Approximate Noether gauge symmetries of Bardeen model

We investigate the approximate Noether gauge symmetries of the geodesic Lagrangian for the Bardeen spacetime model. This is accommodated by a set of new approximate Noether gauge symmetry relations for the perturbed geodesic Lagrangian in the spacetime. A detailed analysis to the spacetime of Bardeen model up to third-order approximate Noether gauge symmetries is presented.


Introduction
The black holes predicted in the framework of Einstein's gravity theory appear to exist in the Universe [1]. The existence of singularities inside the black holes was predicted by singularity theorems, and they indicate breakdown of the classical theory [2]. A regular (i.e. non-singular) black hole which is characterized by the fact that the spacetime metric as well as the curvature invariants R, R ab ,R ab , R abcd R abcd do not present singularities anywhere was firstly derived from Einstein's field equations by Bardeen [3] which was initially approximately obtained. As it was shown in Ref. [4], the metric of Bardeen model solves Einstein equations with a self-gravitating nonlinear magnetic monopole as a source. The Bardeen model includes two parameters in the solution, namely the parameters m and e. As usual from the Schwarzschild metric, the parameter m can be interpreted as the mass parameter for the system. The other parameter e can be interpreted as the magnetic charge of the geometry. Although Bardeen model is always a regular black hole at r = 0 for e = 0, it represents a regular black hole spacetime only when the inequality 27e 2 ≤ 16m 2 holds.
Let us briefly give some information about spacetime symmetries. A conformal Killing vector (CKV) K have to satisfy £ K g ab = 2ψ(x a )g ab , where g ab is the metric tensor, £ K is the Lie derivative operator along K and ψ(x a ) is a a e-mail: ucamci@akdeniz.edu.tr conformal factor. When ψ ;ab = 0, the CKV field is said to be proper [5]. The vector field K is called the special conformal Killing vector (SCKV) field if ψ ;ab = 0, the homothetic Killing vector (HKV) field if ψ , a = 0, e.g. ψ is a constant on the manifold, and the Killing vector (KV) field which gives the isometry if ψ = 0. The set of all CKV (respectively SCKV, HKV and KV) form a finite-dimensional Lie algebra. The maximum dimension of the CKV algebra on the manifold M is fifteen if M is conformally flat, and it is seven if the spacetime is not conformally flat.
If a Lagrangian L for a given dynamical system admits some symmetry, this property should strongly be related with Noether symmetries which describe physical features of differential equations possessing a Lagrangian L in terms of first integrals admitted by them [6,7]. This can actually be seen in two ways. First of all, one can consider a strict Noether symmetry approach [8][9][10] which yields £ X L = 0, where £ X is the Lie derivative operator along X. On the other side, one could use the Noether gauge symmetry (NGS) approach [11][12][13][14][15][16][17] which is a generalization of the strict Noether symmetry approach in the sense that the Noether symmetry equation includes a gauge term in such a way that the vector field X = ξ ∂ s + η a ∂ ∂ x a is called a NGS of a Lagrangian L if there exists a gauge function, A(s, x a ), such that the Noether symmetry condition holds [6,7] X [1] where X [1] = X + η a s ∂ ∂ẋ a is the first prolongation operator of X, η a s = D s η a −ẋ a D s ξ , and D s = ∂ ∂ s +ẋ a ∂ ∂ x a is the total derivative operator. The approximate symmetries are as useful as exact ones in a variety of problems arising from physics and applied mathematics. We will discuss how the approximate Noether gauge symmetries (ANGSs) of the perturbed geodesic Lagrangian are related to the geometrical symmetries of spacetimes in the following section.
