Method of the Self-Consistent Field in General Relativity and its Application to the Gravitational Geon

Concentrations of radiation held together for a long time by their own gravitational attraction ("geons") have been studied for nearly a decade. We extend the previous analyses to the case where gravitational waves are the source of the geon's mass energy. To analyze these solutions of the free-space Einstein equations with persistent features, we develop an approximation method to treat small ripples on a strongly curved background metric. The background metric describes the large-scale persistent features of the geon and is taken to be spherically symmetric. The waves superimposed on this background have an amplitude small enough so that their dynamics can be analyzed in the linear approximation; however, their wavelength is so short, and their time dependence so rapid that their energy is appreciable and produces the strongly curved background metric in which they move. The Einstein equations are investigated in this limit of short wavelength. It is found that the large-scale features of thin-shell spherical gravitational geons---in fact, of thin-shell spherical geons constructed from any field of zero rest mass---are identical to those of the spherical electromagnetic geons analyzed previously.