Carroll-Hawking Effect in Carrollian Black Holes

• Carroll-Hawking effect is a quantum phenomenon where a non-vanishing asymptotic energy density arises in Carroll black hole backgrounds, defined by Carroll temperature and extremal surfaces.
• The derivation uses dual approaches combining anomalous energy-momentum tensors and Ward identities to confirm a 2d Stefan-Boltzmann law behavior without an outgoing flux.
• This effect redefines traditional Hawking radiation, emphasizing thermality through stationary state data rather than radiative evaporation in Carrollian gravity.

The Carroll-Hawking effect is a Hawking-like quantum phenomenon on Carroll black hole backgrounds in which quantum matter develops a non-vanishing asymptotic energy density rather than an outgoing radiative flux. In the formulation introduced for the Carroll-Schwarzschild background, the effect is tied to the Carroll temperature of the geometry, the existence of a Carroll extremal surface, and the ultra-relativistic $c\to 0$ kinematics of Carrollian physics, where ordinary energy transport is strongly constrained. Its canonical realization is that a free massless scalar field on a Carroll black hole acquires an asymptotic energy density obeying the 2d Stefan-Boltzmann law, while the usual Unruh-type evaporative interpretation is absent (Aggarwal et al., 2024, Ecker, 13 Mar 2026).

1. Definition and nomenclature

The term Carroll-Hawking effect denotes the appearance of a non-vanishing asymptotic energy density for quantum matter on a Carroll black hole background. In the thesis that explicitly names the phenomenon, the definition is operational: “For the Carroll-Schwarzschild black hole we find a non-vanishing asymptotic energy density. We refer to this phenomenon as the Carroll-Hawking effect” (Ecker, 13 Mar 2026). The effect is Hawking-like because the thermal properties of the background geometry are reflected in the quantum state of matter, but it is Carrollian because the response is not encoded as an outgoing energy flux.

This terminology distinguishes the effect from standard Hawking radiation in Lorentzian black-hole physics. In ordinary settings, Hawking radiation is commonly characterized by particle production and flux at infinity. In the Carrollian setting studied in the 2024 analysis, the central observable is instead an asymptotic thermal energy density, since “energy flux cannot propagate in the usual way” and the Ward identities force the flux to vanish. The result is therefore a form of thermality without evaporation (Aggarwal et al., 2024).

A common misconception is to identify the Carroll-Hawking effect with ordinary Hawking emission in an unusual gauge. The literature does not support that identification. The defining statements are instead that Carroll black holes admit a temperature already at the classical level, that a quantum scalar field on such a background acquires a nonzero asymptotic energy density, and that this energy density matches the Stefan-Boltzmann law when the temperature is identified with the Carroll temperature (Aggarwal et al., 2024).

2. Carroll black holes and thermodynamic background

Carroll black holes are not defined by event horizons in the standard Lorentzian sense. In the 2026 thesis, they are defined as “C-thermal states with finite entropy that have a Carroll extremal surface” (Ecker, 13 Mar 2026). This replacement is essential because, in the Carroll limit $c\to 0$, the causal and horizon structure differs from that of relativistic black holes.

The underlying geometry is Carrollian. In flat space, the metric data become degenerate,

$\lim_{c\to 0}\eta = \dd \vec{x}^2 = h_{\mu\nu}\dd x^\mu \dd x^\nu, \qquad \lim_{c\to 0} c^2\eta^{-1} = -\partial_t\otimes\partial_t = -v^\mu v^\nu \partial_\mu\partial_\nu ,$

with

$v^\mu h_{\mu\nu}=0, \qquad v^\mu \tau_\mu = -1.$

The Carroll boost ambiguity acts as

$$\tau_\mu \to \tau_\mu - \lambda_\mu,\qquad \lambda_\mu v^\mu=0.$$

In two-dimensional Carroll dilaton gravity, the black-hole sector is described in terms of a conserved Casimir $M$, and the thermodynamic variables satisfy

$$E = \frac{k}{2\pi}M, \qquad T = \frac{w'(X_{\rm min})}{2\pi}, \qquad S = k\,X_{\rm min}, \qquad \delta E = T\,\delta S.$$

For the Carroll-Schwarzschild model, the thesis gives the scaling

$$E = \frac{k}{2\pi}M, \qquad T \propto \frac{1}{M}, \qquad S \propto M^2,$$

with negative specific heat (Ecker, 13 Mar 2026).

