史瓦西几何中的真时与真长
Proper time and length in Schwarzschild geometry
摘要：
研究了Kruskal-Schwarzschild永恒质量时空$M$的宇宙（$U$）和反宇宙（$\bar{U}$）扇区中静止观测者的本征时间（$τ$）间隔，以及黑洞（BH）和白洞（WH）扇区中的本征长度（$ρ$）.事实上，在渐近平坦的区域，坐标时间$t$在无穷远是适当的时间，导致过去的定向Kruskal时间$T$在$\bar{U}$。在BH和WH扇区中，最大固有长度与这些区域中的最大固有时间间隔$πM$相一致，即与径向自由下落（弹射）到（从）奇点的固有时间从（在）静止在视界处开始（结束）。
英文摘要：
We study proper time ($τ$) 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 ($ρ$) 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, $πM$, 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.
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