On the 1+3 Formalism in General Relativity

We present in this paper the formalism for the splitting of a four-dimensional Lorentzian manifold by a set of time-like integral curves. Introducing the geometrical tensors characterizing the local spatial frames induced by the congruence (namely, the spatial metric tensor, the extrinsic curvature tensor and the Riemann curvature tensor), we derive the Gauss, Codazzi and Ricci equations, along with the evolution equation for the spatial metric. In the present framework, the spatial frames do not form any hypersurfaces as we allow the congruence to exhibit vorticity. The splitting procedure is then applied to the Einstein field equation and it results in an equivalent set of constraint and evolution equations. We discuss the resulting systems and compare them with the ones obtained from the 3+1 formalism, where the manifold is foliated by means of a family of three-dimensional space-like surfaces.