Study of gyratonic pp-waves by using the Noether symmetry approach

This paper is devoted to study the gyratonic pp-waves using the Noether symmetry approach. We consider gyratonic pp-waves that travel in the z direction with unknown profile function H(t, z, rho, phi) and the gyratonic source function J(t, z). We evaluate velocities, kinetic energy and angular momentum of free particles for different cases of unknown metric coefficients of the spacetime. We show that the kinetic energy may decrease or increase for arbitrary choices of unknown metric coefficients of gyratonic pp-wave spacetime. The change of the kinetic energy of the free particle depends on the choices of the unknown functions under certain condition that the profile function H(t, z, rho, phi) should be harmonic in the Cartesian coordinates and J(t, z) should be function of t and z coordinates only. We also investigate the components of angular momentum in each case and observe that the total amount of the angular momentum per unit mass of the particles decreases with time and oscillatory with the polar angle phi. Therefore, there is a transfer of energy and angular momentum between the gravitational field and the free particles.