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cianfa72

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- Einstein definition of simultaneity for Langevin observers

Hi,

reading this old thread Second postulate of SR quiz question I'd like to ask for a clarification on the following:

Is the above correct ? thank you.

reading this old thread Second postulate of SR quiz question I'd like to ask for a clarification on the following:

Here the Einstein definition of simultaneity to a given event on the Langevin observer's worldlineFor example, consider the family of Langevin observers, who are all moving in circular trajectories about a common origin, with the same angular velocity . In the accelerated coordinates in which these observers are at rest (we use cylindrical coordinates here to make things look as simple as possible), the metric is

$$ds^2 = - \left( 1 - \omega^2 r^2 \right) dt^2 + 2 \omega r^2 dt d\phi + dz^2 + dr^2 + r^2 d\phi^2$$

If we look at a spacelike slice of constant coordinate time in this metric, we find something unexpected: it is Euclidean! The metric of such a slice is simply ##dz^2 + dr^2 + r^2 d\phi^2##, which is the metric of Euclidean 3-space in cylindrical coordinates. Why, then, is it always said that "space" is not Euclidean for such observers?

The answer is that, although the observers are at rest (constant spatial coordinates ##z,r,\phi##) in this chart, the spacelike slices of constant coordinate time ##t## arenotsimultaneous spaces for those observers. That is, the set of events all sharing a given coordinate time ##t## are not all simultaneous (by the Einstein definition of simultaneity) for the observers.In fact, the set of events which are simultaneous, by the Einstein definition of simultaneity, to a given event on a given observer's worldline do not even form a well-defined spacelike hypersurface at all.So we can't even use that obvious definition of "space" for such observers.

*locally*means take the events on the 3D spacelike orthogonal complement to the worldline's timelike tangent vector at that point. Since the Langevin congruence does rotate (i.e. its vorticity is*not*null) then it is not hypersurface orthogonal, hence there is not a spacelike foliation of Minkowski spacetime orthogonal in each point to the Langevin's worldlines.Is the above correct ? thank you.