# It’s roomy in here

Recently I was talking to somebody about the expansion of the Universe and they asked me a very good question,

“If space goes on forever, how can it be expanding?”

This is related to one I’ve been asked quite often,

“What’s the Universe expanding into?”

It seems obvious at first thought that if something is expanding it must be doing so into some sort of surrounding space, some ‘outside’. So something without an edge, like infinite space, has no ‘outside’ to expand into and therefore can’t expand. The problem here is that our personal experiences in the world tell us one thing, but physics and maths tell us another, and to relate these is very tricky.

The expansion of the Universe is a metric expansion, which means that stationary objects in space, without moving, find themselves further and further apart as time passes.

The Multiverse. Mmmmm!

One handy analogy is a loaf rising in the oven. In this case, let’s imagine it’s a loaf in the bakery at Waitrose (other posh supermarkets are available). This is a ciabatta with olives and sun blushed tomatoes mixed through the dough, which could represent galaxies (or, more correctly, clusters of galaxies). As the loaf rises, the olives and tomatoes grow further and further apart whilst remaining in-place within the dough. It’s the dough which is expanding. Of course, we’re still faced with the issue of the loaf expanding into the space inside the oven. It might be better then to think about an infinitely large loaf and imagine ourselves as being one of the olives inside, observing this expansion.

But here we face the issue highlighted in the first question above. If we can imagine an infinitely large loaf, how can it possibly be expanding and getting bigger? Surely infinitely big things can’t get any bigger.

Well, it depends what we mean by ‘bigger’.

## Number Magic

Let’s think about the whole numbers for a minute. Imagine we have the whole numbers, the integers, in an infinitely long line with us standing beside zero. Off to our left are the negative integers (-1, -2, -3, …) and off to our right are the positive integers (1, 2, 3, …). They stretch off to infinity in both directions. We could walk in either direction and would never reach the end of the line, as there is no ‘last’ number.

What is this magic?

Now a maths wizard comes along (not just someone who is good at maths, an actual wizard… who specialises in maths magic). He surveys our number line. He looks a bit like Christopher Lee. A wry smile spreads across his face and he raises his staff, utters a spell and summons forth a small plastic rectangle with buttons and a screen, which he calls a “magical calculating machine”. Impressive.

With a long, bony finger he types ‘× 2’ and, after a dramatic pause, hits ‘=’. The ground shakes very slightly and there’s a kind of magical sound. “Behold!” he booms in a Shakespearean manner (Sir Christopher, is that you?). But what has he done?

He’s only gone and multiplied all the numbers in our number line by two!

Ok, so it’s not his best trick, but it is convenient for this discussion, so we politely thank him and pop some spare change in his upturned wizard’s hat.

Our numbers have all doubled. They haven’t moved anywhere, but every one of them has been multiplied. That means we still have just as many as we did before. We had infinitely many, and we still have the same number; infinitely many. This might be a bit weird at first thought, but it’s actually a very important mathematical truth.

It was Georg Cantor, a maths wizard of the less becloaked variety, who showed that these sets of numbers, {…, -2, -1, 0, 1, 2, …} and {…, -4, -2, 0, 2, 4, …}, have the same number of members/elements – the same cardinality. Indeed, any other infinite set of integers does too. They are said to be countably infinite. This is because we could take our before and after sets and match them all up one-to-one and clearly every one would have a partner. In other words, we could count them.

Sets of rational numbers (the fractions) are also countably infinite. We can always match the elements of such sets with each other in a one-to-one relationship, whether the sets are related by addition, subtraction, multiplication, division, or any combination of mathematical operations.

## Infinite and still going

Now let’s relate this maths back to the expansion of an infinite universe.

Imagine we have our original set of numbers again, the set of all integers. However, this time they’re not in a line in front of us, but instead represent coordinates on a map. Clearly this map must be infinite, since there are infinitely many coordinates, and these coordinates are marked in at regular intervals in distance. This means that if you’re standing at coordinate (2) and I’m at coordinate (0), we’re 2 units of distance apart. Miles, cubits, furlongs; it doesn’t matter what the units are here, only that the co-ordinates represent distance.

Now the wizard comes back at coordinate (10). This time he’s looking for at least enough change for a cup of coffee, so he has a better spell up his massive sleeve. He uses his fancy machine to multiply the coordinate numbers by two, just like before, but this has an interesting consequence for us.

While I’m still at coordinate (0), you find yourself now at (4). Since these represent distances, we’re 4 units of distance apart; twice as far as we were before! Indeed, every point on the map is now twice as far away from every other point. The wizard is at (20), so from being 8 units from you and 10 from me, he’s now 16 and 20 units away respectively. But, just as with our number line before where the numbers didn’t move, neither have we. The distance between us has increased, but we haven’t moved. This is a metric expansion.

So the map is still infinitely big, like before, in exactly the same way that we saw with the number lines. We could match up all the new coordinate points with old coordinate points, one-to-one, so there are the same number of coordinate points now as there were before. The map is the same size of infinitely big, even if distances between points have increased, and it always was and always will be the same size of infinitely big, no matter how much of this maths magic the wizard unleashes.

This is the same phenomenon we see occurring in space, albeit in more dimensions. The distances between well separated galaxy clusters is increasing as the Universe undergoes a metric expansion, which is perfectly fine even if space is infinite in extent. It would just mean that, for as long as space has existed and will exist, it always was and always will be spatially infinite.