Cosmological Black hole Candidates: A Detailed Analysis of McVittie, Culetu, Sultana-Dyer, and Glass-Mashhoon Spacetimes

This paper analyzes trapping horizons in various dynamical spacetimes to conclude that while the McVittie and Glass-Mashhoon solutions fail to represent cosmological black holes, the Culetu and Sultana-Dyer metrics can successfully describe them in a matter-dominated early universe under specific energy conditions.

The Gist

Imagine the universe as a giant, expanding balloon. Now, imagine sticking a heavy, dense marble (a black hole) onto the surface of that balloon. As the balloon inflates, the space around the marble stretches. The big question physicists have been asking for decades is: Does the marble stay a black hole, or does the stretching of the balloon tear it apart or change its nature?

This paper is like a team of detectives (the authors from the University of Tehran) investigating four different "crime scenes" (mathematical models of the universe) to see if they can find a Cosmological Black Hole—a black hole that survives and behaves correctly while the universe expands around it.

To solve this, they use a special flashlight called Hayward's Formalism. Instead of looking at the whole universe at once (which is too hard to see), they look for specific "trapping horizons." Think of these horizons as invisible fences:

• The Black Hole Fence (FOTH): A future boundary where light tries to escape but gets pulled back in. This is the "event horizon" of a real black hole.
• The Universe Fence (PITH): A past boundary where light is being pushed apart by the expansion of the universe (like the edge of our observable universe).

For a model to be a "Cosmological Black Hole," it needs both fences to exist at the same time.

Here is the breakdown of the four suspects they investigated:

1. The Classic Suspect: McVittie Spacetime

The Story: This is the oldest and most famous model, proposed in 1933. It's like the "standard model" of a black hole in an expanding universe.
The Investigation: The detectives looked closely at the "fences." They found that while the universe expands, the black hole fence (FOTH) never actually forms.
The Verdict: Not a Cosmological Black Hole.

• Analogy: Imagine trying to build a house (the black hole) on a beach while the tide is coming in. In the McVittie model, the water rises so fast that the foundation for the house never gets laid. You get a "ghost" of a black hole, but it lacks the essential "future outer" fence required to be a real black hole in this framework. It only has "past" fences, which act more like white holes (things that spit stuff out) or just cosmological horizons.

2. The Conformal Twins: Culetu and Sultana-Dyer Spacetimes

The Story: These are newer models. They are like "conformal" versions of the classic Schwarzschild black hole, meaning they are stretched and squeezed to fit the expanding universe, but they keep the same basic shape.
The Investigation: The detectives found that these models do have both fences!

• They found a "Black Hole Fence" (FOTH) that traps light.
• They found a "Universe Fence" (PITH) that expands with the cosmos.
  The Catch: There is a problem with the "fuel" (energy conditions). In the very early universe, the matter making up these black holes behaves a bit strangely. It violates some rules of physics (like the energy conditions) in certain zones, meaning the "stuff" holding the black hole together is a bit exotic or unstable.
  The Verdict: Yes, they are Cosmological Black Holes.
• Analogy: Think of these as two different types of "floating islands." They successfully have a lighthouse (black hole) and a coastline (universe expansion) existing together. However, the soil on the island is a bit wobbly (violates energy conditions) in the early days. But structurally, they work as black holes in an expanding universe.

3. The Complex Generalization: Glass-Mashhoon Spacetime

The Story: This is the "Swiss Army Knife" of models. It's a huge, flexible mathematical family that includes the McVittie model as just one tiny special case. It was designed to describe a star collapsing while the universe expands.
The Investigation: The authors dug deep into the math, checking every possible setting of the "knobs" (parameters) in this model. They checked the density, pressure, and the "fences."
The Verdict: No, it is not a Cosmological Black Hole.

• The Twist: Even though this model is very flexible, it turns out that no matter how you tune the knobs, it never creates the "Black Hole Fence" (FOTH). It only creates "Universe Fences" (Past Horizons).
• Analogy: Imagine a shape-shifting alien. It can turn into a rock, a tree, or a cloud. But no matter what form it takes, it can never become a bird. Similarly, the Glass-Mashhoon solution can look like many things, but it can never become a true cosmological black hole because it lacks the specific "future" horizon needed to trap light.

