![]() ![]() If you focus on what actually happens there isn't a paradox. And then you could travel to the other side of the earth and sit and wait for the image of you landing to finally arrive from the long way around.Īnd you never see a paradox. What happens is that you see two earths, the one you are at, and an image of an older earth where light left long ago, wrapped all around the universe and just now got to you.Īs you move towards it the image of the far away gets bluer an you see the images of the people on earth celebrating holidays arrive more often than you age.Īs you move away from the nearby earth, you see the image of the nearby earth get redder and you see the images of the people on earth celebrating holidays arrive less often than you age.Įventually the image of the earth that was initially farther away is an earth that is actually closer, and eventually you land on it. The equations of general relativity can be solved with a universe hat wraps around, even if it is flat. In general, we mean by a curved space simply one in which the rules of Euclidean geometry break down with one sign of discrepancy or the other. I'm thinking perhaps that travelling through curved space itself constitutes acceleration, since in a way you're changing direction? In which case the travelling twin wouldn't remain in an inertial reference frame and we could solve the 'paradox' as before?Īnd therefore he would be able to return to Earth without having changed inertial reference frame, and then we would actually have a paradox? In general, a two-dimensional world will have a curvature which varies from place to place and may be positive in some places and negative in other places. Almost positive curvature on an irreduciblecompact rank 2 symmetric space Jason DeVito and Ezra Nance Abstract A Riemannian manifold is said to be almost positively curved if thesets of points for which all 2-planes have positive sectional curvatureis open and dense. ![]() However, if it was positive, does this mean that the travelling twin, if he travelled long enough, would loop back to Earth eventually? And therefore he would be able to return to Earth without having changed inertial reference frame, and then we would actually have a paradox? ![]() As far as I'm aware, we're quite sure it's flat. Now, there are three possible curvatures the universe could have - positive, negative, or flat. The 'paradox' can be resolved by realising that the travelling twin does not remain in an inertial frame for all time, since he has to turn around at the planet and hence accelerate. However, from the perspective of the travelling twin, it was the Earth that was moving and so the Earth twin should be the younger one. I think this would make it harder to solve the Friedmann equations compared to Standard cosmology, since you would have boundary conditions for $a(t)$ at the beginning and everytime the time component loops, which for example forbids exponential growth as in a dark energy dominated universe.I'm reading about the twin paradox in special relativity - if there are two identical twins, one of whom who sets off in a high speed rocket to a planet, and then heads back, will find the twin who remained on Earth to have aged comparatively more due to time dilation. You could think of a universe where time loops, being a circle, I think this would correspond to your idea of time having $k=1$ (which, again, it has not, since mathematically a 1D circle still has no curvature). Mathematically you cannot assign a curvature to ONLY the time component, because a one-dimensional space cannot have curvature (there are no angles between different points that you could measure). The idea of assigning ONE curvature to all four space-time coordinates, including time, does not make a lot of sense in a universe that we expect to be different at different points in time, since including the time dimension, the universe is NOT homogenous and isotropic (it DOES matter at which time you look, and it DOES matter if you look in the direction of the past or towards the future). This leaves the three possibilities of $k=1,0,-1$, if all spatial coordinates are normalised correspondingly, because these three spatial metrics fulfill the conditions. In general, a two-dimensional world will have a curvature which varies from place to place and may be positive in some places and negative in other places. observation, that the spatial part of the universe at a certain instance in time (!) should be homogenous and isotropic. The idea of constant spatial curvature comes from the idea, i.e. ![]()
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