Effects of quintessence dark energy on the action growth and butterfly velocity

In this work we investigate the effects of dark energy quintessence on the evolution of the computational complexity in the context of AdS/CFT correspondence. We use complexity− action conjecture to write the action for a charged black hole gravity and by considering the dark energy counterpart to the model, we try to find how the complexity change by time? Does it satisfy the Lloyd bound. We also investigate evolution of the complexity when boundary region is perturbed by a small amount of energy. Actually we want to find how dark energy could be effective in the shock wave geometry.


Introduction
In the language of gauge/gravity duality all dynamical evolution in the antide Sitter spacetime has a dual boundary field theory [1]. Actually this duality acts as a dictionary for all field theory characteristics in the language of black hole physics in AdS spacetime. One of important conjectures in the holographic context is computational quantum complexity which implies the minimum quantum gates that are necessary to produce such states associated with boundary complexity from the reference state. The old conjecture states complexity−volume (CV), for which the volume of a maximal space-like slice in the black hole interior is assumed to be connected with computational complexity of dual boundary conformal field theory [2]. This conjecture is a result of the behavior of the interior volume of black hole which grows linearly with time, so it could be translated with the growth of computational complexity on the boundary with time [3,4]. However, if the bulk contains a shock wave then the interior volume of the black hole shrinks for a period of time and shows an opposite behavior. In the new conjecture namely complexity − action (CA), computational complexity of a holographic state 1 E-mail: hghafarnejad@semnan.ac.ir 2 E-mail: mhdfarsam@semnan.ac.ir 3 E-mail: eyaraie@semnan.ac.ir on the boundary, is the on-shell action in a patch which is created by the light rays from two boundaries of two-sided eternal black hole which is called Wheeler-DeWitt patch [5,6]. This new CA conjecture has some preferences with respect to the old one (CV). Lloyd shows in [7] the growth rate of the bulk action or computational complexity has an upper bound which is related to the energy of the quantum state on the boundary, E, the equality case happens for the neutral black hole and any charge makes the growth rate slower. For charged black holes which have more than one event horizons, "E" must be the difference between the energy at the outer and inner horizons [8]. This is check for static shells and shock waves in ref. [6]. The action growth in this case shows interestingly that the black holes are the fastest computers in the nature. Actually shock waves near the horizon of an AdS black hole describes chaos in a thermal CFT [9,10,11] and represents interesting aspects of the boundary complexity. From the view point of holography dictionary the spreading of the shock wave near the horizon has a field theory butterfly effect description. This effect could be seen with a sudden decay after scrambling time t * = β 2π lnS, with "S" stands for the black hole entropy. It is a necessary time where the black hole can be the fastest scramblers to render the density matrix of a small thermal subsystem. The spreading of the shock wave could be happened by the injection of a small amount of energy into the boundary. This causes to throw a few quanta into the horizon and destroy the thermofield double state of the system. The growth of perturbation or the shock wave on the boundary due to back-reaction effects is identified with butterfly velocity "v B " which could be obtained by solving the equation of motions of perturbed geometry. Complexity growth rate and the effects of butterfly effect on it, is investigated on various gravity models for the bulk. For instance in [5,6,8] the authors investigated action growth for various AdS black holes and tested the Lloyd bound by considering the effects of charge. The growth of holographic complexity is studied in massive gravity [12], and in a more variety of other works [13][14][15][16][17]. On the other hand some works have done by studying the shock wave geometry for different gravity models in the bulk [9,10,11,18,19] and were investigated the effects on the action growth by obtaining butterfly velocity and comparing with other simple models [20,21]. In the present work we consider the effects of quintessence dark energy on the black hole geometry, and therefore we will see how it change the action growth rate and butterfly velocity in the shock wave geometry. Quintessence dark energy is a canonical scalar field which is one of the successful theories to explain the acceleration phase of the universe [22,23,24]. In this model introduced by Kislev [25] an energy-momentum tensor must be added to the Einstein equation as G = κ(T matter + T quintessence ). The effects of quintessence has studied in a wide range of works and thermodynamics of the various black holes are investigated when they are surrounded by the dark energy [25][26][27][28][29][30]. It would be challenging to see how it affects the holographic characteristics as well. Layout of the paper is as follows. We first study the action growth in the presence of quintessence dark energy in section 2 and found Lloyd bound [7] is satisfied with new charge associated with this dark energy. In section 3 we calculate the butterfly velocity and compare the action growth in the presence of shock wave geometry in our gravity model. At last section 4 denotes to summarize of the work and some outlooks.

