Energy momentum complex

Abstract

We show that the definition of the energy-momentum complex given by Møller using Weitzenböck spacetime in the calculations of gravitational energy gives results which are different from those obtained from other definitions given in the framework of general relativity.Keywords: Møller's energy-momentum complex, Weitzenböck space-time, energy-momentum complexes in general relativity.Since the advent of Einstein's energy-momentum complex, used for calculating energy and spatial momentum in a general relativistic system, many definitions of the energymomentum complexes have been found for instance Landau and Lifshitz, Papapetrou and Weinberg [1]. All these definitions of the energy-momentum complexes mainly depend on the metric tensor g µν . The major difficulty of these definitions is that they are coordinate dependent. Møller [2] has constructed an energy-momentum complex which enables one to evaluate energy and spatial momentum in any coordinate system. However, Møller's [3] definition of this energy-momentum complex suffer some criticism. Teleparallel theories of gravity have been considered long time ago in connection with attempts to define the energy of gravitational field [4,5]. It is clear from the properties of the solutions of Einstein field equation of an isolated system that a consistent expression for the energy density of the gravitational field would be given in terms of second order derivatives of the metric tensor. It is well known that there exists no covariant, nontrivial expression constructed out of the metric tensor, both in three and four dimensions that contain such derivatives. However, covariant expressions that contain second order derivatives of the tetrad fields are feasible. Thus it is legitimate to conjecture that the difficulties regarding the problem of defining the gravitational energymomentum is related to the geometrical description of the gravitational field rather than being an intrinsic drawback of the theory [6]. Møller later showed that Weitzenböck spacetime [7] description of gravitation allows a more satisfactory treatment of the energy-momentum complex than does general relativity [8]. It is the aim of this brief report to show that the energy-momentum complex given by Møller in 1978 gives results which are different from the energy-momentum complexes given in the framework of general relativity.