The equation of motion and gravitational radiation of a small extended spherically symmetric mass

Abstract

On the basis of an approximation method developed in a previous paper the motion of an extended small mass on a gravitational background\(\begin{array}{*{20}c} {(in)} \\ {g_{\mu \nu } } \\ \end{array} = \eta _{\mu \nu } + \begin{array}{*{20}c} {(in)} \\ {\gamma _{\mu \nu } } \\ \end{array} \) is investigated. The mass is described by a spherically symmetric rest mass distribution with some form of “rigidity;” the smallness of the mass is defined by the assumption that the radius of the mass is small compared with the change of the background\(\begin{array}{*{20}c} {(in)} \\ {\gamma _{\mu \nu } } \\ \end{array} \). The equation of motion is yielded by integrating Einstein's conservation law of energy and momentum over the world tube of the mass. In the lowest mixed order (mixed of the background\(\begin{array}{*{20}c} {(in)} \\ {\gamma _{\mu \nu } } \\ \end{array} \) and the retarded potentials of the mass in lowest order) this equation is identical with the geodesic line linearized in\(\begin{array}{*{20}c} {(in)} \\ {\gamma _{\mu \nu } } \\ \end{array} \). In the case when the motion on a static background generated by a localized matter distribution is finite, the gravitational radiation of the mass in lowest order is given.