GENERALIZED SPLITTING OF SPACETIME

Normally, when a spacetime splitting is considered the ADM 3 + 1 split is brought to mind. In this paper, the idea of spacetime splitting is extended to include an m + n splitting of spacetime. The global spacetime has dimension (m + n) and the foliating spaces have dimension m. There are n independent normals to each of these foliating spaces, thus giving n different extrinsic curvatures. The generalised Gauss-Weingarten and the generalised Gauss-Codazzi equations associated with this splitting are derived. These generalised equations reduce to the familar ADM equations when a 3 + 1 split is considered. The generalised equations are found to have a particularly elegant form when an orthogonal splitting of spacetime is examined.