Bondi-type accretion onto a Kerr black hole in the kinetic regime

This paper derives an exact kinetic solution for stationary Bondi-type accretion onto a Kerr black hole, providing explicit formulas for mass, energy, and angular momentum accretion rates to analyze black hole growth and spin-down under constant or cosmologically decreasing ambient energy densities.

The Gist

Imagine a black hole not as a terrifying vacuum cleaner, but as a giant, spinning whirlpool in a cosmic ocean. Now, imagine that ocean isn't made of water, but of invisible, ghostly particles called "dark matter" or "kinetic gas" that don't bump into each other like billiard balls, but instead glide past one another like a swarm of silent, non-colliding fireflies.

This paper by Patryk Mach, Mehrab Momennia, and Olivier Sarbach is like a master chef's recipe book for predicting exactly how fast that whirlpool eats these fireflies, and how the whirlpool's spin changes as it does so.

Here is the story of their discovery, broken down into simple concepts:

1. The Problem: A Spinning Top in a Storm

For decades, physicists have tried to figure out exactly how a Kerr black hole (a black hole that spins) eats matter.

• The Old Way: Previous attempts assumed the gas acted like a thick, sticky fluid (like honey) or moved in very simple, non-spinning ways. These models were either too simple to be real or too mathematically messy to solve exactly.
• The New Approach: The authors decided to treat the gas as a "collisionless" swarm. Think of it like a school of fish that never touches another fish; they only react to the gravity of the black hole. Because these particles don't crash into each other, the math becomes surprisingly clean and solvable.

2. The Solution: A Cosmic Map

The team derived an exact solution. In the world of physics, "exact" is a huge deal. It means they didn't just guess or approximate; they wrote down a precise mathematical map that tells you exactly where every particle goes.

• The Trap: Some particles are like moths to a flame; they spiral in and get swallowed by the black hole (the "absorbed" orbits).
• The Escape: Other particles are like birds flying near a storm; they get pulled in, swing around the black hole, and then get flung back out into space (the "scattered" orbits).

The authors calculated exactly how many particles fall into each category based on how fast the black hole is spinning and how hot the gas is.

3. The Spin-Down Effect: The Ice Skater Analogy

Here is the most fascinating part of their discovery. When a spinning black hole eats this gas, something counter-intuitive happens: it slows down.

Imagine an ice skater spinning with their arms out. If they pull their arms in, they spin faster. But imagine if the skater was eating heavy weights thrown at them from the side. The impact of the weights would actually make the skater spin slower.

• The Finding: As the black hole swallows the gas, it gains mass (it gets heavier), but it loses its spin speed. The gas acts like a brake.
• The Result: Over billions of years, a super-fast spinning black hole will gradually slow its rotation as it feeds on this gas, eventually becoming more like a non-spinning (Schwarzschild) black hole.

4. Two Scenarios: The Big Bang and the Neighborhood

The authors used their new formulas to test two different cosmic scenarios:

Scenario A: The Primordial Black Hole (The Baby Black Hole)
Imagine a tiny black hole formed right after the Big Bang, surrounded by a hot, dense soup of particles.

• The Prediction: Even if this baby black hole is spinning like a top, the heat and density of the early universe are so intense that it will eat so much matter so quickly that it will grow massive and stop spinning in a relatively short cosmic time. It's a "robust" eater.

Scenario B: The Supermassive Black Hole (The Giant in the Neighborhood)
Think of the giant black hole at the center of our galaxy (or M87). It is surrounded by a very cold, thin cloud of dark matter.

• The Prediction: For this black hole to grow significantly by eating dark matter, that dark matter must be extremely cold. If the particles are too hot (moving too fast), they will zip right past the black hole without getting caught.
• The "M87" Test: They applied this to the famous M87 black hole. They found that for it to grow as much as we think it has, the dark matter around it must be "colder" than almost anything else in the universe. It's like trying to catch a bullet with a net made of spider silk; the bullet (particle) has to be moving incredibly slowly to get caught.

The Takeaway

This paper is a major step forward because it gives us a precise, mathematical "rulebook" for how spinning black holes eat and change.

• Before: We had rough sketches.
• Now: We have a high-definition blueprint.

It tells us that black holes aren't just static monsters; they are dynamic engines that change their own spin and size depending on what they are eating. And perhaps most importantly, it suggests that the "temperature" of the invisible dark matter in our universe is a key factor in how the giants at the centers of galaxies evolve over time.

Technical Summary

1. Problem Statement

The search for an exact analytical solution describing Bondi-type accretion (stationary, nearly spherically symmetric inflow) onto a rotating Kerr black hole has been a long-standing challenge in general relativity.

• Previous Limitations: Existing exact solutions, such as the 1988 work by Petrich, Shapiro, and Teukolsky, rely on highly idealized assumptions: zero-vorticity potential flow and an "ultra-hard" equation of state. These models do not generalize well to realistic fluids or collisionless gases.
• The Gap: There was no exact solution for a collisionless kinetic gas (described by the Vlasov equation) accreting onto a Kerr black hole. While numerical simulations and approximate models exist, they lack the rigorous analytical foundation required to derive precise accretion rates and characteristic time scales for black hole evolution.

2. Methodology

The authors derive an exact solution by modeling the accreting matter as a collisionless gas governed by the Vlasov equation (collisionless Boltzmann equation) in the Kerr spacetime.

