Existence of a constant mean curvature foliation in the extended Schwarzschild spacetime

We construct a $T$-axisymmetric, spacelike, spherically symmetric, constant mean curvature (CMC) hypersurface foliation in the Kruskal extension with properties such that the mean curvature varies in each slice and ranges from minus infinity to plus infinity. This family of hypersurfaces extends the CMC foliation discussions by Malec and Ó Murchadha in 2009 [Phys. Rev. D 80, 024017 (2009)].

Figure 1

The Kruskal extension and the Schwarzschild spacetimes.

Figure 2

Each point on the graph of ${\tilde{k}}_H(r)$ determines a TSS-CMC hypersurface and its $T$ intercept in the Kruskal extension.

Figure 3

The curve $\gamma (c)$ is the graph of a decreasing function and each point on $\gamma (c)$ corresponds to a TSS-CMC hypersurface ${\mathrm{\Sigma }}_{H(c),c}$ in the Kruskal extension. (a) For $H\le 0$, the curve $\gamma (c)$ is a graph of a function $y(r)$ which satisfies all properties in Proposition 1. (b) For $H\ge 0$, the curve $\gamma (-c)$ is a graph of a function $y(r)$ which satisfies all properties in Proposition 1. (c) TSS-CMC hypersurfaces are shown in the Kruskal extension.

Figure 4

${\mathrm{\Sigma }}_{H(c_1),c_1}$ and ${\mathrm{\Sigma }}_{H(c_2),c_2}$ are disjoint by comparing with ${\mathrm{\Sigma }}_{H,c}$.

Figure 5

Proof of the family $\{{\mathrm{\Sigma }}_{H(c),c}\}$ covering the Kruskal extension.