General relativity and symbolic algebra

Applications have so far been centred around the field equations. These equations express the relationship between the geometrical structure of the four dimensional space and its material content, that is, the distribution of mass and energy. These field equations are extremely complicated even in physical situations that are apparently simple; their solution is always a difficult and at present frequently an impossible task. In very general terms they may be interpreted in three ways. In the first case a complete definition of the geometrical structure of the space to investigate is supposed and it is proceeded via the field equations, to compute the distribution of matter. Another view is to assume a given distribution of matter and determine from the field equations the possible geometrical configurations that are compatible with that distribution. A course of action midway between these alternatives is to impose certain geometrical constraints upon the space (e.g. axial symmetry) and other constraints on the distribution of matter (e.g. irrotational flow) and then discover from the field equations those solutions consistent with the imposed conditions