Twist and higher modes of a complex scalar field at the threshold of collapse

This study extends the investigation of critical collapse in axisymmetric massless complex scalar fields to higher angular modes ($m=1, 2$) using a novel $m$-cartoon symmetry reduction, revealing that while discrete self-similarity and universality persist within each mode, the critical exponents depend explicitly on $m$ and extremal black holes are excluded at the threshold.

The Gist

The Big Picture: The Edge of a Black Hole

Imagine you are standing on the edge of a cliff. If you take one step back, you are safe; if you take one step forward, you fall. In the universe, there is a similar "cliff" for matter. If you squeeze a cloud of energy (in this case, a complex scalar field) just right, it collapses into a black hole. If you squeeze it just a tiny bit less, it just puffs out and disappears.

The "edge" of this cliff is called the threshold of collapse. For decades, physicists have studied what happens right at this tipping point. They found that nature behaves in a very strange, magical way here: it repeats itself like a fractal (a pattern that looks the same no matter how much you zoom in) and follows strict mathematical rules.

This paper asks a new question: What happens if the collapsing matter is spinning?

The Setup: Spinning vs. Stationary

In previous studies, scientists looked at matter collapsing without spinning (like a perfectly round, still ball of dough). They found a very specific, universal pattern.

In this paper, the team (led by Krinio Marouda and colleagues) decided to look at matter that is spinning. Think of it like a spinning top made of energy.

• The Challenge: Spinning makes the math incredibly messy. It's like trying to balance a spinning top on a needle while someone is shaking the table.
• The Tool: They used a super-powerful computer code called bamps (which is like a high-tech kitchen simulator for gravity). They invented a new trick called the "m-cartoon" method.
  • Analogy: Imagine trying to film a spinning dancer. Instead of filming the whole 3D room, you film a 2D slice of the dancer and use math to "guess" what the rest of the dance looks like based on how many times they spin (the "m" value). This saved them from needing a supercomputer the size of a city.

The Experiment: Two Different Spins

They tested two specific types of spinning:

1. Mode 1 (m=1): The matter spins like a standard top.
2. Mode 2 (m=2): The matter spins in a more complex, double-loop pattern.

They cranked the "squeeze" dial up and down, trying to find the exact moment the black hole forms.

The Findings: What They Discovered

1. The "Echoing" Pattern (Discrete Self-Similarity)

When matter collapses at the threshold, it doesn't just collapse smoothly. It "echoes."

• Analogy: Imagine dropping a pebble in a pond. The ripples get smaller and smaller, but they look exactly the same, just scaled down.
• The Result: They found that for Mode 1, the echoes happened at a specific rhythm (a period of about 0.42). For Mode 2, the rhythm was much faster (about 0.09).
• The Takeaway: Nature is universal, but the "song" it sings depends on how the matter is spinning. A spinning top sings a different song than a still ball, but both follow a strict musical score.

2. The "Critical Exponent" (How Fast Things Grow)

When you are just barely on the side that forms a black hole, the size of the black hole grows according to a power law (a specific mathematical rule).

• The Result: For the slower spin (m=1), the growth rule was one number. For the faster spin (m=2), the growth rule was a completely different, smaller number.
• The Takeaway: The "speed limit" for how fast a black hole forms changes depending on the spin mode.

3. The "Extremal" Question (The Big Mystery)

Recently, some mathematicians proved that if you squeeze charged matter just right, you can create an extremal black hole—a black hole spinning at the absolute maximum speed allowed by physics (like a car hitting the speed limit of the universe).

• The Hypothesis: The authors wondered: "If we spin our matter really fast, will we naturally fall into this 'extremal' black hole trap?"
• The Reality Check: No.
  • Analogy: Imagine trying to balance a pencil on its tip. You might think that if you spin the pencil fast enough, it will stand up perfectly. But in this experiment, as they got closer to the black hole threshold, the "spin" of the resulting black hole actually disappeared.
  • The black holes formed were "boring" (non-extremal). The angular momentum (spin) scaled down faster than the mass. It's as if the universe refuses to let a black hole form at the "speed limit" in this specific setup.

