Energy Of General 4-dimensional Curved Spacetime

The definition of an energy and/or momentum distribution in curved space-times is one of the interesting and most controversial problems of contemporary theoretical physics. However, some authors try to define the energy-momentum density for the gravitational field, and these definitions lead to the prescriptions that are not true tensors. There have been much attempt to solve the energy problem in Einstein's theory of the general relativity. It is well known that the space-times metric in the general relativity describes both the background space-times structure and the dynamical aspects of the gravitational field. There is no natural way to decompose it into its "background" and "dynamical" components. Thus, the notion of the energy cannot be found locally. Some authors defined the pseud-tensor of energy-momentum density and the quasilocal energy for the gravitational field. However, these definitions have some defects. Say, if we consider the rotating space-times, the quasilocal energy can only be applied to the case of the slow rotating space-times. Therefore people are making attempts all the while to find the expression of the gravitational energy-momentum density in the form of a true space-times and gauge tensor.In the context of the teleparallel equivalent of the general relativity (TEGR), we first derive out the field equation with the cosmological constant and matter fields. Then, we generalize the definition of the gravitational energy into the case with the cosmological constant and the matter fields, and obtain a general expression of the energy for the 4-dimensional stationary axisymmetric space-times within an arbitrary spacelike two-spheres. The expression is analytical and very simple in the stationary axisymmetric spacetime. It is a integral form of the constraints equations of the formalism naturally and only related to components of g_rr, g_θθ and g_φφ, without any restriction on the metric parameters except the stationary axisymmetric condition.Then, as examples, we calculate the energy of the KN and KN-AdS spacetimes,the stationary Kaluza-Klein and the rotating Cvetic-Youm spacetimes enclosed by a arbitrary two-sphere. The expressions of the energy are simple, exact and analytic. Furthermore, for comparing with known results obtained by using other method, we discuss the gravitational energy in some special cases (such as r —*■ oo, r = r+, o-tfl and slow rotating approximation) and find that these results coincide with ones obtained by using quasilocal energy method.