Covariant conservation equations and their relation to the energymomentum concept in general relativity
Abstract
A set of unique conservation equations is constructed, and their relation to the energymomentum concept in general relativity is discussed. First of all, the spacetime metric is decomposed as the sum of two secondrank 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 spacetime, and tensors on the tangent bundle to spacetime, it is seen that these conservation equations satisfactorily describe the total angular momentum content of a finite threevolume, but not the total linear momentum content. The tangentbundle 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 spacetime admits a Killing vector, a certain component of the gravitational energymomentum content necessarily vanishes. The matter field energymomentum content agrees with Dixon's definitions. The conclusion of the paper is that for a topologically Euclidean region of an arbitrarily curved spacetime the energymomentum concept is meaningful.
 Publication:

Physical Review D
 Pub Date:
 December 1978
 DOI:
 10.1103/PhysRevD.18.4399
 Bibcode:
 1978PhRvD..18.4399P
 Keywords:

 Conservation Equations;
 Gravitation Theory;
 Kinetic Energy;
 Momentum Theory;
 Relativity;
 SpaceTime Functions;
 Coordinate Transformations;
 Einstein Equations;
 Euclidean Geometry;
 Field Theory (Physics);
 Riemann Manifold;
 Tensors;
 Vector Spaces;
 Astrophysics