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Energy of the Bardeen model using an approximate symmetry method

In this paper, we investigate the energy problem in general relativity using approximate Lie symmetry methods for differential equations. This procedure is applied to the Bardeen model (the regular black-hole solution). Here, we are forced to evaluate the third-order approximate symmetries of the orbital and geodesic equations. It is shown that energy must be re-scaled by some factor in the third-order approximation. We discuss the insights into this re-scaling factor.


Introduction
Energy-momentum is an important conserved quantity in any physical theory whose definition has been under investigation for a long time from the general relativity viewpoint. Unfortunately, there is still no generally accepted definition of energy-momentum for gravitational fields. The main problem is with the expression defining the gravitational field energy part. To obtain a meaningful expression for energy, momentum and angular momentum for a general relativistic system, Einstein proposed an expression, called Einstein's energy-momentum complex [1]. After that many complexes have been found, for instance, Landau-Lifshitz [2], Tolman [3], Papapetrou [4], Møller [5], Weinberg [6] and Bergmann [7]. Some of these definitions are coordinate dependent while others are not. Also, some of these expressions cannot be used to define angular momentum.
Virbhadra [8] gave marvellous idea about coincidence of some complexes which attracted many researchers. Aguireregabiria et al. [9] proved that Einstein, Landau-Lifshitz, Papapetrou, Weinberg prescriptions yielded the same distribution of energy for any metric of the Kerr-Schild class if calculations are performed in Kerr-Schild coordinates. Vagenas [10] and Sharif [11] proved that these four prescriptions coincide for the (2 + 1)-dimensional rotating BTZ black hole (named after Banados, Teitelboim and Zanelli) and regular black hole solution respectively. Xulu [12] extended this investigation and found the same energy distribution for the dyadosphere of the Reissner-Nordström (RN) black hole. However, all these prescriptions have their own drawbacks.
The concept of approximate symmetry theories comes from the combination of Lie group theory and perturbations. Using this idea two different so-called "approximate symmetry" theories have been constructed. The first theory was proposed by Baikov, Gazizov and Ibragimov [13] and the second was presented by Fushchich and Shtelen [14]. It may be mentioned here that a manifold having no exact symmetry can possess approximate symmetries. As compared to exact symmetry, one can acquire more interesting information from a slightly broken (approximate) symmetry. This idea of symmetries would be more accurate as it does not depend on pseudo-tensors and hence agrees with the principles of general relativity. In order to discuss the approach of approximate symmetries, we use the concept of Lie symmetries which is substantiated in spacetime isometries as well as in Noether symmetries.
Kara et al. [15] used approximate Lie symmetry method [13] to discuss the conservation laws of the Schwarzschild spacetime. Later, using the same procedure, in the second-order approximation, the lost conservation laws for RN [16], Kerr-Newmann [17] and Kerr-Newmann AdS [18] spacetimes were recovered and some energy re-scaling factors were obtained. In a recent paper [19], we have discussed re-scaling of energy in the stringy charged black hole solutions using approximate symmetries. In going from the Minkowski spacetime to non-flat regular black hole solution, the recovered conservation laws (recovered in first-order approximation) are lost. It is expected that by minimizing the charge, one should recover all lost conservation laws. In this paper, we use the same procedure to formulate energy re-scaling factor for regular black hole solution.
The format of the paper is as follows. In the next section, we review the basic material about approximate symmetry methods for the solution of differential equations (DEs) and also give exact symmetries of the Minkowski spacetime and the first-order approximate symmetries of the Schwarzschild spacetime. In section 3, we study approximate symmetries of the regular black hole solution. Finally, we summarize and discuss the results.

Mathematical Formalism
Symmetry is a point transformation (a transformation which maps one point (x, y) into another (x * , y * )) under which the form of DE does not change. These transformations form a group known as point transformation group. Symmetries are important for their direct connection with the conservation laws through the Noether theorem [20]. It has been mentioned that if, for a given system of DEs, there is a variational principle, then a continuous symmetry invariant under the action of the given functional provides corresponding conservation law [21,22].
A system of p ODEs, each of n th -order [23,24] E α (s; x(s), x ′ (s), x ′′ (s), ..., x (n) (s)) = 0, (α = 1, 2, 3.., p) (here x is dependent variable, s is independent variable and x ′ , x ′′ , ..., x (n) denote the first, second and so on n th -order derivative of x w.r.t. s) under the point transformation: (s, x) −→ (ξ(s, x), η(s, x)) can admit a symmetry generator if and only if, on the solution of the ODEs, E α = 0, the following symmetry condition is satisfied. Here n th -order extension of the symmetry generator given by Eq.
(2) can be written as and the corresponding prolongation coefficients are The k th -order approximate symmetries of a perturbed ODE is given by the generator if the following symmetry condition holds [25] Here E 0 is the exact part of ODE, E 1 , E 2 are respectively the first and second order approximate parts of the perturbed ODE and so on. X 0 represents the exact symmetry generator and X 1 , X 2 denote respectively the first and second order approximate parts of the symmetry generator and so on. For the k th -order approximate symmetry generator, the terms involving ǫ (k+1) and its higher powers can be substituted to zero so that RHS of Eq.(8) becomes zero. As compared to exact symmetries, these approximate symmetries do not necessarily form a Lie algebra rather do form the so-called "approximate Lie algebra" (up to a specified degree of precision) [26]. The perturbed equation always admits an approximate symmetry ǫX 0 called a trivial symmetry. If a symmetry generator X = X 0 + ǫX 1 exists with X 0 = 0 and X 1 = kX 0 , k is an arbitrary constant, then it is called non-trivial symmetry [27]. It has been mentioned [28] that 10 Killing vectors (Appendix A) of Minkowski spacetime provide conservation laws for energy and linear momentum as well as for angular and spin angular momentum. The algebra calculated from the geodesic equations contains some additional symmetries which provide no conservation law [29]. The Schwarzschild spacetime has four isometries Y 0 , Y 1 , Y 2 , Y 3 which correspond to conservation laws of energy and angular momentum only. The symmetry algebra of the Schwarzschild spacetime (calculated through the geodesic equations) consist of four isometries and the dilation algebra. In the limit of small mass of point gravitating source, ǫ = 2M [15], all lost conservation laws would be recovered as first order trivial approximate symmetries. For the orbital equation, the first-order approximate symmetries are given in Appendix B.

