2.4 Redshift 57
Although the derivation above has been performed in a local inertial frame, it can
be applied piecewise to a general geodesic photon trajectory. The result is therefore
valid in curved spacetime as well. However, the interpretation of the redshift as a
Doppler shift is not applicable for distances larger than the curvature scale. In this
limit, as we have pointed out, distance and relative velocity do not have an invariant
meaning, so the notion of Doppler shift becomes ill defined.
Redshift of peculiar velocitiesThe peculiar velocities of massive particles (veloc-
ities with respect to the Hubble flow) are also redshifted as the universe expands.
The peculiar velocity of a particle,w(t 1 ),measured by observer “1” at timet 1 ,
is different from the peculiar velocity of the same particle,w(t 2 ),measured by
observer “2”, by the relative Hubble speed of the observers:v=H(t)l. Hence,
w(t 1 )−w(t 2 )≈v=H(t)l. (2.49)Given that the particle needs timet=t 2 −t 1 =l/wto make the journey be-
tween the two observers, we can rewrite this equation as
w ̇=−H(t)w. (2.50)Once again we have the solution
w∝ 1 /a. (2.51)Thus, the expansion of the universe eventually brings particles to rest in the co-
moving frame.
The temperature of a nonrelativistic gas of particles is proportional to the peculiar
velocity squared,
Tgas∝w^2 ∝ 1 /a^2 , (2.52)and therefore, if the gas and radiation are decoupled, gas will cool faster than
radiation.
For the same reasons as in the case of radiation, the above derivation for peculiar
velocities is rigorous and applicable in curved spacetime. This can also be verified
directly by solving the geodesic equations for the particles.
Problem 2.13Show that the geodesic equation
duα
ds+βγα uβuγ= 0 (2.53)can be rewritten as
duα
ds−
1
2
∂gβγ
∂xα
uβuγ= 0. (2.54)