General Solutions to the Linearized Einstein Equations and Initial Data for Three Dimensional Time Evolution of Pure Gravitational Waves

Abstract

General solutions to the linearized gravitational waves under the gauge condition of constant lapse function and zero shift vector in the [3+ l]-formalism of the Einstein equations are given. Using the above general solution, we give the initial data of the three dimensional time evolution of pure gravitational waves. The three dimensional space is conformally fiat. The momentum constraint equations are solved analytically. The Hamiltonian constraint equation is solved numerically by using discrete spherical harmonics. As for the time evolution of the initial data, we give regularity conditions which metric tensors in the spherical polar coordinates should satisfy at any time.