Physical Foundations of Cosmology

2.4 Redshift 55

Draw the conformal diagram for the eternal black hole and identify the types of
infinities. The Schwarzschild radius of the black hole is located atv^2 =u^2.

2.4 Redshift

The expansion of the universe leads to a redshift of the photon wavelength. To
analyze this effect, let us consider a source of radiation with comoving coordinate
χem, which at timeηememits a signal of short conformal durationη(Figure 2.9).
According to (2.5), the trajectory of the signal is

χ(η)=χem−(η−ηem)

and it reaches a detector located atχobs=0 at timeηobs=ηem+χem. The confor-
mal duration of the signal measured by the detector is the same as at the source,
but the physical time intervals are different at the points of emission and detection.
They are equal to

tem=a(ηem)η and tobs=a(ηobs)η

respectively. Iftis the period of the light wave, the light is emitted with wavelength
λem=tembut is observed with wavelengthλobs=tobs,so that

λobs

λem

=

a(ηobs)

a(ηem)

. (2.45)

χ

h

hobs+ ∆h

hobs

hem ∆h

χem

Fig. 2.9.