4. No "Bifurcation" (No Splitting Centers)

In previous experiments with non-spinning but lumpy matter, the collapse sometimes got confused. Instead of one center collapsing, it would split into two or three competing centers, breaking the universal pattern.

• The Result: With the spinning matter, this confusion did not happen. The matter stayed focused on a single center.
• Why? The spin acts like a stabilizer. It forces everything to spiral neatly into the center, preventing the "fight" between matter and gravitational waves that causes the split.

The Conclusion: Why This Matters

This paper is a victory for our understanding of the universe's rules.

1. Universality is Robust: Even when you add the complexity of spinning, the "fractal" nature of black hole formation holds up. It's a fundamental law of nature.
2. Spin Matters: The specific "song" (the numbers) changes based on the type of spin, but the "music" (the existence of the pattern) remains.
3. No Extremal Black Holes Here: If you want to create a black hole spinning at the absolute maximum speed, you can't just use this specific type of spinning energy. You'd need a different recipe (perhaps involving massive particles or different initial conditions).
4. Stability: Spinning actually makes the collapse more orderly, preventing the chaotic splitting seen in non-spinning scenarios.

In a nutshell: The authors built a new mathematical "lens" to watch spinning energy collapse. They found that while the rhythm of the collapse changes with the spin, the universe still follows a strict, predictable, and beautiful fractal pattern, and it refuses to create "super-spinning" black holes in this specific scenario.

Technical Summary

1. Problem Statement

The paper investigates the critical phenomena (threshold of gravitational collapse) of a massless complex scalar field in axisymmetric spacetimes with non-zero angular momentum (twist).

• Context: Previous work established that in spherical symmetry ($m=0$), collapse exhibits universal discrete self-similarity (DSS) and power-law scaling (Type II critical phenomena). However, in axisymmetry without twist (real or complex fields), universality was observed to break down, often due to a competition between matter collapse and vacuum gravitational wave thresholds, leading to bifurcating collapse centers.
• Gap: It was unclear whether introducing angular momentum (via a specific azimuthal ansatz) would restore universality or lead to the formation of extremal black holes (where spin $J/M^2 \to 1$) at the threshold, a conjecture supported by recent mathematical proofs in other systems (e.g., charged systems).
• Goal: To determine if universality and DSS hold for complex scalar fields with angular momentum ($m \neq 0$) and to analyze the scaling of angular momentum and mass near the critical threshold.

2. Methodology

The authors employed high-resolution numerical relativity simulations using the pseudospectral code bamps.

• Physical Model: A massless complex scalar field $\psi$ minimally coupled to General Relativity (Einstein-Klein-Gordon system).
• Symmetry Ansatz: The field is constrained by the ansatz $\psi(\rho, \phi, z, t) = e^{im\phi}\psi_m(\rho, z, t)$, where $m$ is the azimuthal winding number. This imposes a specific angular mode while maintaining axisymmetry in the metric but allowing for non-vanishing twist and angular momentum.
• Numerical Innovations:
  • m-cartoon Method: A generalization of the "cartoon method" for symmetry reduction. Instead of evolving the full 3D domain, the code evolves on a 2D $(x, z)$ plane (specifically $y=0$) and reconstructs $\phi$-derivatives analytically using the winding number $m$. This handles the $\phi$-dependence of the scalar field while keeping the metric axisymmetric.
  • Generalized Horizon Finder: The apparent horizon finder (AHloc3d) was adapted to handle spacetimes with non-zero twist, allowing for the calculation of angular momentum on the horizon.
  • Initial Data: Smooth, regular initial data were constructed using the Extended Conformal Thin-Sandwich (XCTS) formulation to solve the constraint equations. The data consisted of pure spherical harmonic modes ($l=|m|$) with added symmetric pulses to ensure regularity at the origin for the pseudospectral code.
• Simulation Strategy:
  • Simulations were performed for $m=1$ and $m=2$ modes.
  • $hp$-adaptive mesh refinement was used to resolve the increasingly small scales near the critical threshold (DSS echoing).
  • Criticality was approached by tuning a parameter $a$ in the initial data to find the threshold value $a^*$ separating dispersion (subcritical) from black hole formation (supercritical).

