In cosmology, there's a parameter called "curvature". If the curvature were slightly less than 1, the topology would be "closed", meaning it would be finite, and traveling in a straight line ends back where you started. This is like the surface of the Earth, but in 3 dimensions instead of 2. Triangle angles add up to greater than 180 degrees. If the curvature were equal to 1, the universe would be "flat", meaning it would be infinite and triangles would add up to 180. If the curvature were greater than 1, it would be "hyperbolic", meaning it would be infinite and triangles would add up to less than 180.
If I recall correctly, currently the best measurement of the curvature is 1 +/- .02, so it's probably flat. What's interesting is if the shape was flat and the universe were truly infinite, there would be an infinite number of perfect copies of Earth, the closest of which would be within 10^10^23 meters, as noted by Max Tegmark (source: http://space.mit.edu/home/tegmark/crazy.html)
Assuming you are in the Star Trek teleporter camp, which believes that your consciousness continues if you get destroyed somewhere and recreated somewhere else, then if the universe is infinite your consciousness might continue forever.
If the universe is infinite, with infinite matter in all possible configurations, then if you happened to be disintegrated one day, your perfect copy is already waiting for you somewhere out there.
Who knows, maybe you already died a bunch of times, but picked up somewhere else, never noticing that you left. You could still witness others dying of course. Nothing would seem out of the ordinary.
Even the surroundings of your new location in the universe would be the same as accurately as you are able to tell. Information about the planets of the solar system is part of your brain – the new location must also have those planets, otherwise you would not be identical.
Think about a torus-chess-board [0]. While it is locally flat, it is finite, unbounded and you would travel back where you started. Curvature cannot give a sound argument as to whether the universe is finite or infinite.
>Level I: A generic prediction of cosmological inflation is an infinite ergodic universe, which contains Hubble volumes realizing all initial conditions - including an identical copy of you about 10^10^29 meters away.
> If the curvature were slightly less than 1, the topology would be "closed", meaning it would be finite, and traveling in a straight line ends back where you started.
Doesn't this require the assumption that the spacetime is embedded in a higher dimensional spacetime?
It's been a while, but IIRC there are perfectly valid topologically open cosmologies that have locally sphere-like curvatures (in other words, there's no necessary connection between curvature and topology without adding an embeddability constraint), right?
It's also possible to have a flat universe (no curvature), where a straight line can return to the same point. An example is a torus (donut shape). You can see this by noticing that a torus can be projected onto a flat sheet (or created from one) without messing with any relative scales or angles, while the surface of a sphere cannot. (For a sphere, you need Mercator or some other projection which does not conserve an honest representation of relative sizes - e.g. Greenland vs. Africa in Mercator.)
Curvature is ofcourse the way we could time travel. Because the light of events would end up at the same place years later. And travelig faster than light would mean we could see events that will happen in the future ;)
If I recall correctly, currently the best measurement of the curvature is 1 +/- .02, so it's probably flat. What's interesting is if the shape was flat and the universe were truly infinite, there would be an infinite number of perfect copies of Earth, the closest of which would be within 10^10^23 meters, as noted by Max Tegmark (source: http://space.mit.edu/home/tegmark/crazy.html)