Horizon Multipole Moments of a Kerr Black Hole

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Imagine a black hole not as a terrifying, invisible void, but as a spinning, cosmic basketball. Like any spinning ball, it has a shape and a "spin" (angular momentum). In physics, we often try to describe the shape and spin of objects using multipole moments. Think of these as a "fingerprint" or a "recipe" that tells you everything about how the object is shaped and how it spins, without needing to see the object itself.

For a long time, physicists had two different ways to write down this recipe for a black hole's event horizon (the point of no return). This paper is like a detective story where the authors, Eric, Alexandre, and Marc, decide to test both recipes on the most famous black hole in the universe: the Kerr black hole (a spinning black hole).

Here is the breakdown of their investigation in simple terms:

1. The Two Different Recipes

The authors looked at two existing methods to measure the black hole's "fingerprint":

Recipe A (The Symmetry Method): This method assumes the black hole is perfectly symmetrical, like a spinning top. It's like trying to describe a basketball by assuming it's a perfect sphere. It works great if the ball is perfectly round and spinning smoothly. This method was proposed back in 2004.
Recipe B (The Generic Method): This is a newer method (from 2022) that doesn't assume the black hole is perfectly symmetrical. It's like describing a basketball that might be slightly squashed or wobbly. It's more flexible and can handle "messier" situations, like when a black hole is being squeezed by a neighbor star.

2. The Experiment: Measuring the Spinning Ball

The authors took the mathematical equations for a spinning black hole (the Kerr metric) and applied both recipes to it. They wanted to see:

Do both recipes give the same answer?
How do these "horizon fingerprints" compare to the "field fingerprints" (the gravity felt far away from the black hole)?

3. The Big Surprise: They Don't Match!

If you were baking a cake and used two different measuring cups, you'd expect the same amount of flour. But here, the authors found that Recipe A and Recipe B give different results for almost every measurement, except for the very basic ones (like the total mass or the very first spin measurement).

The Analogy: Imagine you are trying to measure the "bounciness" of a trampoline.
- Recipe A measures it by assuming the trampoline is a perfect circle.
- Recipe B measures it by looking at the actual fabric, which might be stretched unevenly.
- Even though it's the same trampoline, the two methods give you different numbers for how bouncy the corners are.

The paper shows that as you look at more complex details of the black hole's shape (higher "multipole moments"), the two recipes diverge further and further apart. It's like two people describing the same mountain: one says "it's a cone," the other says "it's a jagged peak," and they keep disagreeing the more details they add.

4. The "Small Spin" Connection

There is one special case where the recipes agree: when the black hole is spinning very slowly (almost not at all). In this "lazy" state, the black hole looks almost like a perfect sphere, so both recipes give similar answers.

However, as the black hole spins faster, the differences become huge. The authors found that the "Generic Method" (Recipe B) predicts a much more complex shape than the "Symmetry Method" (Recipe A) when the spin is high.

5. Why Does This Matter?

You might ask, "So what? They just have different numbers."

This is crucial for the future of astronomy:

Gravitational Waves: When two black holes crash into each other, they create ripples in space-time called gravitational waves. To understand these waves, we need to know exactly what the black holes look like before they merge.
The "Tidal" Effect: Imagine the Earth being pulled by the Moon; the Earth gets slightly squashed. Black holes do this too! If a black hole is near another star, it gets distorted.
- Recipe A (the old one) breaks down if the black hole gets distorted because it requires perfect symmetry.
- Recipe B (the new one) is built to handle distortion.

The authors conclude that Recipe B is the better tool for the real universe, where black holes are rarely perfectly symmetrical. It opens the door to understanding how black holes react when they are being "squeezed" by their neighbors, which is essential for interpreting the data from gravitational wave detectors like LIGO and Virgo.

Summary

This paper is a rigorous mathematical check-up. It proves that while we have two ways to measure a black hole's shape, they aren't interchangeable. The "Symmetry" way is a nice, simple approximation, but the "Generic" way is the real deal that works even when the black hole is messy, distorted, or spinning wildly. This helps physicists build better models of the violent, chaotic dance of black holes in our universe.

1. Problem Statement

The paper addresses the definition and computation of horizon multipole moments for a Kerr black hole. While the asymptotic (field) multipole moments of stationary spacetimes are well-understood (e.g., via the Hansen formalism, where $M_\ell + iS_\ell = M(ia)^\ell$ ), the definition of source multipole moments localized on the event horizon has been a subject of recent development.

Two distinct definitions exist in the literature for non-expanding horizons (NEHs):

Axisymmetric Definition (2004): Proposed by Ashtekar et al., applicable only to axisymmetric isolated horizons. It relies on a unique "unit round metric" constructed from the rotational Killing vector.
Generic Definition (2022): Proposed by Ashtekar et al., applicable to generic (non-axisymmetric) non-expanding horizons. It relies on a conformal decomposition of the horizon metric to a unit round metric, fixed by the condition of a vanishing area dipole moment.

The Core Question: Do these two definitions yield the same multipole moments for the Kerr black hole (which is both axisymmetric and a Killing horizon)? Furthermore, how do these horizon moments relate to the asymptotic field moments, particularly in the regime of small and large spin?

Prior to this work, closed-form expressions for these moments were either limited to low orders ( $\ell \le 8$ ) or restricted to specific definitions.

2. Methodology

The authors perform a rigorous comparative study using the Kerr metric in advanced Kerr coordinates $(v, r, \theta, \phi)$ .

