Post
by apodman » Mon Apr 13, 2009 11:15 pm
If I'm wrong about this, somebody please correct me.
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Suppose I live on a spherical planet. The 2D surface I walk on is curved positively. If I walk from point A to point B to point C and back to point A, taking the shortest possible (straight, within the rules of the positively curved geometry) route from each point to the next, I trace a triangle whose angles add up to more than 180 degrees. Suppose further that this spherical planet has no hills. I can travel anywhere on the surface of this planet in a line that is straight with respect to the spherical geometry. (Now suppose that this spherical planet has hills. In that case, my shortest route between points with an intervening hill is curved - compared to a straight line in the spherical geometry - to a degree determined by the size of the hill.)
Suppose instead that I live on a Euclidean plane. The 2D surface I walk on is flat. If I walk from point A to point B to point C and back to point A, taking the shortest possible (straight, within the rules of the flat geometry) route from each point to the next, I trace a triangle whose angles add up to exactly 180 degrees. Suppose further that this Euclidean plane has no hills. I can travel anywhere on this surface in a line that is straight with respect to the flat geometry. (Now suppose that this Euclidean plane has hills. In that case, my shortest route between points with an intervening hill is curved to a degree determined by the size of the hill.)
Suppose in yet a third case that I live on a hyperbolic paraboloid (an unbounded saddle shape). The 2D surface I walk on is curved negatively. If I walk from point A to point B to point C and back to point A, taking the shortest possible (straight, within the rules of the negatively curved geometry) route from each point to the next, I trace a triangle whose angles add up to less than 180 degrees. Suppose further that this hyperbolic paraboloid has no hills. I can travel anywhere on this surface in a line that is straight with respect to the negatively curved geometry. (Now suppose that this hyperbolic paraboloid has hills. In that case, my shortest route between points with an intervening hill is curved - compared to a straight line in the hyperbolic geometry - to a degree determined by the size of the hill.)
The difference among the three cases is Euclid's fifth postulate, the "parallel postulate". In flat geometry, there is exactly one line parallel to a given line passing through a given point not on the first line. In positively curved geometry, there are no lines parallel to a given line passing through a given point not on the first line. In negatively curved geometry, there are infinitely many lines parallel to a given line passing through a given point not on the first line. There are other ways of telling whether your geometry is flat, spherical (positively curved), or hyperbolic (negatively curved), but the differences in parallelism and the angles in a triangle are the traditional telltale signs. (Note that hills have the same type of effect on all three geometries.)
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Now suppose that the universe is spherical space-time. The 4D space-time a photon travels through is curved positively. If a photon travels from point A to point B to point C and back to point A, taking the shortest possible (straight, within the rules of the positively curved geometry) route from each point to the next, it traces a triangle whose angles add up to more than 180 degrees. Suppose further that this spherical space-time has no objects with mass, or has no gravity, or has no General Relativity. A photon can travel anywhere in this space-time in a line that is straight with respect to the spherical geometry. (Now suppose that this spherical space-time has objects with mass, has gravity, and has General Relativity. In that case, the photon's shortest route between points with an intervening object is curved - compared to a straight line in the spherical geometry - to a degree determined by the mass of the object.)
Now suppose instead that the universe is Euclidean space-time. The 4D space-time a photon travels through is flat. If a photon travels from point A to point B to point C and back to point A, taking the shortest possible (straight, within the rules of the flat geometry) route from each point to the next, it traces a triangle whose angles add up to exactly 180 degrees. Suppose further that this flat space-time has no objects with mass, or has no gravity, or has no General Relativity. A photon can travel anywhere in this space-time in a line that is straight with respect to the flat geometry. (Now suppose that this flat space-time has objects with mass, has gravity, and has General Relativity. In that case, the photon's shortest route between points with an intervening object is curved to a degree determined by the mass of the object.)
Now suppose in yet a third case that the universe is hyperbolic space-time. The 4D space-time a photon travels through is curved negatively. If a photon travels from point A to point B to point C and back to point A, taking the shortest possible (straight, within the rules of the negatively curved geometry) route from each point to the next, it traces a triangle whose angles add up to less than 180 degrees. Suppose further that this hyperbolic space-time has no objects with mass, or has no gravity, or has no General Relativity. A photon can travel anywhere in this space-time in a line that is straight with respect to the hyperbolic geometry. (Now suppose that this hyperbolic space-time has objects with mass, has gravity, and has General Relativity. In that case, the photon's shortest route between points with an intervening object is curved - compared to a straight line in the hyperbolic geometry - to a degree determined by the mass of the object.)
The three cases are analogous to the three 2D cases described above, only with more dimensions. (Note that the presence of objects with mass along with gravity and General Relativity has the same type of effect on all three geometries, just as the presence of hills had the same type of effect on all three geometries in the 2D cases described above.)
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There are a lot of unclear descriptions out there regarding flat versus curved space-time, and there are a lot of unclear descriptions out there regarding the effect of objects with mass, gravity, and General Relativity on the curvature of a photon's path through space-time. These descriptions are so unclear that I can't even tell if the authors are mixing two different concepts, but it appears to me that they are indeed mixing them up.
I think the question about the flatness or curvature of space or space-time and the question about the way mass, gravity, and General Relativity affect the paths of photons are two different questions. I think an opinion that space is flat is not a contradiction of an opinion that a photon's path curves in accordance with the principles of General Relativity.
As I said at the beginning, somebody please correct me if I'm wrong.