Spacetime and Geometry · Problem 5

More General Black Holes — Sean Carroll

Physics — General Relativity · schwarzschild-redshift-infalling-beacon

Problem

Consider a comoving observer sitting at constant spatial coordinates

(r_*, \theta_*, \phi_*)

, around 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

\lambda_{\rm 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

\lambda_{\rm obs}

, measured by the observer at

r_*

, as a function of the radius

r_{\rm em}

at which the radiation was emitted. (d) Calculate the time

t_{\rm obs}

at which a beam emitted by the beacon at radius

r_{\rm em}

will be observed at

r_*

. (e) Show that at late times, the redshift grows exponentially:

\lambda_{\rm obs}/\lambda_{\rm em} \propto e^{t_{\rm obs}/T}

. Give an expression for the time constant

T

in terms of the black hole mass

M

Browse the problem library