[Paper Review] The "Swiss cheese" cosmological model has no extrinsic curvature discontinuity: A comment on the paper by G.A. Baker, Jr. (astro-ph/0003152)

C. C. Dyer, Chris Oliwa|arXiv (Cornell University)|Apr 6, 2000

Cosmology and Gravitation Theories2 references3 citations

TL;DR

This paper rigorously demonstrates that the 'Swiss cheese' cosmological model—where Schwarzschild spacetime is matched to an expanding FLRW universe across a spherical boundary—exhibits no extrinsic curvature discontinuity. By explicitly constructing the matching using Darmois junction conditions, the authors prove continuity of both the first and second fundamental forms (intrinsic metric and extrinsic curvature), resolving a claimed inconsistency in a prior study.

ABSTRACT

Contrary to a claim, the Schwarzschild solution insertion in an expanding universe model, the so called "Swiss cheese" model, does not possess an extrinsic curvature discontinuity. We show that both the intrinsic metric and the extrinsic curvature are continuous, and point out the error that led to the claim.

Motivation & Objective

• To resolve a conflicting claim in the literature regarding extrinsic curvature discontinuity in the Swiss cheese cosmological model.
• To rigorously verify the smooth matching of Schwarzschild and FLRW spacetimes across a time-evolving spherical boundary.
• To identify and correct the mathematical error in Baker's (2000) analysis that led to the false claim of curvature discontinuity.
• To confirm that the pressure is continuous across the matching hypersurface, ensuring physical consistency.

Proposed method

• Explicitly constructs the matching between FLRW and Schwarzschild metrics across a spherical hypersurface Σ with fixed coordinate radius in FLRW but time-evolving in Schwarzschild coordinates.
• Applies the Darmois junction conditions by verifying continuity of the first fundamental form (intrinsic metric) and second fundamental form (extrinsic curvature) on Σ.
• Uses coordinate transformations and unit normal vectors to compute the extrinsic curvature in both frames, ensuring consistency via the general form of the second fundamental form.
• Derives the required relations between time derivatives of T(u) and ρ(u) using the matching conditions, ensuring continuity of the metric and curvature components.
• Verifies pressure continuity by showing that the Einstein field equations in the FLRW region yield zero pressure, consistent with the Schwarzschild interior.
• Identifies the error in Baker's work: the incorrect application of a simplified extrinsic curvature formula (eq. 13) that assumes ρ=constant in Schwarzschild coordinates, which is incompatible with the time-evolving boundary.

Experimental results

Research questions

• RQ1 Does the 'Swiss cheese' model exhibit an extrinsic curvature discontinuity at the matching hypersurface between FLRW and Schwarzschild spacetimes?
• RQ2 Can the Darmois junction conditions be satisfied for a time-evolving spherical boundary in the Schwarzschild frame while maintaining continuity of the intrinsic metric and extrinsic curvature?
• RQ3 What is the correct method for computing the extrinsic curvature in the Schwarzschild frame when the matching surface is not a coordinate surface (i.e., ρ ≠ constant)?
• RQ4 Why does Baker's (2000) claim of curvature discontinuity fail, and what is the specific mathematical error in his derivation?
• RQ5 Is the pressure continuous across the matching hypersurface, and does this support the physical validity of the Swiss cheese model?

Key findings

• The intrinsic metric (first fundamental form) is continuous across the spherical boundary Σ, as confirmed by matching the FLRW and Schwarzschild line elements under the condition K²r₀² = ρ².
• The extrinsic curvature (second fundamental form) is continuous on Σ, with ΩFαβ = ΩSαβ derived from the general definition, not the simplified formula.
• The pressure is continuous and vanishes on Σ, as required by the Darmois conditions, with p = 0 derived from the Einstein field equations in the FLRW region.
• The claimed discontinuity in extrinsic curvature by Baker (2000) arises from an incorrect use of the simplified extrinsic curvature formula (eq. 13), which assumes ρ = constant in the Schwarzschild frame.
• The correct derivation shows that the boundary is not a coordinate surface in the Schwarzschild frame, invalidating the use of the simplified formula and resolving the apparent contradiction.
• The model is physically admissible, as the energy density μ = 6M/(r₀³|R|³) is positive, confirming a non-vacuum, physically meaningful matching.