The Theory of High Frequency Gravitational Radiation, and its Application to Cosmology

In Chapter 1 a brief introduction to the theory of gravitational radiation in general relativity is presented, and an outline of the variety of different methods that have been used to study it is given. In the second chapter, a single theoretical approach, upon which to base the subsequent treatment, is chosen. This approach, developed by R. A. Isaacson (1968a,b), involves obtaining approximate gravitational wave solutions to the vacuum Einstein equations by supposing that the radiation is of high frequency. The work of Isaacson is reviewed to show how the high frequency approximation leads to a tensor representation of the gravitational field energy. In Chapter 3, the Isaacson theory is extended, by the present writer, so that it may be applied to situations in which gravitational radiation is present in a matter filled manifold. The work of J. Madore, who has also considered gravitational radiation in a material fluid, is discussed to show that his results may be found, as are Isaacson’s, as special cases within the proposed general formalism. Provided that certain assumptions are made, the wave energy in matter is shown to be of the same form as that found in vacuum. It is in Chapter 4 that the cosmological applications of the general formalism of Chapter 3 are first considered. Radiation travelling through a cosmological 'background' space-time is examined, with the intention of discovering how this background geometry interacts with the radiation. A Friedmann line element, with the space curvature constant K = 0, is used to represent the cosmological background upon which the radiation propagates. Employing this example I am able to show that test particles, located in a plane perpendicular to the direction of propagation of a monodirectional gravitational wave, experience acceler­ations due to the wave superimposed upon the effects of the model's cosmological expansion. Further, the theory, for constructing an isotropic gravitational radiation field is developed. It is shown that the energy tensor of such a field may be represented by a perfect fluid energy tensor, with an equation of state in which the radiation pressure is one-third of the energy density. In Chapter 5, the manner in which the radiation modifies the cosmological background is considered. The work is motivated by a model of the Universe, containing matter and gravitational radiation, proposed by Isaacson and Winicour (1972, 1973). They assumed that matter was converted into an isotropic field of gravitational radiation. However, as is shown in Chapter 5, the model can lead to a negative mass density. This difficulty is overcome by bringing the conversion to an end at some pre-assigned instant in cosmic time. This ensures that the energy distribution in the model remains physically acceptable throughout the model's development. The cosmological equations for the model are solved, by numerical methods where necessary, and a number of examples of the resulting universes are given in diagrammatic and tabular form. Finally, in an appendix, the possibility that gravit­ational radiation is generated during the ’fireball' era of the Universe is briefly considered.