Carroll Black Holes: Geometric Insights

• Carroll black holes are configurations in Carroll geometry defined by extremal surfaces and thermal properties that replace traditional event horizons.
• They reveal novel near-horizon structures where Carroll symmetries govern conservation laws, membrane dynamics, and the thermodynamic behavior of ultra-relativistic limits.
• Recent studies employ both dilaton and magnetic formulations to analyze geodesic motion, effective potentials, and rotational characteristics in Carrollian frameworks.

Carroll black holes are black-hole-like configurations associated with Carroll geometry, the ultra-relativistic $c\to 0$ contraction of Lorentzian geometry in which the metric becomes degenerate and a preferred vector field $v^\mu$ spans its kernel, $v^\mu h_{\mu\nu}=0$. In the literature, the term is used in two closely related senses. In intrinsic Carroll gravity, a Carroll black hole is a massive vacuum solution with a Carroll extremal surface and Carroll thermal properties, replacing the Lorentzian event horizon by a distinguished degeneration locus of the Carroll structure. In horizon and near-horizon physics, ordinary non-extremal black holes are described by emergent Carrollian or String-Carroll structures on null hypersurfaces and in their Rindler neighborhoods, with Carroll symmetries governing membrane dynamics, probe motion, and aspects of black-hole thermodynamics (Ecker et al., 2023, Donnay et al., 2019, Bagchi et al., 23 Feb 2026).

1. Definition and conceptual scope

The modern definition of a Carroll black hole was formulated to isolate the black-hole-like sector of Carroll gravity despite the absence of a Lorentzian lightcone structure. The defining statement is

$$\text{Carroll black hole}=\text{Carroll extremal surface}+\text{Carroll thermal properties}.$$

In two-dimensional Carroll dilaton gravity, the Carroll extremal surface is characterized in second-order variables by

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

where $X$ is the dilaton and $e^\mu$ is the spatial inverse vielbein. In higher-dimensional magnetic Carroll constructions, the distinguished surface is typically the locus $f(r_0)=0$, often described as a Carroll extremal surface (CES). This surface is not an event horizon in the Lorentzian sense; it is the place where the Carrollian geometry degenerates in the appropriate way (Ecker et al., 2023, Ecker, 13 Mar 2026, Tadros et al., 2024).

This redefinition is necessary because Carroll spacetimes do not support the usual causal notion of “no escape to infinity.” The relevant surviving structures are instead thermodynamic and geometric. In the two-dimensional dilaton models, physically relevant solutions are the linear dilaton vacua labeled by a conserved mass/Casimir $M$, while constant dilaton vacua are not regarded as Carroll black holes because they do not have finite entropy in the required sense. In four-dimensional and higher-dimensional magnetic Carroll gravity, Carroll black holes arise as ultra-relativistic limits of familiar Lorentzian families such as Schwarzschild, Schwarzschild-(A)dS, Reissner–Nordström, and BTZ, together with genuinely Carrollian generalizations (Ecker et al., 2023, Hansen et al., 2021, Tadros et al., 2024).

A useful conceptual distinction runs through the subject. Some works study intrinsic Carroll black holes as solutions of Carroll gravity itself. Others study Carrollian black-hole horizons or near-horizon Carroll geometries associated with ordinary Lorentzian black holes. The two programs are continuous rather than disjoint: the near-horizon null surface is naturally Carrollian, and the intrinsic Carroll black-hole solutions are designed to retain the thermodynamic and geometric data that survive the $c\to 0$ limit (Donnay et al., 2019, Ecker, 13 Mar 2026).

2. Horizon Carroll geometry, conservation laws, and Love numbers

A central structural result is that a black-hole horizon is naturally a Carroll manifold. In null Gaussian coordinates, the near-horizon metric may be written as

$v^\mu$0

with $v^\mu$1 the horizon and $v^\mu$2 a stretched horizon. The identification

$v^\mu$3

shows that approaching the horizon is an ultra-relativistic $v^\mu$4 limit. In this limit, the induced geometry becomes Carrollian, with spatial metric $v^\mu$5, lapse $v^\mu$6, and temporal connection $v^\mu$7 (Donnay et al., 2019).

