Complexity Conjecture of Regular Electric Black Holes
aa r X i v : . [ g r- q c ] F e b Complexity Conjecture of Regular Electric Black Holes
B. Bahrami Asl , , S. H. Hendi , , and S. N. Sajadi , Department of Physics, School of Science, Shiraz University, Shiraz 71454, Iran Biruni Observatory, School of Science, Shiraz University, Shiraz 71454, Iran Canadian Quantum Research Center 204-3002 32 Ave Vernon, BC V1T 2L7 Canada
Recently, the action growth rate of a variety of four-dimensional regular magnetic black holes in F frame is obtained in [1]. Here, we study the action growth rate of a four-dimensional regularelectric black hole in P frame that is the Legendre transformation of F frame. We also investigatethe action growth rates of the Wheeler-De Witt patch for such black hole configurations at the latetime and examine the Lloyd bound on the rate of quantum computation. We show that althoughthe form of the Lloyd bound formula remains unaltered, the energy modifies due to a non-vanishingtrace of the energy-momentum tensor and some extra terms may appear in the total growth action.We also investigate the asymptotic behavior of complexity in two conjectures for static and rotatingregular black holes. PACS numbers:
I. INTRODUCTION
AdS/CFT correspondence [2–4] is a leading formalism providing a consistent relation between gauge and gravitationtheories and has given us a rather deep insight on possible unification of them. Such a correspondence captures anequivalence between the conformal field theory and its dual theory of gravity in AdS space with a strong/weakduality point of view. It is believed that it provides a non-perturbative formulation access to strongly coupled regimesof quantum field theories which cannot be accessed by the traditional perturbation approach. Indeed, AdS/CFTcorrespondence is one of the important examples of entanglement between gravity and quantum information [5–8].Complexity has recently been introduced as a complement to entanglement providing additional information on thequantum properties of black holes.In order to calculate the holographic duality of quantum computational complexity one may follow two proposals.The complexity = volume (CV) [9, 10] and complexity = action (CA) [11, 12] conjectures. The former one states thatthe complexity of the boundary state is proportional to the maximum volume of codimension hypersurface boundedby the CFT slices and the latter conjecture specifies that complexity on the boundary CFT is proportional to theon-shell action of the Wheeler-DeWitt(WDW) patch in the bulk. According to these conjectures, Lloyd showed thatthe growth rate of complexity for the Schwarzschild like black holes is bounded by the total energy of the system [13].Cai, et.al proved that the so-called Lloyd bound should be modified for charges or rotating black holes [14]. Althoughthere are many cases of black holes that confirm the mentioned upper bound [15–21], its validity is an open questionsince there are different types of black hole violating the Lloyd bound [22–24]. Accordingly, the validity of CV andCA conjectures and their possible modifications deserve further study.In this direction, here, we focus on the holographic complexity of nonlinearly charged regular black holes in P frame. Indeed, the motivation for studying this black hole is three folds: regularity, nonlinearity and complexity. Thefirst item is the regularity of the black hole with known action, exactly. The second one is nonlinearity which has thekey role to avoid the singularity. Although such a nonlinear electrodynamics is an ad hoc model near the black hole,it behaves like the Maxwell field far from the horizon radius. The third one is the complexity of black holes which isreasonable for regular black holes since the singularity may violate the interpretation of physical quantities such asentangled entropy and complexity.The subject of singularity itself and its possible solution (leading to regularity) are highly interesting from differentviewpoints: mathematical and geometrical points of view, classical gravity and its quantum standpoints, cosmologicalside, effective field theory and AdS/CFT correspondence features, string theory and supergravity aspects, etc. Indeedthere are two important singularities that their natures are not yet fully understood: the singularity that is coveredby an event horizon of black holes and the big bang singularity. So one of the substantial questions is how one canprevent the singularity. Although it is believed that quantum gravity can be able to smooth out singularities, thereis no consistent theory of quantum gravity overcoming this issue despite many attempts. Thus, the question shouldagain be asked how we can preclude the singularity in an alternative theory.Regarding the Einstein theory of gravity coupling minimally with an appropriate nonlinear electrodynamics theory,various regular black hole metrics are constructed (for an incomplete list, please take a look at [25–31]). The exactregular solution of f ( R ) gravity has been studied in [30]. Gravitational lensing [32], dynamical [33] and thermody-namical stability [34] of regular black hole has been studied. A new Smarr-type formula for the black hole in nonlinearelectrodynamics has been obtained in [35, 36]. In this paper, we concentrate on the complexity of a simple static androtating regular black hole with nonlinear electrodynamics as a source. The rest of the paper is organized as follows.In Sec. II, we briefly review the complexity and thermodynamics of regular black hole, especially deriving its Smarrformula. Section III is devoted to study the complexity of large-extremal-static regular black hole in the framework ofCA and CV conjectures and comparing the results in two frameworks. In Sec. IV, we repeated the same computationof complexity for the large-extremal rotating regular black holes. After that, conclusions are given. In the appendix,we also investigate two other regular cases and calculated thermodynamic quantities, complexity and complexity offormation numerically. II. BASIC FORMALISM
