Buchdahl limits in theories with regular black holes

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Imagine the universe as a giant, cosmic construction site. For decades, physicists have been trying to understand how heavy, dense objects like stars are built and how small they can get before they collapse into a black hole.

This paper is like a new set of blueprints that updates the old rules of construction. The authors, a team of theoretical physicists, are asking: "What happens to the rules of star-building if we change the laws of gravity itself?"

Here is a simple breakdown of their findings using everyday analogies.

1. The Old Rule: The "Buchdahl Limit"

In our current understanding of gravity (Einstein's General Relativity), there is a hard limit to how compact a star can be. Think of a star as a balloon filled with air.

As you squeeze the balloon (increase its density), the pressure inside goes up.
In Einstein's gravity, there is a point where you can't squeeze it any tighter. If you try, the pressure at the very center becomes infinite, and the balloon pops (or rather, the math breaks down).
This limit is called the Buchdahl Limit. It says a star must be at least 12.5% larger than its "event horizon" (the point of no return for a black hole). You can't have a star that is too close to being a black hole without it collapsing.

2. The New Playground: "Quasi-Topological" Gravity

The authors decided to test this rule in a different universe. They used a modified version of gravity called Quasi-topological (QT) gravity.

The Analogy: Imagine Einstein's gravity is a smooth, flat trampoline. QT gravity is like a trampoline made of super-elastic, high-tech springs that get stiffer the more you push them.
In this new gravity, empty space (vacuum) is special. If you try to make a black hole, the "singularity" (the infinitely dense point at the center) doesn't happen. Instead, the black hole is regular—it has a smooth, finite center, like a marble instead of a sharp needle.

3. The Big Discovery: Stars Can Be "Super-Compact"

The team asked: If the black holes in this new gravity are smooth and safe, are the stars also safe? Can they get smaller than the old limit?

The Answer: Yes, but with a catch.

The Catch: In this new gravity, stars can indeed get much smaller and denser than Einstein's rules allow. They can be more compact than in our universe.
The Pressure Problem: However, to hold these super-dense stars together, the "air" inside (the pressure) has to behave strangely.
- In normal stars, pressure pushes out to fight gravity.
- In these ultra-dense QT stars, the pressure sometimes has to push in (become negative) to keep the star from collapsing. This is like trying to hold a balloon together by sucking the air out of the center while the outside pushes in.
- If the star gets too dense, the pressure at the center still goes to infinity, just like in Einstein's gravity. The "smoothness" of the black holes doesn't automatically save the stars.

4. The "Exotic" Matter Zone

The paper maps out a landscape of possible stars.

Ordinary Stars: These are like normal balloons. They have positive pressure pushing out.
Exotic Stars: These are the super-compact ones. They require "exotic matter" (stuff with negative pressure) to exist. It's like a star made of anti-gravity glue.
The Danger Zone: If you try to make a star too dense without using this exotic glue, the center of the star becomes a singularity (a point of infinite curvature), even though the empty space black holes in this theory are supposed to be smooth.

5. The "Curvature" Surprise

One of the most interesting points is about curvature (how much space is bent).

In these QT theories, empty black holes have a "speed limit" on how much they can bend space. They can't bend it infinitely; there's a maximum cap.
The Twist: The authors found that ordinary stars (made of normal stuff) can break this cap. They can bend space more than the black holes do, reaching infinite curvature at their centers.
The Lesson: Just because the "empty" universe is safe and smooth doesn't mean the "stuff" inside it is safe. To keep the universe smooth everywhere, you need to put strict rules on the matter itself (like the "Dominant Energy Condition," which basically says "don't use weird negative pressure glue").

Summary: What Does This Mean?

Think of the universe as a video game.

Einstein's Gravity is the original game. Stars have a size limit, and if you break it, the game crashes (singularity).
Quasi-Topological Gravity is a "mod" (modification) where the game engine is upgraded so that black holes don't crash the game; they just become smooth, round objects.
The Paper's Finding: Even with this upgraded engine, if you try to build a star that is too small, the game still crashes at the center of the star unless you use special "cheat codes" (exotic matter) or follow strict building rules.

The Bottom Line:
Making gravity "nicer" (removing black hole singularities) doesn't automatically make stars "nicer." Stars can still become incredibly dense and dangerous, potentially even more so than in our current universe, unless the matter inside them follows very specific, strict rules. The authors have drawn a new map showing exactly where these limits are, revealing a much richer and stranger landscape of possible stars.

1. Problem Statement

The paper investigates the generalization of Buchdahl's theorem to $D$ -dimensional gravity theories involving higher-curvature corrections.

Context: In General Relativity (GR), Buchdahl's theorem establishes a maximum compactness limit ( $R > \frac{9}{8}r_S$ ) for perfect-fluid stars with positive, outward-decreasing energy density. Beyond this limit, the central pressure diverges, and the star becomes a black hole.
The Gap: The authors study Quasi-topological (QT) gravities, a class of theories where the vacuum spherically symmetric solutions are regular black holes (RBHs). In these vacuum solutions, curvature invariants are bounded by universal, mass-independent maximum values (satisfying Markov's limiting curvature hypothesis).
The Question: Does the presence of regular vacuum solutions imply that matter-filled stars in these theories also possess a universal bound on curvature and compactness? Specifically, can ordinary matter stars in QT theories reach arbitrarily high curvatures, violating the vacuum bounds, or does the theory enforce a new, stricter compactness limit?

