Actual Infinities

[Alex K]

Time dilation and length contraction are two aspects of the same thing - that the speed of light remains the same for all observers. We had this example before - if you travel to Andromeda at near the speed of light, seen from the outside, 2 million light years are travelled in 2 million years. From the inside, only a few light years are travelled in a few years time. Both reductions have to be present for the ratio of distance and time, which is near the speed of light, to be the same for both observers.

That conversation earlier made me think of this in relation to the concept of 'actual infinities' but I missed the bit about length contraction. Although now that I think about it, I do remember hearing about it before but never really understood it.

Can we use this to argue that 'actual infinities' cannot actually exist because that requires it to be absolute and everything is relative within space-time?

I feel like such an idiot trying to speculate on all of this.

Not everything is relative - the time measured by a clock on its journey from A to B (what is called the proper time) is obviously independent of the observer. But there is a more profound problem - we are assuming special relativity, and indeed in special relativity, all inertial frames are equivalent and there is indeed no absolute time standard. However, in general relativity, and more concretely in the Big Bang cosmology, this is not true any more: if it were, you should in the past have wondered how we can say that the universe is 13.7 billion years old - isn't that an entirely relative statement? No it isn't, because in the actual cosmos out there, we have a special local reference frame, namely that where the cosmic microwave background radiation has the same average frequency in all directions. This is basically the frame in which on average matter in the region is at rest, and this is the frame in which the age measurement applies. The "creation" of radiation and matter after the big bang limits the relativity of space and time to a certain extent because there is one special choice of time axis in which the CMB is maximally isotropic and the stuff in the universe at rest on average.