Question: Consider a comoving observer sitting at constant spatial coordinates (r *, ?*, P* ), around a Schwarzs child black hole of mass M. The observer drops a beacon into the black hole (straight down, along a radial trajectory). The beacon emits radiation at a constant wavelength ?em (in the beacon rest frame).

(a) Calculate the coordinate speed dr / dt of the beacon, as a function of r.

(b) Calculate the proper speed of the beacon. That is, imagine there is a comoving observer at fixed r, with a locally inertial coordinate system set up as the beacon passes by, and calculate the speed as measured by the comoving observer. What is it at r = 2GM?

(c) Calculate the wavelength ?obs measured by the observer at r*, as a function of the radius rem at which the radiation was emitted.

(d) Calculate the time lobs at which a beam emitted by the beacon at radius rem will be observed at r *.

(e) Show that at late times, the redshift grows exponentially: ?obs/?em proportional e^(t_obs/T). Give an expression for the time constant T in terms of the black hole mass M.