The Carroll extremal surface is defined in the second-order formulation by

$e^\mu \partial_\mu X = 0, \qquad X>0.$

In the Carroll-Schwarzschild background used in the 2024 derivation, the relevant metric function is

$f(r)=1-\frac{r_s}{r},$

and the surface $c\to 0$0 is the Carroll extremal surface. This geometry carries an associated Carroll temperature

$c\to 0$1

which is later recovered from the quantum energy density (Aggarwal et al., 2024).

These constructions provide the classical thermodynamic input for the effect. A plausible implication is that the Carroll-Hawking effect should be viewed not as an isolated QFT calculation, but as a semiclassical manifestation of an already well-defined Carroll black-hole thermodynamics.

3. Scalar fields, Carroll stress tensors, and anomaly derivation

The explicit derivation studies a free massless scalar field on a Carroll black hole background and combines Carroll boost Ward identities, diffeomorphism Ward identities, and a Weyl/conformal anomaly in a method inspired by Christensen-Fulling (Aggarwal et al., 2024). The analysis uses two Carrollian scalar contractions.

The electric scalar is governed by

$c\to 0$2

This theory is ultralocal in space. The magnetic scalar is

$c\to 0$3

with a momentum field acting as a Lagrange multiplier enforcing time-independence. A regulator-like “electromagnetic” scalar action containing both pieces is introduced for the anomaly derivation.

The Carroll energy-momentum tensor is written in terms of one-forms $c\to 0$4 and $c\to 0$5, with gauge-invariant form

$c\to 0$6

Its Ward identities are

$c\to 0$7

$c\to 0$8

and, classically,

$c\to 0$9

Quantum mechanically, the Weyl identity becomes anomalous. The anomaly is the standard two-dimensional result

$\lim_{c\to 0}\eta = \dd \vec{x}^2 = h_{\mu\nu}\dd x^\mu \dd x^\nu, \qquad \lim_{c\to 0} c^2\eta^{-1} = -\partial_t\otimes\partial_t = -v^\mu v^\nu \partial_\mu\partial_\nu ,$0

On the Carroll-Schwarzschild background, the Ward identities plus the anomaly determine the expectation value of the energy-momentum tensor up to integration constants. After the Carroll limit is taken carefully, the resulting Carroll energy-momentum tensor is compatible with Carroll boost invariance and diffeomorphisms, and its asymptotic energy density is

$\lim_{c\to 0}\eta = \dd \vec{x}^2 = h_{\mu\nu}\dd x^\mu \dd x^\nu, \qquad \lim_{c\to 0} c^2\eta^{-1} = -\partial_t\otimes\partial_t = -v^\mu v^\nu \partial_\mu\partial_\nu ,$1

Using

$\lim_{c\to 0}\eta = \dd \vec{x}^2 = h_{\mu\nu}\dd x^\mu \dd x^\nu, \qquad \lim_{c\to 0} c^2\eta^{-1} = -\partial_t\otimes\partial_t = -v^\mu v^\nu \partial_\mu\partial_\nu ,$2

this becomes

$\lim_{c\to 0}\eta = \dd \vec{x}^2 = h_{\mu\nu}\dd x^\mu \dd x^\nu, \qquad \lim_{c\to 0} c^2\eta^{-1} = -\partial_t\otimes\partial_t = -v^\mu v^\nu \partial_\mu\partial_\nu ,$3

This is exactly the 2d Stefan-Boltzmann law (Aggarwal et al., 2024).

The same asymptotic energy density is recovered in a second derivation using the magnetic action directly, where the anomalous effective action yields a family of Carroll energy-momentum tensors parameterized by an integration constant $\lim_{c\to 0}\eta = \dd \vec{x}^2 = h_{\mu\nu}\dd x^\mu \dd x^\nu, \qquad \lim_{c\to 0} c^2\eta^{-1} = -\partial_t\otimes\partial_t = -v^\mu v^\nu \partial_\mu\partial_\nu ,$4. Imposing finiteness at the Carroll extremal surface fixes $\lim_{c\to 0}\eta = \dd \vec{x}^2 = h_{\mu\nu}\dd x^\mu \dd x^\nu, \qquad \lim_{c\to 0} c^2\eta^{-1} = -\partial_t\otimes\partial_t = -v^\mu v^\nu \partial_\mu\partial_\nu ,$5 uniquely and reproduces

$\lim_{c\to 0}\eta = \dd \vec{x}^2 = h_{\mu\nu}\dd x^\mu \dd x^\nu, \qquad \lim_{c\to 0} c^2\eta^{-1} = -\partial_t\otimes\partial_t = -v^\mu v^\nu \partial_\mu\partial_\nu ,$6

This agreement shows that the result is not an artifact of a particular limiting prescription (Aggarwal et al., 2024).

