ASYMPTOTIC BEHAVIOR 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 is necessarily infinite. It has been proven that the proper length of the Qadir–Wheeler suture model goes to infinity,1 while its proper volume shrinks to zero, and the asymptotic behavior of the length and volume has 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 behavior of the Schwarzschild black hole near the singularity. It is proven here that, unlike the behavior for the suture model, for the Schwarzschild essential singularity Δs ~ K1/3 ln K and V ~ K-1 ln K, where K is the mean extrinsic curvature, or the York time.