{r}+\frac{r^2}{\ell^2},$$6 and $$ds^2 = -c^2 f(r)\,dt^2 + \frac{dr^2}{f(r)} + r^2 d\Omega_2^2, \qquad f(r)=1-\frac{r_0}{r}+\frac{r^2}{\ell^2},$$7 fixed and simply sends $$ds^2 = -c^2 f(r)\,dt^2 + \frac{dr^2}{f(r)} + r^2 d\Omega_2^2, \qquad f(r)=1-\frac{r_0}{r}+\frac{r^2}{\ell^2},$$8, then

$$ds^2 = -c^2 f(r)\,dt^2 + \frac{dr^2}{f(r)} + r^2 d\Omega_2^2, \qquad f(r)=1-\frac{r_0}{r}+\frac{r^2}{\ell^2},$$9

while the entropy remains finite: $\tau=ct$0 The first law therefore contracts to

$\tau=ct$1

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 (Xu et al., 30 Apr 2026).

To obtain a finite, nontrivial limit, the time generator and Newton’s constant are rescaled as

$\tau=ct$2

Because the relevant charges are linear in the generator, one finds

$\tau=ct$3

$\tau=ct$4

so all first-law terms scale homogeneously: $\tau=ct$5 Finite phase-space contractions therefore require

$\tau=ct$6

This is the central scaling result of the AdS phase-space construction (Xu et al., 30 Apr 2026).

The endpoint $\tau=ct$7 corresponds to

$\tau=ct$8

with $\tau=ct$9 fixed. This yields the ordinary non-degenerate Lorentzian finite-clock normalization, not a Carrollian geometry, because

$$\xi_t=\partial_t,\qquad g(\xi_t,\xi_t)=-c^2 f(r)\xrightarrow[c\to 0]{}0,$$0

stays finite (Xu et al., 30 Apr 2026). Genuine Carrollian finite first laws lie on the segment $$\xi_t=\partial_t,\qquad g(\xi_t,\xi_t)=-c^2 f(r)\xrightarrow[c\to 0]{}0,$$1, with $$\xi_t=\partial_t,\qquad g(\xi_t,\xi_t)=-c^2 f(r)\xrightarrow[c\to 0]{}0,$$2.

A particularly simple representative is the strong-gravity Carroll point

$$\xi_t=\partial_t,\qquad g(\xi_t,\xi_t)=-c^2 f(r)\xrightarrow[c\to 0]{}0,$$3

There

$$\xi_t=\partial_t,\qquad g(\xi_t,\xi_t)=-c^2 f(r)\xrightarrow[c\to 0]{}0,$$4

while

$$\xi_t=\partial_t,\qquad g(\xi_t,\xi_t)=-c^2 f(r)\xrightarrow[c\to 0]{}0,$$5

yet the products in the first law remain finite: $$\xi_t=\partial_t,\qquad g(\xi_t,\xi_t)=-c^2 f(r)\xrightarrow[c\to 0]{}0,$$6 This explicitly realizes finite Carrollian black-hole thermodynamics as a competition between vanishing intensive quantities and divergent extensive ones (Xu et al., 30 Apr 2026).

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

On the finite first-law line with $$\xi_t=\partial_t,\qquad g(\xi_t,\xi_t)=-c^2 f(r)\xrightarrow[c\to 0]{}0,$$7, the norm of the generator behaves as

$$\xi_t=\partial_t,\qquad g(\xi_t,\xi_t)=-c^2 f(r)\xrightarrow[c\to 0]{}0,$$8

so the geometry is Carrollian. At the same time

$$\xi_t=\partial_t,\qquad g(\xi_t,\xi_t)=-c^2 f(r)\xrightarrow[c\to 0]{}0,$$9

hence

$\,=0$0

Similarly,

$\,=0$1

and $\,=0$2 stays finite (Xu et al., 30 Apr 2026). This regime is characterized by Carrollian geometry, strong-gravity scaling $\,=0$3, 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: $\,=0$4 (Ecker et al., 2023). 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 (Ecker et al., 2023).

Examples illustrate that the specific heat can be finite or divergent depending on the model. For Carroll JT,

$\,=0$5

and

$\,=0$6

(Ecker et al., 2023). By contrast, Carroll CGHS and Carroll Witten black holes have constant temperature and infinite specific heat because $\,=0$7 (Ecker et al., 2023).

