Dust collapse and horizon formation in Quadratic Gravity

This paper demonstrates that the spherically symmetric collapse of a uniform dust star in Quadratic Gravity inevitably leads to horizon formation and a singularity that forms faster than in General Relativity, despite the theory's rich phase space of static solutions.

The Gist

The Big Picture: Fixing the Engine of Gravity

Imagine General Relativity (Einstein's theory of gravity) as a very successful car engine that has powered our understanding of the universe for a century. It works great for planets, stars, and even black holes. However, physicists know that if you push this engine to its absolute limit—like inside a black hole or at the very beginning of the Big Bang—it starts to sputter and break down mathematically.

Quadratic Gravity is like a "tuned-up" version of that engine. The authors of this paper added some extra parts (mathematical terms involving the square of curvature) to the engine to make it more robust and mathematically "renormalizable" (meaning it doesn't break when you do quantum calculations).

The big question they asked is: If we use this new, upgraded engine, what happens when a star collapses? Does it still turn into a black hole with a horizon (a point of no return), or does it turn into something weird like a naked singularity (a point of infinite density visible to the whole universe) or a wormhole?

The Experiment: A Perfectly Uniform Star

To test this, the authors didn't look at a messy, rotating, lumpy star. They imagined the simplest possible scenario: a giant, perfectly round ball of "dust" (stars with no pressure, just falling inward) that is perfectly uniform inside. Think of it like a giant, perfectly smooth snowball collapsing under its own weight.

They assumed the inside of this snowball stays perfectly smooth and the same in every direction (homogeneous and isotropic). Because of this perfect symmetry, one of the complicated extra parts of their new engine (the "Weyl" term) doesn't actually do anything inside the ball. It's like having a turbocharger that only kicks in when the car is turning corners; since the snowball is falling straight down, the turbo stays off.

The Results: What Happened?

Here is what they found when they ran the simulation:

1. The Collapse is Faster and Faster

In standard Einstein gravity, a collapsing star takes a certain amount of time to crush itself into a singularity. In this new "Quadratic Gravity," the collapse happens faster.

• Analogy: Imagine two cars driving off a cliff. One is a standard car (General Relativity), and the other has a new, powerful engine (Quadratic Gravity). Both fall, but the new engine makes the second car plummet toward the ground slightly quicker. The math shows the "speed" of the collapse increases more violently in the new theory.

2. The "Point of No Return" Still Forms

This is the most important finding. In some exotic theories, people hoped that the new physics might prevent a black hole from forming, leaving behind a "naked singularity" (a singularity without a horizon, visible to everyone).

• The Result: The authors proved that a horizon still forms.
• Analogy: Even with the new engine, the "force field" (the event horizon) that traps light still appears. The star gets crushed, but it gets trapped behind a wall of no escape just like in standard gravity. This means that "horizonless" objects (like the mysterious "2-2 holes" or wormholes) cannot be the result of a simple, uniform star collapsing. If you see a uniform star collapse, it must become a black hole.

3. The Outside World is Messy (For Now)

In standard Einstein gravity, there is a rule (Birkhoff's Theorem) that says: "If a star collapses, the space outside it immediately looks like a perfect, static black hole." It's like a magician instantly swapping a messy room for a pristine one.

In Quadratic Gravity, this rule doesn't exist.

• The Result: The space outside the collapsing star cannot be static (still) immediately. It has to be "wiggly" and changing with time.
• Analogy: Imagine dropping a stone into a pond. In Einstein's world, the water instantly becomes a perfect, still circle. In this new theory, the water ripples, splashes, and wiggles for a long time before it finally settles down into a calm, static circle. The "static" black hole is only the final destination after a long period of chaotic settling.

The "No-Go" Theorems: What We Can't Have

The authors proved several "No-Go" theorems. Think of these as traffic signs that say "Do Not Enter" for certain types of solutions.

• You cannot match a collapsing star to a static, unchanging black hole solution immediately.
• You cannot match it to a solution where the "Ricci scalar" (a measure of how much space is curved) is constant.
• You cannot match it to a solution where the space inside and outside are perfectly symmetrical in a simple way.

