Carroll black holes

This paper defines Carroll black holes as solutions to Carroll gravity that exhibit thermal properties and possess Carroll extremal surfaces, demonstrating these features in various examples including Carroll versions of Schwarzschild, Reissner-Nordström, BTZ, and generic 1+1 dimensional dilaton gravity black holes.

The Gist

Imagine the universe as a giant, bustling city. In our normal reality (which physicists call "Lorentzian" or "Relativistic"), this city has strict traffic rules. Nothing can go faster than the speed of light. This speed limit creates "light cones"—invisible bubbles that dictate what events can influence each other. If you are inside a black hole, the traffic rules are so twisted that everything is forced to move toward the center, and you can never escape. This is the standard definition of a black hole: a place where the "exit" is blocked by the speed of light.

Now, imagine a different version of this city. Let's call it Carroll City. In this city, the speed of light has dropped to zero.

The Carroll City Paradox

In Carroll City, the rules of physics are flipped. Because the speed of light is zero, the "traffic cones" that usually define cause and effect collapse. Everything is stuck in place; nothing can move through space, but time still flows. It's like a city where everyone is frozen in a statue pose, but their internal clocks are ticking.

Here is the big puzzle the paper solves: If nothing can move, how can you have a black hole? Usually, a black hole is defined by an "event horizon"—a point of no return where you get trapped because you can't outrun the pull of gravity. But in Carroll City, you can't run anyway! So, by the old rules, Carroll black holes shouldn't exist.

The New Definition: The "Carroll Black Hole"

The authors of this paper say, "Wait a minute. Just because the old definition doesn't work, doesn't mean the feeling of a black hole is gone."

They propose a new way to spot a black hole in this frozen world. Instead of looking for a "point of no return" (which doesn't exist here), they look for two special ingredients:

1. The "Frozen Center" (Carroll Extremal Surface): In a normal black hole, there's a special surface where time and space swap roles. In Carroll City, this becomes a specific spot where the "clock" of the universe (the time direction) effectively stops or changes its nature. It's like the center of a frozen pond where the ice is thinnest. The authors call this a Carroll Extremal Surface. It's the "heart" of the black hole.
2. The "Heat" (Thermal Properties): Even though the city is frozen, these special spots still have "temperature" and "entropy" (a measure of disorder). It's like finding a warm spot in a block of ice. If a solution to the equations has this "frozen center" AND it has heat/entropy, the authors declare: "This is a Carroll Black Hole!"

The Analogy: The Wormhole Garden

To understand what these black holes look like, imagine a garden with a tunnel (a wormhole) connecting two sides.

• In our normal world: The tunnel is a shortcut through space.
• In Carroll City: The tunnel is still there, but because the speed of light is zero, the "tunnel" looks like a flat, frozen landscape. The "throat" of the tunnel (the narrowest part) is exactly where the Carroll Extremal Surface sits.

The paper shows that famous black holes from our world (like the Schwarzschild black hole around a star) have "Carroll twins." If you take a normal black hole and slow the speed of light down to zero, it doesn't disappear. It transforms into a Carroll black hole. It looks like a wormhole that stretches infinitely, with a special "throat" in the middle that acts as the black hole's core.

Why Does This Matter?

You might ask, "Why study a universe where nothing moves?"

1. The Edge of the Universe: Carroll physics naturally appears at the very edges of our universe (at "infinity") and right on the surface of black hole horizons. It helps us understand the "boundary conditions" of reality.
2. Quantum Gravity: When we try to combine gravity with quantum mechanics (the physics of the very small), things get weird. Carroll physics might be the key to understanding how black holes store information (entropy) without needing the usual rules of motion.
3. New Math: The authors had to invent new math to describe these "frozen" surfaces. They found that even in a world where nothing moves, you can still have "heat," "mass," and "entropy." It turns out that the "frozen center" of a Carroll black hole holds the same amount of information as a normal black hole, just packaged differently.

Summary

Think of this paper as a detective story.

• The Crime: "Black holes can't exist in a universe where the speed of light is zero."
• The Clue: "But wait, these solutions have a special 'frozen center' and they have heat."
• The Verdict: "We redefine what a black hole is. It's not about being trapped by speed; it's about having a special, frozen core with thermal properties."

The authors successfully mapped out these "Carroll Black Holes," showing that even in a universe where time flows but space is frozen, the mysterious, heavy, hot objects we call black holes still exist—they just look and feel very different. They are the "ghosts" of normal black holes, haunting the frozen corners of the mathematical universe.

Technical Summary

1. Problem Statement

The paper addresses a fundamental conceptual gap in the study of Carrollian gravity. Carrollian spacetimes are characterized by a degenerate metric with a one-dimensional kernel (signature $(0, +, +, \dots)$), arising in the ultra-relativistic limit where the speed of light $c \to 0$.

• The Conundrum: While solutions to Carroll gravity (e.g., limits of Schwarzschild or BTZ black holes) exhibit black hole-like properties (such as specific thermodynamic behaviors), the standard definition of a black hole relies on an event horizon (a null surface separating causally disconnected regions). In Carrollian geometry, lightcones collapse, and the concept of a null horizon becomes ill-defined or trivial.
• The Goal: The authors aim to define "Carroll black holes" rigorously without relying on event horizons, providing a definition that applies to both intrinsic Carroll gravity models and limits of relativistic gravity.

