Covariant conservation equations and their relation to the energy-momentum concept in general relativity

A set of unique conservation equations is constructed, and their relation to the energy-momentum concept in general relativity is discussed. First of all, the space-time metric is decomposed as the sum of two second-rank symmetric tensors, the first having vanishing Riemann tensor and constructed from the world function. The conservation equations derived from this bimetric formalism reduce to equations for the conservation of the Einstein pseudocomplex in normal coordinates. Using the local isomorphism between bitensors on space-time, and tensors on the tangent bundle to space-time, it is seen that these conservation equations satisfactorily describe the total angular momentum content of a finite three-volume, but not the total linear momentum content. The tangent-bundle analysis suggests alternative linear momentum equations. The essential point about the (unique) expressions for linear momentum is that they are not generated by infinitesimal coordinate transformations, and therefore differ in an intrinsic way from the Einstein (or indeed any other) pseudocomplex expressions. It is then shown that when space-time admits a Killing vector, a certain component of the gravitational energy-momentum content necessarily vanishes. The matter field energy-momentum content agrees with Dixon's definitions. The conclusion of the paper is that for a topologically Euclidean region of an arbitrarily curved space-time the energy-momentum concept is meaningful.