Time Evolution of Pure Gravitational Waves

Abstract

Numerical solutions to the Einstein equations in the case of pure gravitational waves are given. The system is assumed to be axially symmetric and non-rotating. The time symmetric initial data and the conformally flat initial data are obtained by solving the constraint equations at t=0. The time evolution of these initial data depends strongly on the initial amplitude of the gravitational waves. In the case of the low initial amplitude, waves only disperse to null infinity. By comparing the initial gravitational energy with the total energy loss through an r=constant surface, it is concluded that the Newman-Penrose method and the Gibbon-Hawking method are the most desirable for measuring the energy flux of gravitational radiation numerically. In the case that the initial ratio of the spatial extent of the gravitational waves to the Schwarzschild radius (M/2) is smaller than about 300, the waves collapse by themselves, leading to formation of a black hole. The analytic solutions of the linearized Einstein equations for the pure gravitational waves are also shown.