The Localized Energy Distribution of Dark Energy Star Solutions

Abstract

We examine the question of energy localization for an exact solution of Einstein's equations with a scalar field corresponding to the phantom energy interpretation of dark energy. We apply three different energy-momentum complexes, the Einstein, the Papapetrou, and the Møller prescriptions, to the exterior metric and determine the energy distribution for each. Comparing the results, we find that the three prescriptions yield identical energy distributions.

1. Introduction

The question of energy localization has been a pressing issue in general relativity since Einstein formulated his field equations. Einstein sought to include the energy and momenta of gravitational fields along with those of matter and non-gravitational fields as a well-defined locally conserved quantity. To that end, he proposed an energy-momentum complex that follows conservation laws for those quantities [1].

One concern about the Einstein energy momentum is that its value depends on the coordinate system used. Specifically, it favors the use of quasi-Cartesian (perturbations of a flat background) coordinates. Accordingly, it does not transform as a tensor. Because it is asymmetric in its indices, it does not conserve angular momentum.

To address these issues, a number of other theorists have proposed alternative definitions for energy-momentum complexes. These include prescriptions by Papapetrou [2], Landau and Lifshitz [3], Weinberg [4], and others, each designed to conserve angular momentum in addition to energy momentum. A prescription by Møller [5, 6] offers the special advantage of being coordinate system independent.

While at first the multiplicity of energy-momentum complexes discouraged theorists from using them, important results in the 1990s suggested an underlying unity that served to revive interest. In 1990, Virbhadra found consistency in the application of distinct energy-momentum prescriptions to the same metric [7]. Six years later, a paper by Aguirregabiria et al. revealed that all metrics of the Kerr-Schild class, including the Schwarzchild, Reissner-Nordström, and other solutions, yield identical energy and momentum distributions for a variety of prescriptions [8]. Then in 1999, a paper by Virbhadra indicated that the energy-momentum complexes of Einstein, Papapetrou, Weinberg, and Landau and Lifshitz produced similar results for metrics more general than the Kerr-Schild class if they are constructed with Kerr-Schild Cartesian coordinates [9].

There have subsequently been many attempts to apply the various energy-momentum complexes to black holes and related astronomical objects. In recent years, for example, Radinschi, along with various colleagues, has investigated the energy-momentum distribution for stringy black holes [10, 11], Hor̂ava-Lifshitz black holes [12], for charged black holes in generalized dilaton-axion gravity [13], for asymptotically de Sitter spacetimes [14], and in a Schwarzschild-quintessence spacetime [15]. Vagenas has examined the energy distribution in the dyadosphere of a Reissner-Nordström black hole [16]. Berman has looked at the energy and angular momentum of dilation black holes [17]. Sharif and Jawad have examined the energy of rotating spacetimes in teleparallel gravity [18]. Halpern has investigated the energy distribution of a charged black hole with a minimally coupled scalar field [19].

In this paper, we consider yet another scenario: a static, electrically neutral, and spherically symmetric massive object (such as a star or black hole) embedded in a space filled with phantom energy. Phantom energy is an especially potent type of dark energy proposed by Caldwell [20] that possesses an equation of state parameter 𝑤<−1. It and other forms of dark energy have been hypothesized as possible agents for the acceleration of the universe, as recorded in supernova surveys [21, 22].

The interior and exterior metrics of a dark energy star were calculated by Yazadjiev [23]. The metrics depend on a mass parameter 𝑚, the radius 𝑟, and an additional parameter 𝛽. From these, Yazadjiev determined an overall mass 𝑀 (that includes the rest mass as well as dark energy) and a dark charge 𝐷 through the following relationships: 𝑀=cosh𝛽𝑚,𝐷=sinh𝛽𝑚. (1)

Note that 𝛽=0 corresponds to the situation of a conventional Schwarzschild solution without phantom energy. The conventions that 𝑐=1 and 𝐺=1 are followed.

