Wigner Rotation in Flat Space, Hyperbolic Space and the Outer Space of a Schwarzschild Black Hole

Abstract

We have represented Lorentz transformation and its consequences, i.e. time dilation and length contraction. The velocity addition formula has also explained. The formula of the transformation of orbital angular momentum and spin angular momentum has been derived. The numerical values of orbital angular momentum and spin angular momentum have been calculated. The orbital angular momentum versus light velocities and spin angular momentum versus light velocities have been plotted. We have introduced the Wigner rotation in two different ways. The formula of Wigner rotation has been derived in flat space in two different ways. The numerical values of Wigner rotation have been calculated for different cases. The graph of the Wigner rotation with respect to different velocities has been plotted. Rapidity is a hyperbolic angle that differentiates two frames of reference in relative motion. We have observed that the rapidity space provides a geometric approach to the Wigner rotation and the Thomas precession. We have also derived the formula of Wigner rotation in hyperbolic space. The numerical values of Wigner rotation in hyperbolic space have also been calculated. A graph of Wigner rotation versus boost angle for different boost velocities has been plotted. We have presented different types of Black holes such as Miniature Black hole, Stellar Black hole, Intermediate-mass Black hole, Supermassive Black hole, Schwarzschild Black hole, Kerr Black hole and Reissner- Nordstrom Black hole. We have derived the line element in Schwarzschild Black hole and the line elements of different Black holes have been represented. The formula of Wigner rotation in the outer space of Schwarzschild Black hole has been derived. We have also calculated the numerical values of Wigner rotation in the outer space of Schwarzschild Black hole. The graph of the Wigner rotation with respect to different velocities of a moving object has been plotted.