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

Abstract

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.