Consider a comoving observer sitting at constant spatial coordinates (r*, e*, a Schwarzschild 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 Aem (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 obseryer 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 itatr=2GM?
(c) Calculate the wavelength Aabs, measured by the observer at r*, as a function of the radius rem at which the radiation was emitted.
(d) Calculate the time tabs 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: Aabs/Aem ex etobs/T. Give an expression for the time constant T in tenns of the black hole mass M.