Electrically charged black hole solutions can be studied through an alternative form of nonlinear electrodynamicsobtained by the Legendre transformation. The action of nonlinear electrodynamics in P frame minimally coupled tothe Einstein gravity with the York-Gibbons-Hawking surface term and the two-dimensional joint term is given by A = 14 π Z d x √− g (cid:20)
14 (
R − − (cid:18) P µν F µν − H ( P ) (cid:19)(cid:21) + 18 π Z d x √− h K + 18 π Z d x √− γ a, (2.1)where g is the determinant of the metric, R is the Ricci scalar and the negative cosmological constant is denotedby Λ = − l . In addition, the anti-symmetric tensor P µν = L F F µν in which L F ≡ ∂ L ∂ F , F = F µν F µν and F µν isthe Faraday tensor. Furthermore, the structure function is H ( P ) = 2 FL F − L with P ≡ P µν P µν = FL F since d H = L − F d (cid:0) FL F (cid:1) = H P d P , where H P = d H d P = L − F . Moreover, h and γ are, respectively, the determinant of theinduced metrics h µν in three-dimensions and two-dimensional γ µν . Also, K is the trace of the extrinsic curvature ofthe induced metric h µν K = n µ ; µ = 1 √− g ∂ µ ( √− gn µ ) , (2.2)where n µ is the normal vector. The integrant a is defined as a = ln (cid:18) − N. ¯ N (cid:19) , in which N is the future-directed null normal to the left-moving null surface and ¯ N denotes the future-directed nullnormal to the right-moving null surface.The first integral of Eq. (2.1) represents the bulk action while the second and third ones stand for the boundaryand joint parts of the WDW patch [37]. The WDW patch is used to obtain the rate of temporal change of the action.A typical WDW patch of a black hole with two horizons is shown in Fig. 4 which is evolved in time from t to t + δt .According to this figure, the contribution of different parts is bulk region V and V , and null-null surface joints A , B , C and D (for more details, see [37]). Hence, we can write ∂ A = 14 πG Z V √− g L dtdrdθdφ − πG Z V √− g L dtdrdθdφ + 18 πG Z B √− γ a B dθdφ − πG Z A √− γ a A dθdφ + 18 πG Z D √− γ a D dθdφ − πG Z C √− γ a C dθdφ , (2.3)where L is the Lagrangian of Einstein-nonlinear electrodynamics. In order to compute the contribution of joints, weuse the following transformation of N and ¯ NN α = − b ∂ α ( t − r ∗ ) , ¯ N = b ∂ α ( t + r ∗ ) , (2.4)in which b and b are two arbitrary positive constants and the tortoise coordinate r ∗ is defined as r ∗ = R drf ( r ) . So,the contribution of joints is calculated as S B − S A = δt (cid:18) rf ( r ) (cid:20) ln (cid:18) f ( r ) b b (cid:19) + rf ′ ( r )2 f ( r ) (cid:21)(cid:19) r B r A , (2.5) (a) (b) FIG. 1: Closed hypersurfaces consisting of a past spacelike surface B , a truncated null cone N and a future spacelike surface A . (a) Same direction of normal vectors, (b) Opposite direction of normal vectors. S D − S C = δt (cid:18) rf ( r ) (cid:20) ln (cid:18) f ( r ) b b (cid:19) + rf ′ ( r )2 f ( r ) (cid:21)(cid:19) r A , (2.6)where prime denotes derivative with respect to r . In Fig. 1, we assume close cylindrical hypersurfaces consisting ofa past spacelike surface B , a truncated null cone N , a future spacelike surface A and boundaries for intersection ofspacelike and null surfaces x and y .The contribution of surfaces A and B canceled each other in Fig. (1a) due to the same direction of normal vectors.While for Fig. (1b) according to opposite direction of normal vectors, contributions of boundaries of intersection ofsurfaces, x and y cancel each other. Therefore depending on what we choose for direction of normal vector, boundaryterms or joint contributions will be vanished [37].It is notable that applying the variational principle to the action (2.1), one finds the field equations as G νµ + Λ δ νµ = T νµ = 12 (cid:2) H P P µλ P νλ − δ νµ (2 PH P − H ) (cid:3) , (2.7) ∇ ν P µν = 0 , (2.8)where G µν is the Einstein tensor. By integrating equation (2.8) with the assumption of spherical symmetry spacetime,we obtain the following static solution ds = − f ( r ) dt + dr f ( r ) + r d Ω , f ( r ) = 1 − r Z ρ ( r ) r dr, (2.9) P µν = 2 δ t [ µ δ rν ] qr , or P = − q r , (2.10)where q is an integration constant and H = 2 ρ ( r ).In the following, for the importance of Smarr relation, we will obtain it for the case of spherically symmetric staticcharged regular black holes. In order to do that we use the Komar formula for the mass of black holes as follows M = − π I ∞ ∇ α ξ β dS αβ , (2.11)where ξ is a timelike Killing vector which satisfies Killing equation and dS αβ is a two-dimensional surface element ofthe boundary at infinity. Since the spacetime has two boundaries, we will write the Komar integral for the mass as asum of an integral over a closed null surface at the horizon H and an integral on the spacelike hypersurface Σ whichis bounded by the horizon and infinity as M = − π I H ∇ α ξ β dS αβ − π Z Σ ∇ β ∇ α ξ β d Σ α . (2.12)For the first term, we have − π I H ∇ α ξ β dS αβ = κA π , (2.13)where A is the horizons surface area, κ denotes the surface gravity that is constant at the event horizon and satisfies ξ α ; β ξ α = κξ β . In order to evaluate the second term in equation (2.12), we consider the Stokes theorem for anantisymmetric tensor field B αβ as follows I S B αβ dS αβ = 2 Z Σ B αβ ; β d Σ α . (2.14)For the antisymmetric tensor B αβ = ∇ α ξ β of the bulk term Σ, we have B αβ ; β = ( ∇ α ξ β ) ; β = − ( ∇ β ξ α ) ; β = − (cid:3) ξ α , (2.15)where (cid:3) = ∇ α ∇ α . Recalling that ∇ ρ ∇ µ ξ ν = R νµρσ ξ σ , we get I S ∇ α ξ β dS αβ = − Z Σ R αβ ξ β d Σ α . (2.16)Now, by using the Einstein field equation R αβ = 4 π (2 T αβ − g αβ T ), we obtain I S ∇ α ξ β dS αβ = − π Z Σ (cid:18) T αβ − δ αβ T (cid:19) ξ β d Σ α . (2.17)On the other hand, considering equation (2.7), the nonzero components of energy-momentum tensor are T tt = T rr = − H π , T θθ = T φφ = 18 π (2 PH P − H ) . (2.18)By rewriting the RHS of Eq. (2.17), one can obtain − π Z Σ (cid:18) T αβ − δ αβ T (cid:19) ξ β d Σ α = − π Z Σ (cid:18) T αβ − δ αβ T (cid:19) ξ β d Σ α + 4 π Z Σ T δ αβ ξ β d Σ α = − Z Σ qr Ed Σ α + 4 π Z T √− hdrdθdφ. (2.19)since the timelike Killing vector is ξ β = δ βt and E = − PH P with E = q/r H P identically. Finally, by insertingabove results in Eq. (2.12), we obtain M = κA π + q Φ − Z T √− hdrdθdφ, (2.20)where h is the trace of the induced metric of spacelike hypersurface. III. THE COMPLEXITY GROWTH OF REGULAR ELECTRIC BLACK HOLES
