The expanding universe
First published 4 June 2026
In Step 2.1 (The block universe) we saw that light paths are always on light cones which are the same for all observers, even if they are moving with respect to the light cone. In the previous post, we saw that the local speed limit of the universe is the speed at which light travels. Now I’m going to throw a spanner in the works by showing you the coordinates of a uniformly expanding universe where light cones are not all the same and the speed of light is exceeded.
Key to the apparent violation of the rules in the diagram is that the diagram is not strictly a Minkowski diagram. It is a first-pass representation of a consequence of Einstein’s equations for general relativity, that the universe is expanding, famously observed by Edwin Hubble and others before Einstein himself accepted the implications of his equations. Try not to think of “space expanding” – it doesn’t, because there is no such thing as space: space isn’t a thing, it’s not stuff, it’s not a stretchy rubber sheet like you see in the popular accounts. What expands is the so-called scale factor, which is a measure of how the distance between two objects – galaxies, say – might change with time. Once again, this is a consequence of the mathematical structure which is our universe – see all of the sub-steps in Step 2 (The universe is purely a mathematical structure).
The time grid lines are drawn to intersect our “universe now” space axis at intervals separated by 2 billion light-years.
The diagram shows the special case where the scale factor is expanding uniformly with time: that is, where the distance between any two objects is proportional to the age of the universe. So, for instance, if two galaxies are separated by one billion light-years when the universe is five billion years old, then they will be separated by two billion light-years after the universe has doubled its age to ten billion years, and so on. So, the standard Minkowski diagram no longer applies, because, in those diagrams the space gridlines have to be regular and parallel (as shown by a few indicative vertical grey lines in the diagram). The divergence of the space gridlines in the expanding universe means that the light cones are generally no longer at 45° to the horizontal, but, instead, progressively “tip over” at increasing distances from each other.
I have drawn light cones at intervals along the space axis in the figure, and you can see how they slant increasingly towards the right the further away they are from us (marked by “We are here” in the diagram). If you look at the light cones close to us – say the one next to our own – then, as usual, the left side of the “V” of that light cone shows the path of a light beam travelling towards us, having been emitted in our direction from, say, a neighbouring galaxy. However, there comes a point – for example, the fourth light cone from the left shown on the diagram – where the left side of the “V” is nevertheless tilted so far to the right that the light is travelling away from us – even though it has been emitted towards us by the galaxy!
When we get to the situation that I just described, where light emitted in our direction by a galaxy is nevertheless receding from us, then that galaxy must be receding from us faster than the speed of light. This is why it was appropriate to qualify the speed limit in the universe as a local speed limit – you can’t have a signal or another object passing you faster than the speed of light, but the same restriction doesn’t apply non-locally. (If you try to construct an argument similar to that in Step to show that the speed limit applies when the two objects are separated by a large distance, you find that there are always insurmountable objections to the thought experiment.)
This diagram is another perspective of the previous one. The distance from us to where galaxies are receding from us at exactly the speed of light is called the Hubble distance. In the previous diagram, it is the distance to the light cone where the left arm of the “V” is vertical, meaning that the speed of light that was emitted towards us from a galaxy is exactly balanced by the speed with which the galaxy is receding from us. In a uniformly expanding universe, the Hubble distance should be just the speed of that galaxy multiplied by the time it has been travelling away from us, 13.8 billion light-years. The light cone with the vertical left arm looks slightly further away than this in the diagram, but that is simply because of the relatively large size that I have drawn the light cones.
We are on the central vertical axis (our world line) in the uniformly expanding universe, and the Hubble distance increases uniformly with time. The Hubble distance expands all around us in a perfect sphere, which I have shown as expanding dashed circles.
I have also drawn the Milky Way and a distant galaxy which lies beyond the Hubble sphere and is therefore travelling away from us faster than the speed of light (like the light cones to the right of the Hubble distance in the previous diagram).
An important point to note about the galaxies is that they themselves are not stretched by the “expansion” of space, and neither are their constituent stars or the atoms within them.
You might think that the sizes of atoms and galaxies are unaffected by the expansion of space because atoms, and even galaxies, are small when compared with the cosmological distances over which we see the expansion. That is not the reason, though. The expansion applies at all scales. However, the force required to resist the expansion on a local scale, even on a galactic scale, is overwhelmingly supplied by the electromagnetic forces within atoms and, indeed, by the gravitational forces between stars in a galaxy. So, atoms and individual galaxies are not observably stretched by the expansion of space.
In the next post, I shall use these diagrams to give you what is, hopefully, an intuitive idea of the cosmological particle horizon.