Advances in High Energy Physics

Volume 2016, Article ID 9049308, 9 pages

http://dx.doi.org/10.1155/2016/9049308

## Einstein and Møller Energy-Momentum Complexes for a New Regular Black Hole Solution with a Nonlinear Electrodynamics Source

^{1}Department of Physics, Gh. Asachi Technical University, 700050 Iasi, Romania^{2}Department of Mathematics, Jadavpur University, Kolkata, West Bengal 700 032, India^{3}Department of Civil Engineering, University of Thessaly, 383 34 Volos, Greece

Received 9 June 2016; Accepted 15 September 2016

Academic Editor: Sally Seidel

Copyright © 2016 Irina Radinschi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

A study about the energy and momentum distributions of a new charged regular black hole solution with a nonlinear electrodynamics source is presented. The energy and momentum are calculated using the Einstein and Møller energy-momentum complexes. The results show that in both pseudotensorial prescriptions the expressions for the energy of the gravitational background depend on the mass and the charge of the black hole, an additional factor coming from the spacetime metric considered, and the radial coordinate , while in both prescriptions all the momenta vanish. Further, it is pointed out that in some limiting and particular cases the two complexes yield the same expression for the energy distribution as that obtained in the relevant literature for the Schwarzschild black hole solution.

#### 1. Introduction

Energy-momentum localization plays a leading role in the theories advanced over the years in relation to General Relativity. There is a major difficulty, however, in formulating a proper definition for the energy density of gravitational backgrounds. Indeed, the key problem is the lack of a satisfactory description for the gravitational energy.

Many researchers have conducted extensive research using different methods for energy-momentum localization. Standard research methods include the use of different tools, such as super-energy tensors [1–4], quasilocal expressions [5–9], and the famous energy-momentum complexes of Einstein [10, 11], Landau and Lifshitz [12], Papapetrou [13], Bergmann and Thomson [14], Møller [15], Weinberg [16], and Qadir and Sharif [17]. The main problem encountered is the dependence on the reference frame of these pseudotensorial prescriptions. An alternative method used in many studies on computing the energy and momentum distributions in order to avoid the dependence on coordinates is the teleparallel theory of gravitation [18–26].

As regards pseudotensorial prescriptions, only the Møller energy-momentum complex is a coordinate independent tool. Schwarzschild Cartesian coordinates and Kerr-Schild Cartesian coordinates are useful to compute the energy-momentum in the case of the other pseudotensorial definitions. Over the past few decades, despite the criticism directed against energy-momentum complexes concerning mainly the physicalness of the results obtained by them, their application has provided physically reasonable results for many spacetime geometries, more particularly for geometries in (), (), and () dimensions [27–58].

There is an agreement between the Einstein, Landau-Lifshitz, Papapetrou, Bergmann-Thomson, Weinberg, and Møller prescriptions, on the one hand, and the definition of the quasilocal mass advanced by Penrose [59] and developed by Tod [60] for some gravitating systems, on the other hand (see [61] for a comprehensive review). Several pseudotensorial definitions “provide the same results” for any metric of the Kerr-Schild class and for solutions that are more general than the Kerr-Schild class (see, e.g., the works of Aguirregabiria, Chamorro and Virbhadra, Xulu in [27–36], and Virbhadra in [62]). Furthermore, the similarity between some of the aforementioned results and those obtained by using the teleparallel theory of gravitation [63–68] cannot be overlooked. In fact, the history of energy-momentum complexes should include their definition and use, as well as the attempts for their rehabilitation [69–72].

The present work has the following structure: in Section 2 we describe the new spherically symmetric, static, charged regular black hole solution with a nonlinear electrodynamics source [73] under study. Section 3 is focused on the presentation of the Einstein and Møller energy-momentum complexes used for performing the calculations. Section 4 contains the calculations of the energy and momentum distributions. In the Discussion and Final Remarks given in Section 5, we make a brief description of the results of our investigation as well as some limiting and particular cases. Throughout the article, we use geometrized units (), the signature chosen for our purpose is (), and the calculations are performed using the Schwarzschild Cartesian coordinates for the Einstein prescription and the Schwarzschild coordinates for the Møller prescription. Also, Greek indices range from to , while Latin indices run from to .

#### 2. Description of the New Regular Black Hole Solution with a Nonlinear Electrodynamics Source

In this section, we present a new spherically symmetric, static, charged regular black hole solution with a nonlinear electrodynamics source recently developed by Balart and Vagenas [73].

A brief but interesting discussion about the regular black hole solutions that have been obtained by coupling gravity to nonlinear electrodynamics theories is presented in [73] (see Section 1 and the references therein for more details). Further, in an interesting and similar work, the horizon entropy of a black hole is determined as a function of Komar energy and the horizon area [74].

In order to develop the new charged regular black hole solution, the authors of [73] considered the Fermi-Dirac-type distribution. For this purpose, they generalized the methodology developed in Section 2 of their paper by considering distribution functions raised to the power of a real number greater than zero. We notice that the methodology presented in Section 2 consists in constructing a general charged regular black hole metric for mass distribution functions that are inspired by continuous probability density distributions. The corresponding electric field for each black hole solution is also constructed in terms of a general mass distribution function. The metric function is given bywhere is a normalization factor and the function corresponds to any one of the mass functions listed in Table 1 of [73], but with the coordinate multiplied by an additional factor .

The new spherically symmetric, static, charged regular black hole solution with a nonlinear electrodynamics source given by equation in [73] is obtained, as we pointed out above, using the Fermi-Dirac-type distribution, and the metric function becomes nowMoreover, when the mass function . The distribution function satisfies the condition when . This solution is a generalization of the Ayón-Beato and García black hole solution [75].

The corresponding electric field has the expression In order to construct the extremal regular black hole metric for this example, some values of and the corresponding charges are listed in Table of [73].

Finally, the new charged regular black hole solution with a nonlinear electrodynamics source is described by the metricwith , .

Figure 1 shows that two horizons exist at the points where meets the ()-axis and for four different values of the parameter . We have chosen . Note that the positions of the inner and the outer horizon remain unaffected for various values of .