The motivation for today’s post comes from some different articles I have read around inflationary cosmology, which has highlighted to me a common misconception when considering the basics of expansion. Expansion is interesting; and I would be bold enough to say the most important result to come out of cosmology. It has allowed us to make calculations, extrapolate understanding and identify “hidden” energy and matter – all because of observable phenomena.

The universe is expanding; this is true – however it is not the case the further away an object is the faster it is moving. We must strive to be more delicate than this, that’s only sort of true, but that’s the mistake i’ve seen made in a few places. I thought I would clear things up a little bit. The formula we want to consider is the following, the Hubble relationship;

v = Hd

In the above v represents the apparent velocity, H represents the Hubble constant and d represents the distance from the observer. The takeaway here is that the apparent speed is not the same as the speed. This is a simple case of relative speed – but a very important one. Now let us also state that for all discussions today the universe expands uniformly. There are theorems around non-uniform expansion near the beginning of the universe which are widely accepted to be accurate, however right now consider us undergoing uniform expansion, which is an appropriate way to consider things in the vast majority of circumstances.

Now the “things” in the universe… me, you, planets are not expanding because they are held together by forces. Nuclear forces, electrostatic forces… forces which the expansion is nothing against. To illustrate the idea of uniform expansion, we are going to go on a little journey to the kind of world you might find in Flatland – a two dimensional world where the universe is a long thin elasticated band in two dimensions. We will consider the expansion from the perspective of a citizen at point C. Now remember C is in a galaxy where the matter within it is not expanding because it is held together by forces. The stretchy band however is expanding uniformly, outward and outward. If you ever get bored, you only need an elastic band, a ruler and a very poor social life and you can do this at home.

Time-phase 1: At the first time phase there are two galaxies we can observe from C – galaxy A and galaxy B. At this timephase, we have made some calculations from C and galaxy A appears to be 5 units from us, and galaxy B is 10 units from us. In other words – B is five units further away from A.

Time-phase 2: A billion years has passed us by and now the distance from C to A is 10 units away. This means for every 1 unit space it has stretched, or expanded to 2. So what of galaxy B? Now remember how the space between A and B was five units – this space between the two points must too have uniformly expanded, there is nothing to keep it together, so in line with our stretchy band universe this distance is now 10. I hope you can see where we are getting to with this – from the perspective of the viewer at C – the distance between A and B has doubled. In the same time step B had managed to get further away – and so, if we knew no better we would say that the further something is away the faster it is moving.

What we are therefore saying when we view a distant object is that the further away an object is, the faster it appears to be moving because the space between that object and other reference points is increasing. If we were able to play God and observer the universe from a loft perspective above, we would be able to drop a reference frame on the world and see things much clearer. But alas, we cannot play God, for we are trapped in a small drop in a vast and mysterious ocean doing our best to make sense of what we see.

You should be able to see why it is okay to say that the further away something is the greater its apparent speed – this is true as we have just demonstrated in Flatland, and in my opinion very interesting. Apologies to my more advanced readers, who will have no doubt found this a very simple post.