Relativistic contraction and related effects in noninertial frames

Abstract

Although there is no relative motion among different points on a rotating disc, each point belongs to a different noninertial frame. This fact, not recognized in previous approaches to the Ehrenfest paradox and related problems, is exploited to give a correct treatment of a rotating ring and a rotating disc. Tensile stresses are recovered, but, contrary to the prediction of the standard approach, it is found that an observer on the rim of the disc will see equal lengths of other differently moving objects as an inertial observer whose instantaneous position and velocity are equal to that of the observer on the rim. The rate of clocks at various positions, as seen by various observers, is also discussed. Some results are generalized for observers arbitrarily moving in flat or curved spacetime. The generally accepted formula for the space line element in a nontime-orthogonal frame is found inappropriate in some cases. Use of Fermi coordinates leads to the result that for any observer the velocity of light is isotropic and is equal to c, providing that it is measured by propagating a light beam in a small neighborhood of the observer.

Proper Co-Ordinates of Non-Inertial Observers and Rotation

“…However, in flat space-time, proper non-inertial co-ordinates can be constructed in an alternative way, more useful for practical calculations [2]. Here I present an elegant form of this construction introduced in [3].…”

“…For motivation, let us first review the problems [3] of the standard resolution [4,5] of the Ehrenfest paradox. We study a rotating ring in a rigid non-rotating circular gutter with radius r. One introduces the co-ordinates of the rotating frame S ′…”

“…However, this does not imply that an observer is not allowed to use polar co-ordinates, for example. The most general co-ordinate transformations that correspond merely to a redefinition of the co-ordinates of the same physical observer are the so-called restricted internal transformations [3], i.e. the transformations of the form where t, x i are proper non-inertial co-ordinates.…”

“…However, we want to have a definition that is appropriate for any case and to consider a stationary rotation only as a special case of arbitrary motion. In flat space-time, as shown in [3], if physics is described by proper non-inertial co-ordinates modulo restricted internal transformations, a more appropriate definition of the space line element is (13).…”

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“…No interference fringe change was observed between the light beams in air and in other media configurations. However, the change was observed when the media was stationary and the platform was rotated.The Sagnac effect have been attempted by several authors to explain: using the General Relativity [6,7]; from the special relativistic Doppler effect at the mirrors [8]; by coupling the momentum of interfering particles to rotation [9]; introducing various types of relativistic transformations for rotating frames [10]; using the restricted formula of the 3-space element [11]; considering the non-invariance of the light speed for the rotating observers [4,12].To make a round-trip on the disk the photon must be reflected by several mirrors and its path has the shape of a polygon. For example, in the classical Sagnac experiment [1] the light path was quadratic.…”