The Big Picture Conclusion

The paper concludes with a clear map of the landscape:

1. McVittie: The classic model fails. It's a black hole that never quite "turns on" its trapping horizon in an expanding universe.
2. Glass-Mashhoon: The fancy, generalized model also fails. It's too broad and lacks the necessary "future" horizon.
3. Culetu & Sultana-Dyer: These are the winners! They are the only ones in this study that successfully describe a black hole living inside an expanding universe. They have the right "fences" (horizons). However, they require some "exotic" matter to function, which is a bit of a physical hiccup, but mathematically, they are the only true candidates.

In simple terms: If you want to build a black hole in an expanding universe, you can't just use the old blueprints (McVittie) or the super-flexible ones (Glass-Mashhoon). You have to use the specific, slightly wobbly blueprints of Culetu or Sultana-Dyer to get the job done.

Technical Summary

1. Problem Statement

The paper addresses the challenge of defining and identifying cosmological black holes (BHs) within expanding universes. Traditional black hole definitions rely on stationary, asymptotically flat metrics and global event horizons, which are ill-suited for dynamical spacetimes where the background is expanding (e.g., FLRW universes).

• The Core Issue: Determining whether specific dynamical spacetime solutions (McVittie, Culetu, Sultana-Dyer, Glass-Mashhoon) genuinely represent a black hole embedded in a cosmological background.
• The Framework: The authors adopt Hayward's quasi-local formalism, which defines dynamical black holes via Trapping Horizons (THs) rather than global event horizons.
  • A Cosmological Black Hole in this framework requires the simultaneous existence of:
    1. A Future Outer Trapping Horizon (FOTH): Representing the black hole horizon (area non-decreasing, associated with the inner curvature).
    2. A Past Inner Trapping Horizon (PITH) (or generally a Past Trapping Horizon, PTH): Representing the cosmological horizon (associated with the expansion of the universe).
• The Goal: To rigorously test these candidate metrics against the existence of these specific horizon types and the validity of Energy Conditions (ECs).

2. Methodology

The authors perform a systematic analysis of four distinct spacetime metrics using a multi-step approach:

• Coordinate Systems: They analyze the metrics in various coordinate systems tailored to each solution to ensure robustness:
  • Isotropic Coordinates: Used for McVittie and Glass-Mashhoon spacetimes.
  • Conformal Painlevé-Gullstrand-Schild (PG-Schild): Used for Culetu spacetime.
  • Conformal Kerr-Schild: Used for Sultana-Dyer spacetime.
  • Painlevé-Gullstrand (PG) Coordinates: Applied to all metrics to provide a unified perspective on null expansions and horizon signatures.
• Mathematical Tools:
  • Null Expansions ($\theta_\pm$): Calculation of outgoing ($\theta_+$) and ingoing ($\theta_-$) null expansion parameters.
  • Trapping Horizon Classification: Determining the nature of horizons based on the signs of $\theta_\pm$ and their Lie derivatives ($L_\pm \theta_\mp$).
    • FOTH: $\theta_+ = 0, \theta_- < 0, L_+ \theta_- < 0$.
    • PITH: $\theta_- = 0, \theta_+ > 0, L_+ \theta_- > 0$.
  • Misner-Sharp-Hernandez (MSH) Mass: Calculated to determine the location of marginal surfaces ($R = 2M_{MSH}$).
  • Energy Conditions (ECs): Verification of the Null (NEC), Weak (WEC), Dominant (DEC), and Strong (SEC) energy conditions on and near the horizons to ensure physical viability.
• Specific Cases Analyzed:
  • McVittie: Spatially flat and curved variants.
  • Culetu: Conformal Schwarzschild in PG coordinates.
  • Sultana-Dyer: Conformal Schwarzschild in Kerr-Schild coordinates.
  • Glass-Mashhoon: A generalized class of spherically symmetric solutions (including McVittie as a special case).

3. Key Contributions and Results

A. McVittie Spacetime (Flat and Curved)

• Result: Fails to describe a cosmological black hole.
• Analysis:
  • In a matter-dominated, spatially flat universe, the spacetime possesses only Past Trapping Horizons (PTHs).
  • Specifically, it contains a Past Outer Trapping Horizon (POTH) (white-hole type) and a Past Inner Trapping Horizon (PITH) (cosmological type).
  • Crucially, no Future Outer Trapping Horizon (FOTH) exists. Without an FOTH, the spacetime cannot represent a black hole in the Hayward sense.
  • This result holds in both isotropic and PG coordinates. The PG analysis confirms that the absence of an FOTH is a coordinate-independent feature for the non-negative MSH mass case.
  • Curved McVittie: Even with spatial curvature, the analysis shows only PTHs exist. While the number and nature of PTHs vary with curvature parameters, the absence of an FOTH persists.

