2nd order approximate Noether and Lie symmetries of Gibbons–Maeda–Garfinkle–Horowitz–Strominger charged black hole in the Einstein frame

In this paper, approximate Noether and Lie symmetries of 2nd order for Gibbons–Maeda–Garfinkle–Horowitz–Strominger (GMGHS) charged black hole in the Einstein frame are analyzed comprehensively. To explore the approximate Noether symmetries of 2nd order, Noether symmetries of Minkowski spacetime are used which forms a 17 dimensional Lie algebra. It is observed that no new approximate Noether symmetry is obtained at 1st and 2nd order. To examine the 1st and 2nd order approximate Lie symmetries of the GMGHS black hole spacetime, 35 Lie symmetries (exact) of the Minkowski spacetime are used which forms an algebra sl(6, R). It is shown that no new approximate Lie symmetry exists at 1st and 2nd order and only exact 35 symmetries are recouped as trivial approximate Lie symmetries at both orders. Furthermore, no energy rescaling factor is seen in this spacetime.

are limited and problematic to find. Henceforth, the study of the exact solutions of nonlinear PDEs plays an important role in the investigation of nonlinear physical phenomena. The classical symmetry method is a well-known and recognized method for finding the exact solutions of the DEs. This method is also known as group analysis and was invented in 1881 by Lie [17]. In Mathematics and Physics, the study of symmetries is one of the most important and fundamental methods. However for few nonlinear problems, symmetries do not give us the meaningful solutions. This fact was the inspiration for construction of some generalizations of the classical Lie group method. One of the methods extensively used in examining the nonlinear problems is the perturbation analysis. Perturbation theory contains mathematical techniques. These techniques are applied to find an approximate solution to the problems which can not be solved exactly. Certainly, this method is performed by expanding asymptotically the dependent variables in connection with a small perturbation parameter. The idea of approximate symmetry theories originates by combining Lie group theory and perturbations. Hence, two contrasting theories of approximate symmetry have been developed. To examine approximate symmetries, the first method was exhibited by Baikov, Gazizov and Ibragimov [18,19] and another one was presented by Fushchich and Shtelen [20]. In the present work, we will use the method of Baikov et al. to study the approximate Lie symmetries and to study the approximate Noether symmetries we will use the definitions given in Sect. 2.
To analyze the 2nd order approximate Noether and Lie symmetries, we will introduce charge Q and mass M of GMGHS BH as a small perturbation parameter . To examine the approximate Lie symmetries, we will create the 2nd order perturbed geodesic equations by preserving the 2nd powers of the perturbation parameter. Firstly, we will study the exact and 1st order approximate Lie symmetries. In the absenteeism of BH parameters (Q and M), the metric (given in Sect. 3) reduces to the flat Minkowski spacetime. The exact Lie symmetries for Minkowski spacetime are calculated through Maple software (given in Appendix A). The Minkowski spacetime has 35 Lie symmetry generators which form an algebra sl(6, R). We will use these 35 symmetry generators as exact symmetries to study the approximate Lie symmetries of 1st order and 2nd order for the GMGHS BH.
It is well-known, the Minkowski spacetime is maximally symmetric as it contains 10 killing vectors. These Killing vectors give the conservation laws for angular momentum, energy, spin angular momentum and linear momentum. It has been observed that few of the conservation laws disappear, if we move from Minkowski spacetime to other curved spacetimes for example Schwarzschild, Reissner-Nordström (RN), Kerr-Newmann (KN) spacetime etc. Kara et al. used the approximate Lie symmetry technique to recuperate all the missing conserved quantities in the Schwarzschild spacetime. These conserved quantities were same as the exact symmetry generators that were vanished due to the presence of gravitational field [21]. They used the 10 killing vectors of Minkowski metric to explore the approximate symmetries and conserved quantities for the Schwarzschild spacetime. Using the same method the missing conserved quantities were also recuperated for RN [22], RN with quintessence [23,24], KN [25], KN AdS [26], BTZ [27], slowly-rotating Horava Lifshitz [28] Bardeen [29] spacetimes, and gravitational wave spacetime [30] and an important expression for their energy was obtained from 2nd order approximate Lie symmetries.
In the present paper, we will discuss the approximate Noether and Lie symmetries of 2nd order and energy content for the GMGHS BH in the Einstein frame, using approximate symmetries method. After applying the symmetry condition to the perturbed geodesic equations (given in Sect. 5) we will get a system of 70 PDEs. In the set of 70 equations the coefficient of ∂ ∂τ do not collect a re-scaling factor (discussed in Sect. 6), where τ is the proper time.
The rest of the paper is structured as follows: In Sect. 2, we summarize the definitions relating to approximate Noether symmetries. In Sect. 3 we discuss the 2nd order approximate Noether symmetries of GMGHS BH. In Sect. 4, we summarize the main formulas relating to approximate Lie symmetries to be used in the paper. In Sect. 5 we discuss the 2nd order approximate Lie symmetries of GMGHS BH. In Sect. 6, we summarized the obtained results. We take units such that 8π G = c = 1, metric signature (−, +, +, +) and Greek indices vary from 0 to 3.