The geodesic Lagrangian describing the motion of the massive or massless particles under the influence of a potential function V (x c ) is where the dot represents derivative with respect to the geodetic parameter s. The significance of an ANGS comes from the fact that a manifold having no exact symmetry can possess approximate ones. In this context, using the approximate symmetries of perturbed geodesic Lagrangian for spacetimes, one can recover the lost symmetries and conservation laws corresponding to the perturbed Lagrangian [18][19][20][21][22]. For the Schwarzschild metric, Kara et al. [20] have considered the approximate symmetries and the conservation laws of geodesic equations. Following the use of approximate symmetries for the Schwarzschild spacetime, Hussain et al. [21] have investigated the exact and approximate symmetries of the system of geodesic equations for the Reissner-Nordström (RN) spacetime. Using the ANGS for the Schwarzschild and Kerr metrics, Hussain et al. [22] have recovered all the lost conservation laws as trivial first-order approximate conservation laws in first-order approximate case. Using approximate Lie symmetries the energy content of colliding plane waves have studied by Sharif and Waheed [23]. By analyzing the behavior of the effective potential for the particles and photons, it is investigated the time-like and null geodesic structures for the Bardeen metric [24]. In a recent paper, Sharif and Waheed [25] have applied the approximate Lie symmetry method to investigate the energy re-scaling factor for the Bardeen's regular black-hole solution. They only evaluated the third-order approximate symmetries of the orbital and geodesic equations using approximate Lie symmetry methods for differential equations. Our main aim here is to extend our mathematical understanding of equation (1) by exploring its geometrical properties. We have obtained the geometrical nature of ANGSs up to secondorder perturbed Lagrangian and applied this method to the Schwarzschild and RN spacetimes [26]. It should be stressed that we deal with third-order ANGSs of the perturbed geodesic Lagrangian which are not studied previously as far as we know.
The rest of the paper is organized as follows. In the following section, we present an analysis of third-order ANGSs of first-order geodesic Lagrangian according to perturbed spacetime geometry. In section 3, we apply the obtained results of section 2 to the spacetime of Bardeen model. Conclusions and discussions are presented in last section 4.

Third Order Approximate Noether gauge symmetries
Calculation method of exact symmetries and the first-order approximate symmetries of a Lagrangian are available in the literature [18][19][20]. Hussain et al. [21,22] have extended the procedure of calculating the approximate symmetries of a perturbed Lagrangian to the second-order. In this study we introduce a new geometrical method to calculate the ANGSs of the first-order perturbed Lagrangian extending the procedure of obtaining ANGSs until the third-order.
The third-order ANGSs of the first-order perturbed Lagrangian is given by the ANGS generator up to the gauge function A(s, where X 0 is the exact NGS generator, X 1 , X 2 and X 3 are the first-order, second-order and third-order ANGS generators, respectively, which are defined as The above perturbed Lagrangian (3) yields a third order (in ε) perturbed system of ordinary differential equations (ODEs).
If X j are the ANGSs corresponding to the perturbed Lagrangians L j (s, x a ,ẋ a ), then are the first integrals associated with ANGSs X j (j=0,1,2,3). The spacetime metric g ab can be decomposed as follows: which means by (2) and (3) that the exact and perturbed geodesic Lagrangians of motion have the form where γ ab , h ab , σ ab and k ab are the exact, the first-order, the second-order and the third-order perturbed metrics, respectively. The metric γ ab should be non-degenerate (i.e., det(γ ab ) = 0). But the other metrics h ab , σ ab and k ab can be degenerate (i.e., det(h ab ) = 0, det(σ ab ) = 0, det(k ab ) = 0) or nondegenerate, and they represent the slight deviations from flat spacetime geometry if the metric γ ab represents flat geometry.

Application to Bardeen Model
