Cosmologically Coupled Black Holes with Regular Horizons

This paper presents a new, exact solution in Einstein's gravity that describes the cosmological embedding of static, spherically-symmetric objects, including black holes, within an FLRW universe while fully accounting for backreaction and ensuring the event horizon remains free of curvature singularities.

The Gist

The Big Idea: Black Holes in an Expanding Universe

Imagine the universe as a giant, rising loaf of raisin bread. As the dough expands (the universe grows), the raisins (galaxies and black holes) move further apart.

For a long time, physicists have asked a tricky question: What happens to a black hole when it sits inside this expanding dough?

Does the black hole just sit there, unaffected? Or does it "couple" to the expansion, growing larger as the universe grows, even without eating any new stars or gas? This paper says yes, black holes can grow with the universe, but there was a major problem with previous theories that this paper solves.

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The Problem: The "Cracked Window"

In the past, scientists tried to build a mathematical model of a black hole inside an expanding universe. They used a famous recipe from the 1930s (the McVittie solution).

Think of this like trying to fit a heavy, rigid statue (the black hole) into a stretching rubber sheet (the expanding universe).

• The Issue: In all previous models, when you tried to fit the black hole into the expanding universe, the math broke down right at the black hole's "event horizon" (the point of no return).
• The Analogy: Imagine trying to stretch a rubber band over a glass jar. If you stretch it too hard in the wrong way, the glass shatters. In these old models, the "glass" (the geometry of space) shattered at the horizon, creating a curvature singularity. It was a mathematical "crack" where the laws of physics stopped working. This made the models physically impossible to trust.

The Solution: A Flexible, Smart Fit

The authors of this paper (Cadoni, de Lima, et al.) found a new way to stitch the black hole into the expanding universe without breaking the glass.

1. The "Backreaction" Concept

Previous models treated the universe as a passive stage and the black hole as a static actor. This paper treats them as partners in a dance.

• The Analogy: Imagine the black hole isn't just sitting on the rubber sheet; it's actually pushing back against the stretching. The local gravity of the black hole and the global expansion of the universe push and pull on each other. This is called backreaction.
• By accounting for this push-and-pull, the authors found a mathematical "seam" that holds together perfectly.

2. The "Anisotropic Fluid" (The Stretchy Glue)

To make this work, they had to change the "stuff" filling the space around the black hole.

• Old View: They assumed the space was filled with a fluid that pushes equally in all directions (like water in a balloon).
• New View: They used an anisotropic fluid.
• The Analogy: Think of a piece of kneaded dough. If you pull it, it stretches easily in one direction but resists in another. The "fluid" around the black hole acts like this stretchy dough. It allows the space to expand around the black hole without snapping the horizon. This flexibility prevents the "crack" (singularity) from forming.

The Result: A Smooth Horizon

Because of this new "stretchy dough" approach and the backreaction:

1. No More Cracks: The event horizon of the black hole remains smooth and regular. The math works perfectly right up to the edge of the black hole.
2. A New Black Hole: They derived a brand-new version of the Schwarzschild black hole (the simplest type). It is distinct from the old McVittie version. It's like finding a new, better blueprint for a house that fits perfectly on a sloping hill, whereas the old blueprint kept sliding off.
3. Growth Without Eating: This model supports the idea that black holes can gain mass simply because the universe is expanding, without needing to "eat" matter.

Why Does This Matter?

• Solving a Mystery: It fixes a major theoretical headache that has plagued physicists for decades.
• Testing Reality: If black holes do grow with the universe (as some recent observations suggest), this paper provides the mathematical proof that such a thing is physically possible without breaking the laws of physics.
• Energy Conditions: The paper also checks if this "stretchy dough" violates the laws of energy. They found that while it behaves a bit strangely near the horizon (requiring some "exotic" properties), it doesn't break the fundamental rules of the universe in a fatal way.

Summary in One Sentence

The authors have built a new mathematical model that lets a black hole grow along with the expanding universe without breaking the laws of physics at its edge, effectively fixing a "crack" that existed in all previous theories.

Technical Summary

1. Problem Statement

The paper addresses a critical theoretical inconsistency in the Cosmological Embedding (CE) of Black Holes (BHs) and other compact objects within General Relativity (GR).

• Context: There is growing interest in whether astrophysical black holes are "cosmologically coupled" (CC), meaning their mass $M$ evolves with the cosmic scale factor $a(t)$ (e.g., $M \propto a^k$), independent of accretion. This idea was originally proposed by McVittie (1933) and recently revisited to explain observational data.
• The Issue: All previously constructed CC BH models (including McVittie's solution and recent extensions) suffer from a curvature singularity at the event horizon. While the central singularity is expected, the appearance of a singularity at the would-be event horizon is a pathological feature that renders these solutions physically problematic.
• Goal: To derive the most general, exact solution for a static, spherically-symmetric object embedded in an arbitrary Friedmann-Lemaître-Robertson-Walker (FLRW) universe that removes the curvature singularity at the horizon while maintaining consistency with Einstein's equations.

