Conservation of vector-valued forms and the question of the existence of gravitational energy-momentum in general relativity

Abstract

Recently, Dalton has shown that conserved vector-valued currents have nullcovariant (rather than partial) derivatives and has discussed the corollary that there is no gravitational energy-momentum in general relativity (GR). An equivalent statement on conservation of vector-valued currents was given by E. Cartan in 1922. Given the potential implications of the Dalton corollary, Cartan's contribution to this problem is reproduced here in more modern but very powerful and relatively simple language. For completeness, his accompanying work on the current of GR is also included. The question then arises of whether or not the absence of a gravitational source in GR is a problem. For the purpose of comparison, we consider the gravitational sector of teleparallelism-withtorsion theories. These exhibit field equations of the formGμν =Tμν, whereGμνis the Einstein tensor for the Levi-Civita (part of the total) connection, which is conserved. The source now has two distinct parts, one depending on both the metric and the torsion and the other metric independent. They can be viewed as being respectively gravitational and non-gravitational. In addition, the conservation law of these theories is global. It thus follows that several unesthetic (and certainly controversial) features of GR should be viewed as real problems, and not as necessary concomitants of the geometrization of the physics.