Basically, the math is much stricter here than in Einstein's theory. The "junction" (the surface where the star meets the empty space) has to satisfy 10 different conditions instead of just a few. It's like trying to fit a square peg into a round hole, but the hole has 10 different locks that all have to click open at the exact same time.

The Conclusion: Black Holes Win (For Now)

The main takeaway is this: Even with this fancy new theory of gravity, a simple, uniform star collapsing still ends up as a black hole with a horizon.

The "naked singularities" and "wormholes" that exist as static solutions in this theory (solutions that just sit there) cannot be formed by a normal star collapsing. They are like "impossible shapes" that can exist in a math book but can't be built in the real world through a standard collapse.

However, there is a "But..."
The authors admit they made a simplifying assumption: the star was perfectly uniform. In the real world, stars are lumpy, rotating, and messy. If the star isn't perfect, that "extra engine part" (the Weyl term) might kick in and act like a repulsive force, potentially stopping the collapse or changing the outcome.

So, while this paper says "Black holes win for simple stars," the door is still open for "weird physics" if the star is messy, rotating, or if we eventually add quantum mechanics to the mix.

Summary in One Sentence

By simulating a perfectly smooth star collapsing under a new, upgraded theory of gravity, the authors found that it still inevitably forms a black hole with an event horizon, proving that the "weird" horizonless solutions found in this theory are likely not the result of real-world stellar collapse.

Technical Summary

1. Problem Statement

The paper addresses a fundamental open question in modified gravity: What are the physical endpoints of gravitational collapse in Quadratic Gravity (QG)?

While QG (an extension of General Relativity adding $R^2$ and $C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}$ terms to the action) admits a rich phase space of static, spherically symmetric solutions—including Schwarzschild black holes, Schwarzschild-Bach black holes, wormholes, and "2-2 holes" (horizonless ultra-compact objects with naked singularities)—it remains unclear which of these can actually form via a dynamical process. Previous work focused on static solutions; this paper investigates the dynamical evolution of a collapsing star to determine if horizons form or if naked singularities/wormholes can be the final state.

2. Methodology

The authors generalize the classic Oppenheimer-Snyder (OS) model of gravitational collapse to the framework of Quadratic Gravity.

• Model Setup: They consider a spherically symmetric star composed of a uniform ball of dust (pressureless fluid).
• Interior Geometry: The interior spacetime is assumed to respect the symmetries of the matter distribution (homogeneity and isotropy), adopting a Friedmann-Lemaître-Robertson-Walker (FLRW) metric.
  • Crucial Simplification: Because the FLRW metric is conformally flat, the Weyl tensor vanishes ($C_{\mu\nu\rho\sigma} = 0$). Consequently, the $C^2$ term in the action does not affect the interior dynamics. The interior evolution is driven solely by the Ricci scalar ($R$) and the $R^2$ term.
• Exterior Geometry: The exterior is described by a general time-dependent, spherically symmetric metric $ds^2 = -A(t,r)dt^2 + B(t,r)dr^2 + r^2 d\Omega^2$.
• Matching: The interior and exterior regions are matched at the stellar surface using generalized junction conditions derived for QG. These include the standard Darmois-Israel conditions plus additional constraints arising from the higher-order derivatives in the field equations.
• Analysis:
  • Numerical: The interior field equations (a third-order differential equation for the scale factor $a(\tau)$) are solved numerically.
  • Analytical: Late-time behavior is analyzed via asymptotic expansion.
  • Horizon Detection: The formation of horizons is determined by calculating the expansion parameters ($\theta_\ell, \theta_k$) of null geodesic congruences to identify trapped surfaces.
  • No-Go Theorems: The authors derive theorems to constrain the possible forms of the exterior metric based on the junction conditions.

