Physical Foundations of Cosmology

1.1 Hubble law 5

what happens. After we have reached the limits of validity of Newtonian theory,
we turn to a proper relativistic treatment.
In an expanding, homogeneous and isotropic universe, the relative velocities of
observers obey theHubble law: the velocity of observerBwith respect toAis

vB(A)=H(t)rBA, (1.1)

where the Hubble parameterH(t) depends only on the timet, andrBAis the vector
pointing fromAtoB.Some refer toHas the Hubble “constant” to stress its
independence of the spatial coordinates, but it is important to recognize thatHis,
in general, time-varying.
In a homogeneous, isotropic universe there are noprivilegedvantage points
and the expansion appears the same to all observers wherever they are located. The
Hubble law is in complete agreement with this. Let us consider how two observersA
andBview a third observerC(Figure 1.1). The Hubble law specifies the velocities
of the other two observers relative toA:

vB(A)=H(t)rBA, vC(A)=H(t)rCA. (1.2)

From these relations, we can find the relative velocity of observerCwith respect
to observerB:

vC(B)=vC(A)−vB(A)=H(t)(rCA−rBA)=HrCB. (1.3)

The result is that observerBsees precisely the same expansion law as observerA.
In fact, the Hubble law is theuniqueexpansion law compatible with homogeneity
and isotropy.

A B

VC(B) VC(A)

rCB

rBA VB(C)

rCA

C

Fig. 1.1.