Møller's Energy in the Kantowski-Sachs Space-Time

Abstract

We have shown that the fourth component of Einstein's complex for the Kantowski-Sachs space-time is not identically zero. We have calculated the total energy of this space-time by using the energy-momentum definitions of Møller in the theory of general relativity and the tetrad theory of gravity.

1. Introduction

Since the birth of the theory of general relativity, this theory has been accepted as a superb theory of space-time and gravitation, as many physical aspects of nature have been experimentally verified in this theory. However, this theory is still incomplete theory; namely, it lacks definition of energy and momentum. In this theory many physicists have introduced different types of energy-momentum complexes [1–5], each of them being a pseudotensor, to solve this problem. The nontensorial property of these complexes is inherent in the way they have been defined and so much so it is quite difficult to conceive of a proper definition of energy and momentum of a given system. The recent attempt to solve this problem is to replace the theory of general relativity by another theory, concentrated on the gauge theories for the translation group, the so called teleparallel equivalent of general relativity. We were hoping that the theory of teleparallel gravity would solve this problem. Unfortunately, the localization of energy and momentum in this theory is still an open, unsolved, and disputed problem as in the theory of general relativity.

Møller modified the theory of general relativity by constructing a gravitational theory based on Weitzenböck space-time. This modification was to overcome the problem of the energy-momentum complex that appears in Riemannian space. In a series of paper [6–8], he was able to obtain a general expression for a satisfactory energy-momentum complex in the absolute parallelism space. In this theory the field variable are 16 tetrad components ℎ_𝑎^𝜇, from which the Riemannian metric arises as

𝑔_𝜇𝜈=𝜂_𝑎𝑏ℎ^𝑎_𝜇ℎ^𝑏_𝑣. (1.1)

The basic purpose of this paper is to obtain the total energy of the Kantowski-Sachs space-time by using the energy-momentum definitions of Møller in the theory of general relativity and the tetrad theory of gravity.

The standard representation of Kantowski and Sachs space-times is given by [9]

𝑑𝑠²=𝑑𝑡²−𝐴²(𝑡)𝑑𝑟²−𝐵²(𝑡)(𝑑𝜃²+sin²𝜃𝑑𝜙²), (1.2) where the functions 𝐴(𝑡) and 𝐵(𝑡) are function in 𝑡 and determined from the field equations.

For more detailed descriptions of the geometry and physics of this space-time (see [9–11]).

2. Fourth Component of Einstein's Complex

Prasanna has shown that space-times with purely time-dependent metric potentials have their components of total energy and momentum for any finite volume (𝑇⁴_𝑖+𝑡⁴_𝑖) identically zero. He had used the Einstein complex for the general Riemannian metric

𝑑𝑠²=𝑔_𝑖𝑗(𝑥⁰)𝑑𝑥^𝑖𝑑𝑥^𝑗 (2.1) and concluded the following: for space-times with metric potentials 𝑔_𝑖𝑗 being functions of time variable alone and independent of space variable, the components (𝑇⁴_𝑖+𝑡⁴_𝑖) vanish identically as a consequence of conservation law.

Unfortunately the conclusion above is not the solution to the problem considered, in the sense that it does not give the same result, for all metrics have form of (2.1), using Einstein complex. If (2.1) is given in spherical coordinates, then Prasanna's conclusion is correct by using Møller's complex but not correct for all metrics by using Einstein's complex. Because Møller's complex could be utilized to any coordinate system, Einstein's complex gives meaningful result if it is evaluated in Cartesian coordinates. In the present paper we have found that the total energy for the Kantowski-Sachs space-time is identically zero by using Møller's complex, but not zero by using Einstein's complex.

