Rotating Carroll Black Holes: A No Go Theorem

This paper establishes a no-go theorem demonstrating that stationary and axisymmetric Carrollian black holes in dimensions greater than three are necessarily static, while highlighting that rotating solutions can exist in three dimensions via topological re-identification and that accelerating black holes remain possible in four dimensions.

The Gist

The Big Idea: The "Frozen" Universe

Imagine you have a universe where the speed of light is not just slow, but zero. In this world, nothing can move sideways. If you are standing at a point in space, you are stuck there forever. You can experience time passing, but you cannot travel from point A to point B. This is the world of Carrollian physics.

It's like being in a movie where the camera is locked in place, but the actors can still age and the plot can unfold. Space is "absolute" (fixed), but time is "relative" (flowing).

The authors of this paper asked a big question: Can you have a spinning black hole in this frozen universe?

In our normal universe (Einstein's relativity), black holes can spin. They drag space around them like a whirlpool. But in this "frozen" Carroll universe, the authors prove that spinning black holes are impossible (with one tiny exception).

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The Main Discovery: The "No-Go" Sign

The paper presents a mathematical "No-Go Theorem." Think of it like a traffic sign that says: "No Rotating Black Holes Allowed."

Here is the logic, broken down:

1. The Setup: The researchers looked at the rules of gravity for this frozen universe (Carrollian General Relativity). They tried to build a black hole that spins.
2. The Conflict: To make a black hole spin, you need a specific kind of "twist" in the geometry of space and time. However, the rules of this frozen universe are very strict. They act like a rigid cage.
3. The Result: When they tried to force a spin into the equations, the math forced the spin to disappear. The universe essentially says, "If you want to be a black hole here, you must stand perfectly still."
4. The Conclusion: Any black hole in this universe (in 4 dimensions or higher) must be static (non-rotating). It doesn't matter how you try to set it up; the laws of physics will force it to stop spinning.

The Analogy: Imagine trying to spin a top on a sheet of ice that has instantly turned into solid concrete. No matter how hard you flick it, the top cannot rotate because the surface won't let it. The "concrete" is the rigid structure of the Carroll universe.

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The One Exception: The "3D Magic Trick"

There is one special case where a spinning black hole can exist: in a 3-dimensional universe (2 space + 1 time).

Why? Because in 3D, the rules are a bit looser. The authors show that you can create a spinning black hole, but it's not spinning because the matter inside is moving. Instead, it's spinning because of a topological trick.

The Metaphor: Imagine a cylinder (like a toilet paper roll). If you draw a line around the edge and connect the top to the bottom, you get a loop. Now, imagine twisting the paper before you tape the ends together. When you walk around the loop, you end up slightly "shifted" in time.

In this 3D Carroll universe, the black hole spins because the universe itself is "twisted" globally, like that paper cylinder. It's a global rotation rather than a local spin. It's like a carousel where the horses aren't moving, but the whole platform is twisted in a way that feels like rotation.

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What About Other Stuff? (Matter and Fields)

The researchers also asked: "What if we add electricity, magnetism, or other weird particles to the mix?"

They checked if adding these ingredients would allow a black hole to spin.

• The Answer: No. Even with extra "matter" (like electric fields or axions), the "No-Go" rule still holds. The universe is so rigid that adding more ingredients doesn't loosen the cage enough to let the black hole spin.

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Why Does This Matter?

You might wonder, "Who cares about a universe where nothing moves?"

1. Black Hole Horizons: Real black holes have "event horizons" (the point of no return). Near these horizons, the physics starts to look a lot like this frozen Carroll universe. Understanding Carroll physics helps us understand the extreme edges of real black holes.
2. Holography: Scientists believe our universe might be a "hologram" projected from a lower-dimensional boundary. This boundary often behaves like a Carrollian universe. If we want to understand the hologram, we need to know the rules of the boundary.
3. The Limits of Physics: This paper tells us that "rotation" is a very fragile concept. It requires a specific type of spacetime to exist. If you change the rules of the universe too much (by making light speed zero), rotation vanishes.

Summary

• The Universe: A place where space is frozen and time flows.
• The Problem: Can you have a spinning black hole here?
• The Verdict: No. In 4D and higher, the laws of physics force all black holes to be perfectly still.
• The Loophole: In 3D, you can have a "twisted" black hole, but it's a global trick, not a local spin.
• The Takeaway: Rotation is a luxury that only exists in universes with specific rules. In the "frozen" limit, everything must stand still.

Technical Summary

1. Problem Statement

The paper addresses a fundamental open question in Carrollian General Relativity (CGR): Do rotating black hole solutions exist in dimensions $d > 3$?
While Carroll black holes (massive solutions in magnetic CGR) have been studied, all known examples in $d > 3$ are static. Previous attempts to construct a Carrollian analogue of the Kerr metric (rotating black hole) resulted in inconsistencies. The authors investigate whether this failure is due to a lack of appropriate coordinate systems (as suggested by the foliation requirements of the Carroll limit) or if it is a fundamental property of the theory itself.

