Gravitational collapse in the vicinity of the extremal black hole critical point

This paper numerically investigates the threshold of gravitational collapse in spherically symmetric Einstein-Maxwell-Vlasov systems, revealing a transition from stationary horizonless shells to extremal black holes at a critical charge-to-mass ratio and suggesting a potential mechanism for forming extremal spinning black holes.

The Gist

Imagine the universe as a giant, cosmic tug-of-war. On one side, you have gravity, the invisible force trying to pull everything together into a tight, dense ball. On the other side, you have repulsion (like electric charge pushing things apart) and spin (angular momentum), which try to keep things spread out.

This paper, written by William East, is a story about what happens right at the very edge of this tug-of-war, specifically when we try to create a Black Hole.

The Big Question: How do you make a Black Hole?

Usually, if you pile enough matter together, gravity wins, and a black hole is born. But what if you are just shy of the amount needed? Or what if you have just a tiny bit too much repulsion?

Physicists have long known that there are two ways this "tipping point" can happen:

1. The "Type II" Way (The Classic): Like a glass of water that suddenly freezes. If you add a tiny bit more heat, it stays liquid; a tiny bit less, and it freezes instantly. In black hole terms, if you are just below the limit, the matter just bounces back and flies away. If you are just above, a black hole forms. The size of the black hole can be infinitesimally small.
2. The "Type I" Way (The "Stuck" Way): Imagine trying to balance a ball on top of a hill. It's possible to balance it there, but it's incredibly unstable. If you nudge it slightly one way, it rolls down the hill (forming a black hole). If you nudge it the other way, it rolls back down the other side (dispersing). In this scenario, the "threshold" isn't a tiny point; it's a specific, stationary state (like a shell of matter hovering in place) that is unstable.

The Discovery: A New "Critical Point"

The author of this paper was studying a specific type of matter: charged particles (like a cloud of electrons) falling inward. He wanted to see what happens when you tweak the charge and the spin of these particles.

He discovered a fascinating phase transition, similar to how water turns to ice or steam:

1. The "Hovering Shell" Zone:
   At first, he found a region where the matter could form a stationary shell. Imagine a hollow ball of charged particles hovering in space, perfectly balanced between gravity pulling in and electric repulsion pushing out. It's like a soap bubble that refuses to pop or collapse.

   • However, this bubble is unstable. If you add a tiny bit more charge, it flies apart. If you take a tiny bit away, it collapses into a black hole.
   • As he increased the charge-to-mass ratio of these shells, they got smaller and smaller, getting closer and closer to the size of a black hole.

2. The "Critical Point":
   Eventually, he reached a Critical Point. This is like the "critical temperature" where water can no longer tell the difference between liquid and gas.

   • At this exact point, the hovering shell becomes so compact that it essentially becomes a black hole, but one with zero surface gravity (an "extremal" black hole).
   • In physics, "surface gravity" is like the temperature of the black hole. An extremal black hole is "absolute zero."

3. The "Extremal Black Hole" Zone:
   Beyond this critical point, the rules change. The "threshold" solution is no longer a hovering shell; it is now an Extremal Black Hole itself.

   • If you have too much charge, the matter flies apart (disperses).
   • If you have just enough or too little charge, it collapses into a black hole.
   • The Magic: The time it takes for the matter to either collapse or fly apart follows a very specific mathematical pattern (a "power law") as you get closer to this critical point. It's like the universe is whispering a secret code about how it transitions from one state to another.

The Analogy: The "Perfectly Balanced" Seesaw

Think of the matter as a child on a seesaw.

• Normal Black Hole Formation: You push the child down, and they hit the ground (the black hole forms).
• The "Hovering Shell" (Type I): You try to balance the child perfectly in the middle. It's possible, but the slightest breeze (a tiny change in charge) makes them fall to one side or the other.
• The Critical Point: You keep adjusting the weight until the seesaw is balanced on a knife-edge.
• The "Extremal" Zone: You push past the knife-edge. Now, the "balanced" state isn't a seesaw anymore; it's a black hole that is "frozen" in time (extremal). If you add even a tiny bit of extra weight (charge), the child flies off the seesaw (disperses). If you remove a tiny bit, they fall into the hole.

