Conformally invariant orthogonal decomposition of symmetric tensors on Riemannian manifolds and the initial-value problem of general relativity

Abstract

It is shown that an arbitrary symmetric tensor ψab (or ψab) of any weight can be covariantly decomposed on a Riemannian manifold (M,g) into a unique sum of transverse-traceless, longitudinal, and pure trace parts. The summands involve only linear operators and are mutually orthogonal in the global scalar product on (M,g). Each summand transforms separately into itself if the decomposition is carried out properly in a conformally related space (M,g). The decomposition is therefore determined by a conformal equivalence class of Riemannian manifolds. This property makes the decomposition ideally suited to the initial-value problem of general relativity, which becomes, as a result, a well-defined system of elliptic equations. Three of the four initial-value equations are linear and determine the decomposition of a symmetric tensor. The fourth equation is quasilinear and determines the conformal factor. The decomposition applied to the space of symmetric tensors on (M,g) can be written in terms of a direct sum of orthogonal linear spaces and gives a framework for treating and classifying deformations of Riemannian manifolds pertinent to the theory of gravitation and to pure geometry.