Proper time and length in Schwarzschild geometry

TL;DR

This paper analyzes proper time and length measurements in different sectors of Schwarzschild spacetime, revealing relationships between coordinate and proper times and lengths in black hole and white hole regions.

Contribution

It provides a detailed examination of proper time and length in Schwarzschild geometry, clarifying their values in various spacetime sectors and their relation to coordinate time.

Findings

Proper time in the universe and anti-universe sectors is related to Kruskal time.

Maximal proper lengths in black hole and white hole sectors equal $\, ext{pi} imes M$.

Proper lengths correspond to proper time of radial free fall from the horizon.

Abstract

We study proper time ($\tau$) intervals for observers at rest in the universe ($U$) and anti-universe ($\bar{U}$) sectors of the Kruskal-Schwarzschild eternal spacetime of mass $M$, and proper lengths ($\rho$) in the black hole (BH) and white hole (WH) sectors. The fact that in asymptotically flat regions, coordinate time $t$ at infinity is proper time, leads to a past directed Kruskal time $T$ in $\bar{U}$. In the BH and WH sectors maximal proper lengths coincide with maximal proper time intervals, $\pi M$, in these regions, i.e. with the proper time of radial free falling (ejection) to (from) the singularity starting (ending) from (at) rest at the horizon.