Bardeen Model
Reissner-Nordström solution is a static, spherically symmetric and asymptotically flat solution of the Einstein-Maxwell field equations having a singularity structure. There arise some problems when one applies the laws of physics at the singularity point (origin of the radial coordinate) of the RN solution. To resolve this problem, Bardeen (1968) constructed a singularity free solution [30,31] using the energy-momentum tensor of non-linear electrodynamics as the source of the field equations. The Bardeen model is known as a regular black hole solution because for a particular ratio of mass to charge, this represents a black hole and a singularity free structure. After that many regular black hole solutions have been discussed by various authors, for example, [32]. These metrics, being analogues to the RN metric with magnetic charge, have a quite similar global but regular structure as that for the RN spacetime.

Isometry Algebra and Orbital Equation of Motion
The line element representing Bardeen model is given by [31] The asymptotic behavior of the solution can be observed by the simple expansion of the term g tt as follows where m is the mass of the configuration. When r −→ ∞, the parameter e does not vanish as the Columbian term 1 r 2 but vanishes as the term 1 r 3 . Therefore, one cannot recognize the parameter e as the Coulomb charge Q of the RN spacetime for which g tt is given by Indeed, in the Bardeen model, the parameter e represents the magnetic charge of the non-linear self-gravitating monopole [33] while we have taken the gravitational units, i.e., G = c = 1. For e = 0, it turns out to be the Schwarzschild spacetime while for m = 0 = e, it becomes Minkowski spacetime. The simultaneous solution of the Killing equations for the Bardeen model is given by This solution corresponds to the generators Y 0 , Y 1 , Y 2 and Y 3 which form an algebra so(3)⊕R implying the conservation laws for energy and angular momentum only while the linear and spin angular momentum conservation laws are lost. We would check whether one can recover these lost conservation laws in the limit of small gravitational mass and charge or not. The set of geodesic equations for the spacetime (9) can be written as θ + 2 rṙθ − sin θ cos θφ 2 = 0, φ + 2 rṙφ + 2 cot θθφ = 0.
Using the spacetime, four geodesic equations and the symmetries of the equatorial plane, we obtain the following relativistic equation of motion (orbital equation of motion) where h is the classical angular momentum per unit mass and u = 1 r

