An open universe

The Universe is said to be open if it does not contain enough matter
to stop it expanding6. In this case things get a little trickier to
visualize. For a start, since this type of universe does not close in
on itself, then the only way for it to avoid having an edge is for
it to be infinite7. The simplest shape that such a universe could
have would be for it to be flat, the three-dimensional analogue of
the rubber sheet that would extend out infinitely in all directions.
But for the Universe to have no curvature at all would be a very special case. It would be like the example of the ball rolling up
the slope and managing to reach the top just as it runs out of
puff and having no more energy to roll along the flat top. It is
much more likely, if it is not going to roll back down again, that
it would have some energy left over to keep going along the top.
A universe corresponding to such a scenario will not be flat but
curved. However, this time we say it has negative curvature.
So, by throwing away one of the dimensions of space we
can make sense of the different types of curvature the Universe
might have. If a positively curved universe corresponds, in a
lower dimension, to the surface of a sphere and a flat universe
corresponds to a flat two-dimensional sheet, what shape is a
negatively curved two-dimensional surface? This one is not so
easy. The correct mathematical name for such a shape is a righthyperboloid,
or hyperbolic, surface and is impossible to visualize
properly. Very roughly, it has the shape of a saddle
The difference between the positive curvature of a sphere and the
negative curvature of the saddle is that, whereas in the former
any two paths on the surface that cross each other at right angles
curve round in the same direction, such paths in the case of the
saddle will curve in opposite directions. And the reason why
the saddle is not an accurate depiction of a hyperbolic surface is
that, as you move away from the centre of the saddle the surface
gets flatter, whereas for a right-hyperboloid, the surface must have
the same curvature everywhere. It is impossible to sketch such a
surface.Since the shape of an open universe is something very difficult
to visualize, even in a lower dimension, let us see if we can do any
better at understanding another puzzling feature. Namely, if the
Universe is open and infinite then what does it expand into? By
infinite I mean that space extends on forever in all directions. It
would seem impossible that it can expand at all since all of space
is used up and included within the Universe. Again, we can see
the problem more clearly in two-dimensional space. For the case
of a closed universe (the surface of the balloon), we can imagine
the expansion to be outwards into a higher dimension, but for a
flat sheet which has an infinite area, the expansion will always
be in the plane of the sheet, and we cannot make use of the third
dimension (off the sheet) as somewhere for it to expand into.
To solve this problem I need to explain a little mathematics.
No one is comfortable thinking about infinity. I remember as a
child being told that when we die we go to Heaven and we stay
there for ever and ever. The thought of this use to depress me
since I was uncomfortable thinking about something that would
just go on and on without end however nice it was meant to be.
Despite the difficulty most of us have contemplating the infinite,
some mathematicians have made a living out of studying it. In
fact, there are even different types of infinity.
Think of the sequence of integer numbers (or whole numbers):
1, 2, 3, 4, . . . which goes on forever. We say that there is an infinite
number of integers. But how about the sequence of even integers:
2, 4, 6, 8, . . .? Surely this sequence is also infinitely long. And since
there are twice as many integers in total as there are even ones, we
have two infinities with one seemingly twice as large as the other.
What about the number of all numbers, not just the integers? For
instance, let us consider the numbers:
0, 0.1, 0.2, 0.3, . . . , 0.9, 1.0, 1.1, 1.2, 1.3, . . .
and so on to infinity. This infinite sequence contains ten terms
for every one in the sequence of integers. Is the infinity of
terms in this sequence therefore ten times as big as the infinity
of the integers? In mathematics, there is a whole subject
devoted to the study of infinity. It turns out that the above three sequences all belong to the same class of infinity. But
there are others. Consider the sequences of all numbers (called
the set of real numbers) which includes all the fractions in between
the integers. Even the interval between two consecutive
integers such as 0 and 1 will contain an infinity of numbers
(0, . . . , 0.00103, . . . , 0.36252, . . . , 0.9997, . . . , 0.999999, . . . , 1),
since we can always think of a new fraction to slot in, however
many decimal places it may have. There will likewise be an infinite
number of fractions contained between 1 and 2, and between
2 and 3, and 763 and 764, and so on. Thus, we have a set containing
an infinity of integers and an infinity of fractions between
consecutive integers. This overall infinity is much ‘stronger’ than
the infinity of integers, despite both being never ending! It turns
out that there are in fact an infinite number of different infinities!
Where is all this leading us? The cosmologist Igor Novikov,
considered by many to be Russia’s answer to Stephen Hawking,
uses the idea of different infinities to explain how an infinite
universe is nevertheless still able to expand. Imagine that you
check into Hotel Infinity, which has an infinite number of rooms—
I’ve stayed in a few hotels that came close to this and I have
certainly been lost in a few. You are told at the front desk that
they are very busy that night and that there are already an infinite
number of guests so all the rooms are occupied. You complain to
themanagementthatyouhada reservationandinsist that theyfind
you a room for the night. “No problem” says the management, “in
Hotel Infinity there is always room for more”. They then proceed
to move the person in room 1 into room 2, the person in room 2
into room 3, and so on, all the way to infinity. You are then given
room 1.
What if an infinite number of guests arrive at once? Still no
problem (forget for the moment about the infinitely long queue
at the check-in desk). The management now move the person in
room 1 into room 2, the person originally in room 2 into room 4,
the person in room 3 into room 6, 4 to 8, and so on until all the
guests are moved. Now all even numbered rooms are occupied.
Since there is an infinite number of these rooms, all original guests
are accommodated. This then leaves the infinity of odd numbered
rooms now vacated and available for the new arrivals.We can relate this example of the hotel guests to the space
occupied by an infinite universe. It does not matter thatnewguests
are arriving all the time. The hotel, being infinite, can always
accommodate them. In the same way, an infinite space can always
expand.
We now come to probably the most confusing feature of an
infinite universe. If something is growing in size, then it would, by
definition, take forever to become infinite. Thus, if our Universe is
infinite in size today then it must also have been infinite in the past.
In fact, it must already have been infinite in size at the moment of
the Big Bang! This really flies in the face of the common notion of
the Big Bang as the event when all of space was squeezed down to
a point of zero size. This can at least be visualized for the case of a
closed universe by dropping down a dimension and considering
the example of the balloon. But an open universe was never zerosized.
The only way to think about this is to imagine that the Big
Bang happened everywhere at once in an already infinite universe.
Of course at any point in such an infinite universe the density
would have been infinite too.
Another way to think of it is if the big bang of an open universe
is like an infinitely long line. Even though it has an infinite number
of points on it (since a point has zero extent) it still has zero volume
over all. We could then imagine that our Visible Universe grew
out from just one point (one big bang) of the line. I wouldn’t push
this analogy too hard though.
Finally, just to make sure you are totally confused, whatever
shape the Universe has now, even if it is almost completely flat, it
would have been infinitely curved at the Big Bang!