Mathematical formalism of approximate Noether symmetries
For a vector field P, Noether symmetries, or symmetries of a Lagrangian, are defined as follows [31] Since the corresponding Euler-Lagrange equations are 2nd order ODEs therefore, we take the 1st order Lagrangians. In particular, we consider L(τ, x α ,ẋ α ), where " . " denotes derivative with respect to the arc length parameter τ , which gives the 2nd order DEs. Then P is said to be a Noether point symmetry of the Lagrangian if there exists a gauge function A(s, x α ,ẋ α ) such that P [1] L + (D s ξ)L = D s A, (2.2) where, and The importance of Noether symmetries is evident from the below mentioned Noether theorem [32].
is a constant of motion associated with the symmetry generator P.
For a system of 2nd order DEs, the 2nd order Noether symmetries of the 1st order Lagrangian can be defined by the following theorem [32].

7)
and the functional V dτ under the one-parameter group of transformations with approximate Lie symmetry generator

8)
with gauge function

Approximate Noether symmetries of the GMGHS charged BH
In this section we calculate the 2nd order approximate Noether symmetries of the GMGHS BH. The metric representing a 4-D charged dilatonic GMGHS BH in Einstein frame is given below [33] When Q M is sufficiently small then the solution (3.1) represents the BH with mass M and charge Q [33]. As Q → 0, the above metric represents the Schwarzschild BH which further reduces to the Minkowski spacetime in the absence of the mass M. For the value of charge Q = √ 2M the event horizon r = 2M becomes singular. Since the metric is the same as that of the Schwarzschild BH, for fixed θ and φ, therefore r = 2M represents a regular event horizon when Q < √ 2M [33]. To examine the approximate Noether symmetries of the underlying spacetime we take the mass M and charge Q of the BH in terms of the small perturbation parameter i.e. 2M = , Q 2 = k 2 and k ∈ 0, 1 4 . Hence the Lagrangian for the metric (3.1) is To calculate the 2nd order approximate Noether symmetries of the 1st order Lagrangian, we first calculate the exact Noether symmetries of the Minkowski spacetime. As , 2 → 0, the Lagrangian of the GMGHS BH reduces to the Lagrangian of the Minkowski spacetime. Therefore, using (2.11), we get the 4.3 PDEs with 6 unknown functions i.e. ξ, η 0 , η 1 , η 2 , η 3 and A. Solving this system of equations we get 17 exact Noether symmetry generators P 0 , P 1 , . . . , P 16 (given in Appendix A) which form a 17 dimensional Lie algebra. Now letting 2 → 0 in (3.3) and using 17 exact Noether symmetry generators we calculate the 1st order approximate Noether symmetries by applying the definition given in (2.12) and we get a system of 4.3 determining equations. In these Eq. 3.2 of the 4.1 constants appear which corresponds to the exact symmetry generators. After solving this system, the 15 constants disappear from the determining equations and the system of equations reduces to the system of Minkowski spacetime and hence no non-trivial 1st order approximate Noether symmetry is obtained and we recover the exact Noether symmetry generators as trivial 1st order approximate symmetries i.e. P 0 , P 1 , . . . , P 16 (given in Appendix A). Now to calculate the 2nd order approximate Noether symmetries of the Lagrangian of the GMGHS BH given in (3.3), we use the definition given in (2.13), exact (unperturbed) and the 1st order approximate Noether symmetries and we obtain a set of 4.3 determining equations again. In these Eq. 3.2 of the 4.1 constants appear. After solving this system all these constants disappear for the consistency of determining equations and the resulting system become homogeneous. Thus for the GMGHS BH, no non-trivial approximate Noether symmetry is obtained and we recover the 17 exact and 1st order approximate Noether symmetries as trivial 2nd order approximate Noether symmetries (given in Appendix A).