Geometric Framework: They review the geometry of null hypersurfaces and non-expanding horizons, focusing on the Weyl scalar $\Psi_2$ , which encodes the horizon's shape and angular momentum structure.
Axisymmetric Case (Sec. IV & VI B):
- They utilize the rotational Killing vector $\eta^a$ to construct a privileged unit round metric $\mathring{q}^{axi}_{ab}$ .
- They derive a closed-form expression for the multipole moments $I^{axi}_\ell$ and $L^{axi}_\ell$ for all spherical harmonic degrees $\ell \in \mathbb{N}$ .
- The calculation involves integrating $\Psi_2$ against spherical harmonics defined on this specific metric.
Generic Case (Sec. V & VI C–E):
- They apply the 2022 definition, which requires finding a conformal factor $\psi$ such that $\mathring{q}_{ab} = \psi^2 q_{ab}$ is a unit round metric.
- They impose the vanishing area dipole moment condition to select a unique "canonical" metric from the 3-parameter family of conformal round metrics.
- They derive a closed-form expression for the conformal factor $\psi$ and the associated "electric" ( $E$ ) and "magnetic" ( $B$ ) potentials.
- They compute the resulting multipole moments $I_\ell$ and $L_\ell$ . Since the integral cannot be solved in terms of standard functions for arbitrary spin, they develop a SageMath numerical code capable of arbitrary-precision evaluation.
Small Spin Analysis: They perform asymptotic expansions for small spin parameters ( $a \ll M$ ) to derive analytical closed-form expressions for the generic case, identifying a sequence of rational coefficients $\alpha_\ell$ .

3. Key Contributions

Closed-Form Generalization: The authors provide the first closed-form expression for the axisymmetry-based horizon multipoles for arbitrary $\ell$ , generalizing previous results limited to $\ell \le 8$ .
Correction of Literature: They identify and correct a typo in the expression for the conformal factor $z(\theta)$ (or $Z$ ) found in the 2022 paper [22], providing a simpler and correct closed-form solution for the Kerr horizon.
Canonical Metric Derivation: They explicitly derive the unique unit round metric and the associated electric/magnetic potentials for the Kerr horizon under the generic definition.
Numerical Framework: They provide a publicly available numerical code (SageMath notebooks) to compute the generic horizon multipoles to arbitrary precision for any spin value.
Comparative Analysis: They systematically compare the two definitions and the horizon moments against the asymptotic Hansen field moments.

4. Key Results

A. Differences Between Definitions

The two definitions do not yield the same results for the Kerr black hole, even though the Kerr horizon is axisymmetric.

Discrepancy: The multipole moments differ for all $\ell \ge 1$ (for finite spin $a$ ) and for $\ell \ge 2$ in the small-spin limit.
Divergence: The ratio between the generic moments and the axisymmetric moments diverges as $\ell \to \infty$ .
Monopole and Dipole: Both definitions agree for the monopole ( $\ell=0$ ) and, in the small-spin limit, for the dipole ( $\ell=1$ ).

B. Comparison with Field Moments (Hansen Formula)

Scaling: In the small-spin limit ( $a \ll M$ ), the horizon multipoles scale as $(ia)^\ell$ , similar to the Hansen field moments.
Coefficients: However, the numerical coefficients differ.
- For the axisymmetric definition, the coefficient matches the field moment only for $\ell=0, 1$ .
- For the generic definition, the coefficients differ from the field moments for all $\ell \ge 2$ , even in the small-spin limit.
Large Spin: As the spin $a$ approaches the extremal limit ( $a \to M$ ), the deviation between horizon multipoles and field multipoles becomes significant (e.g., up to 40% for the quadrupole).

C. Specific Formulas

Axisymmetric Moments:
$I^{axi}_\ell + iL^{axi}_\ell = 2^{\ell-1}\sqrt{(2\ell+1)\pi} \frac{\ell!(\ell+2)!}{(2\ell+1)!} (i\hat{a})^\ell G_\ell(\hat{a}^2)$
where $G_\ell$ is a hypergeometric function bounded between 0 and 1.
Generic Moments (Small Spin):
$I_\ell + iL_\ell \sim \sqrt{(2\ell+1)\pi} (i\hat{a})^\ell \alpha_\ell$
where $\alpha_\ell$ is a sequence of rational numbers (e.g., $\alpha_2 = 6/5$ , $\alpha_3 = 44/35$ ) that grows slowly with $\ell$ .

D. Geometric Potentials

The authors provide explicit closed-form expressions for the electric potential $E = \ln(R\psi)$ and magnetic potential $B$ on the Kerr horizon, showing how the horizon geometry deviates from a perfect sphere as spin increases.

5. Significance and Future Prospects

Numerical Relativity: The axisymmetric definition is widely used in numerical relativity to diagnose black hole horizons in binary mergers. This paper clarifies that while useful, it represents a specific gauge choice that differs from the more general "canonical" definition.
Tidal Love Numbers (TLNs): The paper highlights that the generic definition is essential for defining surficial Tidal Love Numbers for spinning black holes. Unlike the axisymmetric definition, the generic definition does not assume axisymmetry, making it suitable for studying black holes under arbitrary tidal perturbations (e.g., in binary systems).
Fundamental Physics: The results demonstrate that in General Relativity, the "source" multipoles (defined on the horizon) and "field" multipoles (defined at infinity) are distinct due to the nonlinearity of the Einstein equations. The horizon geometry cannot be fully characterized by the asymptotic field moments alone.

In conclusion, this work provides the definitive mathematical framework for calculating horizon multipoles of Kerr black holes, resolving ambiguities between existing definitions and establishing the tools necessary for future studies of black hole tidal deformability and horizon dynamics.