Introduction to Cosmology

Relativistic Distance Measures 37

velocities) with clocks running at different local time, but from our vantage point
we would like to have a time value applicable to all of them. If one postulates, with
Hermann Weyl(1885–1955), that the expansion is so regular that the world lines of
the galaxies form a nonintersecting and diverging three-bundle of geodesics, and that
one can define spacelike hypersurfaces which are orthogonal to all of them. Then each
such hypersurface can be labeled by a constant value of the time coordinate푥^0 ,and
using this value one can meaningfully talk about cosmic epochs for the totality of the
Universe. This construction in space-time does not imply the choice of a preferred
time in conflict with special relativity.

2.3 Relativistic Distance Measures

Let us consider how to measure distances in our comoving frame in which we are
at the origin. Thecomoving distancefrom us to a galaxy at comoving coordinates
(휎, 0 , 0 )is not an observable because a distant galaxy can only be observed by the light
it emitted at an earlier time푡<푡 0. In a space-time described by the Robertson–Walker
metric the light signal propagates along the geodesic d푠^2 =0. Choosing d휃^2 =d휙^2 =0,
it follows from Equation (2.36) that this geodesic is defined by

푐^2 d푡^2 −푎^2 (푡)d휒^2 = 0.

It follows that휒can be written

휒=푐

∫

푡 0

푡

d푡

푎(푡)

. (2.38)

The time integral in Equation (2.38) is called theconformal time.

Proper Distance. Let us now define theproper distance푑Pat time푡 0 (when the cos-
mic scale is푎(푡 0 )=1) to the galaxy at(휎, 0 , 0 ). This is a function of휎and of the intrinsic
geometry of space-time and the value of푘. Integrating the spatial distance d푙≡|dl|in
Equation (2.32) from 0 to푑Pwe find

푑P=

∫

휎

0

d휎

√

1 −푘휎^2

=√^1

푘

sin−^1 (

√

푘휎)=휒. (2.39)

For flat space푘=0 we find the expected result푑P=휎. In a universe with curvature
푘=+1andscale푎then Equation (2.39) becomes

푑P=푎휒=푎sin−^1 휎 or 휎=sin(푑P∕푎).

As the distance푑Pincreases from 0 to^12 휋푎,휎also increases from 0 to its maximum

value 1. However, when푑Pincreases from^12 휋푎to휋푎,휎decreases back to 0. Thus,
travelling a distance푑P=휋푎through the curved three-space brings us to the other end
of the Universe. Travelling on from푑P=휋푎to푑P= 2 휋푎brings us back to the point of
departure. In this sense a universe with positive curvature is closed.