Mathematical formalism of approximate Lie symmetries
The method of approximate Lie symmetries of the perturbed DEs was presented by Baikov, Gazizov and Ibragimov in 1980s [18,19]. In this method an approximate generator is calculated to find the approximate solutions and the Lie operator is expanded in a perturbation series other than perturbation for dependent variables. The definition of approximate symmetries given by Baikov a 2nd order approximate Lie symmetry is a vector field for a system of 2nd order perturbed DEs if the following condition is fulfilled In (4.2) P 0 is the exact symmetry generator for the system of the exact equations Q 0 , for = 0.
Since we are determining the approximate Lie symmetries for the system of 2nd order DEs therefore, we use the 2nd prolongation of the symmetry generator P.
The 2nd prolongation for the vector field P is as follows: (4.5) In (4.2) the vector fields P 1 and P 2 are the 1st order and 2nd order approximate parts of the approximate symmetry generator P. Similarly Q 1 and Q 2 are the 1st order and 2nd order perturbed parts of the system of DEs Q, respectively. We will use the symmetry condition given in (4.4) to examine the approximate Lie symmetries of GMGHS BH (discussed in next section).

Approximate Lie symmetries of the GMGHS charged BH
In this section we examine the 2nd order approximate Lie symmetries for GMGHS BH. By introducing the perturbation parameter in Eqs. (3.1) and (3.2), we get the following perturbed metric: The GMGHS BH spacetime in the Einstein frame has four killing vectors: These killing vectors form an algebra so(3)⊕R, which corresponds to the energy-momentum conservation. Now we construct a system of 2nd order perturbed geodesic equations for the metric (3.1). By ignoring the higher powers of in the perturbed geodesic equations and preserving up to 2nd power, we obtain the system of approximate geodesic equations of order two, given below 3) It is noted, when 2 → 0, the above system of approximate equations gives the perturbed system of geodesic equations for the Schwarzschild spacetime [21]. This system further gives the system of equations for the Minkowski metric as → 0. It is clearly understood that to examine the 2nd order approximate Lie symmetries for the above system of Eqs. ((5.3)-(5.6)), we need to study the exact and 1st order approximate symmetries first for the unperturbed and the 1st order perturbed system of equations. By letting → 0, 2 → 0, in the above system, we get the unperturbed equations of the Minkowski metric. For this system the symmetries (exact) are calculated using Maple software and we get 35 exact symmetry generators (given in Appendix A). Out of these exact 35 symmetries, 12 symmetries ( P 0 , P 1 , . . . , P 11 ) form an algebra so(1, 3) ⊕ R ⊕ d 2 .
In the isometry algebra of Minkowski metric, so(1, 3) is isomorphic to so(3) ⊕ so (3), which corresponds to the conservation of energy, angular momentum, linear momentum and spin angular momentum while d 2 = P 10 = ∂ ∂τ , P 11 = τ ∂ ∂τ corresponds to the dilation algebra. Now to study the 1st order approximate Lie symmetries, we let 2 → 0, in the above system of equations and obtain the 1st order perturbed system of equations. Applying the definition of symmetry given in (4.4), to the set of Eqs. ((5.3)-(5.6)), we obtain the following equations P [2] where j = 1, 2, 3, 4. Now using the exact 35 symmetry generators, geodesic Eqs. ((5.3)-(5.6)) and prolongation coefficients and retaining terms with only we find a set of determining equations. In the set of determining equations 30 constants appear. It is observed that for consistency of the determining equations these constants disappear. With the disappearance of these 30 constants, the system of determining equations becomes homogeneous i.e. it transforms to the system of Minkowski spacetime (exact case) and its solution is given in Appendix A. Hence, in 1st order approximation, (when we retain terms with ), no non-trivial approximate symmetry generator is obtained. Now to examine the 2nd order approximate Lie symmetries we again apply the symmetry definition given in (4.4), to the set of Eqs. ((5.3)-(5.6)), and get the following equations P [2] Now making use of the exact and the 1st order approximate Lie symmetries and geodesic equations in the above set of equations and keeping terms with 2 , we find a system of determining equations again. Like in the case of first approximation here we also get 30 constants in the set of determining equations that disappear for the consistency of the determining equations and the system reduces to the system of equations in the exact case. Therefore, in the 2nd approximation, when we keep terms quadratic in , the new symmetry generator is not obtained and we get the same 35 symmetry generators given in Appendix A as trivial approximate symmetry generators.
According to Neother's theorem [32] the conservation of energy is connected to time translational symmetry. Equation (4.1) gives an impression that ξ is the coefficient of ∂/∂τ . It is worth noting that in the set of determining equations for the 1st and 2nd order approximation of the GMGHS BH, the terms containing ξ τ = k 0 , cancel out automatically without collecting any re-scaling factor. Hence, like the other spacetimes [22][23][24][25][26][27][28][29], here we do not get any energy re-scaling factor. Therefore, we can not conclude any result about the energy of this stringy-charged BH using the approximate Lie symmetry methods.