Here, we consider a class of known regular electric black holes in AdS spacetime and calculate its complexity growthrate. For the sake of completeness, we will point out other regular black hole solutions in the appendix. + q FIG. 2: The behavior of q in terms of r + for M = l = 1. dotted line (first case), dashed line (second case) and black line (thirdcase). The motivations of considering the forthcoming kind of H function are its simplicity and its agreement to thecorrespondence which indicates such a modified nonlinear theory should reduce to the usual Maxwell theory in theweak-field limit. The ansatz of structure function which is inspired by the exponential distribution is defined as [38] H = P e − γ
32 ( −P ) 142 54 χ ! + 6 l , (3.1)in which for the weak field limit ( P ≪
H ≈ P + 6 l + O ( P ) . (3.2)Considering the mentioned structure function with Eqs. (2.10) and (2.9), we can find the following sphericallysymmetric exponentially black hole solution [34, 38–41] − g tt = f ( r ) = 1 + r l − χr e (cid:16) − γ χr (cid:17) . (3.3)Using the series expansion of this solution for large values of r , it is noticeable that its asymptotical behavior canbe found by the following expression − g tt ≈ r l − χr + γ r + O (cid:18) r (cid:19) , (3.4)which is the Reissner-Nordstr¨om-AdS metric function provided χ and γ are associated with the mass and electriccharge of the system, respectively. In other words, the asymptotical behavior of the obtained solution is completelymatched to the Reissner-Nordstr¨om-AdS black hole.In order to check the first law and the Smarr formula in the extended phase space, it is convenient to considerthe cosmological constant as a varying thermodynamic quantity and interpret it as a thermodynamic pressure, as P = πl . In addition, its conjugate variable of the introduced pressure is the thermodynamic volume V = 4 π r , (3.5)where r + is the event horizon radius obtained via f ( r + ) = 0. The temperature would be found straightforwardlythrough the use of surface gravity ( κ ) interpretation with the following explicit relation T = κ π = 14 π df ( r ) dr (cid:12)(cid:12)(cid:12)(cid:12) r = r + = (cid:18) m − q r + (cid:19) e − (cid:16) q mr + (cid:17) πr + r + πl , (3.6)in which for the large black hole event horizon ( r + ≫ l ) becomes T ≈ r + πl + O (cid:18) r + (cid:19) . (3.7)Since we are working in the Einstein gravity, the black hole entropy S pursues the area law, yielding S = A πr . (3.8)In addition, the electrostatic potential Φ can be obtained at the event horizon versus spatial infinity as the referenceΦ = Z ∞ r + Edr = 3 m q − (6 mr + − q ) e − (cid:16) q mr + (cid:17) qr + , (3.9)and the asymptotic limit becomes Φ ≈ qr + + O (cid:18) r (cid:19) . (3.10)For a black hole embedded in AdS spacetime, employing the relation between the cosmological constant and thermo-dynamic pressure would result to interpret the mass of black hole as the enthalpy. The enthalpy can be written interms of thermodynamic quantities as H = m = q r + W (cid:16) q l r ( l + r ) (cid:17) , (3.11)where W is the Lambert W function. In the case of large black hole event horizon, we have H ≈ r l + r + O (cid:18) r + (cid:19) . (3.12)It is straightforward to check that these thermodynamic quantities satisfy the first law of black hole thermodynamicsin the enthalpy representation dH = T dS + V dP + Φ dq, (3.13)and the Smarr relation is given by H P V − T S − q Φ2 + 14 Z wdv = 0 , (3.14)where the last term of Eq. (3.14) comes from the fact that the energy-momentum tensor is not traceless, as Z wdv = 12 Z ∞ r + T µµ r dr = m − (cid:18) m + q r + (cid:19) e − (cid:16) q mr + (cid:17) , (3.15)and in the case of r + ≫ l becomes Z wdv ≈ q l r + O (cid:18) r (cid:19) . (3.16) FIG. 3: The left hand side of Eq. (3.14) versus r for M = 0 . , q = 0 . , l = 1 . Now, we want to calculate the complexity growth rate in the above background. For this aim, we should calculatethe Ricci scalar for the solution (3.3) as R = − l + q e − (cid:16) q mr (cid:17) mr , (3.17)whereas by using of Eq. (3.1), the Lagrangian of nonlinear electrodynamics is given by L = − q (4 mr − q ) e − (cid:16) q mr (cid:17) mr − l . (3.18)Using Eqs. (3.17) and (3.18), we can calculate the bulk action growth as d A bulk dt = 14 Z r + r − r (cid:18) R + 6 l − L (cid:19) dr = m (cid:20) e (cid:16) − q mr (cid:17) (cid:21) r + r − , (3.19)while the growth rate of the surface term (Eq. (2.1)) is given by [1] d A boundary dt = 18 π Z ∂ M ( √− h K ) d Ω = 12 (cid:20)p f ( r ) ∂∂r ( r p f ( r )) (cid:21) ∂ M (3.20)= (cid:20) r l + r − (cid:18) m q r (cid:19) e (cid:16) − q mr (cid:17) (cid:21) r + r − , where we should note that in the calculation of boundary term we have used √− h = √ f r sin( θ ) and n µ = (0 , √ f , , S D − S C = δt (cid:18) rf ( r ) (cid:20) ln (cid:18) f ( r ) b b (cid:19) + rf ′ ( r )2 f ( r ) (cid:21)(cid:19) r C . (3.21)After some manipulations, we can find S B − S A = δt me − ( q mr ) − q e − ( q mr ) r + 2 r l + (cid:18) r − me − ( q mr ) + 2 r l (cid:19) ln − mr e − ( q mr ) + r l b b r A , (3.22) S D − S C = δt me − ( q mr ) − q e − ( q mr ) r + 2 r l + (cid:18) r − me − ( q mr ) + 2 r l (cid:19) ln − mr e − ( q mr ) + r l b b r C . (3.23)Since at the late time r A and r C approach to r − and r + , respectively, while f ( r ) vanishes, one finds Eqs. (3.22) and(3.23) reduce to S B − S A = δt me − (cid:16) q mr (cid:17) − q e − (cid:16) q mr (cid:17) r + 2 r l r A , (3.24) S D − S C = δt me − (cid:16) q mr (cid:17) − q e − (cid:16) q mr (cid:17) r + 2 r l r C . (3.25)So, the total growth rate of the action for such a black hole configuration within WDW patch at late time approx-imation is simplified as d A dt = ( r + − r − ) (cid:20) l ( r + r − + r + r − ) (cid:21) + (cid:18) q r − + m (cid:19) e (cid:16) − q mr − (cid:17) − (cid:18) q r + + m (cid:19) e (cid:16) − q mr + (cid:17) . (3.26)In order to describe Eq. (3.26) in terms of r − and r + , we can use the redefinitions m and q in terms of r − and r + as m = r − ( r − + l ) ln( A ) A (cid:16) r + r + − r − (cid:17) r + − r − ) l W ln( A ) A (cid:18) r + r + − r − (cid:19) r − ( l + r − ) r + ( r + − r − )( l + r ) , q = ln( A ) r − ( l + r − ) r + ( r + − r − ) l e r + ln( A ) r −− r + , (3.27)where A = r + ( l + r ) r − ( l + r − ) . (3.28)Although it is straightforward to rewrite Eq. (3.26) in terms of r − and r + , we ignore its explicit relation for the sakeof brevity. Finally, by using the above calculated thermodynamic quantities, it is easy to show that d A dt ≤ (cid:18) m − qφ + 12 Z wdv (cid:19) + − (cid:18) m − qφ + 12 Z wdv (cid:19) − , (3.29)indicating that the action growth rate of the black hole in the WDW patch has been bounded. For instance, for thevalues m = 0 . , q = 0 . , l = 1 this bound is about 1 . . A. Extreme case: r + ≈ r − Now, we focus on the extremal solutions. By introducing α = l/r + , ǫ = 1 − r − /r + and in the case of extremal,large black hole, equation (3.26) becomes d A dt ≈ (cid:18) r + α − r + α + 7 r + (cid:19) ǫ + O ( ǫ ) + O ( α ) , (3.30)in terms of r ± leads to d A dt ≈ l ( r − r − r ) + 74 ( r + − r − ) + O (cid:18) r + (cid:19) . (3.31)By using of equations (3.5), (3.12) and ( r + ≈ r − ) one can rewrite above equation as follows [42] d A dt ≈ P ( V + − V − ) , r + ≫ l, (3.32)which shows that in the large black holes the rate of complexity is proportional to P ∆ V , i.e, is controlled by thermo-dynamical volume. Besides, it is notable that one can obtain such a relation in terms of entropy for the static blackholes since in the static case the thermodynamic volume is not independent of entropy ( V = 4 / (3 √ π ) S / ).In the case of extremal black hole with small degenerate horizon radius, we have d A dt ≈ (cid:18) α + 3 (cid:19) ǫ O (cid:18) α (cid:19) + O ( ǫ ) . (3.33)By using of CV conjecture, the rate of complexity is [43] d C V dt = 4 πGl p − f ( r min ) r min , (3.34)where r min is the turning point of maximal surface. The late time limit of d C V /dt becomes d C V dt = 4 πGl p − f (ˆ r min )ˆ r min , t −→ ∞ , (3.35)where ˆ r min is the extreme value point of p − f ( r min ) r min . For the present case we need to obtain the zeros of thefollowing equation l (6 M ˆ r min + q ) exp (cid:18) − q M ˆ r min (cid:19) − r min (2 l + 3ˆ r min ) = 0 . (3.36)in the case of small q , it becomesˆ r min ≈ r min + εr min = r + q l ε M l − r − r l , (3.37)where r = (108 M l + 4 √ l + 729 l M ) − l (108 M l + 4 √ l + 729 l M ) . (3.38)In the case of extremal black hole with large horizon radius, we can writeˆ r min r + ≈ . − . ε + O ( ε, α ) , (3.39)and correspondingly, for the late time of growth of complexity, we obtain d C V dt ≈ πα ( r + − r − ) ≈ P ∆ V. (3.40)The comparison of the complexity growth from the CV and CA dualities are R rate = d C A /dtd C V /dt = 9 P ∆ V P ∆ V = 0 . . (3.41)0 FIG. 4: Penrose diagram for charged black holes.
B. Complexity of formation
The complexity of formation is the difference between complexity in the process of forming the entangled TFDstate and preparing two individual copies of the vacuum state of the left and right boundary CFTs [44, 45]∆ C A = 1 π ( A BH − A AdS ) . (3.42)For the case of charged black hole with two horizons, the complexity of formation in CA conjecture becomes [44]∆ C A = 1 π (∆ A bulk + A joint,meet ) . (3.43)Now, we evaluate the action for the obtained black hole solutions. The tortoise coordinate for the both horizons ofthe solution is r ∗ ( r ) = l r + ln (cid:18) | r − r + | r + r + (cid:19) r + l − ( l + r ) W (cid:18) l q r ( l + r ) (cid:19) + l r − ln (cid:18) | r − r − | r + r − (cid:19) r − + l − ( l + r − ) W (cid:18) l q r − ( l + r − ) (cid:19) , (3.44)The point where the ingoing null rays from the two asymptotic regions meet inside the black hole between the twohorizons ( r − < r meet < r + ) can be calculated numerically using Eqs. (3.44)-(6.41) which reads r ∗ ( r meet ) = 0 . (3.45)1 FIG. 5: The behavior of r meet in terms of r + (left) and q (right) for M = l = 1. red line (first case), black line (second case)and blue line (third case). We show the results for r meet in Fig. 5 for the considered solution. According to the Fig. 5a, by increasing eventhorizon radius, r meet decreases from extremal value to r − . Then, by increasing the electric charge, r meet increases(Fig. 5b). By using of numerical curve fitting and choosing the appropriate branches of the logarithms, one canobtain the following relation for r meet ( r + ) r meet ≈ . r − . r + + 1 . , (3.46)The above approximation functions are necessary to obtain the A bulk and A joint in the following.The bulk action of the black hole is A bulk,BH =4 4 π πG Z r max r meet (cid:18) R + 6 l − L (cid:19) dr Z v ∞ − r ∗ ( r )0 dt = 2 q G Z r max r meet dr exp (cid:18) − q M r (cid:19) r ( v ∞ − r ∗ ( r )) ≈ . r Gl − . r + Gl + 0 . r + Gδl + O (1) , (3.47)where v ∞ = lim r →∞ r ∗ ( r ) is the constant defining the future null boundary, r max = l δ − δ + Mδ l − q δ l and δ is asome cutoff surface associated with a UV divergence. In order to obtain the action of AdS spacetime, one uses f AdS = 1 + r l , (3.48)in which the tortoise coordinate for AdS spacetime is r ∗ ( r ) = l arctan (cid:16) rl (cid:17) , v ∞ = πl . (3.49)Thus the bulk action of AdS vacuum is A AdS = 4 4 π πG Z r max dr (cid:18) R + 6 l (cid:19) Z v ∞ − r ∗ ( r )0 dt = − Gl Z r max dr ( v ∞ − r ∗ ( r )) = − G (cid:20) − l ln (cid:18) r max l (cid:19) + r max (cid:18) πr max l − r max l arctan( r max l ) (cid:19)(cid:21) , (3.50)where R ads = − /l and r max = l δ − δ . Now, we should calculate the following relation for the solution∆ A bulk = A bulk,BH − A AdS . (3.51)2The action at the meet point r meet is A joint = 2 G r meet ln[ f ( r meet )] , (3.52)which in the case of large black hole becomes A joint ≈ . (cid:18) . r l (cid:19) r + O ( r ) . The total action is the sum of the bulk (3.51) and joint (3.52) terms. Substituting the numerical solution for r meet ,we obtain the interesting result plotted in Fig. 6. According to these figures, one finds the bulk action of black holeand complexity of formation are linear functions of entropy. For the large black hole, we have∆ C A ≈ . C T Sl + O ( S / ) , (3.53)where C T = 3 l /π G is the boundary central charge.Here, we want to obtain the complexity of formation by using of CV conjecture. To do that, we consider themaximal volume functional for the t = 0 timeslice (the straight line connecting the two boundaries through thebifurcation surface in the Penrose diagram) as follows [44, 46] V = 8 π Z r max r + dr r √ f = 4 πlr + 2 πl + πlδ πl ln l p δr + ! + O (cid:18) l r (cid:19) , (3.54)where the corresponding contribution from two copies of the vacuum AdS background is V AdS = 8 π Z r max dr r √ f AdS = 4 πl " r max p l + r max − l ln r max + p l + r max l ! = 4 πl δ − πl ln (cid:18) lδ (cid:19) − πlδ . (3.55)The complexity of formation is obtained by subtracting from (3.54) the volume of the AdS vacuum∆ C V = V − V AdS G N R = 8 πG N R "Z r max r + r dr p f ( r ) − Z r max r dr √ f AdS = 41 . C T Sl + O ( S / ) . (3.56)To understand the behaviour of complexity of formation, we obtain the plot of ∆ C V as a function of r + in Fig. 6.It is interesting, of course, to compare the results of CA and CV duality. The ratio of the complexity of formationfor large black holes r + ≫ l is R form = ∆ C A ∆ C V ≈ . , (3.57)and by using (3.41) and (3.57) one gets R rate − R form = 0 . . (3.58)As can be seen, the differences between two ratios is small. So, the two ratios agree very well and this comparisonsuggests that the two holographic approaches to complexity are consistent. IV. GENERALIZATION TO ROTATING SOLUTION:
The metric of rotating charged regular black hole in the Boyer-Lindquist coordinates is obtained as dS = − ∆ r Σ (cid:18) dt − a sin ( θ )Ξ dφ (cid:19) + Σ∆ r dr + Σ∆ θ dθ + ∆ θ sin ( θ )Σ (cid:18) adt − r + a Ξ dφ (cid:19) , (4.1)where ∆ r = ( r + a )(1 + r l ) − f, Σ = r + a cos ( θ ) , Ξ = 1 − a l , ∆ θ = 1 − a l cos ( θ ) , f ( r ) = M r exp (cid:18) − q M r (cid:19) . (4.2)3 r + D C V r + D C A FIG. 6: Plot of ∆ C V (up) and ∆ C A (down) in terms of r + for static exponential solution.. The outer and inner horizons are determined by the equation ∆( r ± ) = 0, respectively. The required thermodynamicalquantities to check the first law are [34] P = 38 πl , V = r + A πJ M , A = 4 π ( r + a )Ξ , J = M a Ξ , M = M Ξ , Q = q Ξ ,T = − (2 M r + + q ) e − q M r + π ( r + a ) r + + r + ( r + 2 r + a )2 πl ( r + a ) , Φ = qr + r + a , Ω = ar + a (cid:18) r l (cid:19) . (4.3)where M = q r + W (cid:18) l q ( r + a )( r + l ) (cid:19) . (4.4)So, the first law of thermodynamic is satisfied as [34] d M = T dS + P dV + Φ dQ + Ω dJ, (4.5)4Integrating the on-shell Einstein-Hilbert bulk action, directly, in the case of qM ≪ al ≪ d A bulk dt = 18 Z r + r − Z π √− g (cid:18) R + 6 l − L (cid:19) drdθ = − (cid:20) M r − M l q r + 9 l q r + 24 M a r + 24 M a q l r − a l q ml r (cid:21) r + r − , (4.6)since √− g = sin θ Ξ Σ and L = 2 q e − q M r (( M r + q ) a cos θ + ( − M r + q ) a r cos θ + M r ) M r ( r + a cos θ ) . (4.7)The growth rate of the surface term in the case of qM ≪ al ≪ d A boundary dt = 18 π Z ∂ M ( √− h K ) d Ω = " ∆ ′ r r + r − = (cid:20) − M r l + 8 M r ( l + a + 2 r ) + q l M l Ξ r (cid:21) r + r − , (4.8)since the determinant of the induced metric on the null hypersurface r ± is h = − sin θ Σ∆ r Ξ and the trace of extrinsiccurvature can be calculated from Eq. (2.2) and the normal vector is n µ = (0 , q ∆ r Σ , , d A dt = (cid:20) M q l r − q l r − M a q l r + 11 a q l − M l r + 24 M l r + 24 M r M l r (cid:21) r + r − , (4.9)where q = r + ( a l + a r − + r − l + r − ) ln (cid:18) r − ( l + r )( a + r ) r + ( a + r − )( l + r − ) (cid:19) ( r + − r − ) l (cid:18) r − ( l + r )( a + r ) r + ( a + r − )( l + r − ) (cid:19) r + r −− r + . In the case of α ≪ β ≪ ǫ ≪
1, one finds q /r = (cid:18) . α − .
43 + (cid:18) − . α + 140 . (cid:19) β (cid:19) + (cid:18) − . α + 100 .
43 + 100 . β α (cid:19) ǫ + O ( ǫ ) + O ( α ) + O ( β ) , (4.10)while for α ≫ β ≪ ǫ →
0, we have q /r = 2 .
72 + 13 . α − (cid:18) .
15 + 29 . α (cid:19) β − (cid:18) .
72 + 27 . α − . β α (cid:19) ǫ + O ( ǫ ) + O ( β ) + O (1 /α ) . (4.11)In addition, in the case of α ≪ β ≪ ǫ →
0, we obtain d A dt ≈ . r l − . r − r l + O ( r + ) , (4.12)and for α ≪ β ≪ ǫ →
1, one achieves d A dt ≈ . r + α β ǫ + O ( ǫ − ) , (4.13)5since V /r ≈ . × − α β ǫ , ǫ → ,V /r ≈ .
13 + 560 . α , ǫ → . (4.14)As can be seen in both cases ǫ → ,
0, the rate of complexity proportional to the volume of black holes instead ofentropy of black holes.