In a recent paper [12], Gad and Fouad have found the energy and momentum distribution of Kantowski-Scahs space-time, using Einstein, Bergmann-Thomson, Landau-Lifshitz, and Papapetrou energy momentum complexes. In this section we restrict our attention to the Einstein's complex which is defined by [13]

𝜃^𝑘_𝑖=𝑇^𝑘_𝑖+𝑡^𝑘_𝑖=𝑢^[𝑘𝑗]_𝑖,𝑗, (2.2) with

𝑢_𝑖^[𝑘𝑗]=1𝑔16𝜋√−^𝑖𝑛𝑔[−𝑔(𝑔^𝑘𝑛𝑔^𝑗𝑚−𝑔^𝑗𝑛𝑔^𝑘𝑚)]_,𝑚. (2.3) The energy and momentum in the Einstein's prescription are given by

𝑃_𝑖=𝜃⁰_𝑖𝑑𝑥¹𝑑𝑥²𝑑𝑥³. (2.4) The Einstein energy-momentum complex satisfies the local conservation law

𝜕𝜃^𝑘_𝑖𝜕𝑥^𝑘=0. (2.5) The energy density for the space-time under consideration, in the Cartesian coordinates, obtained in [12] is

𝜃⁰₀=18𝜋𝐴𝑟⁴(𝐴²𝑟²−𝐵²), (2.6) and the total energy is

𝐸_𝐸𝑖𝑛=𝑃₀=12𝐴𝑟(𝐴²𝑟²+𝐵²). (2.7)

Following the approach in [12], we obtain the following components of 𝜃^𝑘₀∶

𝜃¹₀𝑥=−8𝜋𝐴²𝑟⁴[𝐴̇𝐴²𝑟²̇̇𝜃+𝐵(2𝐴𝐵−𝐵𝐴)],₂₀𝑦=−8𝜋𝐴²𝑟⁴[𝐴̇𝐴²𝑟²̇̇𝜃+𝐵(2𝐴𝐵−𝐵𝐴)],₃₀𝑧=−8𝜋𝐴²𝑟⁴[𝐴̇𝐴²𝑟²+𝐵(2𝐴𝐵̇−𝐵𝐴̇)]. (2.8) The components (2.6) and (2.8) satisfy the conservation law (2.5).

Hence from (2.6) and (2.8), we have 𝜃^𝑖₀=𝑇^𝑖₀+𝑡^𝑖₀≠0; consequently 𝜃⁰₀=𝑇⁰₀+𝑡⁰₀ is not identically zero.

3. Energy in the Theory of General Relativity

In the general theory of relativity, the energy-momentum complex of Møller in a four- dimensional background is given as [6]

ℑ^𝑘_𝑖=1𝜒8𝜋^{𝑘𝑙𝑖,𝑙}, (3.1) where the antisymmetric superpotential 𝜒_𝑖^𝑘𝑙 is

𝜒_𝑖^𝑘𝑙=−𝜒_𝑖^𝑙𝑘=√−𝑔(𝜕𝑔_𝑖𝑛𝜕𝑥^𝑚−𝜕𝑔_𝑖𝑚𝜕𝑥^𝑛)𝑔^𝑘𝑚𝑔^𝑛𝑙, (3.2)ℑ⁰₀ is the energy density and ℑ⁰_𝛼 are the momentum density components. Also, the energy-momentum complex ℑ^𝑘_𝑖 satisfies the local conservation laws:

𝜕ℑ^𝑘_𝑖𝜕𝑥^𝑘=0. (3.3) The energy and momentum components are given by

𝑃_𝑖=ℑ⁰_𝑖𝑑𝑥¹𝑑𝑥²𝑑𝑥³=18𝜋𝜕𝜒_𝑖^0𝑙𝜕𝑥^𝑙𝑑𝑥¹𝑑𝑥²𝑑𝑥³. (3.4) For the line element (1.2), the only nonvanishing components of 𝜒_𝑖^𝑘𝑙 are

𝜒₁⁰¹𝐵=−²(𝑡)𝜒𝐴(𝑡)sin𝜃,₂₀₂𝜒=−𝐴(𝑡)sin𝜃,₃₀₃=−𝐴(𝑡).sin𝜃 (3.5) Using these components in (3.1), we get the energy and momentum densities as follows

ℑ⁰₀ℑ=0,₀₁=ℑ⁰₃=0,ℑ⁰₂=−𝐴(𝑡)cos𝜃. (3.6) From (3.4) and (3.5) and applying the Gauss theorem, we obtain the total energy and momentum components in the following form:

𝑃₀𝑃=𝐸=0,_𝛼=0. (3.7)

