On the Wigner angle and its relation with the defect of a triangle in hyperbolic geometry (Q1299142)

Three timelike unit vectors \(A,B,C\) in Minkowski space define a triangle on a three-dimensional hyperboloid \({\mathcal{H}}\). On the other hand, each of the pairs \((B,A),(C,B),(A,C)\) defines a Lorentz boost. It is well known that the composition \(L_{AC}L_{CB}L_{BA}\) of these three boosts involves a spatial rotation (Wigner rotation) by a certain angle (Wigner angle). In this brief communication the authors demonstrate that the Wigner angle is equal to the deficit angle of the triangle \((A,B,C)\) with respect to the hyperbolic geometry of \({\mathcal{H}}\).