Lectures on General Relativity

In dealing with physical problems, we are often interested in the solution of field equations with given sources, but with nothing known about initial conditions. Therefore, we cannot solve the Cauchy problem, for although it is a very natural problem for hyperbolic normal equations, its solution requires a detailed knowledge of the field on an initial space-like hypersurface. However, in general, a whole set of fields corresponds to a given distribution of sources, and in order to find a unique solution of the physical problem we must specify some additional conditions. For linear field equations these conditions may consist in prescribing the form of the Green’s function (e.g., retarded, advanced, etc.). If we investigate the field in the whole (unbounded) space-time we can ensure uniqueness by specifying some appropriate boundary conditions at spatial infinity. This latter approach has the advantage of being applicable to nonlinear equations such as Einstein’s gravitational equations. These boundary conditions, first formulated for a periodic scalar field by Sommerfeld [1], have a definite physical meaning. For example, the “Ausstrahlungsbedingung” of Sommerfeld means that the system can lose energy in the form of radiation, but that no radiation is falling on the