Some differential and algebraic consequences of the Einstein field equations

Abstract

Introduction. If a set of four directions in a Riemannian four-dimensional space, V4, is orthogonal, then the ds2 can be expressed in terms of their sixteen parameters, hka(XO, X1l x2, X3), as in Einstein's recent papers. The first purpose of this paper is to set up sixteen invariant linear firstorder partial differential equations in these parameters (?2). The solutions of these equations include all solutions for empty space of the Einstein field equations of 1917. There is a restriction which excludes some special cases. In addition to the ka, these sixteen equations contain four linear combinations of the components of the curvature tensor. These four combinations are to be taken as independent variables, Xk. Since only alternating tensors appear it is convenient to use Cartan's notationt for symbolic differential forms and for their derivatives and products. Covariant differentiation in the sense of the absolute differential calculus is not used, except in ?4. The components of the curvature tensor may be taken as coefficients in the equation of a quadratic line complex in a three-dimensional projective space,P34 ?. The directions hka correspond to the vertices of a tetrahedron in P3. Bivector and simple bivector correspond to linear complex and special complex. Where there is no danger of misunderstanding, the language of V4 will be used interchangeably with that of P3. The second purpose of this paper is the application of some of the theory of quadratic complexes to the study of the curvatures in V4.? The lines which lie in a plane in P3 and which belong to the quadratic complex are tangent to a conic. The envelope of planes for which this conic