A stricter Carroll limit based on Schwarzschild-(A)dS and Schwarzschild-Bach-(A)dS yields yet another regime. There the temperature scales linearly with $\,=0$8,

$\,=0$9

the leading entropy behaves as

$e^\mu\partial_\mu X = 0,\quad X>0,$0

and therefore

$e^\mu\partial_\mu X = 0,\quad X>0,$1

while the energy remains finite (Tadros et al., 2024). In this formulation the specific heat diverges as $e^\mu\partial_\mu X = 0,\quad X>0,$2 and can be positive, negative, or zero depending on parameters such as $e^\mu\partial_\mu X = 0,\quad X>0,$3, $e^\mu\partial_\mu X = 0,\quad X>0,$4, and higher-derivative couplings (Tadros et al., 2024). The authors argue that Carroll black holes then behave as an incompressible thermodynamical system with divergent entropy when the temperature goes to zero (Tadros et al., 2024).

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^\mu\partial_\mu X = 0,\quad X>0,$5, $e^\mu\partial_\mu X = 0,\quad X>0,$6, and $e^\mu\partial_\mu X = 0,\quad X>0,$7 in a fully Carrollian theory; strict Carroll limits of Lorentzian black holes can instead produce $e^\mu\partial_\mu X = 0,\quad X>0,$8 and $e^\mu\partial_\mu X = 0,\quad X>0,$9. 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 $f(r)$0 and the rescaling $f(r)$1, $f(r)$2, one finds

$f(r)$3

so the contracted first law

$f(r)$4

is finite and nonzero iff $f(r)$5 (Xu et al., 30 Apr 2026). No new exponent appears as long as the geometric charge parameter $f(r)$6 is held $f(r)$7 (Xu et al., 30 Apr 2026).

For fixed-rotation Kerr–AdS, one similarly obtains

$f(r)$8

so again

$f(r)$9

is finite only on $f(r_0)=0,$0 (Xu et al., 30 Apr 2026). In the Carrollian regime $f(r_0)=0,$1, the normalized angular velocity tends to zero,

$f(r_0)=0,$2

while the work term $f(r_0)=0,$3 can remain finite because $f(r_0)=0,$4 compensates (Xu et al., 30 Apr 2026). This aligns with the statement that stationary, axisymmetric Carroll black holes are effectively static (Xu et al., 30 Apr 2026).

In arbitrary spacetime dimension within the Schwarzschild–AdS family, the same scaling law persists. With

$f(r_0)=0,$5

the thermodynamic variables obey

$f(r_0)=0,$6

and hence

$f(r_0)=0,$7

so the finite-first-law condition $f(r_0)=0,$8 is dimension-independent (Xu et al., 30 Apr 2026).

A boundary interpretation of this finite line has been developed as a double-scaled low-temperature, large-$f(r_0)=0,$9 ensemble. The same scaling $c\to 0$00 implies

$c\to 0$01

so the Carrollian temperature decreases while the effective number of boundary degrees of freedom grows, leaving the thermodynamic products finite (Xu et al., 24 Jun 2026). 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 (Xu et al., 24 Jun 2026). The Hawking–Page locus is identified with the zero of the chemical potential conjugate to the count of degrees of freedom (Xu et al., 24 Jun 2026).

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 $c\to 0$02, the horizon variables satisfy the Raychaudhuri and Damour–Navier–Stokes equations

$c\to 0$03

$c\to 0$04

which match Carrollian hydrodynamic equations through the dictionary

$c\to 0$05

with

$c\to 0$06

(Redondo-Yuste et al., 2022). In equilibrium the horizon fluid has vanishing energy density and constant negative pressure (Redondo-Yuste et al., 2022). In dynamical settings, the Carrollian fluid describes finite-time relaxation, mode couplings, and teleological equilibration of the horizon (Redondo-Yuste et al., 2022). 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

$c\to 0$07

the Carroll extremal surface is at $c\to 0$08, and the Carroll temperature satisfies

$c\to 0$09

(Aggarwal et al., 2024). The asymptotic energy density of a conformal scalar in the unique regular Carroll Hartle–Hawking state is

$c\to 0$10

precisely compatible with the 2D Stefan–Boltzmann law (Aggarwal et al., 2024). However, the Carroll Ward identities enforce

$c\to 0$11

so there is no net flux and no evaporation (Aggarwal et al., 2024). 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 (Bagchi et al., 2024). 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 (Bagchi et al., 2024). 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 (Aggarwal et al., 2024, Redondo-Yuste et al., 2022, Ecker et al., 2023).

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 $c\to 0$12 under the correlated scaling $c\to 0$13 (Xu et al., 30 Apr 2026). In intrinsic Carroll gravity, finiteness means that $c\to 0$14, $c\to 0$15, and $c\to 0$16 can be defined directly from Carrollian geometry and Noether charges (Ecker et al., 2023). At the horizon, finiteness means that the divergent stretched-horizon stress tensor reorganizes into finite Carrollian momenta and charges obeying Carrollian conservation laws (Donnay et al., 2019). 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.