Page 27 - A Brief History of Time - Stephen Hawking

A Brief History of Time - Stephen Hawking... Chapter 3

Figure 3:4

In this case the separation, shown in Figure 3:4, also starts at zero and increases forever. However, the speed
at which the galaxies are moving apart gets smaller and smaller, although it never quite reaches zero.

A remarkable feature of the first kind of Friedmann model is that in it the universe is not infinite in space, but
neither does space have any boundary. Gravity is so strong that space is bent round onto itself, making it rather
like the surface of the earth. If one keeps traveling in a certain direction on the surface of the earth, one never
comes up against an impassable barrier or falls over the edge, but eventually comes back to where one
started.

In the first kind of Friedmann model, space is just like this, but with three dimensions instead of two for the
earth’s surface. The fourth dimension, time, is also finite in extent, but it is like a line with two ends or
boundaries, a beginning and an end. We shall see later that when one combines general relativity with the
uncertainty principle of quantum mechanics, it is possible for both space and time to be finite without any edges
or boundaries.

The idea that one could go right round the universe and end up where one started makes good science fiction,
but it doesn’t have much practical significance, because it can be shown that the universe would recollapse to
zero size before one could get round. You would need to travel faster than light in order to end up where you
started before the universe came to an end – and that is not allowed!

In the first kind of Friedmann model, which expands and recollapses, space is bent in on itself, like the surface
of the earth. It is therefore finite in extent. In the second kind of model, which expands forever, space is bent
the other way, like the surface of a saddle. So in this case space is infinite. Finally, in the third kind of
Friedmann model, with just the critical rate of expansion, space is flat (and therefore is also infinite).

But which Friedmann model describes our universe? Will the universe eventually stop expanding and start
contracting, or will it expand forever? To answer this question we need to know the present rate of expansion of
the universe and its present average density. If the density is less than a certain critical value, determined by
the rate of expansion, the gravitational attraction will be too weak to halt the expansion. If the density is greater
than the critical value, gravity will stop the expansion at some time in the future and cause the universe to
recollapse.

We can determine the present rate of expansion by measuring the velocities at which other galaxies are
moving away from us, using the Doppler effect. This can be done very accurately. However, the distances to
the galaxies are not very well known because we can only measure them indirectly. So all we know is that the
universe is expanding by between 5 percent and 10 percent every thousand million years. However, our
uncertainty about the present average density of the universe is even greater. If we add up the masses of all

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