Kerroll black holes

This paper constructs rotating black hole solutions in Carroll gravity through two distinct approaches: one yielding an intrinsically Carrollian solution by dressing a Schwarzschild black hole with rotational charge, and another deriving a "Kerroll" black hole as a Carroll analog of the Kerr solution via an odd-power expansion of general relativity in the speed of light.

The Gist

Imagine the universe as a giant, flexible fabric. In our everyday world, this fabric behaves according to Einstein's General Relativity, where space and time are woven together, and nothing can travel faster than the speed of light ($c$).

This paper explores what happens to this fabric if we imagine a universe where the speed of light is effectively zero. This is called "Carroll gravity." In this strange world, time and space completely decouple. Time becomes a rigid, universal clock that ticks the same for everyone, while space becomes a frozen, degenerate sheet where you can't move from one point to another in the usual way.

The authors of this paper tackle a big problem: How do you have a spinning black hole in a universe where nothing can move?

In normal physics, a spinning black hole (like the famous Kerr black hole) drags space around with it. But in a "frozen" Carroll universe, the usual math says spinning is impossible. The authors found two clever "loopholes" to create spinning black holes in this frozen world. They call these new objects "Kerroll" black holes (a pun on Kerr and Carroll).

Here is how they did it, using two different approaches:

Approach 1: Dressing the Frozen Black Hole with "Ghost" Spin

Think of a standard, non-spinning Carroll black hole as a statue. It's heavy, it has a horizon, but it's perfectly still.

In standard physics, spin is a property of the shape of the space itself. But in this specific type of Carroll gravity (called "magnetic"), the authors realized that spin doesn't have to be in the shape of the space; it can be hidden in the "rules" of how things connect.

• The Analogy: Imagine a frozen lake (the space). Usually, if the lake is flat, it's not spinning. But imagine you can paint invisible "currents" or "instructions" on the ice that tell a skater how to turn, even though the ice itself isn't moving.
• The Result: They took a standard, frozen Carroll black hole and "dressed" it with these invisible connection rules. The black hole looks static, but it carries a hidden "angular momentum" charge. It's like a statue that is secretly holding a spinning top. This is a purely Carrollian invention; it has no equivalent in our normal, fast-moving universe.

Approach 2: The "Odd-Power" Expansion (The Kerroll Black Hole)

The second approach is more like taking a movie of a spinning Kerr black hole and playing it in extreme slow motion, frame by frame, to see what happens as the speed of light drops to zero.

• The Problem: When physicists usually slow down the speed of light to zero, they only look at the "even" steps (like $c^2$, $c^4$). They found that if you only look at these even steps, the spin disappears completely. The black hole stops spinning.
• The Discovery: The authors realized they were missing the "odd" steps ($c^1$, $c^3$). It's like a dance where the spin happens on the "and" counts (the off-beats) rather than the main beats.
• The Result: By including these "odd" steps in their math, they found a new type of black hole: the Kerroll black hole.
  • In this version, the rotation isn't a feature of the main shape of space. Instead, the spin is a "subtle echo" or a "ghost" that appears in the lower-order details of the expansion.
  • It's as if the black hole is spinning, but the spin is so faint and hidden in the "odd" layers of reality that you have to look very closely to see it.

What Does This Mean for the Black Holes?

The paper calculates what these objects look like and how they behave:

1. They are real solutions: They aren't just mathematical tricks; they satisfy all the complex equations of this specific type of gravity.
2. They have "charges": Just like a spinning top has angular momentum, these black holes carry a measurable "spin charge."
3. They are different from normal black holes:
   • In a normal spinning black hole, space gets "dragged" around (frame-dragging). In the Kerroll black hole, this dragging effect is different or absent in certain ways because the "frozen" nature of time changes the rules.
   • The paths that particles take (geodesics) around these black holes are unique. For example, in the "dressed" version, the energy of a particle depends on its spin in a way that doesn't happen in our normal universe.

Summary

The authors successfully built two types of spinning black holes in a universe where the speed of light is zero.

1. One is a static-looking black hole that carries a hidden spin charge in its connection rules.
2. The other is a true analog of the Kerr black hole (the Kerroll), where the spin is revealed only by looking at the subtle, "odd-numbered" layers of the math that usually get ignored.

They call this the "Kerroll" black hole, a new object that exists only in the strange, frozen realm of Carroll gravity, proving that even in a world where nothing can move, things can still spin.

Technical Summary

Technical Summary: Kerroll Black Holes

Problem Statement
The construction of rotating black hole solutions within Carrollian gravity has historically been obstructed by a "no-go" theorem [55], which asserts that every stationary axisymmetric solution of magnetic Carroll gravity in dimensions $D > 3$ must be static. This limitation arises because the standard Carroll limit of the Kerr metric (derived via an even-power expansion in the speed of light, $c$) fails to satisfy the spatial Ricci scalar constraint ($R[g_{ij}] \approx 0$) required by magnetic Carroll gravity unless the rotation parameter vanishes. Consequently, existing Carroll black hole models, such as the Carroll–Schwarzschild solution, are restricted to non-rotating configurations. The paper addresses the question of whether rotating geometries can exist in Carroll gravity and, if so, how they can be constructed without violating the theory's constraints.

