Einstein and Gravitation — Question 9

Question 9

A static-clock formula cannot be extended to stationary observers inside a horizon

Outside a nonrotating black hole, a clock held at constant Schwarzschild radius $r$ obeys the supplied rate $d\tau/dt=\sqrt{1-r_s/r}$, where $r_s=2GM/c^2$ and $t$ is time at infinity. This static-clock relation applies only for $r>r_s$.

1. Find the rate at $r=4r_s$ and the radius where it is one half.

2. Explain why an imaginary square root at $r<r_s$ is not a prediction of imaginary time on a falling clock.

Question 9 - Solution

A static-clock formula cannot be extended to stationary observers inside a horizon

(a) Derivation and calculation

For a supported static clock at $r=4r_s$, $$\boxed{\frac{d\tau}{dt}=\sqrt{1-\frac14}=\frac{\sqrt3}{2}\simeq.866025}.$$ For a half-rate clock, $$\sqrt{1-r_s/r}=\frac12\quad\Longrightarrow\quad 1-r_s/r=\frac14,$$ $$\boxed{r=\frac43r_s}.$$ Both radii are outside the horizon and refer to supported, not orbiting, clocks.

(b) Result and interpretation

The formula assumes a worldline at fixed spatial coordinates in the exterior, with $$r>r_s,\qquad 1-r_s/r>0.$$ Extending it to $r<r_s$ violates this static-observer domain. A falling clock does not satisfy the fixed-radius assumption and continues to measure real proper time along its own trajectory.

Physical check

As $r\to\infty$, the static rate tends to one, matching the distant reference time.