Special Relativity in an (in)finite universe

[Alkatran]

We've all heard of the twin paradox, and the idea of an infinite but finite universe, on the "surface" of a hyper-sphere and what not..

Anyways, my question is, what does relativity predict about someone who would travel all the way around this sphere? Starting at Earth and ending at Earth without ever accelerating? (Unless the rotation around the hyper-sphere counts as acceleration)

 

[pervect]

For the standard Friedmann cosmologies, the prediction is that in the best case, a photon emitted at the big bang could just make it around the universe by the time of the big crunch. This happens onlyfor a universe that's totally mater dominated. If the universe were radiation dominated, the photon could only make it halfway around by the time of the big crunch.

So in the standard cosmologies, an observer won't be able to circumavigate a closed universe between the big bang and the big crunch (except for the case of a photon in a purely matter dominated universe, the best case, where it just barely makes it).

If the universe isn't closed, the observer (or photon) just keeps going and never arrives back at his starting point.

Note - this is discussed a little bit in Gravitation pg 734, and applies for a zero cosmological constant. Unfortunately the case for a non-zero cosmological constant is left as an exercise. If the universe expands indefinitely, though, the photon isn't going to ever reach it's starting point, and neither is any form of matter.

 

[R0man]

According to the principles of special relativity, the laws of physics should be the same for all observers in uniform motion. This means that for someone traveling all the way around a hyper-sphere without accelerating, they should experience the same laws of physics as someone who is stationary on Earth.

However, there are some interesting implications when it comes to the concept of time and space in an infinite universe. If the universe is truly infinite, then there is no defined center or edge, and therefore no preferred frame of reference. This means that there is no absolute way to measure the speed or position of an object in the universe.

In the case of traveling around a hyper-sphere, the person would not experience any time dilation or length contraction as they are not accelerating. However, their perception of time and space may be different from someone who is stationary on Earth due to the curvature of the hyper-sphere. This could lead to some interesting effects, such as a longer or shorter journey depending on the path taken and the observer's frame of reference.

It is also worth noting that the concept of an infinite but finite universe is still a theoretical idea and has not been proven. Therefore, any predictions based on this concept should be taken with a grain of salt. As our understanding of the universe continues to evolve, we may gain a better understanding of the implications of special relativity in an infinite universe.