4. Vacuum structure, regularity, and the absence of evaporation

A central part of the Carroll-Hawking analysis is the selection of an admissible semiclassical state. The energy density is required to remain finite at the Carroll extremal surface $\lim_{c\to 0}\eta = \dd \vec{x}^2 = h_{\mu\nu}\dd x^\mu \dd x^\nu, \qquad \lim_{c\to 0} c^2\eta^{-1} = -\partial_t\otimes\partial_t = -v^\mu v^\nu \partial_\mu\partial_\nu ,$7. This condition eliminates the Carroll analogue of the Boulware vacuum, because that choice makes the energy density diverge at the extremal surface (Aggarwal et al., 2024).

The Carroll analogue of the Unruh vacuum is excluded for a different reason. In ordinary Lorentzian black-hole physics, the Unruh state describes an evaporating black hole with outgoing Hawking flux. In the Carrollian case, however, the Ward identities force the flux components to agree, and this is another expression of the fact that no net energy flux can propagate. The Unruh construction is therefore incompatible with Carroll boost invariance and the Carroll Ward identities (Aggarwal et al., 2024).

The remaining admissible state is the Carroll analogue of the Hartle-Hawking vacuum. It is regular at the Carroll extremal surface and describes a thermal equilibrium state. The 2024 analysis therefore concludes that Carroll black holes exhibit a Hawking-like effect, but not through a radiating flux; instead, the effect appears as a thermal energy density in the asymptotic region, fixed by the anomaly and consistent with the Carroll temperature (Aggarwal et al., 2024).

This state-selection problem clarifies the sense in which the effect is thermal. It is not a statement about late-time evaporation or an outgoing particle current. It is a statement that the only consistent semiclassical vacuum is the regular thermal one, and that its observable imprint is the asymptotic energy density. A plausible implication is that “thermality” in Carrollian gravity is encoded more naturally in stationary state data than in transport observables.

5. Relation to Carroll symmetry, null horizons, and Carrollian fluids

The Carroll-Hawking effect sits within a broader body of work linking null or near-null gravitational structures to Carrollian symmetry. In the near-horizon analysis of black holes in general relativity, the event horizon is described by the membrane paradigm as a $\lim_{c\to 0}\eta = \dd \vec{x}^2 = h_{\mu\nu}\dd x^\mu \dd x^\nu, \qquad \lim_{c\to 0} c^2\eta^{-1} = -\partial_t\otimes\partial_t = -v^\mu v^\nu \partial_\mu\partial_\nu ,$8-dimensional fluid with Brown-York stress tensor

$\lim_{c\to 0}\eta = \dd \vec{x}^2 = h_{\mu\nu}\dd x^\mu \dd x^\nu, \qquad \lim_{c\to 0} c^2\eta^{-1} = -\partial_t\otimes\partial_t = -v^\mu v^\nu \partial_\mu\partial_\nu ,$9

and momentum density

$v^\mu h_{\mu\nu}=0, \qquad v^\mu \tau_\mu = -1.$0

For weak perturbations,

$v^\mu h_{\mu\nu}=0, \qquad v^\mu \tau_\mu = -1.$1

while finiteness of the momentum density requires

$v^\mu h_{\mu\nu}=0, \qquad v^\mu \tau_\mu = -1.$2

as the stretched horizon approaches the true horizon. This yields the characteristic behavior of a membrane with nonzero momentum density but vanishing local velocity (Penna, 2018).

The symmetry explanation is that the near-horizon geometry is locally Carrollian. The Carroll group is the $v^\mu h_{\mu\nu}=0, \qquad v^\mu \tau_\mu = -1.$3 contraction of the Poincaré group, with boosts

$v^\mu h_{\mu\nu}=0, \qquad v^\mu \tau_\mu = -1.$4

and a degenerate metric structure such as

$v^\mu h_{\mu\nu}=0, \qquad v^\mu \tau_\mu = -1.$5

The event horizon, being a null surface, locally takes the Carroll form; for the Schwarzschild horizon in ingoing Eddington-Finkelstein coordinates,

$v^\mu h_{\mu\nu}=0, \qquad v^\mu \tau_\mu = -1.$6

The key physical statement emphasized there is that Carroll-invariant particles cannot move, which explains why the membrane fluid velocity vanishes (Penna, 2018).