3. Key Contributions

A. Interior Dynamics and Collapse Rate

• Faster Collapse: The numerical solution shows that the collapse proceeds faster than in standard General Relativity (GR).
• Singularity Formation: The scale factor $a(\tau)$ reaches zero in finite proper time, leading to a curvature singularity.
• Late-Time Behavior: While the GR collapse follows $a(\tau) \propto (\tau_0 - \tau)^{2/3}$, the QG collapse follows $a(\tau) \propto (\tau_0 - \tau)^{1/2}$. This confirms that the $R^2$ term accelerates the collapse toward the singularity.

B. Horizon Formation

• Trapped Surfaces: The authors calculate the expansion of outgoing and ingoing null rays. They demonstrate that during the collapse, the expansion of the outgoing rays ($\theta_\ell$) becomes negative, indicating the formation of a trapped region.
• Apparent Horizon: An inner apparent horizon forms at a finite time $\tau_{ah}$ when the star's radius equals the apparent horizon radius.
• Implication: The formation of an inner horizon necessitates the existence of an outer horizon to maintain causal structure. This effectively rules out horizonless solutions (like 2-2 holes or naked singularities) as the endpoint of this specific collapse scenario.

C. Junction Conditions and Exterior Constraints

• Extended Conditions: The paper explicitly derives the fourth-order junction conditions required for QG. In addition to the standard continuity of the metric and extrinsic curvature, QG requires the continuity of the Ricci scalar ($[R]=0$) and its normal derivative ($[\nabla_\perp R]=0$), as well as continuity of the Ricci tensor components.
• No-Go Theorems: The authors prove four theorems restricting the exterior metric:
  1. The exterior Ricci scalar $R_+$ cannot be constant.
  2. $R_+$ cannot depend solely on $t$ or solely on $r$; it must depend on both.
  3. No Static Exterior: A static exterior metric ($\partial_t A = \partial_t B = 0$) cannot be smoothly matched to a dynamical FLRW interior.
  4. No Single Metric Function: The exterior cannot satisfy $B(t,r) = 1/A(t,r)$ (a condition often found in GR vacuum solutions).
• Conclusion on Exterior: The exterior spacetime must be non-stationary (time-dependent) during the collapse. It can only approach a static black hole metric asymptotically at late times and large distances.

4. Results

1. Horizons Form: A uniform dust collapse with an FLRW interior in QG inevitably leads to the formation of a horizon. The endpoint is a black hole, not a naked singularity or a horizonless object.
2. Acceleration of Collapse: The $R^2$ term acts to accelerate the collapse compared to the standard Oppenheimer-Snyder scenario in GR.
3. Exterior Non-Stationarity: Unlike in GR where Birkhoff's theorem guarantees a static Schwarzschild exterior, QG requires a time-dependent exterior metric during the dynamical phase. The static Schwarzschild solution is only recovered asymptotically.
4. Exclusion of Exotic Solutions: The specific "2-2 hole" solutions and other horizonless static metrics found in the literature cannot be formed via the collapse of a homogeneous, isotropic dust ball.

5. Significance

• Physical Viability of QG Solutions: The paper provides the first dynamical test of QG static solutions. It suggests that while the theory allows for exotic static geometries (naked singularities, wormholes), they may not be the physical endpoints of realistic gravitational collapse starting from regular matter configurations.
• Cosmic Censorship: The results support a version of the Cosmic Censorship Conjecture within this specific QG setup, as the collapse leads to a black hole with a horizon rather than a naked singularity.
• Methodological Framework: The derivation of the full set of junction conditions for QG and the proof of the no-go theorems for static exteriors provide a rigorous framework for future numerical relativity simulations in higher-derivative gravity theories.
• Limitations and Future Work: The authors note that their assumption of a perfectly homogeneous interior suppresses the Weyl tensor. In more realistic scenarios with inhomogeneities, anisotropies, or rotation, the Weyl-squared term (which contains the massive spin-2 ghost) would become active. This term has repulsive properties that could potentially alter the collapse dynamics or prevent singularity formation, a topic left for future investigation. Additionally, the analysis is purely classical; quantum effects near the singularity remain unaddressed.