Advances in High Energy Physics

Volume 2017 (2017), Article ID 7656389, 10 pages

https://doi.org/10.1155/2017/7656389

## Energy-Momentum for a Charged Nonsingular Black Hole Solution with a Nonlinear Mass Function

^{1}Department of Physics, “Gh. Asachi” Technical University, 700050 Iasi, Romania^{2}Department of Civil Engineering, University of Thessaly, 383 34 Volos, Greece^{3}Department of Mathematics, Jadavpur University, Kolkata, West Bengal 700 032, India^{4}School of Applied Mathematics and Physical Sciences, National Technical University of Athens, 157 80 Athens, Greece^{5}Department of Mathematics, Amity Institute of Applied Sciences, Amity University, Major Arterial Road, Action Area II, Rajarhat, New Town, West Bengal 700135, India^{6}Department of Physics, University of Trieste, Via Valerio 2, 34127 Trieste, Italy

Correspondence should be addressed to Irina Radinschi

Received 5 July 2017; Revised 20 September 2017; Accepted 21 November 2017; Published 24 December 2017

Academic Editor: Elias C. Vagenas

Copyright © 2017 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

The energy-momentum of a new four-dimensional, charged, spherically symmetric, and nonsingular black hole solution constructed in the context of general relativity coupled to a theory of nonlinear electrodynamics is investigated, whereby the nonlinear mass function is inspired by the probability density function of the continuous logistic distribution. The energy and momentum distributions are calculated by use of the Einstein, Landau-Lifshitz, Weinberg, and Møller energy-momentum complexes. In all these prescriptions, it is found that the energy distribution depends on the mass and the charge of the black hole, an additional parameter coming from the gravitational background considered, and the radial coordinate . Further, the Landau-Lifshitz and Weinberg prescriptions yield the same result for the energy, while, in all the aforesaid prescriptions, all the momenta vanish. We also focus on the study of the limiting behavior of the energy for different values of the radial coordinate, the parameter , and the charge . Finally, it is pointed out that, for and , all the energy-momentum complexes yield the same expression for the energy distribution as in the case of the Schwarzschild black hole solution.

#### 1. Introduction

The problem of the energy-momentum localization in general relativity has been investigated over the years by using various and different powerful tools such as superenergy tensors [1–4], quasilocal expressions [5–9], and the mostly known pseudotensorial energy-momentum complexes introduced by Einstein [10, 11], Landau and Lifshitz [12], Papapetrou [13], Bergmann and Thomson [14], Møller [15], Weinberg [16], and Qadir and Sharif [17].

As it is well-known, the main difficulty which arises consists in developing a properly defined expression for the energy density of the gravitational background. Until today, no generally accepted meaningful definition for the energy of the gravitational field has been established. However, despite this difficulty, many physically reasonable results have been obtained by applying the aforesaid definitions for the energy-momentum localization. At this point, one cannot but notice the existing agreement between the pseudotensorial prescriptions and the quasilocal mass definition elaborated by Penrose [18] and further developed by Tod [19].

Although the dependence on the coordinate system continues to be the main “weakness” of these tools, a number of physically interesting results have been obtained for gravitating systems in , , and space-time dimensions by using the energy-momentum complexes [20–50]. In fact, the Møller energy-momentum complex is the only computational tool independent of coordinates. In the context of other pseudotensorial prescriptions, in order to calculate the energy and momentum distributions, one introduces Schwarzschild Cartesian and Kerr-Schild coordinates.

An alternative option for avoiding the problem of the coordinate system dependence is provided by the teleparallel theory of gravity [51, 52], whereby one notices the considerable similarity of results obtained by this approach with results achieved by using the energy-momentum complexes [53–57].

Finally, closing this short introduction to the topic of the energy-momentum localization, it is necessary to point out the broadness of the ongoing attempts in order to define properly and, actually, rehabilitate the concept of the energy-momentum complex [58–61].

