Radiation and Boundary Conditions in the Theory of Gravitation

TL;DR

Addresses how to characterize gravitational radiation through boundary conditions at infinity in general relativity and how to define radiated energy. Proposes generalized Sommerfeld-type boundary conditions that use a null vector $k_\nu$ so that $g_{\mu\nu}=\eta_{\mu\nu}+O(r^{-1})$ and $g_{\mu\nu,\rho}=h_{\mu\nu}k_\rho+O(r^{-2})$, with $(h_{\mu\nu}-\tfrac{1}{2}\eta_{\mu\nu}\eta^{\rho\sigma}h_{\rho\sigma})k^\nu=O(r^{-2})$, admitting radiative fields. Demonstrates that the leading $1/r$ piece of the curvature in the wave zone is of Petrov type II (via $R\cong \tfrac{1}{2} k_{[\mu} i_{\nu][\rho} k_{\sigma]}$) and that a nonzero radiative flux $p_\mu$ arises from $\mathfrak{t}_\mu{}^\nu=\tau k_\mu k^\nu+O(r^{-3})$; discusses inclusion of electromagnetism with $\mathfrak{T}_\mu{}^{\nu}+\mathfrak{t}_\mu{}^{\nu}=\bar{\tau} k_\mu k^\nu+O(r^{-3})$. Finally, relates these conditions to Pirani and Lichnerowicz pure-radiation criteria (with $R_{\mu\nu}=\rho k_\mu k_\nu+O(r^{-3})$ in the EM case) and notes that real metrics approach the wave-zone radiation fields asymptotically.

Abstract

The Sommerfeld boundary conditions, applied to an asymptotically weak gravitational field, are shown to imply that the 1/r part of the curvature tensor of a space-time, satisfying the Einstein equations, is of type null in the Petrov classification and that there is then a flux of energy carried away by the outgoing gravitational wave.