On the Møller’s energy complex of the charged dilaton black hole

Using Møller’s energy complex , we obtain the energy distributions of GHS solution and dyonic dilaton black hole solution in the dilaton gravity theory. It is confirmed that the Møller’s energy complex is indeed a 3-scalar under purely spatial transformation in these energy distributions. Some interested properties of the energy distribution of dyonic black hole are disscussed. PASC:04.20.-q, 04.50,+h keywords:Møller’s energy complex, dilaton gravity theory, GHS solution, dyonic dilaton solution E-mail:[email protected] E-mail:[email protected] E-mail:[email protected] 1 The well known Einstein’s energy complex is the foremost definition of energy complex. This idea comes from that the continuity equation ∂T μ ν ∂x = 0 can give out a conserved quantity in the absence of a gravitation field. However, when the gravitational field is present, the conservation law will be generalized to be ∇μT μ ν = 1 √ −g ∂( √ −gT μ ν ) ∂xμ − 1 2 ∂gμσ ∂xν T μσ = 0. (1) It does not generally induce any conservation quantity. In order to determine the conserved total four-momentum, a particular coordinate system where all the first derivatives of the gμν are vanish at some particular points must be chosen. However, the Einstein’s energy complex does not supply a physically satisfactory description of the energy distribution or of the energy contained in a limited part of space. Due to the the condition that the first order derivative of gμν must be equal to zero, one must specify the calculation at the large spatial distance r from the system in Cartestian coordinate. Thus, it has no meaning to speak of a definite localization of gravitation field in this case [1]. Another energy complex which has the property of transforming as a 3 scalar density with respect to the group of purely spatial transformations was defined by Møller [2, 3]. This property was shown that the energy distribution will be the same while calculating in different kinds of spatial coordinate (but the same time coordinate). In this paper, we calculate the energy distributions of two black hole solutions in dilaton gravity theory to examine the property of Møller energy complex. 2 In the dilaton gravity theory in which the gravity is coupled to the electromagnetic and dilaton fields can be described by the four dimensional effective string action [4]. The action can be expressed as