Proper time and length in Schwarzschild geometry
  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. 
O. Brauer, H. A. Camargo, M. Socolovsky
physics.gen-ph