Later work on dynamical horizons develops a full Carrollian fluid description. In this dictionary, for a horizon $v^\mu h_{\mu\nu}=0, \qquad v^\mu \tau_\mu = -1.$7 with intrinsic metric $v^\mu h_{\mu\nu}=0, \qquad v^\mu \tau_\mu = -1.$8, expansion $v^\mu h_{\mu\nu}=0, \qquad v^\mu \tau_\mu = -1.$9, shear $$\tau_\mu \to \tau_\mu - \lambda_\mu,\qquad \lambda_\mu v^\mu=0.$$0, and Hájíček one-form $$\tau_\mu \to \tau_\mu - \lambda_\mu,\qquad \lambda_\mu v^\mu=0.$$1, the fluid variables are identified as

$$\tau_\mu \to \tau_\mu - \lambda_\mu,\qquad \lambda_\mu v^\mu=0.$$2

$$\tau_\mu \to \tau_\mu - \lambda_\mu,\qquad \lambda_\mu v^\mu=0.$$3

with momentum sector tied to $$\tau_\mu \to \tau_\mu - \lambda_\mu,\qquad \lambda_\mu v^\mu=0.$$4 (Redondo-Yuste et al., 2022). In the sourced setting, the null expansion $$\tau_\mu \to \tau_\mu - \lambda_\mu,\qquad \lambda_\mu v^\mu=0.$$5 becomes the fluid energy density, the Hájíček field becomes the momentum density,

$$\tau_\mu \to \tau_\mu - \lambda_\mu,\qquad \lambda_\mu v^\mu=0.$$6

and relaxation to a non-expanding horizon corresponds to fluid equilibration (Hüsnügil et al., 27 Aug 2025).

These horizon-fluid results do not define the Carroll-Hawking effect in the strict sense used in the 2024 paper and the 2026 thesis. They nonetheless provide a closely related structural background: Carrollian geometry suppresses ordinary transport, null or extremal surfaces support degenerate kinematics, and equilibrium is governed by atypical constraints, including teleological ones for event horizons. This suggests a coherent Carrollian interpretation of why a thermal quantity can survive even when an outgoing flux does not.

6. Scope, limitations, and open directions

The explicit evidence for the Carroll-Hawking effect is currently concentrated on Carroll quantum fields on fixed Carroll black hole backgrounds in two dimensions, with the Carroll-Schwarzschild case as the central example (Ecker, 13 Mar 2026). The effect has been derived from two perspectives emphasized in the thesis: a $$\tau_\mu \to \tau_\mu - \lambda_\mu,\qquad \lambda_\mu v^\mu=0.$$7 limit of relativistic Hawking physics and an intrinsic Carroll derivation. In both perspectives, the result is the same: the Carroll-Schwarzschild background yields a nonzero asymptotic energy density (Ecker, 13 Mar 2026).

The effect should therefore not be overstated as a universal theorem for all Carrollian gravities or all matter sectors. The established statement is more specific: for the scalar theories analyzed, the asymptotic vacuum is not empty in the trivial sense, but carries a thermal energy density fixed by the anomaly and compatible with the Carroll temperature (Aggarwal et al., 2024). Whether analogous phenomena persist for other matter content, other Carroll black hole models, or backreacted semiclassical dynamics is not settled by the cited works.

A second limitation concerns the role of symmetry. Near-horizon Carroll symmetry explains frozen kinematics in the membrane paradigm and has been related to vanishing Love numbers for $$\tau_\mu \to \tau_\mu - \lambda_\mu,\qquad \lambda_\mu v^\mu=0.$$8 Schwarzschild black holes in general relativity, where

$$\tau_\mu \to \tau_\mu - \lambda_\mu,\qquad \lambda_\mu v^\mu=0.$$9

But that analysis is careful to note that Carroll symmetry alone is probably not the entire story, because Love numbers are not zero in higher spacetime dimensions, asymptotically AdS spacetimes, or modified theories of gravity (Penna, 2018). By analogy, it would be premature to infer that Carroll symmetry by itself exhausts the explanation of all Hawking-like phenomena in Carrollian gravity.

The broader significance of the Carroll-Hawking effect lies in showing that Carroll black holes are thermal objects in a specifically Carrollian sense. Their thermodynamic properties are imprinted on quantum matter, but the imprint takes the form of an asymptotic vacuum energy density rather than a radiative flux. In that precise sense, the Carroll-Hawking effect is the Carrollian analogue of Hawking radiation: a quantum response to black-hole thermodynamics, reformulated for degenerate causal structure and constrained transport (Aggarwal et al., 2024, Ecker, 13 Mar 2026).