⚛️ general relativity

On a cosmological Oppenheimer-Snyder model: matching McVittie and FLRW spacetimes

This paper establishes that while a smooth, semi-global matching between an expanding McVittie spacetime and a flat FLRW spacetime is mathematically possible via a unique solution to a specific ODE system, the resulting matching hypersurface generically contains spacelike regions, thereby proving that a global isotropic source for the McVittie cosmological black hole does not exist.

Original authors: Brien C. Nolan

Published 2026-07-30

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CC BY 4.0

Original authors: Brien C. Nolan

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, stretching rubber sheet. In the grand story of cosmology, we often picture this sheet as perfectly smooth and uniform everywhere, like a calm ocean. This is the Friedmann-Lemaître-Robertson-Walker (FLRW) model, the standard "background" for our universe. But we also know that gravity can be so strong in certain places that it crumples the sheet into deep, bottomless pits called black holes. For a long time, physicists have been trying to stitch these two ideas together: Can we take a smooth, expanding ocean and seamlessly glue a black hole onto it without creating a jagged tear in the fabric of reality? This is the ultimate puzzle of "matching" spacetimes. It's not just about drawing pretty pictures; it's about understanding how the universe evolves from a regular, smooth beginning into a state containing these cosmic monsters. If we can't stitch them together smoothly, it might mean our current theories of gravity are missing a crucial piece of the puzzle, or that black holes in an expanding universe are fundamentally different from the ones we see in empty space.

Enter the McVittie spacetime, a mathematical model proposed decades ago that attempts to describe exactly this: a black hole sitting inside an expanding universe. It's like a whirlpool in a rising tide. The big question has always been: Where did this whirlpool come from? Did it form from a collapsing star, like the classic Oppenheimer-Snyder model where a ball of dust collapses into a black hole? Or is it something else entirely? In this paper, author Brien C. Nolan sets out to find out if we can construct a "cosmological Oppenheimer-Snyder" model. The goal is to see if we can take a region of smooth, expanding universe (the FLRW part) and match it perfectly to the McVittie black hole region across a boundary, creating a single, seamless universe where the black hole is just the "interior" of a collapsing cloud.

Nolan treats this like a complex jigsaw puzzle where the pieces are made of math. He doesn't just assume the boundary between the two regions is a simple, time-like wall (like a solid shell you could walk through). Instead, he lets the boundary be whatever the math demands: it could be a wall, a light-speed surface, or even a surface that flips between being a wall and a light-speed surface as time goes on. He writes down the strict rules of "junction conditions"—the cosmic equivalent of ensuring the edges of the puzzle pieces fit perfectly without any gaps or overlaps in the curvature of space.

The results of this mathematical detective work are surprising and slightly disappointing for those hoping for a simple origin story. Nolan proves that while you can mathematically stitch an FLRW universe to a McVittie black hole, the "seam" (the matching hypersurface) behaves very strangely. It cannot be a simple, smooth, time-like boundary that stays a wall forever. Instead, the math forces the boundary to be "spacelike" (like a moment in time rather than a place in space) at certain points, particularly near the beginning of the timeline. In fact, the boundary must start at the black hole's past singularity (a point of infinite density at the very beginning of time) and evolve in a way that makes it impossible to view the FLRW region as a simple, spatially bounded "interior" that collapses to form the black hole.

The paper explicitly rules out the existence of a global, isotropic source for the McVittie spacetime. In other words, you cannot build a McVittie black hole by simply taking a ball of expanding dust and letting it collapse; the math breaks down if you try to force the boundary to be a simple, time-like wall everywhere. The boundary must change its nature, becoming "spacelike" at times, which means the FLRW region cannot be considered the "inside" of the black hole in the way the classic Oppenheimer-Snyder model describes a collapsing star. While the authors show that a unique solution exists for the future (meaning the black hole and the universe can coexist smoothly moving forward), the past is problematic: the matching hypersurface hits a snag at the very beginning, failing to provide a clean, regular origin story for the black hole. So, while the McVittie model is a valid description of a black hole in an expanding universe, it doesn't seem to be the result of a simple, smooth collapse from a regular, isotropic cloud of matter. The universe, it seems, is a bit more complicated than a simple ball of dust turning into a black hole.

Technical Summary: Matching McVittie and FLRW Spacetimes

Problem Statement
The paper investigates the existence and nature of a "cosmological Oppenheimer-Snyder (OS) model." Specifically, it asks whether a given expanding McVittie spacetime (an exact solution representing a black hole embedded in an isotropic cosmological background) can be smoothly matched across a boundary hypersurface $\Sigma$ to a spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime. In the classical OS model, a homogeneous dust sphere collapses to form a Schwarzschild black hole, matched across an everywhere-timelike boundary. The authors seek to determine if a similar configuration exists where the FLRW region acts as an isotropic source for the McVittie exterior. Crucially, the study does not presuppose that the matching hypersurface is timelike; instead, it allows the causal character of $\Sigma$ to vary (timelike, spacelike, or null) to see if such a character emerges naturally from the matching conditions.