3. Key Contributions

1. First Pseudospectral Simulations of Twisted Collapse: The study is the first to use a pseudospectral code to investigate the threshold of Kerr black hole formation in axisymmetry with angular momentum.
2. Development of the m-cartoon Method: A novel symmetry reduction technique that efficiently handles complex scalar fields with specific angular modes in axisymmetry, enabling high-precision evolution.
3. Extension to Higher Modes: The work extends previous $m=1$ results to the $m=2$ case, testing the robustness of critical phenomena across different angular modes.
4. Resolution of Extremality Conjecture: The study provides strong numerical evidence against the formation of extremal black holes at the threshold for this specific matter model and initial data family.

4. Key Results

A. Universality and Discrete Self-Similarity (DSS)

• Restoration of Universality: Unlike the twist-free axisymmetric case (where universality breaks down), the twisted cases ($m=1$ and $m=2$) exhibit universal behavior within their respective $m$-sectors.
• Mode-Dependent Critical Parameters: While universality holds within a fixed $m$, the critical exponents depend on the mode:
  • $m=1$: Echoing period $\Delta \simeq 0.42$, critical exponent $\gamma \simeq 0.11$. (Consistent with previous $m=1$ studies).
  • $m=2$: Echoing period $\Delta \simeq 0.09$, critical exponent $\gamma \simeq 0.035$.
  • Significance: The critical solution is not unique to the system but varies with the angular mode $m$.
• DSS Structure: The Kretschmann invariant and scalar field derivatives exhibit clear DSS oscillations (echoes) in both subcritical and supercritical regimes.

B. Scaling of Angular Momentum and Extremality

• Vanishing Spin: The dimensionless spin parameter $\chi_{AH} = J_{AH}/M_{AH}^2$ was found to vanish as the threshold is approached ($\chi_{AH} \to 0$).
  • For $m=1$, $J_{AH} \propto M_{AH}^{2.73}$.
  • For $m=2$, $J_{AH} \propto M_{AH}^{5.66}$.
• Exclusion of Extremal Black Holes: Since $\chi_{AH} \to 0$ rather than approaching 1, the threshold solutions are non-extremal. This contradicts conjectures that extremal Kerr black holes might form at the threshold in this system.
• Lyapunov Exponents: Analysis of the scaling relations using the perturbative model of Gundlach and Baumgarte revealed a negative subdominant Lyapunov exponent ($\lambda_1 < 0$) for the rotational mode. This implies that angular momentum is an irrelevant perturbation at the threshold, decaying faster than the mass.

C. Competition of Thresholds

• No Bifurcation: In contrast to twist-free axisymmetric collapse (where multiple centers of collapse can form), the twisted cases show a single center of collapse.
• Gravitational Wave Content: The ratio of the Weyl invariant (gravitational waves) to the Kretschmann scalar remains constant near the threshold. This suggests that while gravitational waves are present, they do not compete with the matter field to disrupt the universality or cause bifurcation. The angular momentum appears to force the collapse to concentrate around the symmetry axis, suppressing the vacuum threshold competition.

5. Significance

• Theoretical Insight: The results clarify the role of angular momentum in critical collapse. It demonstrates that while universality can be restored in axisymmetry by imposing a specific angular mode, the critical exponents are not universal constants of the theory but depend on the specific symmetry sector ($m$).
• Refutation of Extremality: The study provides numerical evidence that, for massless complex scalar fields with these initial data, the threshold of collapse does not lead to extremal Kerr black holes. This suggests that achieving extremality at the threshold may require different initial data families or massive fields (Type I collapse).
• Numerical Relativity: The successful implementation of the m-cartoon method and the generalized horizon finder in a pseudospectral code sets a new standard for simulating rotating, axisymmetric critical collapse with high precision.
• Future Directions: The authors highlight the need to investigate twisting vacuum axisymmetric spacetimes to fully understand the competition between matter and vacuum thresholds, as the current results suggest the vacuum threshold for rotating black holes might not exist in axisymmetry with a single horizon.