Introduction to Cosmology

36 Special Relativity

The Robertson–Walker Metric. Let us now include the time coordinate푡,thecos-

mic expansion scale푎(푡)and the curvature parameter푘in Equation (2.29). We then

obtain the complete metric derived byHoward RobertsonandArthur Walkerin 1934:

d푠^2 =푐^2 d푡^2 −d푙^2 =푐^2 d푡^2 −푎(푡)^2

(

d휎^2

1 −푘휎^2

+휎^2 d휃^2 +휎^2 sin^2 휃d휙^2

)

. (2.32)

In the tensor notation of Equation (2.14) the components of the Robertson–Walker

metricgare obviously

푔 00 = 1 ,푔 11 =−

푎^2 (푡)

1 −푘휎^2

,푔 22 =−푎^2 (푡)휎^2 ,푔 33 =−푎^2 (푡)휎^2 sin^2 휃. (2.33)

Equation (2.32) is the most general metric for a four-dimensional space-time which
is homogeneous and isotropic at a given time. It thenwill always remain homogeneous
and isotropic, because a galaxy at the point(휎,휃,휙)will always remain at that point,
only the scale of spatial distances푎(푡)changing with time. The displacements will be
d휎=d휃=d휙=0 and the metric equation will reduce to

d푠^2 =푐^2 d푡^2. (2.34)

For this reason one calls such an expanding frame acomoving frame. A metric with

푘=0 is calledflat.

An observer at rest in the comoving frame is called afundamental observer.Ifthe

Universe appears to be homogeneous to him/her, it must also be isotropic. But another

observer located at the same point and in relative motion with respect to the funda-

mental observer does not see the Universe as isotropic. Thus the comoving frame is

really a preferred frame, and a very convenient one, as we shall see later in conjunction

with the cosmic background radiation. Let us note here that a fundamental observer

may find that not all astronomical bodies recede radially; a body at motion relative to

the comoving coordinates(휎,휃,휙)will exhibit peculiar motion in other directions.

Another convenient comoving coordinate is휒, defined by integrating over

d휒=√d휎

1 −푘휎^2

. (2.35)

Inserting this into Equation (2.32), the metric can be written

d푠^2 =푐^2 d푡^2 −푎^2 (푡)[d휒^2 +푆^2 푘(휒)(d휃^2 +sin^2 휃d휙^2 )], (2.36)

where

푆푘(휒)≡휎

and

푆 1 (휒)=sin휒, 푆 0 (휒)=휒, 푆− 1 (휒)=sinh휒. (2.37)

We shall use the metrics in Equations (2.32) and (2.36) interchangeably since both

offer advantages. In so doing I have take great care of not introducing contradictions.

Let us briefly digress to define what is sometimes calledcosmic time.Inanexpand-

ing universe the galaxies are all moving away from each other (let us ignore peculiar