Lorentz transformations as space–time reflections. II. Timelike reflections

Continuing previous work, some features of the covariant factorization of Lorentz transformations, into complementary space–time reflections, are further discussed in terms of timelike reflections. Several properties of timelike reflections are shown, which bear interesting relations with the Lorentz tensor performing active isometric transformations between two inertial observers, while their geometric meaning is also briefly examined. Some product rules for timelike reflections appear, as a background enabling the discussion of the group multiplication laws for Lorentz isometry tensors. This brings into the fore the realization of the restricted Lorentz group attained in the present formalism. The instances of two multiplication laws of the Lorentz tensors are examined. Next, the same problem (i.e., factorization of an ordinary rotation into two complementary reflections) is readily solved for the Euclidean 3-space. Both formalisms (Euclidean and Minkowskian) are essentially the same. Indeed, factorization of an isometric transformation into two complementary reflections is a general property of flat Riemannian geometry. A few concluding remarks are presented.