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cianfa72
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- Extending a 1D spacelike line to a 2D spacelike surface orthogonal to Langevin worldlines at a single radius.
Reading again this old thread Einstein Definition of Simultaneity for Langevin Observers, I'd like to ask about the following.
Consider a disk rotating at fixed angular velocity ##\Omega##. In the inertial coordinate system the center of the disk is at rest, the worldlines of the Langevin observers at any fixed ##r## are timelike helical lines around the cylinder of radius ##r## -- we use a (2+1)d spacetime model charted on 3D euclidean space as usual.
That thread explains how to "unwrap" the cylinder into a flat sheet using the (1+1)d Minkowski metric on it inherited from the "ambient" (2+1)d Minkowski metric. Langevin worldlines are timelike worldlines on it (i.e. parallel straight lines pointing inside the light cone at any point along them). We can draw then the 1D spacelike lines orthogonal to them.
Now, since the Langevin worldlines are "slanted" in the picture on the flat sheet, their (Minkowski) orthogonal lines won't be drawn as intersecting them at 90 degree in the picture. By wrapping again around the cylinder, such spacelike orthogonal lines on the flat sheet picture will become non-closing helical lines.
Repeating again the same for increasing values of ##r##, we'll get a 2D non-achronal spacelike surface which however, as explained there, won't be orthogonal in all directions to the full set of Langevin worldlines (i.e. Langevin congruence) at any point along them. This follows from the fact that the Langevin congruence has non-zero vorticity hence is not hypersurface orthogonal. So far so good.
Consider now a single 1D spacelike non-closing helical line (1D spacelike surface). As @DrGreg explained, one can extend such line to a 2D spacelike surface that's orthogonal to (less than a complete revolution of) the Langevin worldlines at the corresponding/relevant radius (but not orthogonal to Langevin worldlines at a different radius). However it can't be extended too far, otherwise it would meet the same Langevin worldline twice.
My doubt: does the above mean that extending it to far would meet twice the same Langevin worldline apart from/excluding those of Langevin observers at the specific radius used in the construction? Thanks.
Consider a disk rotating at fixed angular velocity ##\Omega##. In the inertial coordinate system the center of the disk is at rest, the worldlines of the Langevin observers at any fixed ##r## are timelike helical lines around the cylinder of radius ##r## -- we use a (2+1)d spacetime model charted on 3D euclidean space as usual.
That thread explains how to "unwrap" the cylinder into a flat sheet using the (1+1)d Minkowski metric on it inherited from the "ambient" (2+1)d Minkowski metric. Langevin worldlines are timelike worldlines on it (i.e. parallel straight lines pointing inside the light cone at any point along them). We can draw then the 1D spacelike lines orthogonal to them.
Now, since the Langevin worldlines are "slanted" in the picture on the flat sheet, their (Minkowski) orthogonal lines won't be drawn as intersecting them at 90 degree in the picture. By wrapping again around the cylinder, such spacelike orthogonal lines on the flat sheet picture will become non-closing helical lines.
Repeating again the same for increasing values of ##r##, we'll get a 2D non-achronal spacelike surface which however, as explained there, won't be orthogonal in all directions to the full set of Langevin worldlines (i.e. Langevin congruence) at any point along them. This follows from the fact that the Langevin congruence has non-zero vorticity hence is not hypersurface orthogonal. So far so good.
Consider now a single 1D spacelike non-closing helical line (1D spacelike surface). As @DrGreg explained, one can extend such line to a 2D spacelike surface that's orthogonal to (less than a complete revolution of) the Langevin worldlines at the corresponding/relevant radius (but not orthogonal to Langevin worldlines at a different radius). However it can't be extended too far, otherwise it would meet the same Langevin worldline twice.
My doubt: does the above mean that extending it to far would meet twice the same Langevin worldline apart from/excluding those of Langevin observers at the specific radius used in the construction? Thanks.
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