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cianfa72
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From a mathematical point of view which are the definitions of expansion and shear for a congruence ?
See this thread: https://www.physicsforums.com/threa...-for-rod-and-hole-relativity-paradox.1010444/cianfa72 said:From a mathematical point of view which are the definitions of expansion and shear for a congruence ?
Yes.cianfa72 said:Sorry, just another piece of the puzzle. We said that the timelike geodesic congruence at rest in standard coordinates in Godel spacetime has zero shear and expansion, however it has not zero vorticity.
In general, yes, we would expect two randomly chosen timelike geodesics to have nonzero geodesic deviation. This will be true in any curved spacetime.cianfa72 said:What about any two arbitrary timelike geodesics in Godel spacetime ? I believe they will not have costant proper distance (i.e. there will be a non-zero geodesic deviation between any two arbitrary timelike geodesics).
The point I would like to make is that if and only if (iff) there is a timelike geodesic congruence hypersurface orthogonal (i.e. zero vorticity) having zero expansion and shear then the underlying spacetime is flat (Minkowski) and the global chart in which the worldlines of the above congruence are at rest is a global inertial coordinate chart for the spacetime.PeterDonis said:In general, yes, we would expect two randomly chosen timelike geodesics to have nonzero geodesic deviation. This will be true in any curved spacetime.
Yes, but...cianfa72 said:In the above case, any two randomly chosen timelike geodesics will always have zero geodesic deviation
...no. For example, consider the worldlines ##x = 0## and ##x = 0.5t##. They are both timelike geodesics and their geodesic deviation is zero (their "relative velocity" is always the same), but the proper distance between them is not constant. This is why discussions of spacetime curvature normally focus on the case of initially parallel geodesics (which the above two are not).cianfa72 said:(i.e. constant proper distance)
Yes.cianfa72 said:each one zero coordinate acceleration
ok, the above two are examples of non proper constant distance timelike geodesics in flat spacetime.PeterDonis said:For example, consider the worldlines ##x = 0## and ##x = 0.5t##. They are both timelike geodesics and their geodesic deviation is zero (their "relative velocity" is always the same), but the proper distance between them is not constant.
"Geodesic deviation" is the wrong term here, it does not apply to worldlines in a congruence. It applies to the spacetime geometry. "Zero geodesic deviation" means flat geometry, independent of any properties that particular congruences of worldlines might or might not have. We have already discussed in this thread how the existence of a geodesic congruence with zero expansion and zero shear is not sufficient for a flat spacetime geometry.cianfa72 said:zero expansion and shear for worldlines in a congruence (i.e. constant proper distance) implies their zero geodesic deviation
The two geodesics I gave cannot be part of the same congruence because they intersect. A congruence is a set of non-intersecting worldlines that fill a region of spacetime.cianfa72 said:however the reverse is not true in general.