Coordinate conditions in a Riemannian space for coordinates based on a subspace

Abstract

It is customary to specify the geometry of a Riemannian N -space by writing down a quadratic line-element, the coefficients being ½ N ( N + 1) functions of the coordinates. But since there is an N -fold arbitrariness in the choice of coordinates, there is an N -fold arbitrariness in the metric tensor, and one expresses this by saying that the metric tensor satisfies N coordinate conditions, so that there are essentially only ½ N ( N - 1) components. If the coordinate system is made definite by constructing it according to some geometrical plan, the coordinate conditions may be made explict; their form is well known for Riemannian coordinates (based on geodesics drawn out from a point) and for Gaussian coordinates (based on geodesics drawn orthogonal to an ( N - 1)-space), and in some other cases. Our purpose is to present in a single argument the coordinate conditions for coordinates based on geodesics drawn orthogonal to a subspace of M dimensions ( M = 0, 1, ..., N - 1). These conditions are very simple in form. They are used to express the metric tensor in terms of integrals of the linear part L ijkm of the covariant Riemann tensor. If in these integrals L ijkm is replaced by any other set of functions E ijkm having the same symmetries as L ijkm , then L ijkm and E ijkm differ only by terms evaluated on the subspace. All the results are applicable to the space-time of general relativity if one puts N = 4.