2. Methodology

The authors employ a rigorous analytical approach within General Relativity, utilizing the following steps:

• Metric and Source: They start with a metric for a spherically symmetric object in an FLRW background using conformal time $\eta$ and comoving radius $r$:
  $$ds^2 = a^2(\eta) \left[ -e^{\alpha(\eta,r)} d\eta^2 + e^{\beta(\eta,r)} dr^2 + r^2 d\Omega^2 \right]$$
  The source is modeled as an anisotropic fluid with energy density $\rho$, radial pressure $p_\parallel$, and transverse pressure $p_\perp$. Crucially, they enforce no radial energy flux (no accretion), meaning the fluid is comoving with the expansion.
• Field Equations: They derive the Einstein field equations coupled with the covariant conservation of the energy-momentum tensor.
• Parametrization via Mass Functions:
  • Instead of solving for metric functions directly, they introduce a local mass function based on the Misner-Sharp (MS) mass.
  • They decompose the energy density and pressure into a homogeneous cosmological part ($\rho_c, p_c$) and a local contribution ($\rho_l, p_{l\parallel}$).
  • They define a Local Object Mass (MLO), $M^l_{MS}$, which subtracts the purely cosmological contribution from the total MS mass.
• Exact Solution Construction:
  • They introduce a new function $M(r, a)$ related to the MLO and the metric functions.
  • By transforming coordinates from $(r, \eta)$ to $(\ell, a)$ (where $\ell = ar$ is the areal radius) and solving the resulting characteristic equations, they derive a general solution (Eq. 20) that depends on an arbitrary mass profile $M(r, a)$ and the cosmological density $\rho_c$.
  • This solution explicitly includes a backreaction term (dependence on $\rho_c$ in the metric functions) that was missing or treated differently in previous models.

3. Key Contributions

A. Resolution of the Horizon Singularity

The primary contribution is the demonstration that the curvature singularity at the event horizon is an artifact of neglecting the backreaction of the local geometry on cosmological dynamics.

• Previous Models: Solutions derived by simply promoting static coordinates to cosmological ones (e.g., Eq. 25 in the paper) result in a curvature singularity at the horizon ($2M(\ell) = \ell$).
• New Solution: The general solution (Eq. 20) incorporates the cosmological density $\rho_c$ directly into the metric structure via the backreaction term. This term acts to cure the singularity, rendering the horizon regular everywhere except for the central singularity (if present).

B. New Cosmological Embedding of the Schwarzschild Black Hole

The authors derive a new CE for the Schwarzschild BH that is distinct from the McVittie solution.

• Inequivalence: Unlike McVittie's solution (which assumes isotropic pressure and vanishing flux in isotropic coordinates), this new solution assumes anisotropic pressure and vanishing flux in comoving areal coordinates.
• Regularity: This new embedding is regular at the horizon, unlike the McVittie solution which is singular there.

C. General Framework for Nonsingular BHs

The framework is applied to nonsingular BH models (e.g., those with a de Sitter core). Previous embeddings of these models also exhibited horizon singularities; the authors show that their method eliminates these pathologies as well.

D. Clarification of Theorems

The work clarifies a theorem by Faraoni and Rinaldi (Ref. [73]) which forbids regular static horizons in CE solutions.

• The authors confirm their solutions comply with this theorem: the horizons in their solutions are apparent horizons (dynamical), not static event horizons. The "singularity" in previous models arose from incorrectly treating the horizon as static while neglecting backreaction.

4. Results

• General Solution (Eq. 20): The paper provides the exact metric functions $e^{-\gamma}$ and $\theta$ in terms of the mass function $M(r, a)$ and cosmological density $\rho_c$.
  • The metric is free of curvature singularities at the horizon for any smooth choice of $M(r, a)$, provided $\rho_c \neq 0$.
  • Far from the object ($r \to \infty$), the solution reduces to the standard FLRW metric.
• Energy Conditions:
  • The authors analyze the Weak (WEC), Null (NEC), and Dominant (DEC) energy conditions.
  • WEC/NEC: Satisfied outside the black hole for standard cosmological fluids.
  • DEC: Typically violated in the near-horizon region. The authors argue this is acceptable because the anisotropic fluid is an effective, coarse-grained description of inhomogeneities, not necessarily a fundamental microscopic violation.
• Comparison with McVittie:
  • McVittie's solution (Eq. 25) is recovered as a limit outside the apparent horizon but fails near the horizon due to the lack of the backreaction term.
  • The new solution (Eq. 28) differs significantly near the horizon, growing with the scale factor $a$, ensuring regularity.

5. Significance

• Theoretical Consistency: This work resolves a decades-old issue in the theoretical modeling of black holes in expanding universes. It proves that cosmologically coupled black holes can exist without pathological singularities at their horizons, provided the backreaction of the local mass on the cosmological expansion is correctly accounted for.
• Observational Implications: By providing a regular, exact solution, the paper strengthens the theoretical basis for testing cosmological coupling ($M \propto a^k$) against observational data (e.g., gravitational wave events or black hole mass distributions at different redshifts).
• New Paradigm for Fluid Sources: The use of an anisotropic fluid is shown to be essential for constructing regular horizons in $K=0$ (flat) cosmologies, extending previous work that relied on isotropic pressure (which could not avoid singularities).
• Physical Interpretation: The results suggest that the "singularity" in previous models was a mathematical artifact of an inconsistent approximation (neglecting backreaction), rather than a physical inevitability.

In summary, Cadoni et al. provide the first general, exact, and regular solution for cosmologically coupled black holes, fundamentally altering our understanding of how local compact objects interact with global cosmological dynamics in General Relativity.