Time-asymmetric structure of gravitational radiation.

Abstract

Gravitational radiation reaction effects in the dynamics of an isolated system arise from the use of retarded potentials for the radiation field, satisfying time-asymmetric boundary conditions imposed at past-null infinity. Part one of this paper investigates the ``antisymmetric'' component, a solution of the wave equation of the type retarded minus advanced, of the linearized gravitational field generated by an isolated system in the exterior region of the system. At linearized order such a component is well defined and is ``time odd'' in the usual post-Newtonian (PN) sense. We introduce a new linearized coordinate system which generalizes the Burke and Thorne coordinate system both in its space-time domain of validity, which is no longer limited to the near zone of the source, and in the post-Newtonian smallness of the linear antisymmetric (``time-odd'') component of the metric, for all multipolarities of antisymmetric waves. These waves (as viewed in the near zone) define a generalized radiation reaction four-tensor potential ${\mathit{V}}_{\mathrm{react}}^{\mathrm{\ensuremath{\alpha}}\mathrm{\ensuremath{\beta}}}$ of the linear theory.At the 2.5 post-Newtonian approximation, the tensor potential reduces to the standard Burke-Thorne scalar potential of the lowest-order local radiation reaction force. At the 3.5 PN approximation, the potential involves scalar (${\mathit{V}}_{\mathrm{react}}^{00}$) and vector (${\mathit{V}}_{\mathrm{react}}^{0\mathit{i}}$) components which are associated with subdominant radiation reaction effects such as the recoil effect. At the higher-order PN approximations, the potential is intrinsically tensorial. A nonlinear exterior metric is iteratively constructed from the new linearized metric by the method of a previous work. Part two of this paper is devoted to the near-zone reexpansion of the nonlinear iterations of the exterior metric. We use a very convenient decomposition of the integral of the retarded potentials into a particular solution involving only ``instantaneous'' potentials, and a homogeneous solution of the antisymmetric type. The former particular solution is ``even'' in the sense that it explicitly contains only even powers of ${\mathit{c}}^{\mathrm{\ensuremath{-}}1}$. The latter homogeneous solution defines a component of the exterior metric which is associated with radiation reaction effects of nonlinear origin. This decomposition of the retarded integral enables us to control the occurrence and the magnitude of ``odd'' terms in any nonlinear iterations of the metric, and to compute explicitly the radiation reaction potential of the nonlinear theory up to the 3.5 PN approximation. Finally we recover and complete a previous work concerning the hereditary modification, of quadratic nonlinear origin, of the radiation reaction potential at the 4 PN approximation.