A. Complexity of formation
In this section complexity of formation for rotating black holes in the limit al ≪ qM ≪ al ≪ qM ≪
1, which in thesecond-order approximation are written the following form L bulk = − r l − r a − a l r + q l ( l + a ) r − a q l r l , L AdS = − r l . For obtaining the complexity of formation we should compute the following relation: π ∆ C A = A bulk − A Ads + A joints = Z (cid:18)Z √− g £ bulk ( r ∗∞ − r ∗ ) dr − Z √− g £ AdS ( r ∗∞ − r ∗ ) dr (cid:19) dθ + A joints , where £ bulk = R + 6 l − L, £ AdS = R + 6 l , r ∗ = Z r g rr g tt dr. Since lim r →∞ r ∗ = 0 and r ∗ in the limit qM ≪ al ≪ r can not be calculated analytically thus it is actually more convenient to use integrationby parts to eliminate the appearance of r ∗ inside this expression, because after that we must integrating with respectto θ therefore the first integral must be calculated analytically, A bulk = Z Z √− g £ bulk ( − r ∗ ) drdθ = Z (cid:18) ( − r ∗ ) Z √− g £ bulk dr + Z (cid:18)r − g rr g tt Z √− g £ bulk dr (cid:19) dr (cid:19) dθ.A Ads is obtained in the same way. After some calculations complexity of formation for slowly rotating black holesin the limit qM ≪ r meet r + ( r meet ) versus r − /r + ( r + ) is plotted in Fig. 8. We find that r meet is approximately a linearfunction of event horizon and the best fitting curve relation is r meet = − . r + + 2 . Complexity of formation in CV conjecture
For calculating the complexity in CV conjecture, we must find the surface with maximal volume. For this purposewe should write the line element in the tortoise coordinate ds = dυ (cid:18) − ∆ r Σ + a ∆ θ sin ( θ )Σ (cid:19) + dr (cid:18) − ∆ r Σ g rr g tt + a ∆ θ sin ( θ )Σ g rr g tt + Σ∆ r (cid:19) +2 drdυ r − g rr g tt (cid:18) ∆ r Σ − a ∆ θ sin ( θ )Σ (cid:19) + Σ∆ θ dθ + dφ (cid:18) − ∆ r Σ a sin ( θ )Ξ + ( a + r ) ∆ θ sin ( θ )Ξ Σ (cid:19) − dφdυ (cid:18) a sin ( θ )ΣΞ (cid:19) (cid:0) r − ( a + r )∆ θ (cid:1) + drdφ r − g rr g tt (cid:18) − r Σ + a ( a + r )∆ θ sin ( θ )ΞΣ (cid:19) , where υ = t + r ∗ , r ∗ = Z r − g rr g tt dr = Z s Σ ∆ r − a ∆ r ∆ θ sin ( θ ) dr. r − /r + D C A FIG. 7: Complexity of formation in the CA conjecture versus r − r + for rotating black hole. To find the maximal surface we describe the surface with the parametric relations r = r ( λ ) and υ = υ ( λ ) with someparameter λ , then the volume of the surface becomes in the following form V = Z √ σd x = 4 π Z Z drdθr vuut(cid:20) − ∆ r Σ + ∆ θ a sin ( θ )Σ (cid:21) ˙ υ + 2 ˙ r ˙ υ s − ∆ θ a sin ( θ )∆ r , (4.15)where dots indicate derivatives with respect to λ and the factor 2 is originated from this fact that the surface iscomposed from two equivalent parts. We are free to choose λ to keep the radial volume element fixed as follow r vuut(cid:20) − ∆ r Σ + ∆ θ a sin Σ (cid:21) ˙ υ + 2 ˙ r ˙ υ s − ∆ θ a sin ( θ )∆ r = 1 , (4.16)this Lagrangian is independent of υ and hence there is a conserved quantity given by E = − ∂ £ ∂υ = (cid:18) ∆ r Σ − ∆ θ a sin( θ ) Σ (cid:19) ˙ υ − s − ∆ θ a sin( θ ) ∆ r ˙ r. (4.17)By substituting ˙ υ from (4.16) into the (4.17), one can obtain E = r (cid:18) ∆ r Σ − ∆ θ a sin ( θ )Σ (cid:19) + r ˙ r s − ∆ θ a sin ( θ )∆ r , with substituting above equations into the relation (4.15) for maximal surface volume one can find that V = 4 π Z r max r min Z π dr ˙ r = 2Ω Z r max r min Z π r q − ∆ θ a sin ( θ )∆ r dθdr q r ( ∆ r Σ − ∆ θ a sin ( θ )Σ ) + E . Here we wish to take r max to be infinity, but this will yield a divergent result in general. A finite result can beobtained by performing a carefully matched subtraction of the AdS vacuum. Here r min is the turning point of thesurface which determined by the condition ˙ r = 0 E + r (cid:18) − ∆ r Σ + ∆ θ sin ( θ )Σ (cid:19) r min = 0 . − /r + r meet /r + + r meet FIG. 8: The behavior of r meet r + versus r − r + (up) and r meet versus r + (down). A simple calculation shows that r min will be on or inside the (outer) horizon, and so we have that, g tt ( r min ) < υ ( λ min ) > ⇒ E < g tt ( r min ) < τ = 0 in this case r min = r + which gives E = 0. Thus, the complexity of formationbecomes∆ C V = V bulk − V ads G N L = 4 πG N L Z r max r + Z π r q Σ∆ r (1 − ∆ θ a sin ( θ )∆ r ) dθdr − Z r Z π r s Σ r (1 + r l ) − M r dθdr , r − /r + D C V FIG. 9: Complexity of formation in the CV conjecture versus r − r + . integrating with respect to θ gives us∆ C V bulk = 8 lπ G N L Z r max r + dr [ r − ( − r + 2 m ) l ] (cid:20)(cid:18) r − r m + (cid:18) − q − a (cid:19) r + 5 a q r (cid:19) l + r − a r
16 + (cid:18) r − mr + (cid:18) − q m − a (cid:19) r + (cid:18) q m − a m (cid:19) r + a (cid:18) m + 7 q (cid:19) r + a q m (cid:19) l (cid:21) − lπ G N L × Z r dr l √ r p ( r − m ) l + r . As the next step, we have to integrate the above relation with respect to r . In this case the integral cannot becalculated analytically, and therefore, we should calculate it numerically which the results are shown in Fig. 9.According to Figs. 7 and 9, it is obvious that complexity of formation in two mentioned conjectures has samebehavior and it is bounded without divergency. In the limit r − r + → × while at the large value of r − r + ∼ .
25, it’s valueapproximately tends to a constant. Since increasing the electric charge leads decreasing (increasing) inner horizon(event horizon), it is easy to find that r − r + is an increasing function of electric charge, q , and therefore, the complexityof formation in both conjectures increases as one increases the electric charge. V. CONCLUSION
In this paper, we studied the complexity of regular static and rotating black holes in the presence of nonlinearelectrodynamics . For the case of the static black hole, by looking at the action growth rates of the Wheeler-De Wittpatch, we showed the Lloyd bound is satisfied and its form remains unaltered . Then, we have obtained the rate ofcomplexity and complexity of formation by using both CA and CV conjectures. From the comparison of the result,we concluded that the two approaches to the complexity are consistent. In the case of slowly rotating one and withassumption qM ≪
1, we obtained the rate of complexity proportional to the volume of the black hole (in the limit α ≪ β ≪ Acknowledgements
The authors thank Shiraz University Research Council.
