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.

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 [

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 [

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 [

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 [

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 [

The interior and exterior metrics of a dark energy star were calculated by Yazadjiev [

Note that

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.

We now apply Einstein’s prescription to Yazadjiev’s dark energy star solution. The exterior metric has the following form:

Einstein’s prescription defines the local energy-momentum density as

This complex has the antisymmetric property that:

The energy-momentum components can be found by integrating the energy-momentum density over the volume under consideration:

Through Gauss’s theorem we can express this as a surface integral:

The localized energy

Therefore, the relevant superpotentials to determine the localized energy are

Taking

Note that

We substitute the metric (

We insert the superpotentials (

It is interesting that the localized energy precisely matches the dark energy star’s total mass

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

Again, we use Gauss’s theorem to express the total energy as a surface integral:

We determine the relevant values of the superpotentials to be

Substituting (

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

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

Inserting the metric components for Yazadjiev’s solution (

One more time, we employ Gauss’s theorem to express the total energy as a surface integral:

Integrating over the full range of coordinates, we find the total energy using the Møller complex to be

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

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

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