Scattering of GravitationalWaves and their Energy

Abstract

This thesis primarily focuses on the study of plane gravitational waves and their collisions, which are exact solutions of the vacuum Einstein field equations. Independently, both Khan and Penrose, as well as Szekeres, derived exact solutions for colliding plane impulsive and sandwich gravitational waves, respectively. A noteworthy feature of these colliding plane waves is that the spacetime, after the collision, develops a curvature singularity. These spacetimes have been extensively studied in the past through various approaches. We investigate the Szekeres colliding sandwich gravitational wave spacetime using the pseudo-Newtonian formalism, which expresses spacetime curvature in terms of forces. We show that no momentum is imparted to test particles at the time and place of the collision. Szekeres had stated that an essential singularity develops after the collision of two sandwich waves, observed as a rapid build-up of force. However, this singularity does not appear in the force when either wave has passed before the collision, leading to the conclusion that it is a topological singularity. We are exploring the question of whether this feature is an artifact of the colliding plane wave or is inherent in the causal structure of the spacetime. Notably, what Szekeres initially labeled as a coordinate singularity after the collision of the waves is found to be a curvature singularity. Khan and Penrose and separately, Szekeres considered the collision of two plane and sandwich gravitational waves of equal strength respectively. In neither case was it really clear what parameter represented the “strength" of the wave. We identify the required parameter and extend the colliding sandwich plane gravitational wave solution to unequal strengths. Since the impulsive plane waves of Khan and Penrose correspond to a limit of the sandwich waves with the pulse duration tending to zero, we have used the procedure to obtain the generalization of the Khan-Penrose solution to unequal strengths. The resulting spacetimes are exact solutions of vacuum field equations. Interestingly, the singularity structure of the spacetime is no longer symmetric, as observed in the case of the collision of equal strengths. This thesis also includes the study of gyratonic pp-waves using the Noether symmetry method. 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 either decrease or increase for arbitrary choices of 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 conditions. 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. This implies a dynamic exchange of energy and angular momentum between the gravitational field and the free particles.