This near-horizon Carroll structure is not merely kinematical. The null Raychaudhuri and Damour equations become Carrollian conservation laws. The horizon expansion and shear,

$v^\mu$8

enter equations that match the $v^\mu$9 limit of the conservation of a Brown–York-type stress tensor on the stretched horizon. The resulting Carrollian variables $v^\mu h_{\mu\nu}=0$0, $v^\mu h_{\mu\nu}=0$1, $v^\mu h_{\mu\nu}=0$2, and $v^\mu h_{\mu\nu}=0$3 play the roles of energy density, effective pressure, dissipative/shear stress, and heat current. The horizon is therefore an ultra-relativistic fluid system, not a Galilean one (Donnay et al., 2019).

The symmetry algebra preserving this horizon Carroll structure contains BMS-like supertranslations and superrotations. On the horizon, the relevant vector fields reduce to

$v^\mu h_{\mu\nu}=0$4

and the associated conserved charges include a generalized angular momentum for non-stationary black holes. This identifies the horizon BMS-like symmetry algebra as the symmetry algebra of the induced Carrollian geometry, rather than a merely formal analogy (Donnay et al., 2019).

Penna’s analysis of Schwarzschild horizons sharpened the physical content of this Carrollian viewpoint. In the membrane paradigm, the stretched horizon behaves as a $v^\mu h_{\mu\nu}=0$5-dimensional fluid with nonzero momentum density but vanishing velocity. For weak perturbations,

$v^\mu h_{\mu\nu}=0$6

and finite momentum density with divergent energy density implies

$v^\mu h_{\mu\nu}=0$7

Hence $v^\mu h_{\mu\nu}=0$8 as the lapse $v^\mu h_{\mu\nu}=0$9, and the horizon fluid cannot respond hydrodynamically in the usual way. Penna related this to an emergent local Carroll symmetry of the horizon and, for $$\text{Carroll black hole}=\text{Carroll extremal surface}+\text{Carroll thermal properties}.$$0 Schwarzschild in GR, showed that in the Binnington–Poisson perturbative gauge the vanishing horizon velocity is equivalent to the vanishing of the gravitoelectric and gravitomagnetic Love numbers,

$$\text{Carroll black hole}=\text{Carroll extremal surface}+\text{Carroll thermal properties}.$$1

This connects the horizon’s Carrollian no-motion property to the puzzling vanishing of Schwarzschild Love numbers in four-dimensional asymptotically flat GR (Penna, 2018).

3. Carrollian dynamics, geodesics, and extremal-surface optics

The systematic derivation of Carroll black-hole dynamics begins with the Carroll expansion of general relativity. In pre-ultra-local variables,

$$\text{Carroll black hole}=\text{Carroll extremal surface}+\text{Carroll thermal properties}.$$2

the Einstein–Hilbert action expands as

$$\text{Carroll black hole}=\text{Carroll extremal surface}+\text{Carroll thermal properties}.$$3

The leading electric Carroll theory has ultralocal evolution and admits momentum and angular-momentum-type initial data but no mass. Mass appears only at next-to-leading order. A consistent truncation of the NLO theory, identified as the magnetic Carroll limit, yields the Carroll–Schwarzschild solution; in isotropic coordinates,

$$\text{Carroll black hole}=\text{Carroll extremal surface}+\text{Carroll thermal properties}.$$4

with Carroll data

$$\text{Carroll black hole}=\text{Carroll extremal surface}+\text{Carroll thermal properties}.$$5

This is the $$\text{Carroll black hole}=\text{Carroll extremal surface}+\text{Carroll thermal properties}.$$6 shadow of Schwarzschild in a coordinate system analytic in $$\text{Carroll black hole}=\text{Carroll extremal surface}+\text{Carroll thermal properties}.$$7 (Hansen et al., 2021).