Methodology
The authors employ the general hypersurface formalism for matching spacetimes, utilizing the framework established by Nolan and others (specifically referencing [10]). The methodology involves:

Geometry Definition: Defining the McVittie metric (parameterized by mass $m$ and Hubble function $H(t)$ ) and the FLRW metric (parameterized by Hubble function $J(\tau)$ ) in area-radius coordinates.
Matching Conditions: Applying the necessary and sufficient conditions for a smooth match (absence of delta-function singularities in the Riemann tensor). These include:
- Preliminary Conditions: Continuity of the induced metric (first fundamental form) and the identification of the area radius $r$ across the boundary.
- Junction Conditions: Continuity of the extrinsic curvature (or rigged fundamental form) across the boundary.
Case Analysis: The problem is decomposed based on the causal character of the hypersurface $\Sigma$ $Σ$ and the behavior of the area radius $r$ $r$ :
- Null hypersurfaces (constant vs. non-constant radius).
- Non-null hypersurfaces (constant vs. non-constant radius).
Dynamical System Formulation: For the most general case (non-null, non-constant radius), the matching conditions are reduced to a system of first-order ordinary differential equations (ODEs) for the variables describing the hypersurface trajectory ( $r(t), \tau(t)$ ) and the FLRW Hubble function $J(\tau)$ , subject to an algebraic constraint (mass continuity).
Existence and Uniqueness Analysis: The authors analyze this ODE system to prove semi-global existence and uniqueness of solutions given a specified McVittie spacetime and initial data.

Key Contributions and Results

Reduction to ODEs: The paper demonstrates that the full set of matching conditions for a general hypersurface is equivalent to a 3-dimensional non-linear system of first-order ODEs subject to a single initial value constraint.
Semi-Global Existence and Uniqueness: Theorem VI.1 establishes that for a given expanding McVittie spacetime (with specific properties regarding $H(t)$ ) and a chosen initial point on the hypersurface, there exists a unique FLRW spacetime and a unique matching hypersurface $\Sigma$ that satisfy the matching conditions globally to the future.
Causal Character of the Hypersurface:
- Past Behavior: The matching hypersurface necessarily originates at the past singularity of the McVittie region ( $r=2m$ ) at a finite time in the past.
- Initial Nature: The hypersurface is initially spacelike near the past singularity.
- Future Evolution: The causal character of $\Sigma$ at late times depends on the equation of state of the background McVittie region. For certain equations of state, the hypersurface remains timelike; for others, it becomes spacelike.
- Time Orientation: The coordinate time $\tau$ of the FLRW region is not monotonically increasing with the McVittie time $t$ along the hypersurface. Specifically, $d\tau/dt$ is initially negative (decreasing) and becomes positive at late times, implying $\tau$ has a minimum.
No-Go Results for Isotropic Sources:
- Non-Null Constant Radius: Matching across a hypersurface of constant radius is shown to be highly restrictive, allowing only a one-parameter family of Hubble functions, ruling out generic McVittie spacetimes.
- Null Hypersurfaces: Matching across a null hypersurface of constant radius is impossible for non-trivial cases.
- Co-moving Hypersurfaces: If the hypersurface is co-moving with the fluid in both regions, the McVittie mass $m$ must be zero (reducing to FLRW). If co-moving in only one region, the McVittie region must be Schwarzschild-de Sitter.
Failure of the Cosmological OS Model: The central result is that a global cosmological Oppenheimer-Snyder model does not exist. Because the matching hypersurface must contain spacelike portions (initially and potentially at late times), the FLRW region cannot be interpreted as a spatially bounded "interior" source for the McVittie exterior in the same way a dust sphere is in the classical OS model. The hypersurface fails to be everywhere timelike, and the matching does not excise the McVittie past singularity from the global spacetime.

Significance and Claims
The paper claims to provide a comprehensive account of the matching between McVittie and FLRW spacetimes. Its primary significance lies in rigorously demonstrating that while a smooth match is mathematically possible (solutions exist and are unique), it does not yield a physical model where an isotropic FLRW region serves as the source for a McVittie black hole in a global, evolutionary sense.

The authors conclude that the "isotropic source" interpretation of the McVittie spacetime fails globally because the required matching hypersurface is not everywhere timelike. This stands in contrast to the Oppenheimer-Snyder model, where the boundary is strictly timelike. The paper suggests that thin-shell models (where the boundary is a singular shell with non-vanishing stress-energy) remain the most viable candidates for providing an isotropic source for McVittie spacetimes, though the global structure of those shells remains an open question. The work clarifies the limitations of attempting to construct McVittie spacetimes from regular initial data via smooth matching to FLRW interiors.

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