4.2.1

Problem 4.2.1: Local time

Find the interval of local time (proper time of stationary observer) at a point $(r,\theta,\varphi)$ in terms of coordinate time $t$, and show that $t$ is the proper time of an observer at infinity. What happens when $r\to r_{g}$?

solution

The proper time of the stationary observer is $d\tau=ds|_{dr=d\theta=d\varphi=0}$: \[d\tau=\sqrt{g_{00}}\;dt=\sqrt{1-\frac{r_{g}}{r}}\;dt.\] At $r\to \infty$ it coincides with $dt$, so the coordinate time $t$ can be interpreted as the proper time of a "remote" observer. At $r\to r_g$ the local time flows slower and asymptotically stops. If one of two twins were to live some time at $r\approx r_g$, he will return to his remote twin having aged less (thought he might have acquired some grey hair due to constant fear of tumbling over the horizon).

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