Energy momentum complex

Energy momentum complex

Gamal G.L. Nashed*

Mathematics Department, Faculty of Science, Ain Shams University, Cairo, Egypt

ABSTRACT

We show that the definition of the energy-momentum complex given by Møller using Weitzenböck space-time 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, Weitzenbock 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 energy-momentum 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 energy-momentum 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 space-time [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.

Now let us consider the solution given by Mikhail et al. [9]. They obtained a spherically symmetric solution in vacuum in Møller tetrad theory of gravitation for a tetrad which has three unknown functions of the radial coordinate r and having spherical symmetry. This solution is given by

where i = is a factor to preserve Lorentz signature and are the contravariant components of the tetrad field. Here and are given in terms of the unknown function () as

As is clear from (1) that the solution contains one arbitrary function of the radial coordinate, i.e., (). In spite of this all previous solutions can be obtained from it [9]. For example when () = 1 or () = we discover the solutions given in [13] which reproduce the Schwarzschild metric. The associated metric, which is defined by , with solution (1) gives Schwarzschild metric in the form

regardless the value of ().

Now we are going to calculate the energy content associated with solution (1) using different definitions of the energy-momentum complex given in general relativity and that given by Møller in 1978.

The Einstein energy-momentum complex is [2]

where comma denotes the differentiation with respect to the coordinates. The energy and spatial momentum are given by

where n_α is the outward unit normal vector over the infinitesimal surface element dS, P₀ and P_α stand for the energy and spatial momentum.

The symmetric energy-momentum complex of Landau and Lifshitz is [10]

L⁰⁰ and L^α0 are the energy and spatial momentum density components. The energy and spatial momentum are given by

The symmetric energy-momentum complex of Papapetrou is [11]

η^ik = diag (–1,1,1,1), Σ⁰⁰ and Σ^α0 are the energy and spatial momentum density components. The energy and spatial momentum are given by

The symmetric energy-momentum complex of Weinberg is [12]

h_ik = g_ik – η_ik and the indices on h_ik or ∂/∂x_i are raised or lowered by η's. W⁰⁰ and W^α0 are the energy and spatial momentum density components. The energy and spatial momentum are given by

The energy-momentum complex given by Møller in 1978 is [13]

where

and the semicolon denotes covariant differentiation with respect to Christoffel symbols. The energy and spatial momentum are given by

Now we proceed to calculate the energy associated with solution (1) using the definitions (6,8,10,12) which are given in general relativity

while if we calculate the energy of the same solution using the definition (15) given by Møller in the Weitzenböck space-time we get

where E_E,E_LL, E_P, E_Wand E_M stand for the energy (P₀) using the definition of the energy-momentum complex given by Einstein, Landau and Lifshitz, Papapetrou, Weinberg and Møller.

The arbitrary function must be non vanishing so that the area of a sphere of constant R is finite [14]. We also assume that () and () satisfy the asymptotic condition

using (18) in (17) we get

As is clear from (16) and (19) that there is a difference between the calculations done using the definitions (6,8,10,12) constructed within general relativity and that constructed within Weitzenböck space-time. This is due to the fact that the definitions given in general relativity mainly depend on the metric tensor g_µν and as is clear from (4) that the metric tensor of this solution depends only on the gravitational mass M and the radial coordinate R. However, the definition (15) depends mainly on the covariant components of tetrad field or the contravariant components.

The difference between the two results (16) and (19) does not means that the calculations using the definitions within the framework of general relativity theory is more accurate than that used in the framework of Weitzenböck space-time. But may be related to some reasons:

i) Dividing the tetrad into two classes, the one in which the components and of the parallel vector fields () tend to zero faster than 1/ for large R and the other, in which those components go to zero as 1/. It was found that the equality of the energy and the gravitational mass holds only in the first class. It is of interest to note that the tetrad structures (1) has such property, i.e., the components () and () go to zero as 1/. So its energy content is different from the energy content given by (16) [14].

ii) Many authors believe that a tetrad theory should describe more than a pure gravitation field. In fact; Møller himself [15] considered this possibility in his earlier trials to modify general relativity. In these theories, the most successful candidates for the description of the other physical phenomenon are the skew-symmetric tensors of the tetrad space, e.g., Φ_µ;ν-Φ_ν;µ. Some authors; e.g; [16,17], believe that these tensors are related to the presence of an electromagnetic field. Others; e.g.; [18] believe that these tensors are closely connected to the spin phenomenon. There are a lot of difficulties to claim that Møller's theory deserves such a wider interpretation. This needs a lot of investigations before arriving at a concrete conclusion.

