Examples of cosmological spacetimes without CMC Cauchy surfaces

Area of Science:

• Mathematical Relativity
• General Relativity
• Differential Geometry

Background:

• Constant mean curvature (CMC) Cauchy surfaces simplify solving Einstein's constraint equations in mathematical relativity.
• Previous studies demonstrated cosmological spacetimes without CMC Cauchy surfaces using specific topologies (e.g., connected sum of two tori).

Purpose of the Study:

• To enlarge the known set of spatial topologies for cosmological spacetimes that do not possess CMC Cauchy surfaces.
• To generalize Bartnik's construction for creating such spacetimes.

Main Methods:

• Generalizing Bartnik's construction for cosmological spacetimes.
• Utilizing the Tolman-Bondi class of metrics.
• Proving gluing results for variable marginal conditions.

Main Results:

• Demonstrated the existence of cosmological spacetimes without CMC Cauchy surfaces for a broader range of spatial topologies.
• Showed that the connected sum of any two compact Euclidean or hyperbolic three-manifolds can serve as such a spatial topology.
• Indicated the possibility of analogous constructions in higher spacetime dimensions.

Conclusions:

• The study significantly expands the understanding of spatial topologies in general relativity, particularly for spacetimes lacking CMC Cauchy surfaces.
• The findings provide new examples and a generalized method for constructing complex spacetimes, advancing the field of mathematical relativity.