Systematic approach to the relativistic Doppler effect based on a test theory

Abstract

Numerous experiments have been carried out to validate the Lorentz transformation or to find possible violations of Lorentz invariance, based on test theories to test special relativity. The test theory of Mansouri and Sexl (MS) provides a general framework for the transformation of inertial frames, presuming a preferred system of reference. Based on the MS framework, this paper systematically approaches the relativistic Doppler effect such that its dependency on transformation coefficients and parameters can be investigated. The Doppler effect is formulated in a complex Euclidean space (CES) where time is represented with imaginary numbers. Two formulae of the Doppler effect which have been derived in an arbitrary transformation within the MS framework are presented: one between an inertial frame and the preferred one and the other between arbitrary inertial frames. It is shown from the former formulation that the Doppler effect is independent of the synchronization of clocks, which implies that the Doppler shifted frequency in the absolute synchronization is the same as that in the standard synchronization. The latter formula can allow us to find Doppler shifts without information on the velocities of inertial frames relative to the preferred frame. Exploiting these theoretical results derived, we examine the transverse and the longitudinal Doppler effects in detail.

Uniqueness of the Isotropic Frame and Usefulness of the Lorentz Transformation

“…The PT interval is measured at the same position while the AT interval is the time difference between different positions. The (1, 1)-entry of ) , ( i j G   T represents the time dilation factor, which makes a connection between PT and AT, and it is given by [12] ) ,…”

“…Proper times and Doppler-shifted frequencies, which are independent of the synchronization of clocks [2,3,11,12], can be obtained by regarding any inertial frame as isotropic. Nonetheless, their values are demonstrated not to be uniquely determined unless the isotropic frame is unique.…”

“…Hence there are an infinite number of isotropic frames. Recently the relationship of time dilation and the formula of the Doppler effect have been derived under the uniqueness of the isotropic frame by exploiting the MS framework [11,12]. The unique isotropic frame is the preferred reference frame.…”

“…represents the time dilation factor, which makes a connection between PT and AT, and it is given by [12] ) , ( | ) , (…”

“…( 6). Consider that a plane wave of frequency l  propagates with a direction vector k are related to [12] i i j…”