The outline of the present paper is the following. In Section 2, we introduce the new static and charged, spherically symmetric, nonsingular black hole solution under study. Section 3 is devoted to the presentations of the Einstein, Landau-Lifshitz, Weinberg, and Møller energy-momentum complexes used for the calculations. In Section 4, the computations of the energy and momentum distributions are presented. In the discussion given in Section 5, we comment on our results and explore some limiting and particular cases. We have used geometrized units , while the signature is . The calculations for the Einstein, Landau-Lifshitz, and Weinberg energy-momentum complexes are performed by use of the Schwarzschild Cartesian coordinates. Greek indices range from to , while Latin indices run from to .

#### 2. The New Charged Nonsingular Black Hole Solution with a Nonlinear Mass Function

The determination of nonsingular black hole solutions by coupling gravity to nonlinear electrodynamics has attracted interest long ago (for a review, see, e.g., [62] for spherically symmetric solutions or [63] and references therein for charged axisymmetric solutions). Recently, Balart and Vagenas [64] constructed a number of new charged, nonsingular, and spherically symmetric, four-dimensional black hole solutions with a nonlinear electrodynamics source. Indeed, start with the static and spherically symmetric space-time geometry described by the line element:with the metric functionwhere the distribution function depends on the mass , the charge , the radial coordinate , and the parameter , and is a normalization factor. Thus one has a nonlinear mass function of the following form:which, at infinity, becomes . It is shown in [64] that for specific distribution functions the curvature invariants (, , ) and the associated nonlinear electric field are nonsingular everywhere.

Based on these results, we have already studied [65] the problem of the localization of energy for a function resembling the form of the Fermi-Dirac distribution. Here, following up that work, we adopt another distribution function given in [64] that is inspired by the form of the probability density function of the continuous logistic distribution [66], such thatwith the nonlinear mass function given by (3).

The associated nonsingular space-time exhibits two horizons, while the nonlinear and nonsingular electric field that asymptotically goes to reads nowWhen , the metric function (4) takes the form ; that is, we get the Schwarzschild black hole geometry.

Thus, in what follows, we are going to investigate the problem of energy-momentum localization for a charged and nonsingular black hole solution with the space-time described by (1), (4) and the nonlinear mass function:

#### 3. Einstein, Landau-Lifshitz, Weinberg, and Møller Energy-Momentum Complexes

The definition of the Einstein energy-momentum complex [10, 11] for a -dimensional gravitating system is given bywhere the von Freud superpotentials are given asand satisfy the required antisymmetric propertyThe components and correspond to the energy and the momentum densities, respectively. In the Einstein prescription, the local conservation law holds:Thus, the energy and the momenta can be computed byApplying Gauss’ theorem, the energy-momentum iswhere represents the outward unit normal vector on the surface .

The Landau-Lifshitz energy-momentum complex [12] is defined aswhere the Landau-Lifshitz superpotentials are given byThe and components represent the energy and the momentum densities, respectively. In the Landau-Lifshitz prescription, the local conservation law readsBy integrating over the 3-space, one obtains for the energy-momentum:By using Gauss’ theorem, we have

The Weinberg energy-momentum complex [16] is given by the following expression:where are the corresponding superpotentials:Here the and components correspond to the energy and the momentum densities, respectively. In the Weinberg prescription, the local conservation law readsThe integration of over the 3-space yields for the energy-momentum:Applying Gauss’ theorem and integrating over the surface of a sphere of radius , one obtains for the energy-momentum distribution the following expression:

The Møller energy-momentum complex [15] is given by the following expression:where the Møller superpotentials areand satisfy the necessary antisymmetric property:Møller’s energy-momentum complex also satisfies the local conservation lawwith and representing the energy and the momentum densities, respectively. In the Møller prescription, the energy-momentum is given byWith the aid of Gauss’ theorem, one gets

#### 4. Energy and Momentum Distributions for the New Charged Nonsingular Black Hole Solution

In order to calculate the energy and momenta by using the Einstein energy-momentum complex, it is required to transform the metric given by the line element (1) in Schwarzschild Cartesian coordinates. We obtain the line element in the following form:

The components of the superpotential in Schwarzschild Cartesian coordinates are

The remaining, nonvanishing, components of the superpotentials in the Einstein prescription are as follows:From the line element (29), the expression (12), and the superpotentials (31), we obtain for the energy distribution in the Einstein prescription (see Figure 1) the following:By using (12) and (30), we find that all the momenta vanish: