Notes on Relativity and Cosmology - Physics Department, UCSB

10.2. DYNAMICS! (A.K.A. TIME EVOLUTION) 287
10.2 Dynamics! (a.k.a. Time Evolution)
So, homogeneity and isotropy restrict the shape of space to be in one of a
few simple classes. That is to say, at any time (to the extent that this means
anything) the shape of space takes one of these forms. But what happens as
time passes? Does it maintain the same shape, or does it change? The answer
must somehow lie inside Einstein’s equations (the complicated ones that we
have said rather little about), since they are what control the behavior of the
spacetime metric.
Luckily, the assumptions of homogeneity and isotropy simplify these equations
a lot. Let’s think about what the metric will look like. It will certainly have
a dt 2 part. If we decide to use a time coordinate which measures proper time
directly then the coefficient of dt 2 will just be −1. We can always decide to make
such a choice. The rest of the metric controls the metric for space 1 , which must
be the metric for one of the three spaces described above. Now, the universe
cannot suddenly change from, say, a sphere to a Lobachevskian space. So, as
time passes the metric for space can only change by changing the the overall
size (a.k.a. ‘scale’) of the space. In other words, the space can only get bigger
or smaller.
What this means mathematically is that the metric must take the general form:
ds 2 = −dτ 2 + a 2 (t)(metric for unit − sized space). (10.1)
The factor a(t) is called the ‘scale factor’ or ‘size of the universe.’ When a is
big, all of the spatial distances are very big. When a is small, all of the spatial
distances are very small. So, a space with small a will have a highly curved space
and very dense matter. Technically, the curvature of space is proportional to
1/a 2 , while the density of matter is proportional to 1/a 3 .
Note that the only freedom we have left in the metric is the single function a(t).
Einstein’s equations must therefore simplify to just a single equation that tells
us how a(t) evolves in time.
10.2.1 Expanding and Contracting Universes
Before diving into Einstein’s equations themselves, let’s first take a moment to
understand better what it means if a changes with time. To do so, let’s consider
a case where a starts off ‘large’ but then quickly decreases to zero:
1 There can also be terms of the form dtdx, dtdy, and dtdz that mix time and space.
However, these three terms can be thought of as giving the x, y, z components of a vector in
space. But space is isotropic, so there is no special direction in which this vector can point.
The only vector with no direction is the zero vector, so in fact this vector must vanish. It
follows that there are in fact no cross terms of this sort.