Intrinsic Carrollian test-particle dynamics can be derived by EFT rather than by naively Carroll-limiting relativistic geodesics. The worldline action

$$\text{Carroll black hole}=\text{Carroll extremal surface}+\text{Carroll thermal properties}.$$8

contains coupled “magnetic” and “electric” sectors. For the Carroll–Schwarzschild background, the radial motion can be written as

$$\text{Carroll black hole}=\text{Carroll extremal surface}+\text{Carroll thermal properties}.$$9

The Newton-like $e^\mu\partial_\mu X=0,\qquad X>0,$0 term depends on the particle energy $e^\mu\partial_\mu X=0,\qquad X>0,$1, and the only circular orbit lies at the CES,

$e^\mu\partial_\mu X=0,\qquad X>0,$2

where it is unstable. For large impact parameter, the deflection angle is half the usual GR light-bending result,

$e^\mu\partial_\mu X=0,\qquad X>0,$3

while for impact parameters near the CES trajectories wind around it, and for small impact parameter the Carroll black hole reflects incoming particles. This underlies the characterization of the Carroll–Schwarzschild black hole as a perfect mirror (Ciambelli et al., 2023).

The same picture generalizes, but nontrivially, beyond Schwarzschild. In Carroll Schwarzschild-(A)dS, nearly tangential particles from infinity wind a finite number of times around the extremal surface, with the winding depending on the impact parameter and the cosmological constant. In Schwarzschild-Bach-(A)dS, by contrast, sufficiently close trajectories wind infinitely many times and do not escape to asymptotic infinity. The Einstein and quadratic-gravity cases are therefore sharply distinguished by finite versus infinite winding near the extremal surface (Tadros et al., 2024).

Charged black holes exhibit an even richer structure. In magnetic-electric Carrollian Reissner–Nordström, the CESs are the roots of

$e^\mu\partial_\mu X=0,\qquad X>0,$4

For the nonextreme case there are two asymptotically flat patches outside the outer CES and a third geodesically complete region inside the inner CES; in the extreme case there is only one asymptotic flat patch. Neutral geodesics are well defined after projection to absolute space, while charged particles in the magnetic-electric background are forced onto fixed-radius circular motion by the Lorentz-force constraint

$e^\mu\partial_\mu X=0,\qquad X>0,$5

This reorganizes the global structure of RN under the ultra-relativistic limit into a CES-separated, geodesically complete Carrollian spacetime (Chen et al., 2024).

4. Thermodynamics and the Carroll extremal surface

In two-dimensional Carroll dilaton gravity, thermodynamics is formulated directly from Noether charges and regularity conditions. For the generic linear-dilaton sector one has

$e^\mu\partial_\mu X=0,\qquad X>0,$6

together with the first law

$e^\mu\partial_\mu X=0,\qquad X>0,$7

Temperature can be obtained either from holonomy/Gauss–Bonnet regularity on a C-thermal manifold or from a Carroll surface gravity $e^\mu\partial_\mu X=0,\qquad X>0,$8, $e^\mu\partial_\mu X=0,\qquad X>0,$9. Entropy is the Noether charge evaluated at the Carroll extremal surface. This is not presented as an analogy: the thermodynamic structure is derived within the Carroll theory itself (Ecker et al., 2023, Ecker, 13 Mar 2026).

These formulas support a broad class of explicit models. The literature analyzes Carroll JT, Carroll–Schwarzschild, Carroll Reissner–Nordström, Carroll BTZ, Carroll CGHS, and Carroll Witten black holes. In each case, the black-hole sector is selected by the existence of a Carroll extremal surface and finite entropy, while charged and rotating examples obey familiar BPS-like inequalities for the existence of inner and outer extremal surfaces. In the Carroll–Schwarzschild example, the thermodynamic behavior reproduces the Schwarzschild pattern in Carroll variables, and the distinguished four-dimensional surface is the throat at $X$0 (Ecker et al., 2023).

