Surveying in rotating systems

Abstract

The principles of radar are applied to the problem of measuring distances in rotating systems. Observations made by a single observer rotating with an inertial angular velocity omega and at an inertial radius r are investigated and it is shown that the radius, as measured by the rotating observer, is given by r'=r(1-r2 omega 2/c2)1/2 where c is the velocity of light in vacuo. The angular velocity according to the rotating observer is shown to be omega '= omega (1-r2 omega 2/c2)-1/2. Also, piecemeal measurements of distances within rotating systems are made by summing an infinite number of infinitesimal, contiguous measurements that have been collated by an observer in the inertial frame of the centre of rotation of the system. Such measurements are used to determine the length of a light path between two points in the rotating system and to measure the shortest distance between two points in the rotating system. These two measurements are found not to be identical. It is also shown that light signals used to measure infinitesimal piecemeal distances in a rotating system are emitted and received, according to an observer rotating with the system, in one and the same direction.