B. Culetu Spacetime

• Result: Succeeds as a cosmological black hole candidate (under specific conditions).
• Analysis:
  • The spacetime admits two distinct horizons:
    1. An FOTH (Future Outer Trapping Horizon) at a smaller radius ($R_+$), representing the black hole.
    2. A PTH at a larger radius ($R_-$), which evolves from a POTH (early times) to a PITH (later times), representing the cosmological horizon.
  • Energy Conditions:
    • On the cosmological horizon (PITH), the NEC, WEC, and SEC are satisfied.
    • On the black hole horizon (FOTH), the WEC and DEC are violated, though NEC and SEC may hold for specific radii.
  • Conclusion: It successfully models a cosmological BH in a matter-dominated early universe, provided the NEC holds on the event horizon.

C. Sultana-Dyer Spacetime

• Result: Succeeds as a cosmological black hole candidate.
• Analysis:
  • Similar to Culetu, it possesses an FOTH (BH-type) and a PTH (Cosmological-type).
  • The FOTH is always spacelike. The PTH transitions from spacelike (POTH) to timelike (PITH) as the universe evolves.
  • Energy Conditions:
    • The NEC and SEC are satisfied on the cosmological horizon.
    • All ECs are violated on the black hole horizon (FOTH) in the matter-dominated case.
  • Conclusion: Like Culetu, it represents a valid cosmological BH model within the Hayward framework, despite EC violations on the BH horizon.

D. Glass-Mashhoon Spacetime

• Result: Fails to describe a cosmological black hole.
• Analysis:
  • This is a generalized solution encompassing McVittie. The authors analyze the general class and specific subcases (including curved McVittie).
  • Key Finding: In an expanding universe, the Glass-Mashhoon solution only possesses Past Trapping Horizons (PTHs).
  • While the mathematical formalism allows for the vanishing of both $\theta_+$ and $\theta_-$, the physical analysis (checking signs of Lie derivatives and asymptotic behavior) reveals that no FOTH is formed.
  • The PG coordinate analysis confirms that the generalized solution lacks the necessary future-directed trapping horizon structure required for a black hole.
  • Conclusion: Despite its complexity and ability to model stellar collapse in specific limits, it does not constitute a cosmological black hole in an expanding background.

4. Significance and Implications

1. Resolution of Ambiguity: The paper provides a definitive classification of these well-known metrics using a rigorous, quasi-local definition of black holes. It clarifies that the mere presence of a mass parameter and an expanding background is insufficient to define a cosmological black hole; the specific causal structure of the trapping horizons is the deciding factor.
2. Validation of Culetu and Sultana-Dyer: It confirms that the Culetu and Sultana-Dyer metrics are the most viable candidates among those studied for modeling black holes in an expanding universe, provided one accepts the violation of certain energy conditions on the black hole horizon (a common feature in dynamical spacetimes).
3. Refutation of McVittie and Glass-Mashhoon: It reinforces the conclusion that the McVittie metric (and its generalization, Glass-Mashhoon) does not describe a black hole in the standard dynamical sense because it lacks a future outer trapping horizon. Instead, it describes a white-hole-like structure or a naked singularity embedded in a cosmological flow.
4. Coordinate Independence: By analyzing the metrics in both isotropic/Kerr-Schild and Painlevé-Gullstrand coordinates, the authors demonstrate that the existence (or non-existence) of FOTHs is a physical property of the spacetime, not an artifact of the coordinate choice.
5. Energy Condition Constraints: The study highlights the tension between modeling realistic cosmological black holes and satisfying all energy conditions. While Culetu and Sultana-Dyer work as BH models, they require violations of the WEC/DEC on the black hole horizon, suggesting that such objects may require exotic matter or specific dynamical phases in the early universe.

In summary, the paper establishes that only the Culetu and Sultana-Dyer spacetimes among the candidates analyzed successfully satisfy the Hayward criteria for a cosmological black hole (simultaneous FOTH and PITH), while the McVittie and Glass-Mashhoon solutions fail due to the absence of a future outer trapping horizon.