Carroll black-hole thermodynamics also extends to Schwarzschild-(A)dS and to higher-derivative Schwarzschild-Bach-(A)dS. In these models the entropy diverges in the strict Carroll limit, the temperature goes to zero, and the system is argued to behave like an incompressible thermodynamic system, with $X$1. The specific heat is divergent and can be positive, negative, or zero depending on the model and parameters. In the pure Schwarzschild-Bach Carroll case it is always negative, but the paper stresses that, unlike in Lorentzian black-hole thermodynamics, negative specific heat does not imply instability because Carroll black holes do not radiate Hawking quanta in the usual sense (Tadros et al., 2024).

A holographic thermodynamic interpretation appears in Carroll partition-function analyses. By analytically continuing the momentum chemical potential and studying the regime $X$2, one obtains Carrollian thermodynamics in which the energy density satisfies

$X$3

in the Carroll regime. For holographic two-dimensional CFTs, the boundary Carrollian field theory is interpreted as living on the horizon of a large AdS black hole, which is pushed to the null boundary in the flat-space limit. In the BTZ case, the Carroll limit $X$4 is matched to the flat-space limit of inner-horizon thermodynamics rather than the outer-horizon entropy (Poulias et al., 26 Mar 2025).

5. Quantum matter, strings, and horizon transport

Quantum matter on Carroll black-hole backgrounds exhibits a Hawking-like effect, but its realization differs sharply from the Lorentzian case. For the two-dimensional Carroll-Schwarzschild background with Carroll temperature

$X$5

scalar-field analysis yields a nonzero asymptotic energy density

$X$6

The effect is encoded in a thermal energy density rather than in an outgoing flux, because Carrollian Ward identities force the two chiral fluxes to coincide and hence forbid net energy transport. Finiteness at the Carroll extremal surface excludes the Carroll analogue of the Boulware vacuum, and local Carroll boost invariance excludes the Unruh analogue. The surviving semiclassical state is a Hartle–Hawking-type equilibrium state. The thesis literature reformulates this result as the Carroll-Hawking effect, namely the emergence of nonvanishing asymptotic energy density in the matter vacuum of a Carroll black hole (Aggarwal et al., 2024, Ecker, 13 Mar 2026).

String probes make the near-horizon Carroll regime especially explicit. For a Schwarzschild black hole viewed by a stationary observer at infinity, the near-horizon expansion

$X$7

is interpreted as a string Carroll expansion with $X$8. In the magnetic sector, the leading-order string freezes on the sphere,

$X$9

and the dynamics reduce to a null geodesic in the longitudinal two-dimensional Rindler space, with subleading oscillations on the sphere. Part II showed that the relativistic solution space near a non-extremal horizon bifurcates into magnetic and electric Carroll strings: magnetic strings shrink to a point on $e^\mu$0 and follow null geodesics or folded-string configurations in Rindler, whereas electric strings wrap the sphere and move as timelike geodesics in the longitudinal sector. The same String-Carroll expansion extends to Kerr and Reissner–Nordström (Bagchi et al., 2023, Bagchi et al., 2024).

The generic formulation of this picture is the String-Carroll geometry of a non-extremal black-hole near horizon. In Gaussian-null coordinates, after scaling $e^\mu$1, the geometry takes the form of a transverse base $e^\mu$2 — typically a sphere or plane — fibred by a two-dimensional Rindler spacetime $e^\mu$3. Particle geodesics and scalar-field equations derived intrinsically in this String-Carroll geometry agree with the near-horizon limits of the corresponding equations in the parent Lorentzian black-hole backgrounds, including Schwarzschild, Kerr, BTZ, AdS black branes, and Lifshitz black holes. This identifies String-Carroll geometry as a universal near-horizon organization of generic non-extremal black objects (Bagchi et al., 23 Feb 2026).

