Finslerian extension of Lorentz transformations (Q5894656)

In this paper a two-dimensional Finslerian metric is studied. This metric is left invariant under the action of a deformed \(\text{SO}(1,1)\) Lorentz group. Namely, starting with the well-known 2-dimensional Lorentz transformation, defined in terms of the Minkowski angle \(\beta=\text{arcth }v\) (with \(c=1\) and \(v\) being velocity) one obtains the relativistic law of velocities \(\beta_3=\beta_1+\beta_2\) and gets a quite intriguing answer to the problem of generalizing the hyperbolic functions of \(\beta\) in such a way that the resulting Lorentz-like transformations would form a group again, preserving the addition law of Minkowski angles. The author's solution consists of a continuous deformation of \(\text{SO}(1,1)\) with respect to a real parameter ``\(g\)'', that generalizes the ``old'' cos and sine hyperbolic functions of \(\beta\). One gets a nontrivial \(g\)-deformation \[ w_3=\frac{w_2+w_1-gw_2|w_1|}{1+w_1w_2} \] of the Einstein addition law of velocities \(w_3=\frac{w_2+w_1}{1+w_1w_2}\).