On Schwarzschild black hole singularity formation

This paper argues that the formation of a Schwarzschild black hole from a non-singular configuration cannot occur as a continuous process, as the geometry inevitably develops a "Minkowski breaking" discontinuity and curvature singularities before the point source forms, implying that black hole emergence requires a noncontinuous, possibly quantized, framework.

The Gist

Imagine you are watching a movie of a star collapsing under its own gravity. In the classic story of General Relativity, this star shrinks down, gets smaller and smaller, and eventually turns into a "Schwarzschild black hole"—a perfect, point-like speck of infinite density at the center, surrounded by an event horizon (the point of no return).

For decades, physicists have assumed this transformation happens smoothly, like a balloon slowly deflating until it's just a tiny dot.

This paper argues that the movie is actually glitching.

The authors, Jorge Ovalle, Roberto Casadio, and Alexander Kamenshchik, suggest that the universe cannot smoothly turn a regular, non-singular star into a Schwarzschild black hole. Instead, the process hits a "hard reset" button. The fabric of spacetime itself tears apart before the black hole can fully form.

Here is the breakdown using simple analogies:

1. The Setup: The "Perfect" Black Hole vs. Reality

In the standard textbook version, a black hole is defined by two rules:

1. Empty Space: There is no matter inside the black hole's gravity field (except at the very center).
2. Perfect Symmetry: It looks the same from every angle.

The problem is that these two rules contradict each other. To have a spherical shape, you usually need a source (like a star) to create it. The standard solution is to say, "Okay, there's a tiny, invisible point-mass right in the middle." But mathematically, this is like saying, "The room is empty, except for this one speck of dust that is infinitely heavy." It's a mathematical trick, not a physical description of how a star actually collapses.

2. The Experiment: Watching the Collapse in Slow Motion

The authors decided to stop looking at the "after" picture (the finished black hole) and instead watched the "during" picture. They took a model of a Regular Black Hole—a theoretical object that has no singularity (no infinite point) and is smooth everywhere—and watched it evolve over time.

Think of this regular black hole as a smooth, squishy ball of jelly. As it collapses, it gets denser. The authors tracked the "shape" of the space inside this jelly ball as it shrank.

3. The Glitch: "Minkowski Breaking"

As the jelly ball collapses, the authors found a terrifying moment where the rules of geometry break down.

• The Analogy: Imagine you are walking on a smooth, flat road (this is "Minkowski space," the normal, flat fabric of our universe). As you walk toward the center of the collapsing star, the road suddenly stops being flat.
• The Break: At a specific moment during the collapse, the road doesn't just get bumpy; it snaps. The value of the "road" at the very center jumps from a smooth 1.0 to a jagged -0.5 instantly.
• The Name: They call this "Minkowski Breaking." It's like the universe trying to transition from a smooth video game to a pixelated, broken one. The smoothness of spacetime is lost.

4. The Disappearing Act: The Cauchy Horizon

Before the black hole can become the "point source" we expect, a safety barrier called the Cauchy horizon (a kind of inner boundary that protects the predictability of physics) shrinks and vanishes.

• The Analogy: Think of the Cauchy horizon as a safety net under a trapeze artist. As the artist (the collapsing star) falls, the net gets smaller and smaller. Just before the artist hits the ground, the net disappears completely.
• The Result: Once the net is gone, the laws of physics that let us predict the future stop working. The smooth evolution of the collapse is impossible to continue past this point.

5. The Sudden Jump: No Gradual Collapse

The most shocking finding is about the mass itself.

• What we expect: The star's mass slowly piles up at the center, like a pile of sand growing grain by grain until it forms a mountain.
• What the paper says: The mass does not pile up gradually. It stays spread out, and then, in a split second, everything suddenly snaps into the center point.

It's not a slow squeeze; it's a quantum leap. The universe refuses to let the mass slowly accumulate. Instead, the transition from a "regular star" to a "Schwarzschild point" is a discrete jump, like flipping a light switch from "Off" to "On" with no dimmer in between.

The Big Conclusion

The paper suggests that the Schwarzschild black hole (the point-like singularity) is not the natural, smooth end-state of a collapsing star.

If you try to build a black hole from a regular star, the universe hits a wall. The spacetime fabric tears ("Minkowski breaking"), the safety nets vanish, and the process forces a sudden, discontinuous change.

The Takeaway:
This implies that our current understanding of gravity (General Relativity) is incomplete. It suggests that to truly understand how black holes form, we need a new theory—perhaps one where spacetime is made of tiny, discrete "pixels" (quantized) rather than a smooth, continuous fabric. The universe might not allow a smooth transition to a singularity; it might require a fundamental "reboot" of reality to create one.

Technical Summary

Here is a detailed technical summary of the paper "On Schwarzschild black hole singularity formation" by Ovalle, Casadio, and Kamenshchik.

1. Problem Statement

The paper addresses a fundamental question in General Relativity: Can the Schwarzschild black hole (a spacetime with a point-mass singularity at $r=0$) emerge as the continuous, dynamical end-state of gravitational collapse from a non-singular configuration?

Standard General Relativity treats the Schwarzschild solution as a static vacuum field generated by a point mass at the origin. However, this ignores the dynamical process of collapse. While numerous "regular" (non-singular) black hole models exist, the mechanisms by which they could evolve into a standard Schwarzschild singularity are poorly understood. The authors investigate whether a smooth, continuous evolution from a regular interior geometry to the singular Schwarzschild geometry is mathematically possible without violating the principles of spacetime continuity.

