DaleSpam said:You could make this same statement about a Euclidean flat space. The metric there has the form of a sphere. This does not imply that the space is curved nor that parallel geodesics will converge.
What is spherical/hyperbolic is a surface of constant distance. Such surfaces are curved, but the space is not.
Yes, I understand that is what they call the "velocity space" of Special Relativity.bapowell said:Minkowski is flat. The hyperboloids that arise are a result of the action of the Lorentz group. The Lorentz transformations leave invariant the interval:
[tex]s^2 = c\Delta t^2 - \Delta x^2[/tex].
Consider in 2D a frame K and a frame K' that we will take to be moving relative to K with some velocity v. If we take a specific point (x',t') in frame K' and boost frame K' across the full range of velocities from -c to c, this point traces out a hyperbola in the frame K. So, for example, if we choose the point (0,1) in K', then it traces out the hyperbola
[tex]x^2 - t^2 = 1[/tex]
in K. I like to think of the Lorentz transformations in Minkowski space as playing an analogous role to ordinary rotations in Euclidean space: rotations in Euclidean space leave invariant the circle ([tex]x^2 + y^2 = 1[/tex]), whereas Lorentz transformations leave invariant hyperbolas. This becomes more transparent when you write the Lorentz transformations in terms of hyperbolic sines and cosines, a form in which the action of the Lorentz transformations as hyperbolic rotations is manifest.
So, Minkowski space is flat, but because of the signature (the minus sign in there), the locus of points traced by its symmetry group (Lorentz) is a hyperbola.
I don't think so. The space-like hypersurfaces of Minkowski space (as you say, freezing time -- looking at the 3-space that is orthogonal to the time direction) are flat. After all, the Minkowski metric is Euclidean on the subspace of spacelike hypersurfaces. Perhaps I'm misunderstanding you.TrickyDicky said:But does what I said in my previous post about the spacelike slice make sense or not really?
bapowell said:I don't think so. The space-like hypersurfaces of Minkowski space (as you say, freezing time -- looking at the 3-space that is orthogonal to the time direction) are flat. After all, the Minkowski metric is Euclidean on the subspace of spacelike hypersurfaces. Perhaps I'm misunderstanding you.
Well, negative in the sense that, eg:dx said:I think its better to say spatial intervals become negative. Changing sign convention of the interval doesn't make spatial distances negative.
Yes, I got that.bapowell said:However, the intrinsic curvature of the full 4D Minkowski space vanishes, and will remain zero under any global diffeomorphism.
bapowell said:Your question regarding the 'hyperbolic nature' of Minkowski space is, however, not related to the Milne coordinatization, but rather corresponds to the orbits of the Lorentz group in Minkowski space.
Minkowski spacetime geodesics are the paths that objects follow in the four-dimensional spacetime of special relativity. They represent the straightest possible paths that objects can take in the presence of gravity or acceleration.
Euclidean geodesics are the shortest paths between two points in three-dimensional space. Minkowski spacetime geodesics, on the other hand, take into account the curvature of four-dimensional spacetime and can include both space and time dimensions.
Minkowski spacetime geodesics are essential in understanding the behavior of objects in the presence of gravity or acceleration. They are used in Einstein's theory of general relativity to describe the paths of particles and the bending of light in the presence of massive objects.
Minkowski spacetime geodesics are calculated using differential equations known as the geodesic equations. These equations take into account the curvature of spacetime and the forces acting on an object to determine its path.
Yes, Minkowski spacetime geodesics can be observed in various phenomena, such as the bending of light by massive objects, the orbit of planets around a star, and the motion of objects in an accelerating frame of reference. However, they are not usually noticeable in everyday life due to the small scale of the effects.