Existence of CMC Cauchy surfaces and spacetime splitting

Galloway, Gregory J.

doi:10.4310/pamq.2019.v15.n2.a2

Cited by 5 publications

(3 citation statements)

References 31 publications

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“…In this section, we present the sources of interest for “no-CMC” cosmological spacetimes, i.e., cosmological spacetimes without any CMC Cauchy surfaces. The flow of the presentation follows [ 12 ], where we refer to for many more details, especially regarding historical developments. Let us begin by recalling the cosmological version of the Hawking–Penrose singularity theorem.…”

Section: Introductionmentioning

confidence: 99%

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Examples of cosmological spacetimes without CMC Cauchy surfaces

Ling,

Ohanyan

2024

Lett Math Phys

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CMC (constant mean curvature) Cauchy surfaces play an important role in mathematical relativity as finding solutions to the vacuum Einstein constraint equations is made much simpler by assuming CMC initial data. However, Bartnik (Commun Math Phys 117(4):615–624, 1988) constructed a cosmological spacetime without a CMC Cauchy surface whose spatial topology is the connected sum of two three-dimensional tori. Similarly, Chruściel et al. (Commun Math Phys 257(1):29–42, 2005) constructed a vacuum cosmological spacetime without CMC Cauchy surfaces whose spatial topology is also the connected sum of two tori. In this article, we enlarge the known number of spatial topologies for cosmological spacetimes without CMC Cauchy surfaces by generalizing Bartnik’s construction. Specifically, we show that there are cosmological spacetimes without CMC Cauchy surfaces whose spatial topologies are the connected sum of any compact Euclidean or hyperbolic three-manifold with any another compact Euclidean or hyperbolic three-manifold. Analogous examples in higher spacetime dimensions are also possible. We work with the Tolman–Bondi class of metrics and prove gluing results for variable marginal conditions, which allows for smooth gluing of Schwarzschild to FLRW models.

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Section: Introductionmentioning

confidence: 99%

“…While this conjecture has been proved over the years under various additional assumptions (we refer to [ 12 ] for a detailed discussion of these developments), the version as stated remains open to this day and is one of the most significant open problems in mathematical General Relativity.…”

Section: Introductionmentioning

confidence: 99%

Examples of cosmological spacetimes without CMC Cauchy surfaces

Ling,

Ohanyan

2024

Lett Math Phys

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“…2], which is the same statement but assuming only (i)-(iii). Bartnik's conjecture can be understood as a rigid generalization of the Hawking-Penrose singularity theorem [11]. The lattes states that a spacetime with compact Cauchy surfaces that satisfies the timelike convergence condition and the generic condition must be causally geodesically incomplete.…”

Section: Introductionmentioning

confidence: 99%

Causal bubbles in globally hyperbolic spacetimes

García-Heveling

Soultanis

2022

Gen Relativ Gravit

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In 1972, Geroch, Kronheimer, and Penrose introduced what is now called the causal boundary of a spacetime. This boundary is constructed out of Terminal Indecomposable Past sets (TIPs) and their future analogues (TIFs), which are the pasts and futures of inextendible causal curves. The causal boundary is a key tool to understand the global structure of a spacetime. In this paper, we show that in a spacetime with compact Cauchy surfaces, there is always at least one minimal TIP and one minimal TIF, minimal meaning that it does not contain another TIP (resp. TIF) as a proper subset. We then study the implications of the minimal TIP and TIF meeting each other. This condition generalizes some of the "no observer horizon" conditions that have been used in the literature to obtain partial solutions of the Bartnik splitting conjecture.

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