Local irregularities in an expanding universe

Abstract

Einstein's theory of gravitation is here applied to a model of the universe in which local irregularities ( h μν ) in the space-time metric are superimposed on the values appropriate to a uniform, isotropic universe. The mass tensor is taken to be that of a perfect fluid. The mass density is considered to fluctuate about a mean ϱ 0, the density in the uniform model, but the fluctuations need not be small with respect to ϱ 0. The effect of pressure is included. The system of simultaneous equqtions derived by Lanczos in 1925, which relates the irregularities in the metric to the irregularities in the mass tensor, is shown to simplify under conditions of astronomical interest. A generalized Poisson equation for h 44 results, in which only the fluctuating component of the mass tensor appears. The solution is obtained with the aid of the Green's function found by Whittaker in 1928, which reduces to the familiar form if space is flat. The equation of motion for the irregularities is derived in comoving coordinates (Eq. (35)). The fourth component of this equation leads to an energy equation involving the kinetic and potential energies of the irregularities and excluding the expanding substratum. This equation should be of significance to cosmogony. The relation of the local inertial coordinate system to the comoving system is examined. An approximate inertial system may be defined in a spatial region within which the square of the expansion velocity is much less than c 2; the expansion velocity need not be small relative to random local velocities. The local inertial system is determined by, and is nonrotating with respect to, the cosmic distribution of matter, so that Mach's principle is satisfied. The equation of motion in the inertial system (Eq. (72)) is derived. Its validity is not restricted to small pressures. For a particle at distance L from the chosen origin only the fluctuating component of ϱ at radii greater than L contributes to the gravitational force, while the total density within this radius has a gravitational effect. When the pressure may be neglected, the equation reduces to Newton's law of gravitation. The inclusion of steady-state cosmology within general relativity appears to lead to inadmissible results when local irregularities are present. Expressions for the cosmic pressure, density, and internal energy are derived in terms of local fluctuations in ϱ and velocity. p 0 has the form (Eq. (96)) that would be expected from the kinetic theory of fluids. The relation between cosmology and cosmogony is discussed.