General Relativity: An Introduction for Physicists · Problem 14.5

The Friedmann–Robertson–Walker geometry — Hobson, Efstathiou & Lasenby

Physics · particle-velocity-expanding-universe

Problem

At cosmic time

t_1

, a massive particle is shot out into an expanding FRW universe with velocity

v_1

relative to comoving cosmological observers. At a later cosmic time

t_2

the particle has a velocity

v_2

with respect to comoving cosmological observers. Show that, at any intermediate cosmic time

t

, the velocity of the particle as measured by a comoving cosmological observer is

v(t) = R(t)\frac{d\chi}{dt}.

Hence show that

\frac{\gamma_{v_2} v_2}{\gamma_{v_1} v_1} = \frac{R(t_1)}{R(t_2)},

where

\gamma_v = (1 - v^2/c^2)^{-1/2}

and

R(t)

is the scale factor at cosmic time

t

. By considering the particle momentum, show that as

v_1 \to c

the photon redshift formula is recovered.