VI. APPENDIX
For the sake of completeness, here, we investigate complexity conjecture of two more examples of static regularblack holes in P . A. Second Case:
Here, we consider another structure function inspired by the log-logistic distribution as the second class of solutionwhich is given by [38, 47] H = P (1 + γ √−P ) + 6 l , (6.1)where γ = r √ γ , and r and γ are two parameters. In the weak field limit ( P ≪
H ≈ P + 6 l + O ( P ) . (6.2)Taking such a structure function into account with H = 2 ρ ( r ) and Eqs. (2.10) and (2.9), we can, directly, achievethe following arctan form of the solution [47, 48] f ( r ) = 1 − χπr (cid:20) arctan ( x ) − x x (cid:21) + r l , r = πγ χ , x = rr , (6.3)with the following asymptotical behavior for large r − g tt ≈ r l − χr + γ r + O (cid:18) r (cid:19) . (6.4)As a result, the obtained solution behaves like Reissner-Nordstr¨om-AdS, asymptotically, when we adjust χ and γ asthe mass and electric charge of the system, respectively.Such as before, we can obtain the following thermodynamic quantities for the recent arctan solution P = 38 πl , (6.5) V = 4 πr , (6.6) S = πr , (6.7) T = m (cid:0) (1 + x ) arctan( x + ) − x + − x (cid:1) π r (1 + x ) + r + πl , (6.8)Φ = Z ∞ r + Edr = 3 mπq (cid:18) π − arctan( x + ) + (1 + x ) x + (1 + x ) (cid:19) , (6.9)where x + = x | r = r + . Having the mentioned thermodynamic quantities, one can directly examine the first law of blackhole thermodynamics in the enthalpy representation dH = T dS + V dP + Φ dq, (6.10)0where H = m = πr + (1 + r l )4 (cid:16) arctan( x + ) − x + x (cid:17) . (6.11)Moreover, the Smarr relation is given by H P V − T S − q Φ2 + 14 Z wdv = 0 , (6.12)where Z wdv = 12 Z ∞ r + T µµ r dr = m − mx + ( x − π (1 + x ) − mπ arctan( x + ) . (6.13)Now, we calculate joint terms according to joint contribution which is described in the first class. After simplification,we can write S B ′ B = 14 δt " mπr (cid:18) arctan( x ) − x x (cid:19) − mπr x r (1 + x ) ! + 2 rl r B ,S C ′ C = 14 δt " mπr (cid:18) arctan( x ) − x x (cid:19) − mπr x r (1 + x ) ! + 2 rl r C . (6.14)Considering the metric function obtained here, it is easy to show that the Ricci scalar is calculated as R = 4 q r (1 + x ) − l , (6.15)and also, the Lagrangian of nonlinear electrodynamics can be calculated at the event horizon with the followingexplicit form L = 16 m (1 − x ) πr (1 + x ) − l . (6.16)Consequently, the growth rate of bulk action and surface term are, respectively, d A bulk dt = (cid:20) mπ (cid:18) arctan( x ) − x x (cid:19)(cid:21) r + r − , (6.17)and d A boundray dt = (cid:20) r − mπ arctan( x ) + mx (3 + x ) π (1 + x ) + 3 r l (cid:21) r + r − . (6.18)Finally, we find that the total growth rate of the action for such black hole configuration within WDW patch atlate time approximation can be collected as d A dt = (cid:20) r − mπ arctan( x ) − mx (1 − x ) π (1 + x ) + 3 r l (cid:21) r + r − . (6.19)Thus, by using the above calculated thermodynamic quantities, it is easy to show that the Lloyd bound is satisfied.As an example, for the values m = 0 . , q = 0 . , l = 1 this bound is about 1 . . B. Third Case:
As the third case, we consider the following structure function which is inspired by the Fermi-Dirac distributionfunction[38, 49] H = P cosh (cid:18) γ ( −P ) χ (cid:19) + 6 l , (6.20)with the following weak field limit ( P ≪ H ≈ P + 6 l + O ( P ) . (6.21)Considering the spherically symmetric spacetime with previous relations for H , the third class of solution is f ( r ) = 1 − χr (cid:20) − tanh (cid:18) γ χr (cid:19)(cid:21) + r l . (6.22)According to the series expansion of the metric function for large r , one can find this solution behaves like theReissner-Nordstr¨om-AdS, asymptotically, as − g tt ≈ r l − χr + γ r + O (cid:18) r (cid:19) , (6.23)whereas χ and γ will be associated with the mass and electric charge of the system, respectively.Using the previous approach in the extended phase space, the following thermodynamical quantities can be obtained,directly, P = 38 πl , (6.24) V = 4 πr , (6.25) S = πr , (6.26) T = m πr (cid:20) − tanh (cid:18) q mr + (cid:19)(cid:21) − q πr (cid:20) − tanh (cid:18) q mr + (cid:19)(cid:21) + r + πl , (6.27)Φ = Z ∞ r + Edr = 3 m q − mr + (cid:18) e q mr + (cid:19) − q e q mr + qr + (cid:18) e q mr + (cid:19) . (6.28)It is evident that these thermodynamic quantities satisfy the first law of black hole thermodynamics as dH = T dS + V dP + Φ dq, (6.29)where in the enthalpy representation, we have H = m. (6.30)In addition, the related Smarr formula is given by H P V − T S − q Φ2 + 14 Z wdv = 0 , (6.31)where Z wdv = 12 Z ∞ r + T µµ r dr = m − mr + (cid:18) e q mr + (cid:19) + 2 q e q mr + r + (cid:18) e q mr + (cid:19) . (6.32)2According to the previous section, joint contribution can be calculated as below S B ′ B = 14 δt (cid:18) mr (cid:20) − tanh (cid:18) q mr (cid:19)(cid:21) − q r (cid:20) − tanh (cid:18) q mr (cid:19)(cid:21) + 2 rl (cid:19) r B , (6.33) S C ′ C = 14 δt (cid:18) mr (cid:20) − tanh (cid:18) q mr (cid:19)(cid:21) − q r (cid:20) − tanh (cid:18) q mr (cid:19)(cid:21) + 2 rl (cid:19) r C . (6.34)Moreover, the Ricci scalar for this solution is simplified as R = q sinh (cid:16) q mr (cid:17) mr cosh (cid:16) q mr (cid:17) − l , (6.35)and the nonlinear Lagrangian is as follows L = q (cid:16) q tanh (cid:16) q mr (cid:17) − mr (cid:17) mr cosh (cid:16) q mr (cid:17) − l . (6.36)It is notable that the growth rate of the bulk action and its related surface term can be written as d A bulk dt = " m e (cid:16) q mr (cid:17) r + r − , (6.37) d A boundray dt = (cid:20) r − ( 3 m q r ) + 3 m (cid:18) q mr (cid:19) + q r tanh (cid:18) q mr (cid:19) + 3 r l (cid:21) r + r − . (6.38)As a result, the total growth rate of action for such a black hole configuration within WDW patch at late timeapproximation is d A dt = (cid:18) r + 3 r l − m (cid:20) − tanh (cid:18) q mr (cid:19)(cid:21) − q r (cid:20) − tanh (cid:18) q mr (cid:19)(cid:21)(cid:19) r + r − . (6.39)For instance, for the values m = 0 . , q = 0 . , l = 1 this bound is about 8 . . C. Complexity of formation
