I am asking about how inherent curvature interacts with relativity of velocity. I am not asking about transforming curvature away. An important question in the study of Riemannian manifolds of positive sectional cur-vature is how to distinguish manifolds that admit a metric with non-negative sectional curvature from those that admit one of positive curvature. The question is, is that an unavoidable feature of curved spaces, or do non-euclidean spaces exist where Galilean relativity holds, and no choice of inertial is more convenient than any other? Certainly observational confirmation of negatively curved space sections would not constitute proof of a multiverse, for that can occur in a single universe. In previous works, we have constructed examples of non-negatively and flag-wise positively curved homogeneous Finsler spaces which do not admit any positively curved homogeneous Finsler. The surface of the sphere has a preferred frame in which objects at rest experience no distortion, and an absolute measure of velocity. Note that the same stroke leads to opposite swimming direction in the positively and negatively curved spaces. POSITIVE AND NEGATIVE FREQUENCY DECOMPOSITIONS IN CURVED SPACE-TIME. The swimmer in the Demonstration uses only internal forces, but can still move by changing and, and then reversing the changes. For the first time we realized two beam interference in negatively curved space. This is analogous to a cat turning without changing its angular momentum and landing on its feet. In contrast, light spreads exponentially on surfaces with constant negative Gaussian curvature. ![]() A surface has positive curvature at a point P if a plane tangent to the. A, of an equilateral triangle in a positive curvature space with radius of curvature, R Hint: Think about great circles on a sphere and the largest. If I choose to give that figure some velocity, however, and animate its points following geodesic that start out parallel, it will get squished as time moves on. On a surface with constant positive Gaussian curvature we observe periodic refocusing, self-imaging, and diffractionless propagation. A curve in 3-dimensional Euclidean space is a vector valued function of a. ![]() As time progresses, it is perfectly happy to stay put, with no distortion, as it's 4-velocity is entirely in the time direction, andit does not move throughthecurved space. ![]() I am explicitly positing no time curvature, only curved space.įor a concrete example, suppose I draw a figure on the positively-curved surface of a sphere.
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