6 research outputs found
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
and , where
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