Similar to the complexity of formation of the first case we would like to study the complexity of formation of secondand third cases. In this way, r ∗ ( r ) and r meet of the second and the third solutions can be obtained as follows: r ∗ ( r ) = πl r ln (cid:18) | r − r + | r + r + (cid:19) ml arctan( r + r ) + 2 πr r + r + 2 r r + ( πr r − ml r − ml r )( r + r ) + πl r − ln (cid:18) | r − r − | r + r − (cid:19) ml arctan( r − r ) + 2 πr − r − + r + 2 r r + ( πr r − − ml r − − ml r )( r − + r ) second case (6.40) r ∗ ( r ) = l r ln (cid:18) | r − r + | r + r + (cid:19) mr + l (cid:18) − tanh (cid:18) q mr + (cid:19)(cid:19) − q l (cid:18) − tanh (cid:18) q mr + (cid:19)(cid:19) + 2 r + l r − ln (cid:18) | r − r − | r + r − (cid:19) mr − l (cid:18) − tanh (cid:18) q mr − (cid:19)(cid:19) − q l (cid:18) − tanh (cid:18) q mr − (cid:19)(cid:19) + 2 r − third case (6.41)3 + FIG. 10: Plot of A joint for second and third cases which are shown in black and dotted line respectively. and r meet ≈ . r − . r + + 1 . , second case (6.42) r meet ≈ − . r − . r + + 0 . . third case (6.43)Accordingly, the bulk action of the black holes are A bulk,BH = 16 M r πG Z r max r meet dr (cid:18) r + r ) (cid:19) ( v ∞ − r ∗ ( r )) second case (6.44) A bulk,BH = 2 q G Z r max r meet dr − tanh (cid:18) q M r (cid:19) r ( v ∞ − r ∗ ( r )) third case (6.45)here r max is the same as the first case. The relation for A joint for large black holes are written in the following form: A joint ≈ . (cid:18) . r l (cid:19) r + O ( r ) . second case (6.46) A joint ≈ . (cid:18) − . r l (cid:19) r + O ( r ) . third case (6.47)The figure of A joint for second and third cases are plotted in Fig. 10. According to Fig. 10, it is obvious that forsmall black hole the contribution of the joints to the action is large for large black holes, this contribution becomessmaller and goes to zero.Finally according to above relations and the results related to the AdS spacetime (3.48)-(3.50), ∆ C A from (3.43)can be obtained. The plot of ∆ C A for second and third cases are shown in Fig. 11.Figure 11 describes that complexity of formation for second and third cases increases for large black holes. Finally∆ C V according to Eq. (3.56) for these two cases can be obtained.Figure 12 shows that ∆ C V for both cases has the same behavior as with ∆ C A , increasing with radius of the blackhole. To summarize the behavior of ∆ C A and ∆ C V for three cases versus r + are plotted in Fig. 13.4 r + D C A − t h i r d c a s e r + D C A − s e c ond c a s e FIG. 11: Plots of ∆ C A for second and third cases are shown in the right and left panels respectively. r + D C V − t h i r d c a s e r + D C V − s e c ond c a s e FIG. 12: Plots of ∆ C V for second and third cases are shown in the right and left panels respectively.[1] H. El Moumni and K. Masmar, Nucl. Phys. B , 114837 (2020). + D C A r + D C V r + D C A r + D C V + D C A r + D C V FIG. 13: Plot of Complexity of formation for the different geometries. For first, second and third cases respectively are shownfrom up to down, and left panels describe ∆ C A and right ones indicate ∆ C V .[2] G. ’t Hooft, Conf. Proc. C , 284 (1993).[3] L. Susskind, J. Math. Phys. , 6377 (1995).[4] J. M. Maldacena, Int. J. Theor. Phys. , 1113 (1999) [Adv. Theor. Math. Phys. , 231 (1998)].[5] S. Ryu and T. Takayanagi, JHEP , 045 (2006).[6] V. E. Hubeny, M. Rangamani and T. Takayanagi, JHEP , 062 (2007).[7] H. Casini, M. Huerta, and R. C. Myers, JHEP , 036 (2011).[8] A. Lewkowycz and J. Maldacena, JHEP , 090 (2013).[9] L. Susskind, Fortsch. Phys. , 24 (2016).[10] D. Stanford and L. Susskind, Phys. Rev. D , 126007 (2014).[11] A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle and Y. Zhao, Phys. Rev. Lett. , 191301 (2016).[12] A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle and Y. Zhao, Phys. Rev. D , 086006 (2016).[13] S. Lloyd, Nature , 6799 (2000).[14] R. G. Cai, S. M. Ruan, S. J. Wang, R. Q. Yang and R. H. Peng, JHEP , 161 (2016).[15] W. J. Pan and Y. C. Huang, Phys. Rev. D , 126013 (2017).[16] P. A. Cano, R. A. Hennigar and H. Marrochio, Phys. Rev. Lett. , 121602 (2018).[17] A. Ovgun and K. Jusufi, [arXiv:1801.09615].[18] K. Meng, Eur. Phys. J. C , 984 (2019).[19] X. H. Feng and H. S. Liu, Eur. Phys. J. C , 40 (2019).[20] J. Jiang, Eur. Phys. J. C , 130 (2019).[21] J. Jiang, Phys. Rev. D , 086018 (2018).[22] E. Yaraie, H. Ghaffarnejad and M. Farsam, Eur. Phys. J. C , 967 (2018).[23] K. Nagasaki, Phys. Rev. D , 126014 (2018).[24] Z. Y. Fan and H. Z. Liang, Phys. Rev. D , 086016 (2019).[25] L. Balart and E. C. Vagenas, Phys. Rev. D , 124045 (2014).[26] E. Ayon-Beato and A. Garcia, Phys. Rev. Lett. , 5056 (1998).[27] K. A. Bronnikov, Phys. Rev. D , 044005 (2001).[28] A. V. B. Arellano and F. S. N. Lobo, Class. Quantum Grav. , 5811 (2006).[29] M. Hassain, Class. Quantum Grav. , 246 (2008). [30] L. Hollenstein and F. S. N. Lobo, Phys. Rev. D , 124007 (2008).[31] L. Balart, Mod. Phys. Lett. A , 2777 (2009).[32] E. F. Eiroa and C. M. Sendra, Class. Quant. Grav. , 085008 (2011).[33] S. Fernando and J. Correa, Phys. Rev. D , 064039 (2012).[34] S. H. Hendi, S. N. Sajadi and M. Khademi, [arXiv:2006.11575].[35] D. Kastor, S. Ray and J. Traschen, Class. Quant. Grav. , 235014 (2010).[36] L. Balart and S. Fernando, Mod. Phys. Lett. A , 1750219 (2017).[37] L. Lehner, R. C. Myers, E. Poisson and R. D. Sorkin, Phys. Rev. D , 084046 (2016).[38] L. Balart and E. C. Vagenas, Phys. Rev. D , 124045 (2014).[39] L. Balart and E. C. Vagenas, Phys. Lett. B , 14 (2014).[40] H. Culetu, Int. J. Theor. Phys. , 2855 (2015).[41] S. G. Ghosh, Eur. Phys. J. C , 532 (2015).[42] J. Couch, W. Fischler and P. H. Nguyen, JHEP , 119 (2017).[43] Y. S. An, R. G. Cai and Y. Peng, Phys. Rev. D , 106013 (2018).[44] S. Chapman, H. Marrochio and R. C. Myers, JHEP , 062 (2017).[45] W. Cottrell and M. Montero, JHEP , 039 (2018).[46] A. A. Balushi, R. A. Hennigar, H. K. Kunduri and R. B. Mann, [arXiv:2008.09138].[47] I. Dymnikova, Class. Quant. Grav. , 4417 (2004).[48] S. N. Sajadi and N. Riazi, Gen. Rel. Grav. , 45 (2017).[49] E. Ayon-Beato and A. Garcia, Phys. Lett. B464