Summary
In this paper, we have discussed the energy content, 2nd order approximate Noether and Lie symmetries of the GMGHS charged BH in Einstein frame, using approximate symmetry technique. Firstly, we have studied the approximate Noether symmetries of the Lagrangian of the GMGHS BH. For this, we have taken the BH parameters i.e. charge Q and mass M in connection with a small perturbation parameter. In the absence of and 2 , the Lagrangian given in (3.3) reduces to the Lagrangian of Minkowski spacetime and as k → 0 and 2 → 0, the Lagrangian reduces to the Lagrangian of the Schwarzschild spacetime. To calculate the approximate Noether symmetries of order two, we have first calculated the exact Noether symmetries of the Minkowski spacetime, using definition (2.11) and obtained 17 exact Noether symmetries. Out of these 17 Noether symmetries, 10 are Killing vectors which correspond to Law of conservation of energy-momentum, linear momentum and spin angular momentum. Using these exact Noether symmetries and the definition given in (2.12), we have examined the 1st order approximate Noether symmetries. It has been seen that for the 1st order approximate case, no new Noether symmetry is obtained and we have recovered the 17 exact symmetries as trivial approximate Noether symmetries of order one. Finally, we have calculated the 2nd order approximate Noether symmetries of the Lagrangian (3.3), by applying the definition given in (2.13). Using the exact and 1st order approximate Noether symmetries in the determining equations, we have calculated the approximate Noether symmetries of order two and recovered the exact Noether symmetries as trivial approximate Noether symmetries of order two.
To study the approximate Lie symmetries, we have constructed a perturbed system of 2nd order geodesic equations. In set of 2nd order perturbed geodesic equations, we have retained the 2nd powers of only. This system of perturbed equations can be reduced in to a system of Minkowski spacetime and Schwarzschild spacetime in the required limits i.e when → 0 and 2 → 0. Next, we have studied the exact symmetries for the exact system of Eqs. (5.3)-(5.6). The exact symmetries are calculated through Maple software and we have obtained 35 symmetry generators (given in Appendix A). Out of these 35 symmetry generators, 10 are the Killing vectors which correspond to the energy and momentum conservation, linear momentum conservation and spin angular momentum conservation. Now using these exact symmetries, we have explored 1st order approximate Lie symmetries for the system of 1st order perturbed geodesic equations. It has been noted that there do not exist any new symmetry generator and we have recovered all the 35 exact symmetries as trivial approximate Lie symmetries of order one. Finally, we have examined 2nd order approximate Lie symmetries for the 2nd order perturbed system of geodesic equations using the symmetry condition for DEs, 2nd prolongation of symmetry generator, exact and 1st order approximate Lie symmetries. In 2nd approximation, when we retain terms up to 2 , we do not obtain any new symmetry generator and recovered all the exact symmetries as trivial 2nd order approximate symmetries. It has been observed that in the set of determining equations for the 1st and 2nd order approximation of the GMGHS BH, the terms containing ξ τ = k 0 , cancel out automatically. This result implies that in the second approximation, we have not found any re-scaling factor. It is clear from (5.1), the term which we have introduced at 2nd order of the perturbation parameter (i.e. Q 2 = k 2 ), is appearing in the metric coefficient g θθ and g φφ . While in rest of the other curved spacetimes [22][23][24][25][26][27][28][29], where the energy re-scaling factor has been seen, this term has appeared in the metric coefficient g tt . In the GMGHS BH spacetime no energy re-scaling factor has been seen and hence we can not conclude any result about the energy content of the underlying spacetime through approximate Lie symmetries. Also, in the GMGHS charged BH the metric coefficient g tt is independent of the charge Q and no 2 is appearing in the system of perturbed geodesic equations, which is actually responsible for the energy re-scaling factor. Therefore, this could be the reason of not getting an energy re-scaling factor using approximate Lie symmetry technique in the present study.
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