The proper characteristics of frame reference as a 4-invariants

Abstract

The paper proposes 4-dimensional equations for the proper characteristics of a rigid reference frame \[ {{{W}'}^{\gamma }}=\Lambda ^{0}_{\;\;i}\frac{d\Lambda ^{\gamma i}}{d{t}}\,\,,\] \[ {{{\Omega }'}^{\gamma }}=-\frac{1}{2}\,{{e}^{\alpha \mu \gamma }}\Lambda _{\;\;i}^{\mu }\frac{d{{\Lambda }^{\alpha i}}}{d{t}}\,\,.\] From these conditions follow the law of motion of the proper tetrad and the equations of the inverse problem of kinematics, i.e., differential equations that solve the problem of restoring the motion parameters of a rigid reference frame from known proper acceleration and angular velocity. In particular, it is shown that when boosted, a moving reference frame that has proper Thomas precession relative to the new laboratory frame will have a combination of two rotations: the new Thomas proper precession and the Wigner rotation, which together give the original frequency of Thomas precession $$ \frac{1-\sqrt{1-v^{2}}}{v^{2}\sqrt{1-v^{2}}}\,\,e^{\beta\mu\nu}v^{\mu}\dot{v}^{\nu}=\omega'^{\beta}_{W} + b^{\beta\alpha}_{W}\,\frac{1-\sqrt{1-v^{*2}}}{v^{*2}\sqrt{1-v^{*2}}}\,\,e^{\alpha\mu\nu}v^{*\mu}\dot{v}^{*\nu}\,.$$