Dust collapse in asymptotic safety: a path to regular black holes

This paper demonstrates that modeling dust collapse within Asymptotically Safe gravity, where the gravitational coupling vanishes at high energies due to a specific matter-geometry coupling function, leads to the formation of regular black holes that are entirely free of singularities.

The Gist

The Big Problem: The "Infinite" Glitch

In our current understanding of the universe (General Relativity), if you squeeze enough matter into a small enough space, it collapses into a black hole. But there's a catch: the math predicts that at the very center, everything gets crushed into a single point of infinite density.

Think of this like a video game glitch. If you try to calculate the physics of a black hole's center using standard rules, the numbers go to infinity, and the game crashes. Physicists call this a "singularity." It's a sign that our rules break down. We need a new rulebook that works even when things get incredibly small and heavy.

The New Rulebook: "Asymptotic Safety"

The authors of this paper propose a new way to look at gravity, based on a theory called Asymptotic Safety.

Imagine gravity is like a rubber band. In our normal world, the more you stretch it (or the more mass you add), the stronger the pull gets. But in this new theory, when you get to the tiniest, most energetic scales (like the center of a collapsing star), gravity starts acting differently. It becomes "antiscreening."

The Analogy: Imagine gravity is a magnet. Usually, the closer you get to the magnet, the stronger the pull. But in this theory, once you get too close (at the Planck scale), the magnet suddenly starts to lose its power. The gravitational force actually vanishes or becomes very weak right at the center.

The Experiment: Collapsing Dust

To test this, the authors modeled a star collapsing. They didn't use a complex, hot, exploding star; they used a simple cloud of "dust" (particles with no pressure, just falling inward).

1. The Setup: They wrote a new equation (a "Lagrangian") that mixes the matter (the dust) with gravity.
2. The Twist: They added a special "coupling function" (let's call it the Magic Switch, or $\chi$). This switch is controlled by the Asymptotic Safety theory.
3. The Result: As the dust cloud collapses and gets denser, the "Magic Switch" turns on. Because of the Asymptotic Safety rules, the gravitational pull weakens as the density increases.

The Outcome: A "Regular" Black Hole

In standard physics, the dust cloud keeps shrinking until it hits zero size (the singularity).

In this paper's model, the dust cloud shrinks, but as it gets tiny, the gravity weakens so much that it stops the collapse.

• The Bounce: Instead of crushing into a point, the cloud reaches a tiny, finite size and stops. It doesn't bounce back out like a spring (in this specific model), but it simply stops shrinking.
• No Glitch: Because the cloud never reaches zero size, there is no "infinite density." The center of the black hole is a tiny, dense, but smooth ball of matter. The "glitch" is fixed.

The Exterior: What the Outside Sees

The paper also looks at what happens outside the collapsing cloud.

• Matching the Puzzle: They used mathematical "sewing" (junction conditions) to attach the inside of the collapsing cloud to the empty space outside.
• The Result: The outside looks almost exactly like a normal black hole (the Schwarzschild solution) far away. But as you get closer to the center, the math changes.
• The Horizon: Depending on the specific settings of their "Magic Switch," the black hole might have:
  • Two horizons (an outer and an inner one).
  • One horizon (the critical point).
  • No horizon at all: If the "Magic Switch" is set a certain way, the collapse stops before an event horizon ever forms. The result is a "scalar remnant"—a tiny, dense object that looks like a black hole but isn't quite one, and definitely has no singularity inside.

The Takeaway

This paper suggests that if we accept the rules of Asymptotic Safety (where gravity fades away at the highest energies), we don't need to worry about the "infinite" singularity problem.

The Metaphor:
Imagine a car driving toward a cliff.

• Old Theory (General Relativity): The car drives off the cliff and falls forever into an infinite abyss.
• This Paper's Theory: As the car gets to the edge of the cliff, the road suddenly turns into thick, sticky mud. The car slows down, stops right at the edge, and never falls in. The "abyss" (singularity) is avoided, and the car (the black hole) remains safe and intact, just very small.

The authors conclude that this provides a consistent, mathematically sound way to describe how a black hole forms without breaking the laws of physics at the center.

Technical Summary

Problem Statement
General relativity predicts that gravitational collapse leads to spacetime singularities concealed within event horizons. The existence of these singularities suggests the need for a theory beyond classical general relativity that ensures geodesic completeness. While various models of "regular black holes" (BHs) exist—often constructed by modifying the static Misner-Sharp mass to ensure convergence to zero at small distances, or by matching non-singular interior models to static exteriors—deriving these solutions from a physically plausible effective Lagrangian remains a significant challenge. Furthermore, many existing approaches struggle to maintain the conservation of energy-momentum tensors or rely on 2D dilaton gravity models that are difficult to generalize to 4D without a motivated Lagrangian formulation.

