Singular hypersurfaces and thin shells in cosmology

Abstract

We analyse spherically symmetric spacetimes obtained by gluing a cosmological region to a Schwarzschild black hole across a singular co-dimension one hypersurface. Assuming an arbitrary homogeneous and isotropic cosmology, and working in spacetime dimensions greater than three with general cosmological constant, we derive the stress-energy tensor required on the hypersurface directly in terms of the cosmological energy density. This general framework yields a new exact solution in four dimensions describing a radiation-filled cosmology matched to vacuum through a pressureless dust shell. A systematic exploration of parameter space reveals twenty-two distinct families of solutions, including bubble-of-cosmology and Swiss-cheese spacetimes with different global and causal structures. We also discuss possible generalisations of the construction and explain why such thin-shell cosmologies are of interest in the context of holography and quantum cosmology. For negative cosmological constant, a subset of these solutions admits a Euclidean continuation compatible with a holographic interpretation developed in related work. In addition, we provide a pedagogical introduction to hypersurfaces in general relativity and a practical framework for constructing thin-shell spacetimes.

Figures (6)

• Figure 1: A sketch of spatial slices of (a) bubble of cosmology spacetime and (b) Swiss cheese spacetime. The cosmological part is shaded in pink, the black hole patch in green, and the thin shell in brown. The thin shell depicted here as a circle is a $(D-1)$-sphere with the other angular dimensions suppressed.
• Figure 2: A schematic depiction of the procedure for constructing a thin shell spacetime. A cosmological patch (pink) taken from the left picture is glued to a black hole patch (green) in the middle picture along a thin shell (brown line) to give the resulting thin shell spacetime on the right. The thin shell moves along a co-moving trajectory in the cosmological spacetime. Note that the Penrose diagram for a thin shell spacetime is only qualitative.
• Figure 3: An assortment of cosmological spacetimes: (a) open monotonically expanding cosmology (b) open time-reversal symmetric cosmology with big bang/big crunch, (c) closed cosmology with big bang/big crunch, and (d) closed bouncing cosmology. The zig-zag lines indicate the big bang/big crunch singularities and the solid black lines represent asymptotic infinities.
• Figure 4: Penrose diagrams of Schwarzschild black holes in (a) Anti-de Sitter, (b) Minkowski and (c) de Sitter backgrounds. The zig-zag lines indicate the black hole singularity, the solid black lines represent null infinities, and the dashed lines represent event horizons (as well as cosmological horizons for dS black holes). The maximally extended dS black holes repeat indefinitely in each direction as denoted by the dotted lines.
• Figure 5: Possible time reversal symmetric thin shell spacetimes