4. Energy in the Tetrad Theory of Gravity

The superpotential of Møller in the tetrad theory of gravity is given by (see [7, 8, 14])

𝑈_𝜇^𝜈𝛽=√−𝑔𝑃2𝜅^{𝜏𝜈𝛽𝜒𝜌𝜎}[Φ^𝜌𝑔^𝜎𝜒𝑔_𝜇𝜏−𝜆𝑔_𝜏𝜇𝛾^𝜒𝜌𝜎−(1−2𝜆)𝑔_𝜏𝜇𝛾^𝜎𝜌𝜒], (4.1) where

𝑃^𝜏𝜈𝛽_𝜒𝜌𝜎=𝛿^𝜏_𝜒𝑔^𝜈𝛽_𝜌𝜎+𝛿^𝜏_𝜌𝑔^𝜈𝛽_𝜎𝜒−𝛿^𝜏_𝜎𝑔^𝜈𝛽_𝜒𝜎, (4.2) with 𝑔^𝜈𝛽_𝜌𝜎 being a tensor defined by

𝑔^𝜈𝛽_𝜌𝜎=𝛿^𝜈_𝜌𝛿^𝛽_𝜎−𝛿^𝜈_𝜎𝛿^𝛽_𝜌, (4.3)

𝛾_𝑎𝑏𝑐 is the con-torsion tensor given by

𝛾_𝜇𝜈𝛽=ℎ_𝑖𝜇ℎ^𝑖_𝜈;𝛽 (4.4) and Φ_𝜇 is the basic vector defined by

Φ_𝜇=𝛾^𝜌_𝜇𝜌, (4.5) The energy in this theory is expressed by the following surface integral:

𝐸=lim_{𝑟→∞𝑟=const.}𝑈₀^0𝛼𝑛_𝛼𝑑𝑆, (4.6) where 𝑛_𝛼 is the unit three vector normal to the surface element 𝑑𝑆.

The tetrad components of the space-time (1.2), using (1.1), are as follows

ℎ^𝑎_𝜇ℎ=[1,𝐴(𝑡),𝐵(𝑡),𝐵(𝑡)sin𝜃],_𝑎𝜇=[1,𝐴⁻¹(𝑡),𝐵⁻¹𝐵(𝑡),⁻¹(𝑡)sin𝜃]. (4.7) Using these components in (4.4), we get the nonvanishing components of 𝛾_𝜇𝜈𝛽 as follows

𝛾₀₁₁=−𝛾₁₀₁̇𝛾=−𝐴(𝑡)𝐴(𝑡),₀₂₂=−𝛾₂₀₂̇𝛾=−𝐵(𝑡)𝐵(𝑡),₀₃₃=−𝛾₃₀₃=−𝐵(𝑡)𝐵̇(𝑡)sin²𝛾𝜃,₂₃₃=−𝛾₃₂₃=−𝐵²(𝑡)sin𝜃cos𝜃. (4.8) Consequently, the only nonvanishing components of basic vector field are

Φ⁰̇=−2{𝐴(𝑡)+̇𝐴(𝑡)𝐵(𝑡)Φ𝐵(𝑡)},₂=cot𝜃𝐵².(𝑡) (4.9) Using (4.8) and (4.9) in (4.1) and (4.6), we get

𝐸=0. (4.10)

5. Summary and Discussion

In this paper we have shown that the fourth component of Einstein's complex for the Kantowski-Sachs space-time is not identically zero. This gives a counterexample to the result obtained by Prasanna [15]. We calculated the total energy of Kantowski-Sachs space-time using Møller's tetrad theory of gravity. We found that the total energy is zero in this space-time. This result does not agree with the previous results obtained in both theories of general relativity [12] and teleparallel gravity [16], using Einstein, Bergmann-Thomson, and Landau-Lifshitz energy-momentum complexes. In both theories the energy and momentum densities for this space-time are finite and reasonable. We notice that the results obtained by using Einstein, Bergmann-Thomson, and Papapetrou are in conflict with that given by Møller's values for the energy and momentum densities if 𝑟 tends to infinity, while Landau-Lifshitz's values are not in conflict.