Methodology
The authors propose a resolution by extending the framework of Markov and Mukhanov, utilizing an effective Lagrangian for Quantum Einstein Gravity (QEG) within the Asymptotically Safe (AS) gravity paradigm. The core methodology involves:

1. Effective Action Formulation: The system is described by an action $S = \frac{1}{16\pi G_N} \int d^4x \sqrt{-g} [R + 2\chi(\epsilon)L]$, where $L = -\epsilon$ is the matter Lagrangian for a dust fluid, and $\chi(\epsilon)$ is a multiplicative coupling function between gravity and matter.
2. Asymptotic Safety Constraints: The functional form of $\chi$ is deduced from the Renormalization Group (RG) flow of the gravitational coupling $G$ near the Reuter ultraviolet (UV) fixed point. A key assumption is that the presence of matter does not significantly deform the RG flow or compromise the fixed point. This leads to a running Newton constant $G(\epsilon) = G_N / (1 + \xi \epsilon)$, where $\xi$ is a dimensionful scale related to the UV fixed point.
3. Field Equations and Dynamics: The variation of the action yields field equations where the effective Newton constant $G(\epsilon)$ and an effective cosmological constant $\Lambda(\epsilon)$ emerge dynamically. For a spherically homogeneous dust collapse ($p=0$), the authors derive the evolution of the scale factor $a(t)$ and the effective Misner-Sharp mass.
4. Junction Conditions: To construct a global spacetime, the collapsing interior is matched to a static exterior geometry using Israel's junction conditions (refined by Senovilla et al.). The continuity of the metric and extrinsic curvature across the boundary determines the exterior mass function $M(R)$.

Key Contributions and Results

• Derivation of a Regular Exterior: The primary result is the derivation of the exterior metric function $M(R)$ directly from the interior collapse dynamics governed by the AS effective Lagrangian. The resulting mass function is:
  $$M(R) = \frac{R^3}{6\xi} \log\left(1 + \frac{6M_0\xi}{R^3}\right)$$
  This expression ensures that the spacetime is regular everywhere. In the limit $\xi \to 0$ (classical limit), it recovers the Schwarzschild solution. In the small $R$ regime, the mass function behaves such that the curvature remains finite, avoiding singularities.
• Singularity Resolution via Antiscreening: The model demonstrates that the "antiscreening" behavior of gravity at high energies (Planckian scales) generates an effective repulsive force. This is mediated by the running cosmological constant $\Lambda(\epsilon)$, which becomes negative and large at high densities, halting the collapse before a singularity forms. The scale factor $a(t)$ approaches zero only as $t \to \infty$ (specifically $a(t) \sim e^{-t^2/4\xi}$), ensuring the spacetime is geodesically complete.
• Conservation Laws: Unlike many alternative approaches, the authors note that in their theory, both the standard energy-momentum tensor and the effective energy-momentum tensor are conserved.
• Horizon Structure: The analysis of the horizon formation reveals a dependence on the parameter $\xi$ relative to a critical threshold $\xi_{cr}$.
  • For $\xi < \xi_{cr}$, the solution possesses both an inner and an outer event horizon.
  • For $\xi = \xi_{cr}$, the horizons merge into a single degenerate horizon.
  • For $\xi > \xi_{cr}$, no event horizons form, leaving a scalar remnant.
  • The apparent horizon in the interior reaches a minimum value and then expands, implying that for certain boundary conditions, the collapse process may not be covered by a horizon at all.

Significance and Claims
The paper claims to offer a new perspective on the formation of regular black holes by demonstrating that a global, singularity-free spacetime can be constructed from a physically motivated effective Lagrangian derived from Asymptotically Safe gravity. The authors emphasize that their approach successfully bridges the gap between a non-singular collapsing interior and a static exterior without ad-hoc modifications to the mass function.

The significance lies in the consistency of the model: it recovers standard general relativity in the low-energy limit, respects energy-momentum conservation, and provides a mechanism (the running of $G$ and the induced $\Lambda$) that naturally resolves the singularity problem through the antiscreening nature of gravity at high energies. The authors acknowledge that while the current model uses an idealized pressureless dust fluid, the framework is designed to be generalizable to more realistic equations of state and accurate RG trajectories in future work.