SPACE, TIME AND FUNDAMENTAL INTERACTIONS

GEOMETRY OF EIGENVECTORS FOR COMPOSITION OF LORENTZ BOOSTS

Kharinov M. V.

The composition \(L_1L_2\) of Lorentz boosts \(L_1\) and \(L_2\) is considered in the Euclidean space inherent to quaternions with the usual Euclidean product \((u, v)\), as well as the pseudo-Euclidean product \((u, \bar v)\) of the vectors \(u\) and \(v\), where the conjugate vector \(\bar v = 2(v, i_0)i_0 - v\), and \(i_0\) is the unit of the quaternion algebra (the unit vector along the time axis). An analytical solution to the problem of eigenvectors and eigenvalues is presented. It is shown that the eigenvectors form a quartet of pseudo-orthogonal, but in general not orthogonal basis vectors \(c_0\), \(c_1\), \(c_2\), \(v\), in which the light-like vectors \(c_0\) and \(c_1\) correspond to mutually inverse eigenvalues, and the vectors \(c_2\) and \(v\) correspond to the eigenvalue \(1\), where the unit vector \(v\) is the direction of the Wigner rotation axis. A geometric method for constructing eigenvectors in a three-dimensional subspace of space-time orthogonal to the Wigner rotation axis is proposed, which, along with some simplification of the formulae, reflects the novelty of this paper.