Inspired by this question here.

The Schwarzschild metric, describing the exterior gravitational field of a planet of mass $M$ and radius $R$, is given by$$ds^2 = -(1 - 2M/r)\,dt^2 + (1 - 2M/r)^{-1}\,dr^2 + r^2(d\theta^2 + \sin^2\theta \,d\phi^2).$$A tower has its base on the surface of this planet ($r = R$) and its top at radial coordinate $r = R_1$. A ball is held at rest by an observer at the top of the tower. It is then dropped and caught by an observer at the bottom of the tower.

What is the speed, $v$, of the ball as measured by the observer who catches the ball, just before the ball is caught?

Here, we are not assuming that $R \gg 2M$ or that $R_1 - R \ll R$. Also, I want the physical speed here, $v$, as would be measured, e.g. by a radar gun, not a coordinate speed, such as $dr/dt$.

Edit. This is not a homework question, and is merely inspired by the question I linked to, I am not sure why it is closed.

  • 1
    $\begingroup$ This is on the verge of being closed (4 close votes at the time of writing) because as stated it's just a worked example and falls afoul of our homework policy. However there is some interesting physics involved because we have (a) the equation of motion for a freely falling object (b) the relative time dilation of the two observers and (c) the length contraction to consider. If you could phrase the question to be less specific it might get a better reception. $\endgroup$ – John Rennie May 19 '16 at 17:32
  • $\begingroup$ It doesn't matter if this is not a 'homework question', it's a 'homework-like question' --- which is what the rule i written for. It's a good topic, but the question needs to be restructured. $\endgroup$ – DilithiumMatrix May 24 '16 at 2:32