The Algebra of the Riemann Curvature Tensor in General Relativity: Preliminaries

In a four-dimensional curved space-time it is well-known that the Riemann curvature tensor has twenty independent components; ten of these components appear in the Weyl tensor, and nine of these components appear in the Einstein curvature tensor. It is also known that there are fourteen combinations of these components which are invariant under local Lorentz transformations. In this paper, we derive explicitly closed form expressions which contain these twenty independent components in a manifest way. We also write the fourteen invariants in two ways; firstly, we write them in terms of the components; and, secondly, we write them in a covariant fashion, and we further derive the appropriate characteristic value equations and the corresponding Cayley-Hamilton equations for these invariants. We also show explicitly how all of the relevant components transform under a Lorentz transformation. We shall follow the very general and powerful methods of Sachs [1]. We shall not point out at every stage of the calculation which equations are due to Sachs, and which equations are new; this is easily ascertained. Generally speaking, however, the equations depending on the Einstein curvature tensor, and the emphasis placed on this tensor, appear to be new.