Asymptotic Behaviour of the Proper Length and Volume of the Schwarzschild Singularity

Though popular presentations give the Schwarzschild singularity as a point it is known that it is spacelike and not timelike. Thus it has a "length" and is not a "point". In fact, its length must necessarily be infinite. It has been proved that the proper length of the Qadir-Wheeler suture model goes to infinity [1], while its proper volume shrinks to zero, and the asymptotic behaviour of the length and volume have been calculated. That model consists of two Friedmann sections connected by a Schwarzschild "suture". The question arises whether a similar analysis could provide the asymptotic behaviour of the Schwarzschild black hole near the singularity. It is proved here that, unlike the behaviour for the suture model, for the Schwarzschild essential singularity $\Delta s$ $\thicksim$ $K^{1/3}\ln K$ and $V\thicksim$ $K^{-1}\ln K$, where $K$ is the mean extrinsic curvature, or the York time.Comment: 13 pages, 1 figur

Uniqueness of Flat Spherically Symmetric Spacelike Hypersurfaces Admitted by Spherically Symmetric Static Spactimes

It is known that spherically symmetric static spacetimes admit a foliation by {\deg}at hypersurfaces. Such foliations have explicitly been constructed for some spacetimes, using different approaches, but none of them have proved or even discussed the uniqueness of these foliations. The issue of uniqueness becomes more important due to suitability of {\deg}at foliations for studying black hole physics. Here {\deg}at spherically symmetric spacelike hy- persurfaces are obtained by a direct method. It is found that spherically symmetric static spacetimes admit {\deg}at spherically symmetric hypersurfaces, and that these hypersurfaces are unique up to translation under the time- like Killing vector. This result guarantees the uniqueness of {\deg}at spherically symmetric foliations for such spacetimes.Comment: 10 page

Foliation of the Kottler-Schwarzschild-De Sitter Spacetime by Flat Spacelike Hypersurfaces

There exist Kruskal like coordinates for the Reissner-Nordstrom (RN) black hole spacetime which are regular at coordinate singularities. Non existence of such coordinates for the extreme RN black hole spacetime has already been shown. Also the Carter coordinates available for the extreme case are not manifestly regular at the coordinate singularity, therefore, a numerical procedure was developed to obtain free fall geodesics and flat foliation for the extreme RN black hole spacetime. The Kottler-Schwarzschild-de Sitter (KSSdS) spacetime geometry is similar to the RN geometry in the sense that, like the RN case, there exist non-singular coordinates when there are two distinct coordinate singularities. There are no manifestly regular coordinates for the extreme KSSdS case. In this paper foliation of all the cases of the KSSdS spacetime by flat spacelike hypersurfaces is obtained by introducing a non-singular time coordinate.Comment: 12 pages, 4 figure