CHAPTER 11 - THE GRAVITATIONAL FIELD EQUATIONS

Abstract

This chapter highlights the gravitational field equations. In a curved space, the parallel displacement of a vector from one given point to another gives different results if the displacement is carried out over different paths. The Einstein equations are nonlinear and are the required equations of the gravitational field—the basic equation of the general theory of relativity. Therefore, for gravitational fields, the principle of superposition is not valid. The principle is valid only approximately for weak fields that permit a linearization of the Einstein equations, particularly the gravitational field in the classical Newtonian limit. The equation of state relates to one another not two but three thermodynamic quantities, for example, the pressure, density, and temperature of the matter. In applications in the theory of gravitation, this point is, however, not important, as the approximate equations of state used here actually do not depend on the temperature. To understand the solution of the Einstein equations for given initial conditions (in the time), the question of the number of quantities for which the initial spatial distribution can be assigned arbitrarily must be considered. The Einstein equations can also be written in an analogous way for the general case of a time-dependent metric.