• Kinetic Framework:

  • The gas is described by a one-particle distribution function $f(x, p)$ defined on the future mass shell.
  • The Vlasov equation is linear, allowing for solutions where $f$ depends only on the constants of motion (energy $E$, angular momentum $L_z$, and Carter constant related parameters).
  • The authors assume an asymptotic distribution function $F_\infty(E)$ (specifically Maxwell-Jüttner distributions) at infinity, representing a gas at rest and homogeneous.

• Phase Space Analysis:

  • The authors utilize the separability of geodesic equations in Kerr spacetime.
  • They introduce adapted phase-space coordinates $(E, Q, \chi)$ to distinguish between unbound orbits originating from infinity.
  • Orbits are categorized into two types:
    1. Absorbed: Orbits that plunge into the black hole (crossing the event horizon).
    2. Scattered: Orbits deflected by the centrifugal barrier and returning to infinity.
  • Bound orbits and white hole emanations are neglected.

• Integration Strategy:

  • The particle current density $J^\mu$ and energy-momentum tensor $T^{\mu\nu}$ are computed by integrating the distribution function over the relevant phase-space regions.
  • The integration limits are determined by a critical parameter $Q_c$, which separates absorbed from scattered orbits. This parameter depends on the black hole spin $a$, radial coordinate $r$, and polar angle $\vartheta$.

3. Key Contributions

1. Exact Analytical Solution: The paper provides the first exact solution for Bondi-type accretion of a kinetic (Vlasov) gas onto a Kerr black hole. The solution expresses physical quantities (current density, accretion rates) as explicit integrals that can be evaluated numerically.
2. Analytic Approximations: The authors derive simplified formulas for accretion rates in the slow-rotation limit (expanding up to third order in the spin parameter $\alpha = a/M$) and in the low/high temperature limits.
3. Accretion Rate Formulas: Explicit expressions are derived for the rates of:
   • Particle number accretion ($\dot{N}$)
   • Energy accretion ($\dot{E}$)
   • Angular momentum accretion ($\dot{J}$)
4. Evolutionary Dynamics: The derived rates are used to formulate differential equations governing the time evolution of the black hole's mass ($M$) and spin parameter ($\alpha$).

4. Key Results

A. Accretion Rates and Spin Dependence

• The accretion rates depend on the asymptotic energy density $\varepsilon_\infty$, the black hole mass $M$, and the spin parameter $\alpha$.
• Spin Effect: The black hole spin has a weak effect on the mass/energy accretion rate. The slow-rotation approximation agrees with exact numerical results to within 4% for energy accretion and 5% for angular momentum accretion, even for high spins ($\alpha \leq 0.99$).
• Angular Momentum: The accretion of angular momentum is significant. The gas carries angular momentum that opposes the black hole's rotation, leading to a spin-down effect.

B. Black Hole Evolution Equations

The time evolution of the mass and spin is governed by:
$$\frac{dM}{dt} = -\dot{E} \propto \varepsilon_\infty M^2 (1 - B\alpha^2)$$
$$\frac{d\alpha}{dt} \propto -\varepsilon_\infty M \alpha (1 - C\alpha^2)$$

• Result: As the black hole accretes mass, its spin parameter $\alpha$ decreases (spin-down). The magnitude of the spin decreases at a rate proportional to the ambient energy density.

C. Application Scenarios

The authors apply these results to two specific cosmological scenarios:

1. Primordial Black Holes (PBHs) in the Early Universe:

   • Assuming accretion of hot dark matter in a radiation-dominated era ($\varepsilon_\infty \propto t^{-2}$).
   • Finding: If the initial mass is sufficiently large ($\kappa > 32/81$), the black hole mass diverges to infinity in a finite time $t_\infty$, while the spin parameter $\alpha$ tends to zero. This suggests that rapid accretion can effectively "spin down" primordial black holes.

2. Supermassive Black Holes (SMBHs) in Galaxies:

   • Considering an SMBH (e.g., M87*) accreting cold dark matter from a reservoir of constant density.
   • Finding: For significant mass growth to occur over cosmological timescales (e.g., $10^9$ years), the dark matter must be extremely cold.
   • Quantitative Estimate: For M87*, the ratio of thermal energy to rest mass energy must satisfy $k_B T / mc^2 \sim 10^{-18}$. This is well below the cosmological upper bound for cold dark matter ($10^{-14}$), implying that standard cold dark matter models are consistent with this accretion scenario, provided the gas is sufficiently cold.

5. Significance

• Theoretical Rigor: This work bridges the gap between idealized fluid models and realistic kinetic descriptions of accretion in rotating spacetimes. It validates the use of kinetic theory for Bondi-type flows in strong gravity.
• Astrophysical Implications:
  • It provides a robust framework for estimating the growth rates of black holes in dark matter halos.
  • It highlights the mechanism of spin-down via kinetic accretion, suggesting that even rapidly rotating black holes may lose their spin over time if surrounded by a dense, collisionless medium.
  • It offers constraints on the temperature of dark matter required to explain the mass growth of supermassive black holes, reinforcing the "cold" nature of dark matter in these environments.
• Future Utility: The explicit integral forms and Mathematica notebooks provided allow researchers to easily compute accretion rates for various distribution functions and spin parameters without resorting to full-scale numerical relativity simulations.