The local speed limit of the universe
In Step 4.4 (Beyond the cosmological particle horizon) I showed that events beyond our cosmological particle horizon may nevertheless be considered as part of our universe. I needed to show this because the argument in Step 4.3 (The multiverse and all of its universes are finite) depended on that supposition. However, I didn’t elaborate on the cosmological particle horizon in that Step because I didn’t want to disturb the flow of the overarching argument too much. In the following posts, I’m going to try to make the idea of the cosmological particle horizon more intuitive. We start by showing that there is a local speed limit in the universe.
In Step 2.1 (The block universe) I referred to light cones in Minkowski diagrams. These are mathematical surfaces containing paths of light which, uniquely for the speed of light, are the same for any observer. Because of the geometry of Minkowski diagrams, we can show that a signal (or, for that matter, another observer) cannot pass an observer faster than the speed of light.
You may be wondering what is so special about light: why not use the speed of sound, for instance, for the light cone, so that it would be a “sound cone” instead? The answer is that it is not so much that light is special; instead, it is that light travels at a special speed – special, because that happens to be the local speed limit of the universe! (The reason for using the word “local” will become clearer in in the following posts – in the meantime, think of the speed limit as saying that the relative speed of a signal, or an observer, passing close to another observer cannot exceed the speed at which light happens to travel.)
To see this, suppose Alice sends such a superluminal signal from her origin at O, towards a point P, and which arrives at P sometime later. We could imagine that the signal is intended to destroy P in an explosion. You can see that the signal is, indeed, travelling faster than light, because P is below the light cone. (So the signal must have travelled more than one space unit in one time unit – faster than the speed of light.)
Suppose that Bob is travelling relative to Alice at a speed of 0.6 c, and happens to pass her at O, just at the moment when Alice emits her signal. Notice how Bob’s space axis is inclined relative to Alice’s space axis, as we saw in Step 2.1 (The block universe). Since P is below Bob’s space axis, it is also in his past. So, in Bob’s reference frame, at the moment when he is at point O, the explosion had already occurred – before the detonation signal was emitted!
If the universe allowed such outrageous violations of causality, then physics would be inconsistent and, indeed, the foundations of logic itself would have been built on sand – nothing would make sense. So, the sequence shown in the diagram cannot happen – because there is a local speed limit in the universe.
So, we see from this diagram that Alice can only influence events in the future that lie within her future light cone. Anything outside it could only be reached by travelling faster than the speed of light, which breaks causality. Equally, anything that lies outside Alice’s past light cone cannot have influenced her here-and-now, because a signal would have had to travel faster than light, breaking causality again.