Applying the prescriptions of Einstein, Papapetrou, and Møller to the exterior metric derived by Yazadjiev, we will examine the localized energy of the dark energy star.

2. Applying Einstein’s Prescription to Dark Energy Stars

We now apply Einstein’s prescription to Yazadjiev’s dark energy star solution. The exterior metric has the following form: 𝑑𝑠²=−1−2𝑚𝑟^𝜅𝑑𝑡²+1−2𝑚𝑟^1−𝜅×𝑑𝑟²1−(2𝑚/𝑟)+𝑟²𝑑𝜃²+sin𝜃²𝑑𝜙², (2) where 𝑀𝜅=cosh𝛽=𝑀²−𝐷². (3)

Einstein’s prescription defines the local energy-momentum density as 𝜃^𝑘_𝑖=1𝐻16𝜋^{𝑖𝑘𝑙,𝑙}, (4) where the superpotentials 𝐻_𝑖^𝑘𝑙 are given by 𝐻_𝑖^𝑘𝑙=𝑔_𝑖𝑛√𝑔−𝑔−𝑔^𝑘𝑛𝑔^𝑙𝑚−𝑔^𝑙𝑛𝑔^𝑘𝑚_,𝑚. (5)

This complex has the antisymmetric property that: 𝐻_𝑖^𝑘𝑙=−𝐻_𝑖^𝑘𝑙. (6)

The energy-momentum components can be found by integrating the energy-momentum density over the volume under consideration: 𝑃_𝑖=𝜃⁰_𝑖𝑑𝑥¹𝑑𝑥²𝑑𝑥³. (7)

Through Gauss’s theorem we can express this as a surface integral: 𝑃_𝑖=1𝐻16𝜋^𝑖0𝛼𝜇_𝛼𝑑𝑆, (8) where 𝜇_𝛼 is the outward unit vector normal to the spherical surface element: 𝑑𝑆=𝑟²sin𝜃𝑑𝜃𝑑𝜙. (9)

The localized energy 𝐸=𝑃₀ can thereby be expressed as: 𝑃₀=1𝐻16𝜋^00𝛼𝜇_𝛼𝑑𝑆. (10)

Therefore, the relevant superpotentials to determine the localized energy are 𝐻₀^0𝛼 with 𝛼 ranging from 1 to 3.

Taking 𝑖=𝑘=0 we find that (5) reduces to 𝐻₀^0𝑙=𝑔_0𝑛√𝑔−𝑔−𝑔^0𝑛𝑔^𝑙𝑚−𝑔^𝑙𝑛𝑔^0𝑚_,𝑚. (11)

Note that 𝑔_0𝑛=0 for all values of 𝑛 except 𝑛=0. Therefore, the only nonzero components of (11) are the ones for which 𝑛=0, namely 𝐻₀^0𝑙=𝑔₀₀√𝑔−𝑔−𝑔⁰⁰𝑔^𝑙𝑚−𝑔^𝑙0𝑔^0𝑚_,𝑚. (12)

We substitute the metric (2) into expression (12) and find that the relevant superpotential values are 𝐻₀⁰¹=4𝜅𝑚𝑥𝑟³,𝐻₀⁰²=4𝜅𝑚𝑦𝑟³,𝐻₀⁰³=4𝜅𝑚𝑧𝑟³. (13)

We insert the superpotentials (13) into the double integral (10). Evaluating this double integral over the full coordinate range, we find the total energy of 𝑠 solution within a sphere of radius 𝑟 to be 𝐸=𝑃₀=𝜅𝑚=𝑀. (14)

It is interesting that the localized energy precisely matches the dark energy star’s total mass 𝑀. This indicates that the Einstein complex well-encompasses the complete energy of the star, including the energy associated with its rest mass 𝑚 as well as its dark energy. Note that if the dark charge 𝐷 is zero, 𝜅 becomes 1, and the localized energy reverts to the Schwarzschild value of 𝑚.

