General Relativity: An Introduction for Physicists · Problem 9.9

The Schwarzschild geometry — Hobson, Efstathiou & Lasenby

Physics · gravitational-blueshift-spherical-shell

Problem

An isolated thin rigid spherical shell has mass

M

and radius

R

. Suppose that a small hole is drilled through the shell, so that an observer

O

at the shell's centre can observe the outside universe. Show that a photon emitted by a fixed observer

E

r = r_E

(where

r_E > R

) and received by

O

is blueshifted by the amount

\frac{\nu_O}{\nu_E} = \left(\frac{1-2\mu/r_E}{1-2\mu/R}\right)^{1/2}.