Introduction to Cosmology

Lorentz Transformations 29

Timelike World line

locations

Particle horizon

x

Past

light cone

Event horizon

Spacelike

locations

Future

light

cone

t 0

tmax

tmin

t

y

Figure 2.1Lightconein푥,푦,푡-space. An event which is at the origin푥=푦=0 at the present
time푡 0 will follow some world line into the future, always remaining inside the future light
cone. All points on the world line are at timelike locations with respect to the spatial origin
at푡 0. World lines for light signals emitted from (received at) the origin at푡 0 will propagate
on the envelope of the future (past) light cone. No signals can be sent to or received from
spacelike locations. The space in the past from which signals can be received at the present
origin is restricted by the particle horizon at푡min, the earliest time under consideration. The
event horizon restricts the space which can at present be in causal relation to the present spatial
origin at some future time푡max.

The redshift is denoted by the letter푧, not to be confused with the coordinate푧in
Equations (2.6) and (2.7).
The ratio of wavelengths actually measured by the terrestrial observer is then

1 +푧=

휆obs

휆rest

=

푅 0

푅(푡)

=

1

푎(푡)

. (2.9)

It should be stressed that the cosmological redshift is not caused by the velocities of
receding objects, but by the increase in scale푎(푡)since time푡. A kinematic effect can be
observed in the spectra of nearby stars and galaxies, for which their peculiar motion
is more important than the effect of the cosmological expansion. This may give rise