Image:Rotating Ring Cyl Cntr2RingRadar.svg

This figure schematically represents the concept of radar distance, one of several distinct operationally significant notions 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″ 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 reaches the central observer at event C′, is sent back, and 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 non-Euclidean 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.

en:Image:Rotating Ring Cyl Cntr2RingRadar.png

2007-04-21

Traced by User:Stannered; original by en:User:Hillman

GFDL original