2. Methodology

The authors employ a geometric, kinematic approach that does not rely on a specific theory of gravity (e.g., they do not solve Einstein's equations for a specific matter Lagrangian) but rather analyzes the constraints imposed by the geometry itself.

• Regular Schwarzschild Metric: They utilize a time-dependent extension of a regular Schwarzschild black hole model (Kerr-Schild class). The metric is defined by a mass function $m(r)$ that transitions from a regular interior to the standard Schwarzschild exterior ($M$) at the horizon $h$.
• Mass Function Construction: The interior mass function is constructed as a superposition of terms parameterized by a set of exponents $\{n_i\}$:
  $$m(r) \propto \sum C_k r^{n_k}$$
  This construction ensures the metric is $C^N$ differentiable (smooth) and satisfies boundary conditions at the horizon.
• Dynamical Evolution: To simulate collapse, the parameters $n_i$ are made time-dependent, $n_i(v)$, where $v$ is the ingoing Eddington-Finkelstein null coordinate. The evolution is constrained by the Null Convergence Condition ($R_{\mu\nu}l^\mu l^\nu \geq 0$), which implies $\dot{m} \geq 0$ (mass accumulation/monotonicity), corresponding to collapse ($\dot{n}_i < 0$).
• Singularity Analysis: The authors track the behavior of the metric function $f(v,r)$ and curvature scalars (like the Kretschmann scalar) as the parameters $n_i$ evolve from regular values ($n > 2$) toward the Schwarzschild limit ($n = -1$).

3. Key Contributions

• Identification of "Minkowski Breaking" (MB): The paper identifies a specific critical threshold where the spacetime structure fundamentally changes. As the evolution parameter $n$ approaches 0, the metric function at the origin, $f(v,0)$, undergoes a discontinuous jump.
• Discontinuity of the Cauchy Horizon: The study demonstrates that the inner (Cauchy) horizon, which exists in regular black holes, shrinks and vanishes precisely at the moment of "Minkowski breaking" ($n=0$).
• Non-Continuous Formation of the Point Source: The authors prove that the formation of the Schwarzschild point mass ($m(v,0) = M$) cannot occur gradually. Instead, the mass remains distributed until a discrete transition occurs, after which it instantaneously collapses to a point.
• Generalization of Collapse Constraints: The work extends previous findings (Ref. [11]) by showing that even with a highly flexible, multi-parameter interior geometry, the transition to a singularity is not smooth.

4. Key Results

The evolution of the system is characterized by three distinct phases based on the parameter $n(v)$:

1. Regular Phase ($n > 2$): The spacetime is a regular black hole with a de Sitter-like core. The metric is smooth, and the Cauchy horizon exists.
2. **Integrable Singularity Phase ($0 < n \leq 2$):** As$n$ decreases, curvature scalars diverge but remain integrable. The Cauchy horizon shrinks.
3. Minkowski Breaking ($n \to 0$):
   • At $n=0$, the metric function at the origin jumps discontinuously:
     $$f(v, 0) = \begin{cases} 1 & \text{for } n = 0^+ \\ -0.5 & \text{for } n = 0 \end{cases}$$
   • This signifies the loss of spacetime smoothness. The local Lorentz frame (Minkowski metric) cannot be defined at the origin.
   • The Cauchy horizon vanishes ($h_c \to 0$).
4. Schwarzschild Formation ($n = -1$):
   • After the discontinuity at $n=0$, the system evolves to $n=-1$, recovering the standard Schwarzschild solution.
   • The mass function at the origin exhibits a second discontinuity:
     $$m(v, 0) = \begin{cases} 0 & \text{for } n = -1^+ \\ M & \text{for } n = -1 \end{cases}$$
   • This indicates that the total mass $M$ does not accumulate gradually at $r=0$; rather, it collapses suddenly into a point source only after the spacetime continuity has already broken down.

Table I Summary (from the paper):

• $n=2, 1$: Regular, locally Minkowski.
• $n=0$: Minkowski Breaking; Cauchy horizon vanishes; Curvature $K \sim r^{-4}$.
• $n=-1$: Standard Schwarzschild (Point source); Curvature $K \sim r^{-6}$.

5. Significance and Implications

• Breakdown of Classical Continuity: The results strongly suggest that the transition from a regular matter distribution to a Schwarzschild singularity is not a continuous process within the framework of classical differential geometry. The spacetime manifold itself "breaks" before the singularity forms.
• Discrete Spacetime Structure: The necessity of a discrete jump (Minkowski breaking) to form a Schwarzschild black hole implies that a continuous description of spacetime is insufficient to describe the final stages of gravitational collapse.
• Quantization of Singularities: The authors propose that the emergence or regularization of gravitational singularities likely requires a non-continuous, possibly quantized framework. The "Schwarzschild point source" may not be a limit of a smooth evolution but a distinct state resulting from a phase transition in spacetime structure.
• Cosmic Censorship: The findings align with the Strong Cosmic Censorship Conjecture by showing that singularities inevitably develop, but they refine the understanding of how they form: not as a smooth limit, but as a catastrophic discontinuity.

In conclusion, the paper argues that the Schwarzschild black hole cannot be the continuous end-state of gravitational collapse. Instead, the formation of the singularity entails a discrete change in the fundamental structure of spacetime, necessitating new physics beyond classical General Relativity to describe the transition.