Physical Foundations of Cosmology

28 Kinematics and dynamics of an expanding universe

τ

r

t= const

χ= const

t=

0,

χ =

∞

Fig. 1.7.

and hence the Minkowski metric reduces to (1.85). A particle with a given comoving
coordinateχmoves with constant velocity

|v|≡r/τ=tanhχ< 1 (1.88)

in Minkowski space and its proper time,

√

1 −|v|^2 τ, is equal to the cosmological
timet.To find the hypersurfaces of constant proper timet, we note that

t^2 =τ^2 −r^2. (1.89)

The hypersurfacet=0 coincides with the forward light cone; the surfaces of con-
stantt>0 are hyperboloids in Minkowski coordinates, all located within the for-
ward light cone. Hence, the Milne coordinates cover only one quarter of Minkowski
spacetime (Figure 1.7).
Despite its obvious deficiencies as a practical model, the Milne universe does
illustrate some useful points. First, it shows the similarities and differences between
an explosion (the popular misconception of the “big bang” ) and Hubble expan-
sion. The Milne universe has a center. It is apparent from the fact that the Milne
coordinates cover only one particular quarter of Minkowski spacetime. The curved
Friedmann universe has no center. Second, the Milne universe reveals the subtleties
in the physical interpretation of recessional velocity. If the recessional velocity of
a particle were defined as|u|≡r/t=sinhχ, it would exceed the speed of light
forχ>1. Of course, there can be no contradiction with the principles of Spe-
cial Relativity and we know that the particle is traveling on a physically allowed,