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

TL;DR

This paper investigates the asymptotic behavior of the proper length and volume of the Schwarzschild singularity, revealing that the length diverges while the volume diminishes near the singularity, contrasting with previous models.

Contribution

It provides a novel analysis of the Schwarzschild singularity's asymptotic length and volume, extending understanding beyond the Qadir-Wheeler suture model.

Findings

Proper length diverges as $K^{1/3}\ln K$ near the singularity.

Proper volume shrinks as $K^{-1}\ln K$ near the singularity.

Contrasts with the suture model where length is infinite and volume goes to zero.

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.