3. Determining the Energy by Use of Papapetrou’s Prescription

We now turn to a second method for determining the local energy, Papapetrou’s prescription. Unlike Einstein’s prescription, Papapetrou’s energy-momentum complex offers the advantage of being symmetric in its indices. Hence, it permits the precise definition of local conservation laws.

The Papapetrou energy-momentum complex is defined as Ω^𝑖𝑘=1𝑁16𝜋^{𝑖𝑘𝑎𝑏,𝑎𝑏},𝑃_𝑖=Ω^𝑖0𝑑𝑥¹𝑑𝑥²𝑑𝑥³, (15) where the functions 𝑁^{𝑖𝑘𝑎𝑏} are given by 𝑁^{𝑖𝑘𝑎𝑏}=√−𝑔𝑔^𝑖𝑘𝜂^𝑎𝑏−𝑔^𝑖𝑎𝜂^𝑘𝑏+𝑔^𝑎𝑏𝜂^𝑖𝑘−𝑔^𝑘𝑏𝜂^𝑖𝑎, (16) and the 𝜂^𝑎𝑏 terms represent the components of a Minkowski metric of signature −2.

Again, we use Gauss’s theorem to express the total energy as a surface integral: 1𝐸=𝜒16𝜋^00𝛼𝜇_𝛼𝑑𝑆, (17) with the superpotentials 𝜒^00𝛼 defined as 𝜒^00𝛼=𝑁_𝑘^00𝑘𝛼. (18)

We determine the relevant values of the superpotentials to be 𝜒⁰⁰¹=4𝜅𝑚𝑥𝑟³,𝜒⁰⁰²=4𝜅𝑚𝑦𝑟³,𝜒⁰⁰³=4𝜅𝑚𝑧𝑟³. (19)

Substituting (16)–(18) into (15) yields 𝐸=𝑃₀=𝜅𝑚=𝑀. (20)

This is identical to the expression obtained using Einstein’s prescription. It is instructive to see that both complexes produce the same result.

4. Determining the Energy by Use of the Møller Prescription

We now turn to a third method for determining the local energy, the Møller prescription, which has the marked advantage of being coordinate system independent. The complex defined by Møller can be expressed as Ξ^𝑘_𝑖=1𝜒8𝜋^{𝑘𝑝𝑖,𝑝}, (21) where 𝜒_𝑖^𝑘𝑙=√−𝑔𝑔_{𝑖𝑝,𝑞}−𝑔_{𝑖𝑞,𝑝}𝑔^𝑘𝑞𝑔^𝑙𝑝. (22)

Inserting the metric components for Yazadjiev’s solution (2) we obtain 𝜒₀⁰¹=4𝜅𝑚𝑥𝑟³,𝜒₀⁰²=4𝜅𝑚𝑦𝑟³,𝜒₀⁰³=4𝜅𝑚𝑧𝑟³. (23)

One more time, we employ Gauss’s theorem to express the total energy as a surface integral: 1𝐸=𝜒16𝜋^00𝛼𝜇_𝛼𝑑𝑆. (24)

Integrating over the full range of coordinates, we find the total energy using the Møller complex to be 𝐸=𝜅𝑚=𝑀. (25)

This is identical to the expression obtained using Einstein’s and Papapetrou’s prescriptions.

5. Conclusion

We have determined the localized energy distribution for Yazadjiev’s solution representing a static, electrically neutral, and spherically symmetric massive object with phantom energy. In applying the Einstein, Papapetrou and Møller energy-momentum complexes to Yazadjiev’s metric, we have found that each yields an identical localized energy equal to the mass 𝑀. This extends earlier results for the Kerr-Schild objects such as the Schwarzschild solution to an interesting case that could bear upon the dark energy question. Our findings help broaden the applicability of energy-momentum complexes, augmenting their use in general relativity as consistent, well-defined ways of describing the local distribution of energy.

Acknowledgment

Thanks are due to K. S. Virbhadra for his helpful advice throughout the years about energy localization and other aspects of general relativity.