Image:Rotating Ring Cyl Cntr2RingRadar.png

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[edit] Summary

This figure schematically represents the concept of radar distance, one of several distinct operationally significant notion of distance which can employed by Langevin observers (and other accelerating observers) in Minkowski spacetime.

This figure depicts two thought experiments using a standard cylindrical coordinate chart.

At left, a Langevin observer with world line passing through events labeled A and A&Prime experimentally determines the radar distance to a static observer with world line on the axis of symmetry r = 0, by sending a radar pulse at A which to event C′ which then returns at A″. To determine the radar distance of the central observer, he divides the elapsed time (measured by an ideal clock which he carries) by two.

At right, the central observer experimentally determines the radar distance to the Langevin observer, by sending a radar pulse at C to A′ which then returns at C″;. To determine the radar distance of the central observer, he divides the elapsed time (measured by an ideal clock which he carries) by two.

These two observers obtain different answers, illustrating the fact that radar distance "in the large" is not symmetric. However, over very small distances, it is symmetric, so it can be used to define the Landau-Lifschitz metric on the quotient manifold obtained using the congruence of world lines of the Langevin observers. This is a three dimensional Riemannian manifold often loosely described as giving the noneuclidean geometry of a rotating disk. However, integrating the Landau-Lifschitz arc length along the track of null geodesics does not in general agree with the corresponding radar distance "in the large" since we are adding up small time intervals belonging to ideal clocks carried by distinct Langevin observers at various radii.

This figure was created by User:Hillman using Xfig to create the png image; the labels were added using the GIMP.

[edit] Licensing

I, the creator of this work, hereby grant the permission to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. Subject to disclaimers.

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