The result given by (19) shows that the calculations of energy depends only on the mass of the system and this is consistence with the principle of energy that the total energy of the system is independent of the coordinate. Also this result shows that the arbitrary function ) has no effect on the calculations of energy in spite that it plays a central role in obtaining the previous known solution by given it a specific form.

The same problem (i.e. the definitions within the framework of general relativity and the definition given by Møller using Weitzenböck space-time will lead to different results) will appear for the solutions obtained before by Shirafuji et al. [14] and [19].

[1] K.S. Virbhadra, Phys. Rev. D60 (1999), 104041.

[2] C. Møller, Ann. of Phys. 4 (1958), 347.

[3] C. Møller, Ann. of Phys. 12 (1961), 118; D. Kovacs., Gen. Relativ. Grav. 17 (1985), 927; J. Novotny, Gen. Relativ. Grav. 19 (1987), 1043.

[4] J. W. Maluf, J. F. Da Rocha-neto, T. M. L. Toribio and K. H. Castello-Branco gr/qc 0204035.

[5] C. Møller, tetrad Fields and Conservation Laws in General Relativity; Proceedings of the International School of Physics Ënrico Fermi" edited by C. Møller, (Academic Press London 1962); Conservation Laws in the Tetrad Theory of Gravitation; Proceedings of the conference on the Theory of Gravtion, Warszawa and Jablonna 1962 (Gauthier-Villars, Paris and PWN-Polish Scientific Publishers, Warszawa, 1964) (NORDITA Publicatiobns No. 136).

[6] S. V. Babak and L. P. Grishchuk, Phys. Rev. D61 (1999), 024038.

[7] R. Weitzenböck, Invariantentheorie (Noordhoff, Groningen, 1923), p. 317.

[8] C. Møller, Mat. Fys. Medd. Dan. Vid. Selsk. 39 (1978), 13.

[9] F.I. Mikhail, M.I. Wanas, E.I. Lashin, and A. Hindawi, Gen. Rel. Grav. 26 (1994), 869.

[10] L.D. Landau, and E.M. Lifshitz The classical Theory of Fields (Pergamon Press, Oxford, 1962), P.341.

[11] A. Papapetrou, Proc. R. Ir. Acad A 52 (1948), 11.

[12] S. Weinberg Gravitation and Cosmology: Principles and application of General Theory of Relativity (Wiley, New York, 1972), p. 165.

[13] F.I. Mikhail, M.I. Wanas, A. Hindawi and E.I. Lashin, Int. J. of Theor. Phys. 32 (1993), 1627.

[14] T. Shirafuji, G.G.L. Nashed, and K. Hayashi, Prog. Theor. Phys. 95 (1996), 665.

[15] C. Møller, Mat. Fys. Medd. Dan. Vid. Selsk. 1 (1961), 10.

[16] A. Einstein, Sitzungsber. Preuss. Akad. Wiss. (1930), 18.

[17] F.I. Mikhail, and M.I. Wanas, Proc. Roy. Soc. Lond. A 356 (1977), 471.

[18] F. Mü ller-Hoissen and J. Nitsch, Phys. Rev. D28 (1983), 718.

[19] G.G.L. Nashed, Phys. Rev. D 66 (2002), 064015.

(Received on 12 January, 2010)

Publication Dates

• Publication in this collection
  27 Sept 2010
• Date of issue
  Sept 2010

History

• Received
  12 Jan 2010