Carrollian transport on the horizon itself can also be nontrivial. On the outer horizon of Kerr–Newman, treated as a $e^\mu$4-dimensional Carroll manifold $e^\mu$5, a massless chargeless particle with anyonic spin, magnetic moment, and doubly extended planar Carroll charges — an “exotic photon” — obeys a Hall-type drift law,

$e^\mu$6

in the background horizon magnetic field. In horizon coordinates $e^\mu$7, the motion satisfies $e^\mu$8 and is purely azimuthal, so the excitation drifts along circles of constant $e^\mu$9. The paper emphasizes that this effect is intrinsic to Carrollian horizon dynamics rather than frame dragging (Marsot et al., 2022).

6. Rotation, no-go theorems, and the status of “Kerroll”

The status of rotation in Carroll black holes is one of the main active points of differentiation in the subject. In magnetic Carrollian general relativity, a strong no-go theorem states that any stationary and axisymmetric solution in dimensions $f(r_0)=0$0 is necessarily static, up to a removable constant angular shift sometimes called topological rotation. In three dimensions, the special case is the Carroll BTZ geometry, where a rotating-looking solution can be obtained from a static one by a Carroll boost and angular re-identification, but the rotation is topological rather than dynamical. The same paper also exhibited a static but non-spherical four-dimensional Carroll C-metric, showing that the no-go theorem excludes genuine rotation, not all nontrivial stationary axisymmetric structure (Kolář et al., 12 Jun 2025).

Framework	Result	Interpretation
Magnetic CGR in $f(r_0)=0$1	Stationary axisymmetric solutions are static	No genuine rotation (Kolář et al., 12 Jun 2025)
Carroll BTZ in $f(r_0)=0$2	Rotating form via boost and re-identification	Topological rotation (Kolář et al., 12 Jun 2025)
Magnetic Carroll gravity with dressed connection	Rotating Carroll–Schwarzschild with charge $f(r_0)=0$3	Rotation stored in connection/momenta (Ecker et al., 14 May 2026)
Extended odd-power theory	Carroll analog of Kerr (“Kerroll”)	Rotation is an odd-power Carroll effect (Ecker et al., 14 May 2026)

A later construction modifies this conclusion in two distinct ways. First, within magnetic Carroll gravity, a rotating Carroll–Schwarzschild solution is produced by “dressing” the static metric with nontrivial momentum/connection data rather than changing the spatial metric itself. With

$f(r_0)=0$4

the conserved charges become

$f(r_0)=0$5

while the Carroll extremal surface remains at $f(r_0)=0$6. In second-order language, the rotation is encoded in the Carroll-compatible connection component

$f(r_0)=0$7

The metric stays Carroll–Schwarzschild; the rotation resides in the connection or canonical momenta (Ecker et al., 14 May 2026).

Second, an odd-power expansion of GR in $f(r_0)=0$8 yields an extended magnetic Carroll theory in which Kerr survives as a Carroll analog, the Kerroll black hole. In this construction the rotational data appear in the odd-power sector, notably the $f(r_0)=0$9 shift and momenta, rather than in the strict even-power magnetic Carroll truncation. The Carroll extremal surface is located at

$M$0

coinciding with the Kerr event horizon, and the charges are

$M$1

This suggests that the status of rotating Carroll black holes depends on the framework: strict magnetic CGR with its staticity theorem, magnetic Carroll gravity with independent connection data, and extended odd-power Carrollian theories do not make the same identification of the physical rotational sector (Kolář et al., 12 Jun 2025, Ecker et al., 14 May 2026).

Across these formulations, the unifying feature remains the replacement of the Lorentzian horizon by Carrollian degeneracy data — a Carroll extremal surface on the intrinsic side, and Carroll or String-Carroll geometry on the near-horizon side. The subject has therefore developed from a horizon-symmetry observation into a broader program in which Carroll geometry organizes black-hole mechanics, probe dynamics, higher-derivative generalizations, and quantum matter, while leaving open the precise scope of rotation and the full classification of genuinely Carrollian black-hole solutions.