2. Methodology

The authors employ a multi-faceted approach combining differential geometry, gauge theory, and thermodynamics:

• Formulations of 2d Carroll Dilaton Gravity:
  • They utilize the first-order formulation (using vielbeins $\tau, e$, spin connection $\omega$, and scalar fields $X, X_H, X_P$).
  • They employ the Poisson Sigma Model (PSM) formulation, which treats the theory as a topological gauge theory. This is crucial for identifying conserved charges and target space symmetries.
  • They derive the second-order formulation by integrating out auxiliary fields, linking the theory to "magnetic" Carroll gravity.
• Limit Procedures:
  • They demonstrate the commutativity of two limits: Spherical Reduction (from $D$-dimensional Einstein gravity to 2d dilaton gravity) and the Carroll Limit ($c \to 0$). This allows them to construct Carroll black holes by taking the limit of known relativistic solutions (Schwarzschild, Reissner-Nordström, BTZ).
  • They also derive these models intrinsically via ultra-relativistic expansions of the action.
• Thermodynamic Analysis:
  • They define energy, temperature, and entropy for these solutions.
  • They introduce the concept of Carroll Thermal (C-thermal) manifolds, relaxing the strict definition of a Carroll manifold to allow for isolated singularities where the temporal vector field vanishes.
  • They use the Gauss-Bonnet theorem and holonomy arguments to fix the periodicity of Euclidean time, thereby defining the temperature.

3. Key Contributions and Definitions

A. Carroll Extremal Surfaces

The central geometric innovation is the definition of a Carroll extremal surface.

• Relativistic Analogue: In Lorentzian gravity, extremal surfaces (where null expansions vanish) correspond to bifurcation surfaces on Killing horizons.
• Carroll Definition: In the PSM target space, a Carroll extremal surface is defined as the locus where the boost-invariant scalar field $X_H = 0$.
• Geometric Meaning: In the second-order formulation, this corresponds to the condition $e^\mu \partial_\mu X = 0$, meaning the spatial derivative of the dilaton vanishes. This locus acts as the "throat" or center of the geometry, replacing the bifurcation sphere.

B. Definition of a Carroll Black Hole

The authors propose the following definition:
$$\text{Carroll Black Hole} = \text{Carroll Extremal Surface} + \text{Carroll Thermal Properties}$$
Specifically, a Carroll black hole is a C-thermal state (a smooth manifold with a boundary diffeomorphic to $S^1$ and isolated structure singularities) that:

1. Possesses finite entropy.
2. Contains a Carroll extremal surface ($X_H=0$).

This definition excludes constant dilaton vacua (which have infinite entropy or no extremal surface) and ensures the solution behaves thermodynamically like a black hole.

C. Thermodynamics

The paper establishes a consistent thermodynamic framework for these objects:

• Energy ($E$): Identified with the conserved Casimir charge (mass parameter $M$) of the PSM.
• Temperature ($T$): Derived via two methods:
  1. Topological: Requiring the absence of conical defects (smoothness) at the extremal surface using the Gauss-Bonnet formula.
  2. Surface Gravity: Defining a Carrollian surface gravity $\kappa$ and showing $T = \kappa / 2\pi$.
• Entropy ($S$): Defined via the Wald entropy formula, yielding $S \propto X_{min}$ (the value of the dilaton at the extremal surface).
• First Law: They verify that $\delta E = T \delta S$ holds for these solutions.

4. Key Results and Examples

The authors apply their framework to several specific models, calculating their thermodynamic quantities (Energy, Temperature, Entropy, Specific Heat):

1. Carroll Jackiw-Teitelboim (CJT):
   • Solutions exist for $M > 0$.
   • Entropy scales as $S \propto \sqrt{E}$, analogous to the Cardy formula for 2D CFTs.
2. Carroll-Schwarzschild (CS) & Tangherlini (CST):
   • Derived from the spherical reduction of $D$-dimensional Schwarzschild.
   • In 4D, the Carroll limit yields a "wormhole" geometry where the extremal surface is the throat.
   • The entropy is proportional to the area of the throat ($S \propto r_s^2$).
3. Carroll CGHS and Witten Black Holes:
   • These models exhibit divergent specific heat, similar to their relativistic counterparts, due to fixed temperature constraints.
4. Charged and Rotating Black Holes:
   • Carroll-Reissner-Nordström (CRN): Includes electric charge. The extremal surface condition leads to a BPS bound ($|q| \leq m$). When saturated, the temperature vanishes.
   • Carroll BTZ: Derived via Kaluza-Klein reduction of the BTZ black hole, treating rotation as a $U(1)$ charge. It also obeys a BPS bound ($|J| \leq M$).

5. Significance and Outlook

• Redefining Black Holes: The work provides a robust, horizon-independent definition of black holes applicable to non-Lorentzian geometries. This is crucial for understanding quantum gravity in ultra-relativistic limits and flat space holography (BMS/CFT correspondence).
• Geometric Singularities: The paper highlights that "Carrollian structure singularities" (where the time vector vanishes) are not pathologies but essential features (the extremal surfaces) required for black hole thermodynamics.
• Holography: By establishing thermodynamic laws and extremal surfaces, the paper lays the groundwork for applying the Island Proposal and Quantum Extremal Surfaces to Carrollian gravity, potentially resolving information paradoxes in non-relativistic limits.
• Future Directions: The authors suggest extending these definitions to supersymmetric Carroll black holes, Galilean black holes (which present different thermodynamic challenges), and fracton gravity models.

In summary, the paper successfully constructs a coherent theory of "Carroll black holes" by identifying the Carroll extremal surface as the geometric anchor and demonstrating that these solutions obey standard thermodynamic laws, despite the absence of lightcones and event horizons.