Gravitational Radiation in the Limit of High Frequency
Isaacson, Richard Allen
Misner, Charles W.
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This dissertation deals with a technique for obtaining approximate radiative solutions to the Einstein equations of general relativity in situations where the gravitational fields of interest are quite strong. In the first chapter, we review the history of the problem and discuss previous work along related lines. In the second chapter, we assume the radiation to be of high frequency and expand the field equations in powers of the small wavelength this supplies. This assumption provides an approximation scheme valid for all orders of 1/r, for arbitrary velocities up to that of light, and for arbitrary intensities of the gravitational field. To lowest order we obtain a gauge invariant linear wave equation for gravitational radiation, which is a covariant generalization of that for massless spin-two fields in flat space, This wave equation is then solved by the W.K.B. approximation to show that gravitational waves travel on null geodesics with amplitude and frequency modified by gravitational fields in exactly the same way as are those of light waves, and with their polarization parallel transported along the geodesics, again as is the case for light. The metric containing high frequency gravitational waves is shown to be type N to lowest order, and some limits to the methods used are discussed. In the third chapter we go beyond the linear terms in the high frequency expansion, and consider the lowest order non-linear terms. They are shown to provide a natural, gauge invariant, averaged effective stress tensor for the energy localized in the high frequency radiation. By assuming the W.K.B. form for the field, this tensor is found to have the same structure as that for an electromagnetic null field. A Poynting vector is used to investigate the flow of energy and momentum in the gravitational wave field, and it is seen that high frequency waves propagate along null hypersurfaces and are not backscattered off by the curvature of space. Expressions for the total energy and momentum carried by the field to flat null infinity are given in terms of coordinate independent integrals valid within regions of strong field strength. The formalism is applied to the case of spherical gravitational waves where a news function is obtained, and where the source is found to lose exactly the energy and momentum contained in the radiation field.