New Insights into the Nature of Nonlinear Gravitational Waves

Abstract

A new nonlinear evolution equation for gravitational waves is derived using the full metric, not the Minkowski metric, to raise and lower indices. It is shown that, to second order, purely transverse-traceless plane waves are unable to satisfy the harmonic gauge restrictions, and purely longitudinal-traceless plane waves are instead the simplest solution to both the gauge restrictions and the final equation. However, this will not provide a true nonlinear evolution equation, and it is shown that in order to derive such an equation, non-plane wave modes with a non-zero trace must be taken into account. These modes must contain both longitudinal and transverse components, as both are permitted by the gauge restrictions.