The Bardeen model describes a singularity-free black hole spacetime which means it is a regular spacetime. In this section, we will apply the above geometrical method to the third-order perturbed spacetime of Bardeen model for finding ANGSs of perturbed geodesic Lagrangian. The line element representing the Bardeen spacetime is given by where m is the mass of configuration and e represents the magnetic charge of the nonlinear self-gravitating monopole, and we have taken the gravitational units, i.e. G = c = 1. The Bardeen black hole metric reduces to the Schwarzschild metric for e = 0 and is flat for m = 0. Using the expansions and the first-order perturbed geodesic Lagrangian of Bardeen model is given by where we take ε = 2m and e 2 = kε 2 , 0 < k ≤ 4/27, in the limit of small mass of a point gravitating source. The above Lagrangian (39) yields where we used γ ab = diag(1, −1, −r 2 , −r 2 sin 2 θ ) which is the Minkowski metric, h ab = diag(−1/r, −1/r, 0, 0), σ ab = diag(−1/r 2 , 0, 0, 0) and k ab = diag(3k/2r 3 , (3k/2−1)/r 3 , 0, 0). Also, the Lagrangian (39) reduces to the geodesic Lagrangian of the Minkowski metric in the limit ε → 0. Thus, we can apply the new set of ANGSs obtained in the previous section to these exact metric γ ab and perturbed metrics h ab , σ ab and k ab for the Bardeen spacetime. For the metric γ ab , we find from the exact ANGS Eqs. (24)-(26) the following 19 equations η 0 0,r − η 1 0,t = 0, η 0 0,θ − r 2 η 2 0,t = 0, η 0 0,φ − r 2 sin 2 θ η 3 0,t = 0, η 1 0,θ + r 2 η 2 0,r = 0, η 1 0,φ + r 2 sin 2 θ η 3 0,r = 0, η 2 0,φ + sin 2 θ η 3 0,θ = 0, Considering both γ ab and h ab , the first-order ANGS Eqs.(27)-(29) yield again 19 equations Using the metrics γ ab , h ab and σ ab , the second-order ANGS Eqs.(30)-(32) give the following equations ξ 2,t = 0, ξ 2,r = 0, ξ 2,θ = 0, ξ 2,φ = 0, )η 1 0,s + r η 1 1,s + r 2 η 1 2,s ] + η 1 3,s + A 3,r = 0, We simultaneously solved exact, the first-order, the secondorder and the third-order ANGS equations (44), (45), (46) and (47) for the constant potential, e.g. V (t, r, θ , φ ) = V 0 . From Eqs.(44), (45), (46) and (47), we find for the exact ANGSs for the first-order ANGSs for the second-order ANGSs and for the third-order ANGSs where a ℓ , b ℓ , c ℓ (ℓ = 1, . . . , 6) and d 1 , . . . , d 9 are the arbitrary constants of integration associated with the ANGSs. It is easily seen from the above solutions that we have 5 exact, first-order and second-order ANGSs and 8 third-order ANGSs which includes also exact ones. It is known that the spacetime of Bardeen model has four KVs, i.e., which correspond to the conservation of energy and angular momentum only. For all order of the ANGSs, we point out here that there is the vector field Y 0 = ∂ s which gives translation in geodetic parameter s and it always exists for a geodesic Lagrangian of the type (2), and remaining ones are the four KVs given in (59) and (60). We found that the exact,the first-order and the second-order ANGS algebras admitted by Bardeen model are five dimensional. In addition to those of five ANGSs, there are three additional third-order ANGSs as the following: with gauge term A 3 = −V 0 s, Thus, it is seen that for the Bardeen model the conservation laws of linear momentum and spin angular momentum in the third-order approximation are lost. In the thirdorder approximate symmetry case, we compute the ANGSs of perturbed geodesic Lagrangian for the Bardeen metric with non-constant potential. In order to determine the ANGSs associated with a given non-constant potential V (t, r, θ , φ ), one has to use the new ANGS Eqs.(44), (45), (46) and (47).
We have collected our results in Table 1. Now, we intend to recover the lost conservation laws. Therefore, it will be considered the lower order than thirdorder, i.e. the second-order, approximate symmetry of the Bardeen model. Thus we will retain the second-order terms in ε and neglect O(ε 3 ) in the geodesic Lagrangian (39) for the Bardeen model. It is noted that the metric of Bardeen model as the second-order perturbed spacetime reduces to the second-order perturbed Schwarzschild solution. So we will use the new set of determining equations up to secondorder to check whether the conservation laws are lost or not. Solving the exact, first-order and second-order ANGS equations (44), (45) and (46) for the constant potential it follows the same components given by (48)-(49) and (50)-(51) for the exact and first-order ANGSs, respectively. But for the second-order ANGSs we have the following components − cos θ (c 6 t + c 9 s + c 10 ) − sin θ s(c 11 sin φ − c 12 cos φ )  Table 1 Third-order Approximate Noether gauge symmetries and associated gauge functions A 0 , A 1 , A 2 and A 3 of the Bardeen metric for some non-constant potentials V (t, r, θ , φ ). Here V 0 is a non-zero constant.  where the generators Z 1 , Z 2 , Z 3 provide the conservation of linear momentum, and Z 4 , Z 5 , Z 6 yield the conservation of spin angular momentum. Using the generator Y 2 one can write s = t 2 or s = r 2 which gives rise to interpretation the four KVs K 0 , K 1 , K 2 , K 3 , and eleven generators Y 0 , Y 1 , Z 1 , . . . , Z 9 are CKVs with conformal factor ψ = (c 1 t 2 + c 2 )/2. It is very well known fact that the Lie algebra of the CKVs for a conformally flat spacetime is 15 dimensional.