Generalized Lorentz transformation for an accelerated, rotating frame of reference

Abstract

An exact, explicit coordinate transformation between an inertial frame of reference and a frame of reference having an arbitrary time-dependent, nongravitational acceleration and an arbitrary time-dependent angular velocity is given. This transformation is a generalization of the Lorentz transformation and is obtained in two steps. First, the Minkowski metric is transformed under an intermediate coordinate transformation to obtain a new set of noninertial metric coefficients in which one can easily identify the Thomas precession, as well as the expression for the acceleration of the moving frame with respect to the instantaneous rest frame in terms of the acceleration as seen from a stationary inertial frame. Second, a rotation of axes is performed to absorb the Thomas precession and to add an ordinary spatial rotation. The coordinate transformation obtained by combining these effects is nonlinear, since certain terms involve time integrals, and leads to the appropriate space-time metric for an accelerated, rotating frame of reference. It is shown that the usual forms of the Lorentz transformation are contained as special cases of this result.