Frames of reference in spaces with affine connections and metrics (Q2718519)

It is known that in a space with affine connection the transition from one connection to another by a (1,2)-tensor \(\overline A\) is possible: \(\Gamma^i_{jk}\to \overline \Gamma^i_{jk}= \Gamma^i_{jk}-\overline A^i_{jk}\). Then the covariant differential operator \(\nabla\) turns into an extended one \(^e\nabla= \nabla- \overline A\). Convenient formulas for covariant calculus in the natural basis \((\partial_i, du^j)\) and in the most general non-holonomic basis \((e_i,e^j)\) are deduced. The different bases and different cases of \(A\) in the so-called \((\overline L_n,g)\)-space are considered. Thus to a vector field \(u\) such that \(\nabla_uu=a\neq 0\) corresponds an operator \(^e\nabla\) such that \(^e\nabla_u u=0\) and a frame exists for which \(^e\nabla=0\). At this time a transition \(AFR\to FRIF\to IFR\) is carried out, \(AFR\) -- accelerated frame of reference, \(FRIF\) -- frame of reference with inertial forces, \(IFR\) -- inertial frame of reference. By \(\overline A={l\over e}\cdot a\) where \(l=g (u,.)\) and \(e=g(u,u)\) a transition \(\nabla\to^e\nabla\) preserving the auto-parallelism \(\nabla_u u=0\Rightarrow^e \nabla_u u=0\) is studied. The Euler-Lagrange equation in variational calculus and the movement equation of a free particle in an external gravitational field in ETG are just autoparallel equations of a special form. The connections with Fermi-Walker and another transports are noted. See also [\textit{S. Manoff}, Int. J. Mod. Phys. A 13, 4289-4308 (1998; Zbl 0952.83013), Acta Appl. Math. 55, 51-125 (1999; Zbl 0976.53029), and Int. J. Mod. Phys. A 15, 679-695 (2000; Zbl 1060.53500)].