Constant scalar curvature hypersurfaces in extended Schwarzschild space-time

We present a class of spherically symmetric hypersurfaces in the Kruskal extension of the Schwarzschild space-time. The hypersurfaces have constant negative scalar curvature, so they are hyperboloidal in the regions of space-time which are asymptotically flat.

Figure 1

The polynomial $p(r)$ and its relation to the line $L$ for different values of $a$ and $c$.

Figure 2

Three classes of hypersurface in different regions of space-time for the case $c=-0.5$. Within each class the parameter $a$ takes the values $a=10$, $a=1$, and $a=0.01$.

Figure 3

The case $0<c<1$

Figure 4

The case $1<c<1+a^2$. We show the hypersurfaces for fixed values of $a$ and $c$, varying their location in space-time.

Figure 5

The case $1+a^2<c$. Hypersurfaces for fixed values of $a$ and $c$, and varying location in space-time.

Figure 6

The case $c=1+a^2$ for $a^2=0.1$

Figure 7

A set of symmetric hypersurfaces for fixed $a=1$ and equidistant intersection with the $T$-axis.