Methodology
The authors employ two distinct approaches to construct rotating Carroll black holes, both utilizing the Hamiltonian formulation of gravity and the ADM decomposition:

1. Canonical Approach in Magnetic Carroll Gravity:
   The authors analyze the asymptotic symmetries and boundary conditions of magnetic Carroll gravity. They exploit the freedom inherent in the Carroll-compatible connection, which is not fully determined by the metric data alone. In the Hamiltonian formalism, the canonical momenta $\pi_{ij}$ act as independent data (Lagrange multipliers) rather than being solvable in terms of the metric, unlike in General Relativity. The authors demonstrate that by dressing a static Carroll–Schwarzschild solution with non-vanishing canonical momenta (specifically $\pi_{r\phi}$), one can induce a non-zero angular momentum charge while maintaining the static metric geometry. This rotation is encoded entirely in the connection degrees of freedom.

2. Odd-Power Expansion of General Relativity:
   The authors propose a new gravitational theory derived from an expansion of General Relativity in odd powers of the speed of light ($c$), contrasting with the standard even-power expansions used to derive Electric and Magnetic Carroll gravity. Starting from the ADM action, they expand the spatial metric $g_{ij}$, its conjugate momentum $\pi_{ij}$, the lapse $N$, and the shift $N^i$ in powers of $c$ (including $c^1$ terms). By truncating this expansion at order $O(c^2)$ while retaining the odd-power fields, they define "Extended Magnetic Carroll Gravity." This theory contains Magnetic Carroll gravity as a subsector but possesses additional physical degrees of freedom associated with the odd-power fields. They then apply this framework to the Kerr metric, expanding its ADM variables to identify a solution that retains rotational information in the subleading odd-power terms.

Key Contributions and Results

• Rotating Solutions in Standard Magnetic Carroll Gravity:
  The authors construct rotating solutions in magnetic Carroll gravity by "dressing" the Carroll–Schwarzschild black hole with a rotational charge.

  • Mechanism: The rotation is not manifested in the spatial metric $g_{ij}$ (which remains static and diagonal) but is encoded in the canonical momenta $\pi_{ij}$ and the associated connection coefficients.
  • Geodesics and Symmetries: They compute the geodesics and Killing fields for these solutions. While the effective potential for geodesics remains similar to the non-rotating case (lacking stable circular orbits), the energy relation for time translation acquires a term dependent on angular momentum. Crucially, they show that frame-dragging effects, present in Lorentzian Kerr geodesics, are absent in this Carrollian construction.
  • Nature of the Solution: These solutions are intrinsically Carrollian; they have no known analog in Einstein gravity because the rotation is carried solely by the connection, a feature specific to the degenerate metric structure of Carroll spacetime.

• The "Kerroll" Black Hole:
  The authors introduce a new solution, the "Kerroll black hole," arising from the Extended Magnetic Carroll Gravity framework.

  • Construction: This solution is obtained by taking the $c \to 0$ limit of the Kerr metric within the odd-power expansion. The rotational parameter $a$ (related to angular momentum $J$) appears in the shift vector $N^i$ and the canonical momentum $\rho_{ij}$ at order $O(c)$.
  • Degrees of Freedom: The extended theory possesses more degrees of freedom than standard magnetic Carroll gravity (e.g., 5 degrees of freedom in $D=4$). The rotation is an "intrinsically odd-power effect," explaining why standard even-power expansions fail to capture rotating limits of the Kerr metric.
  • Charges: The authors compute the boundary charges for the Kerroll black hole. The angular momentum charge $J$ appears as a subleading boundary charge associated with the generator of subleading diffeomorphisms, scaling as $J \sim c J^{(1)}$. The mass/energy charge scales as $E \sim c^2 M$.

• Thermodynamic Outlook:
  The paper briefly discusses the thermodynamic implications. For the Kerroll black hole, the authors suggest that a first law of thermodynamics should emerge from the expansion of the Lorentzian counterpart, consistent with the $O(c^2)$ truncation. They propose relations for the entropy, temperature, and angular velocity expansions but note that an independent derivation within the extended theory remains an open task.

Significance
The paper claims to resolve the apparent contradiction between the existence of rotating black holes in Lorentzian gravity and the no-go theorems for magnetic Carroll gravity.

1. Intrinsic Carrollian Rotation: It demonstrates that rotation in Carroll gravity can be realized without a Lorentzian analog by utilizing the independent connection degrees of freedom, challenging the notion that staticity is a strict requirement for all Carroll solutions.
2. New Gravitational Sector: It establishes "Extended Magnetic Carroll Gravity" as a consistent theory capable of accommodating the Carroll limit of the Kerr metric. This provides a theoretical framework where rotational information is preserved through odd-power expansions, offering a more complete picture of the Carrollian limit of General Relativity.
3. Holographic and Cosmological Relevance: By constructing rotating solutions, the work opens avenues for studying rotating black holes in the context of flat space holography and Carrollian cosmology, areas where Carroll symmetries are increasingly relevant.

The authors conclude that while their constructions successfully bypass the no-go theorem, further investigation is required to fully understand the thermodynamic properties (specifically the first law) and the behavior of these solutions in higher dimensions.