Perturbation Parameters and Approximate Symmetries of the Orbital and Geodesic Equations
The Bardeen model is a singularity free solution and represents black hole for the following inequality Since the vanishing of mass and charge provide the Minkowski spacetime, so the perturbation parameters are to be defined in terms of mass and charge of the black hole to recover the lost conservation laws. The required perturbation parameters are ǫ = 2m, e 2 = kǫ 2 ; 0 < k ≤ 4 27 , (where we have used the inequality defined in Eq. (15)). Introducing these parameters in the Bardeen model, we obtain the following third-order perturbed spacetime Retaining only the terms of first and second order in the above equation and neglecting O(ǫ 3 ), it reduces to the second-order perturbed spacetime for the Schwarzschild solution. Also, it reduces to the Minkowski spacetime in the limit of ǫ = 0. The third-order perturbed orbital equation of motion can be written as For this equation of motion, the exact and first-order parts of the symmetry generator are the same as for the Schwarzschild spacetime given by Eqs.(A10)-(A16). Since there is no term quadratic in ǫ, therefore its secondorder approximate symmetries are zero. Now we calculate its third-order order approximate symmetries. For this purpose, we apply the operator to Eq.(16), then substitute the values of the prolongation coefficients and retain only the terms involving ǫ 3 , we obtain a system of four DEs. The simultaneous solution of these equations is exactly the same as given by Eq.(A10)-(A14). Thus we can conclude that there exist no non-trivial thirdorder approximate symmetries. In order to recover full set of lost conservation laws, we apply the approximate symmetry procedure to the system of geodesic equations given by Eqs.(10)- (13). The third-order perturbed geodesic equations arë θ + 2 rṙθ − sin θ cos θφ 2 = 0, For ǫ = 0, this system of equations reduces to the system for the Minkowski spacetime. For ǫ 2 = 0 with ǫ = 0, this system results in the first-order perturbed system of geodesic equations for the Schwarzschild spacetime. Therefore, its exact and first-order approximate parts of the symmetry generator are the same as for the Minkowski and Schwarzschild spacetimes respectively. For the second-order approximate part of the symmetry generator, we apply the second-order prolongation of the symmetry generator X [2] . This is given by where ξ = ξ 0 + ǫξ 1 + ǫ 2 ξ 2 and η ν = η ν 0 + ǫη ν 1 + ǫ 2 η ν 2 , (ν = 0, 1, 2, 3). The application of this operator to the second-order perturbed system of geodesic equations (i.e., ǫ 3 = 0, ǫ = 0, ǫ 2 = 0) yields the system of 60 DEs. The solution (where we have retained only the terms involving ǫ 2 ) to this system of DEs is 10 Killing vectors and the dilation algebra d 2 .
We have seen that out of four constants corresponding to exact symmetry generators, two have been appeared and canceled out while the six constants corresponding to first and second-order parts of the symmetry generators appear. All these six constants turn out to be zero when we solve this system of equations simultaneously and it results in the system of DEs for the Minkowski spacetime. Its solution provides 10 Killing vectors and the generators corresponding to dilation algebra d 2 . This leads to no non-trivial symmetry in third-order approximation. For this model, the constants corresponding to dilation algebra ξ 0 (s) = c 0 s + c 1 do not cancel automatically but collect some expression for cancelation. This expression is given as follows where we have used the value of k. Since ξ(s) is the coefficient of ∂ ∂s (where s corresponds to proper time) in the symmetry generator given in Eq.(2), so we call this factor as energy re-scaling factor of a test particle due to the presence of charge and mass.

Summary and Discussion
This paper is devoted to study energy of the Bardeen model (regular black hole solution) by using the approximate symmetries. These solutions have the isometry algebra so(3)⊕R while the system of geodesic equations have algebra so(3)⊕R⊕d 2 . The strength of the approximate symmetries (i.e., perturbation parameters) provide the information about the amount of energy contained in the spacetime field. The second-order approximation to symmetry generator yields only the lost conservation laws and hence no information regarding the energy re-scaling factor is obtained. For the energy re-scaling factor, we have to evaluate the third-order approximate symmetries.
Firstly, we have evaluated the third-order approximate symmetries of the orbital equation of motion. The exact and the first-order parts of the symmetry generators are the same as for the Schwarzschild spacetime. However, the second-order approximate part of the symmetry generator is zero. In the third-order approximation, one can obtain the trivial symmetries only (i.e., of Minkowski spacetime) given by Eqs.(A10)-(A14).
Secondly, we have calculated the third-order approximate symmetries of the perturbed geodesic equations. Here the exact, first-order and secondorder approximate parts of the symmetry generator turn out to be the same as for the Schwarzschild spacetime. The third-order approximate symmetries of the third-order perturbed geodesic equations provide no non-trivial symmetry generator. However, we have obtained energy re-scaling factor given by Eq. (22). The re-scaling factor is of great interest as it shows the reduction in energy by the ratio of magnetic self-energy of the source to its gravitational self-energy. For e = 0, it vanishes and coincides with the Schwarzschild spacetime.
When a particle is placed in the field produced by charge gravitating source, some force is exerted on it which should be position dependent. However it is not clear whether the modification in mass (energy) in the presence of charge should be position dependent. The expression for re-scaling of force calculated through the pseudo-Newtonian formalism [34] is given by m r 2 (r 2 + e 2 ) 5/2 (r 3 − 2re 2 ) It is position dependent and provides the reduction in force by a factor depending on the ratio of the magnetic potential energy at a distance r to the rest energy of the gravitational source. However, it is not a convenient expression for relating the magnetic self-energy of the configuration to its gravitational self-energy. It should be position independent where the radial parameter r would be canceled out. Our obtained re-scaling factor is clearly position independent and corresponds to change in mass (energy) due to the presence of magnetic charge. Therefore, it provides a physically more sig-nificant expression for relating these self-energies as compared to the results found through different approaches [35].
We would like to mention here that Bardeen model with magnetic charge is the analogue of the RN spacetime with electric charge for which, in the second-order approximation an energy re-scaling factor is obtained. However, in contrast to this, we are forced to calculate its thirdorder approximate symmetries because the second-order approximation provide no information about the energy re-scaling factor. Clearly both these scaling factors are position independent providing more significant results. For the orbital equation of motion, the second-order approximate symmetries are zero (while for the RN spacetime, there exists non-zero and trivial second-order symmetries). In literature [15,16], a difference between the conservation laws for the full system of geodesic equations and single orbital equation of motion is noted. This difference also holds for the Bardeen solution. It would be worthwhile to investigate the approximate symmetry generators and the concept of energy re-scaling for the rotating stringy black hole solutions which contain electric, magnetic and both charges.