Why Does This Matter?

1. Testing the Laws of Physics: This research helps test the "Third Law of Black Hole Mechanics," which says you can't cool a black hole down to absolute zero (zero surface gravity) in a finite amount of time. This paper suggests a way to do exactly that, potentially breaking the rule or showing us a loophole.
2. Spinning Black Holes: The author speculates that this same "critical point" behavior might happen with spinning black holes (like the ones in the movie Interstellar). If we can understand how charged matter collapses to an extremal state, we might learn how to create or understand the most extreme spinning black holes in the universe.
3. Universality: It shows that the universe has a "grammar" to how things collapse. Whether it's a star, a cloud of gas, or charged particles, they all seem to follow similar rules when they are on the brink of becoming a black hole.

In a Nutshell

William East used a supercomputer to simulate a cosmic tug-of-war. He found a "sweet spot" where matter can hover in a precarious balance before collapsing. As he pushed this balance to its limit, he discovered a new kind of black hole formation that acts like a phase transition (like water freezing). This gives us a new map for understanding the most extreme, "frozen" black holes in the universe and how they might be formed.

Technical Summary

1. Problem Statement

The paper investigates the threshold of gravitational collapse for spherically symmetric distributions of charged, collisionless matter (Vlasov matter) governed by the Einstein-Maxwell-Vlasov equations. The primary goal is to understand the critical phenomena near the formation of black holes, specifically focusing on the transition between two distinct regimes:

1. Type I Critical Collapse: Where the threshold solution is an unstable, stationary, horizonless shell.
2. Extremal Critical Collapse: Where the threshold solution is an extremal black hole (charge $Q$ equals mass $M$).

The study aims to map the phase diagram of these outcomes, characterize the scaling laws of the dynamical timescales near the critical point, and explore the connection between these regimes. A secondary motivation is to determine if this mechanism can be extended to rotating (Kerr) black holes to construct counterexamples to the third law of black hole mechanics.

2. Methodology

The author employs numerical relativity techniques to evolve the Einstein-Maxwell-Vlasov system.

• Physical Model:
  • Matter: A distribution of electrically charged particles with a uniform charge-to-rest-mass ratio ($q_0$) and angular momentum-to-rest-mass ratio ($\ell_0$).
  • Symmetry: Spherically symmetric spacetimes.
  • Equations: The Vlasov equation (describing particle distribution), Maxwell's equations (electromagnetism), and Einstein's equations (gravity).
• Numerical Scheme:
  • Coordinates: Ingoing Eddington-Finkelstein-like coordinates ($ds^2 = -ab^2dt^2 + 2bdtdr + r^2d\Omega^2$) to handle horizon formation and black hole interiors.
  • Particle-in-Cell (PIC): The distribution function is approximated by a collection of macro-particles following the Lorentz force equation.
  • Grid: A compactified radial coordinate $\bar{r} = r/(1+r)$ is used with a grid extending to spatial infinity.
  • Resolution: High-resolution simulations using $\sim 2.8 \times 10^7$ particles and fourth-order Runge-Kutta integration. Convergence studies confirm fourth-order accuracy.
• Initial Data Construction:
  • The process begins with stationary, unstable shell solutions where self-gravity balances charge repulsion.
  • These static shells are evolved in outgoing coordinates (or backwards in time) to allow numerical truncation errors to excite a linear instability, causing the shell to disperse.
  • The dispersed state is then time-reversed to create infalling shell initial data.
  • A one-parameter family of initial data is generated by varying $q_0$ (and sometimes $\ell_0$) while keeping particle positions and momenta fixed, allowing the system to cross the black hole formation threshold.

3. Key Contributions and Results

A. Phase Diagram and Critical Point

The study maps a phase diagram in the parameter space of total charge-to-mass ratio ($Q/M$) versus particle angular momentum ($\ell_0/M$):

• Regime 1 ($\ell_0 > \ell_c$): For higher angular momentum, the threshold solution is a stationary, horizonless shell. As $\ell_0$ decreases toward a critical value $\ell_c$, the shell becomes more compact, and its $Q/M$ ratio approaches unity from below.
• The Critical Point: At $\ell_c/M \approx 0.67$, the stationary shell solutions terminate. Here, the charge-to-mass ratio reaches $Q/M \to 1$, and the instability timescale diverges.
• Regime 2 ($\ell_0 < \ell_c$): Beyond the critical point (lower angular momentum), the threshold solution changes character. The threshold is no longer a horizonless shell but an extremal black hole ($Q=M$).

B. Scaling Laws and Timescales

The paper quantifies how the time to form a black hole ($T_{BH}$) or for the shell to disperse ($T_{disp}$) scales with the distance from the threshold parameter ($\delta = |Q - Q^*|$).

1. Near the Horizonless Shell Threshold ($Q^* < M$):

   • The system exhibits Type I critical collapse behavior.
   • The timescale scales logarithmically with the distance to the threshold:
     $$T \sim -\tau \log |Q - Q^*| + C$$
   • The instability timescale $\tau$ diverges as the critical point is approached:
     $$\tau \approx M [2(1 - Q^*/M)]^{-1/2} \approx \kappa^{-1}$$
     where $\kappa$ is the surface gravity of the near-extremal black hole that would form.

2. Near the Extremal Black Hole Threshold ($Q^* = M$):

   • When the threshold is an extremal black hole, the scaling changes.
   • Dispersion ($Q > M$): The time for the shell to disperse scales as a power law:
     $$T_{disp} \sim M (Q/M - 1)^{-1/2}$$
   • Collapse ($Q \le M$): Black holes form "promptly" (no logarithmic delay), but the matter reaches a minimum radius slightly inside the horizon before expanding (if $Q>M$) or settling (if $Q<M$).
   • The exponent $-1/2$ matches the scaling of the area/radius difference for near-extremal horizons and is derived from the Lyapunov exponent of unstable circular orbits near the horizon.

C. Connection to Phase Transitions

The results draw a strong analogy to thermodynamic phase transitions:

• The transition from horizonless shells to black holes resembles a discontinuous (Type I) phase transition terminating at a critical point.
• Beyond the critical point, the system enters a regime analogous to a supercritical fluid, where the distinction between "dispersing matter" and "black hole formation" becomes continuous in terms of the threshold solution (an extremal black hole), though the dynamical outcome (dispersal vs. capture) remains distinct.

4. Significance and Implications

• Extremal Black Hole Formation: The paper provides a concrete numerical route to forming extremal black holes in finite advanced time. This addresses an open question regarding the scaling relations of extremal critical collapse.
• Third Law of Black Hole Mechanics: The findings offer potential counterexamples to Israel's formulation of the third law (which states surface gravity cannot be reduced to zero in finite advanced time). By tuning parameters to the extremal limit, the surface gravity of the resulting black hole approaches zero in finite time.
• Extension to Rotating Black Holes: The author conjectures that a similar mechanism exists for rotating matter. Stationary solutions of rotating rings (approaching the extremal Kerr limit) are likely unstable. By time-reversing their dispersal, one could theoretically construct an infalling ring that forms an extremal Kerr black hole. This would extend the counterexample to the third law for rotating black holes.
• Universality: The work bridges the gap between Type I (unstable star/shell) and Type II (naked singularity) critical phenomena, showing that Type I solutions can evolve into a regime where the threshold is an extremal black hole, unifying different critical collapse behaviors under a single phase-transition-like framework.

Conclusion

William E. East demonstrates that the threshold of gravitational collapse for charged matter undergoes a qualitative change as the system approaches the extremal limit. The study successfully numerically constructs the transition from unstable horizonless shells to extremal black hole thresholds, derives the specific scaling laws for the dynamical timescales in both regimes, and proposes a pathway to realizing extremal Kerr black holes dynamically, thereby challenging the strict interpretation of the third law of black hole mechanics.