2. Methodology

The authors employ a rigorous geometric and algebraic approach to analyze the constraints of Carrollian General Relativity:

• Carrollian Structure Definition: They define a Carroll manifold using a nowhere-vanishing vector field $v$ (defining time) and a degenerate metric $h$ (defining space) such that $v^\mu h_{\mu\nu} = 0$.
• Pre-Ultralocal (PUL) Parametrization: They utilize the PUL expansion of the Lorentzian metric ($g_{\mu\nu} = -c^2 T_\mu T_\nu + h_{\mu\nu}$) to derive the equations of motion for magnetic CGR.
• Hamiltonian Constraint: A central element of their analysis is the imposition of the Hamiltonian constraint for magnetic CGR:
  $$K_{\mu\nu} \equiv -\frac{1}{2} \mathcal{L}_v h_{\mu\nu} = 0$$
  This constraint is necessary for the theory to retain Carroll boost invariance in stationary solutions.
• General Ansatz: They construct the most general stationary, axisymmetric, degenerate metric $h$ and vector field $v$ in $d$ dimensions.
• Extension to Matter: They extend the analysis to include matter fields, specifically the Einstein-Maxwell-Dilaton-Axion (EMDA) theory, to test the robustness of their findings against the inclusion of electromagnetic and scalar fields.

3. Key Contributions and Results

A. The No-Go Theorem for $d > 3$

The primary result is a No-Go Theorem:

• Statement: In any dimension $d > 3$, every stationary and axisymmetric solution of magnetic Carrollian General Relativity is necessarily static (up to a "topological rotation").
• Mechanism: By imposing the Hamiltonian constraint ($K_{\mu\nu}=0$) on the general stationary axisymmetric ansatz, the authors derive that the "rotation parameter" $V$ (where $v = v_t(\partial_t + V \partial_\phi)$) must be a constant ($V = \text{const}$).
• Coordinate Transformation: Since $V$ is constant, a coordinate transformation $\phi \to \phi + Vt$ can be performed to eliminate the cross-term $dt d\phi$ in the metric. This transforms the solution into a strictly static form.
• Conclusion: The failure to find rotating Kerr-like solutions in the literature is not due to coordinate choices but is an intrinsic feature of CGR in $d > 3$.

B. The Special Case of $d = 3$ (Carroll BTZ)

The situation in three dimensions is distinct:

• Local vs. Global: While the local geometry of a stationary solution in $d=3$ is static (similar to higher dimensions), the global topology allows for rotation.
• Topological Rotation: A rotating Carroll BTZ black hole can be constructed by applying a Carroll boost accompanied by a re-identification of the angular coordinate (a global threading).
• Result: This yields a rotating Carroll BTZ black hole with non-trivial topological angular momentum $J$, analogous to the Lorentzian BTZ case, but the rotation is purely topological rather than dynamical in the local metric structure.

C. Generalization to Matter (EMDA Theory)

The authors investigate whether adding matter fields (Maxwell, Dilaton, Axion) allows for rotation:

• Electric Field Constraint: In the presence of a dilaton field, the field equations force the electric field to vanish ($E_\mu = 0$).
• Staticity Preservation: Even without a dilaton, assuming the magnetic sector (where $E_\mu=0$ is a consistent ansatz), the Hamiltonian constraint still forces the solution to be static.
• Conclusion: The inclusion of EMDA matter fields does not circumvent the no-go theorem; stationary axisymmetric Carroll black holes in this extended theory must also be static.

D. Non-Spherical Stationary Solutions

While rotation is forbidden, the authors demonstrate that Schwarzschild is not the only stationary solution in 4D.

• They construct a Carrollian version of the C-metric (an accelerating black hole).
• This solution is stationary and axisymmetric but not spherically symmetric, proving that the space of static solutions is richer than just the Schwarzschild geometry.

4. Significance

• Fundamental Limits of CGR: The paper establishes a rigorous boundary on the types of black holes that can exist in Carrollian gravity. It clarifies that "rotation" in the sense of a local frame-dragging effect (like in Kerr spacetime) is incompatible with the Hamiltonian constraint of magnetic CGR in $d > 3$.
• Clarification of Carroll Limits: It resolves ambiguity regarding whether rotating Carroll black holes were merely "missing" due to poor coordinate choices. The authors prove they do not exist in the standard formulation.
• Thermodynamics and Holography: By confirming the existence of the static Carroll BTZ and the accelerating C-metric, the paper provides new backgrounds for studying Carrollian thermodynamics and flat-space holography (BMS/Cartan correspondence).
• Dimensional Dependence: The work highlights the unique role of 3D gravity, where topological effects allow for a form of rotation that is forbidden in higher dimensions.

Summary Conclusion

The paper proves that rotating black holes (in the Kerr sense) do not exist in Carrollian General Relativity for dimensions $d > 3$. Any stationary, axisymmetric solution is forced to be static by the Hamiltonian constraint. The only exception is in $d=3$, where a rotating BTZ black hole exists via global topological identifications. This result holds even when extending the theory to include Maxwell, dilaton, and axion fields.