The universe expanding, but what will become of it?
There are three possible solutions to the equations of the General Theory of Relativity which
represent homogeneous and isotropic universes: either it will
continue its expansion forever, or it will
eventually stop and re-contract or it will expand slowing down to a stop
at infinite time. The contents of the universe (matter and radiation)
determine which of these is realized in *our* universe.
In all three cases the shape of space remains the same as the universe
expands (or in the second case, as it expands and contracts).

That the shape of space is determined by the amounts of matter and energy in the universe is not surprising as it is matter and energy which determine the curvature of space (see Sect. 7.9).

- Space in an eternally expanding or
*open*universe is shaped like a 3-dimensional horse saddle. In this case the angles in a triangle add up to*less*than 180^{o}. - Space in a
*closed*universe which will eventually re-contract is shaped like a 3-dimensional sphere. In this case the angles in a triangle add up to*more*than 180^{o}. - Space in a
*flat*universe which expands slowing down to a stop at infinite time is shaped like a 3-dimensional plane. In this case the angles in a triangle add up to 180^{o}.

These three possibilities give the *average* shape of space.
Individual masses produce local bumps and troughs. This is similar to
the way we talk about the Earth: we say it is a sphere, though we know
it is full of bumps (for example, Himalayas) and troughs (the Dead Sea,
for example).

Of these possibilities the one corresponding to our universe is determined by the amount of matter in the cosmos. If there is very little the initial thrust from the Big Bang will never be stopped, if however there is a large amount of matter, the mutual gravitational pull will be sufficient to break the expansion and eventually cause a re-contraction. Hence there is a critical amount of matter such that if our universe has more it will re-contract, if less it will expand forever (if it has precisely the critical amount it will expand forever slowing down to a stop at infinite time). These possibilities are illustrated in Fig. 8.15.

The obvious question is then: how much stuff is in the universe? And to that we can say: we don't know. If we count all the matter that shines (stars and such) we get a number very low compared to the critical value. But, is most of the matter shining? Could it not be that there is a lot of dust out there? The latest results suggest that the universe will expand forever, but at present its ultimate fate is unknown.