2. Methodology

The authors employ a rigorous analytical approach to solve the stellar structure equations in QT gravities:

Theoretical Framework: They consider $D$ -dimensional QT gravities defined by an action with an infinite tower of higher-curvature terms. These theories satisfy a Birkhoff theorem, ensuring unique static, spherically symmetric (SSS) vacuum solutions that are regular black holes.
Stellar Model: They model stars as perfect fluids with stress-energy tensor $T_{ab} = (\rho + p)u_a u_b + p g_{ab}$ . The fluid is minimally coupled to gravity.
Equations of Motion:
- They derive the generalized TOV (Tolman-Oppenheimer-Volkoff) equation and a second-order ODE for the lapse function $\zeta = N\sqrt{f}$ .
- They introduce a "characteristic polynomial" $h(\psi)$ (where $\psi = 1-f/r^2$ ) to encapsulate the theory's specific higher-curvature corrections.
- They define an effective mean density $\bar{\rho}_{\text{eff}}$ which modifies the gravitational source term compared to GR.
Analytical Solutions:
- Constant Density: They solve the equations analytically for incompressible stars ( $\rho = \text{const}$ ), deriving exact expressions for the pressure profile $p(r)$ and metric functions.
- General Density: For stars with monotonically decreasing mean density ( $\bar{\rho}_{,r} \leq 0$ ), they adapt Buchdahl's original inequality arguments to the QT context, utilizing the properties of the function $g(\xi)$ derived from the effective density.
Energy Conditions: They analyze the validity of the Weak (WEC), Null (NEC), Dominant (DEC), and Strong (SEC) energy conditions across different solution regimes.
Curvature Analysis: They compute the Kretschmann scalar ( $K$ ) for the stellar interiors to test if the "Markov limiting curvature hypothesis" (bounded curvature) holds for matter configurations.

3. Key Contributions and Results

A. Generalization of Buchdahl Limits

The authors identify that the space of solutions for QT stars is bounded by three distinct regimes, which intersect at the parameters of an extremal regular black hole:

Divergent Central Pressure: The limit where $p_c \to \infty$ . This corresponds to the most compact stars.
Zero Pressure: Configurations where the pressure vanishes (often associated with critical densities).
Inner-Horizon Limit: Stars whose radius coincides with the inner horizon of the corresponding regular black hole.

B. Compactness in Constant-Density Stars

Density Dependence: Unlike GR, where the Buchdahl limit is universal ( $9/8$ ), in QT theories, the maximum compactness depends on the density.
Higher Compactness: Denser stars can be more compact than their GR counterparts. The ratio $R/r_S$ can be smaller than the GR Buchdahl limit.
New Buchdahl Limit: For a specific subclass of QT theories (where the characteristic polynomial satisfies certain monotonicity conditions), the authors prove that the absolute maximum compactness is attained by a constant-density star with a specific "saturation density" ( $\bar{\rho}_{\text{sat}}$ ). This yields a new, stricter Buchdahl bound:
$f(R_{\text{Buch}}) = \left(\frac{D-3}{D-1}\right)^2 \Delta_{\text{sat}}^2$
where $\Delta_{\text{sat}} < 1$ , implying the star can be more compact than in GR.

C. Violation of Limiting Curvature Hypothesis

This is the paper's most significant finding regarding the interplay between matter and regularity:

Vacuum vs. Matter: While vacuum solutions in QT theories have bounded curvature invariants (regular black holes), ordinary matter stars can violate these bounds.
Arbitrary Curvature: As a star approaches the divergent central-pressure limit, the curvature invariants at the center diverge, reaching arbitrarily high values.
Implication: The regularity of the vacuum sector is insufficient to guarantee regularity for matter-filled stars. To enforce bounded curvature in the presence of matter, additional constraints (such as the Dominant Energy Condition) must be imposed on the fluid. If the DEC is satisfied, the curvature remains bounded; if not, singularities can form even within a theory designed to resolve singularities.

D. Exotic Matter and Negative Pressure

The analysis reveals a region of "exotic" stars with negative pressure ( $p < 0$ ).
These configurations appear when the effective density becomes negative (densities exceeding a critical value).
In many cases, these high-density, negative-pressure stars are covered by horizons, effectively acting as the "exotic core" of a black hole, indistinguishable from the exterior vacuum solution.

E. Specific Examples

The authors apply their formalism to:

Einstein-Gauss-Bonnet (EGB) Gravity: Showing that for large densities, stars are more compact than in GR.
Lovelock Gravity: Analyzing the high-density limit where compactness saturates.
Hayward-like Stars: Using QT theories that admit Hayward regular black holes as vacuum solutions. They map the solution space in $D=4, 5, 6$ , identifying regions of ordinary matter, exotic matter, and unphysical singularities.

4. Significance and Conclusions

Refutation of Naive Expectations: The paper demonstrates that the existence of regular black holes in a theory does not automatically imply that all solutions (including stars) are regular. Matter couplings play a decisive role.
New Compactness Bounds: It establishes that in higher-curvature theories, stars can be significantly more compact than the GR Buchdahl limit, potentially approaching the Schwarzschild radius more closely depending on the density and theory parameters.
Role of Energy Conditions: The work highlights that enforcing the Dominant Energy Condition (DEC) is crucial for maintaining the "limiting curvature" property in the presence of matter. Without such constraints, the theory allows for singular stellar configurations despite having regular vacuum solutions.
Future Directions: The authors suggest that to fully resolve singularities in these theories, one must consider non-minimal couplings between matter and the infinite tower of curvature corrections, rather than just minimal coupling. They also propose extending these bounds to anisotropic fluids and charged stars.

In summary, the paper provides a comprehensive analytical framework for stellar structure in regular gravity theories, revealing that while vacuum regularity is preserved, stellar compactness limits are relaxed, and curvature bounds are easily violated by ordinary matter unless specific energy conditions are enforced.