The rate of action growth in the presence of dark energy
We consider RN-AdS black hole surrounded by quintessence dark energy in four dimension with following action which consists of two parts: The first part for bulk action contains Einstein-Hilbert-Maxwell action in AdS spacetime is given by: in which cosmological constant is related to AdS radius L as Λ = −3/L 2 in four dimension. On the other side boundary part of the action at the late time approximation is: where h stands for the determinant of induced metric on the boundary of anti-de Sitter bulk and K represents the trace of extrinsic curvature. Metric equation of the total action has a spherically symmetric static metric which in a Schwarzschild coordinates system is defined by [25] and for quintessence dark energy regime the state parameter reads −1 < ω < − 1 3 . "a" is the normalization factor related to the density of quintessence via ρ q = −3aω/2r 3(ω+1) . Black hole horizons are determined by solving f (r ± ) = 0 where we call r + (r − ) to be exterior (inner) horizon radius. The electromagnetic tensor F µν = ∂ µ A ν − ∂ ν A µ is defined by 4-vector potential A µ which for the spherically symmetric metric (2.4) reads Substituting the metric solution (2.4) one can obtain Ricci scalar R as for which the action growth of the bulk become: which by setting Ω 2 /4πG = 1 reads To obtain the action growth for boundary part we must evaluate the extrinsic curvature associated with metric as follows: where prime ′ denotes to derivative with respect to "r". By this definition second part of the action leads to the following form. (2.12) Adding (2.9) and (2.11) we can obtain the total growth action for quintessence RN-AdS black hole such that: (2.13) By using horizon solutions f (r ± ) = 0 one can obtain expressions for the charge and mass of black hole, Substituting (2.14) we can rewrite the total growth rate of action (2.13) as follows.
In compare with the work of Brown et al [5,6] we see an extra term added with the rate of action growth due to the presence of quintessence dark energy. So the growth rate of our action within WDW patch at late time approximation could be summarized like: in which µ ± = q E /r ± stands for chemical potential, A ± = −1/2r 3ω ± is conjugated potential for parameter "a". If we take "E" for the average energy of the quantum state related to the ground state then quantum complexity rate satisfies the Lloyd bound [7] as follows: (2.18)

Butterfly effect and shock wave geometry
The shock wave geometry happens when our black hole solution perturbed by a small amount of energy. This phenomenon is understood by studying the butterfly velocity which is the velocity of shock wave near the horizon. To do so we first rewrite the black hole solution (2.4) in Kruskal coordinates system where the apparent horizon remove and so metric components is regular on the apparent horizon such that: where,

2)
and h(u, v) = r 2 . As we know the relationship between Kruskal and spherical coordinates is: where β = 1/k B T and T is Hawking temperature and k B is the Boltzmann constant. dr * = dr/f (r) is the differential of the tortoise coordinate. Near the horizon the tortoise coordinate is given by: Now by rewriting the metric in the new coordinate we can study the effects of disturbance as a shock wave geometry. Suppose that the metric behaves as (3.1) for u < 0 but is changed to a perturbed metric by replacing v → v + α(x i ) [8]. α(x i ) is called the shift function which shows a boundary perturbation in the direction of x i . Applying the following new transformations, in which θ(u) represents the Heaviside step function, metric takes the new form as: where δ(U) is the Dirac delta function. It is easy to see that the above metric reduces to the old one (3.1) for u = U < 0.
Before the injection of disturbance into the boundary which leads to the metric solution (3.1) in Kruskal coordinates the stress-energy tensor, T matter , can be written as follows.
where G is the Einstein tensor. After injection this tensor could be expressed in the new coordinates as the following form: By attention to [11,31] one can consider a massless particle at u = 0 which moves in the v-direction with the speed of light, the stress-energy tensor of this particle which is corresponds to the shock wave stress-energy tensor is: with a dimensionless constant E and a(X) as a local source of perturbation which for simplicity reasons we take it as Dirac delta function, i.e. a(X) = δ(X). By considering the stress-energy tensor of this disturbance the Einstein equation changes to 1 κ G = T matter + T shock and therefore our aim would be the solution of this equation. This equation at the leading order near the horizon is solved as: where, and v B called "butterfly velocity" is given by: In fact, this velocity as it is mentioned before is the spread of the local perturbation on the boundary of spacetime. In our case h(r) = r 2 and so the butterfly velocity reduces to: Now in our model another term related to quintessence dark energy would be added to the stress-energy tensor from matter field sources introduced in (3.7) and (3.8) as G = κ(T matter + T quintessence ), and so affects the propagation of shock wave. In fact butterfly velocity will be depends on quintessence parameters such as normalization factor and the state parameter "ω". This velocity could be calculated by attention to Hawking temperature equation (3.14) It would be useful to study the effect of dark energy on the butterfly velocity for the same gravity model. As we can see dark energy leads to an extra term to v B which is the last term in (3.14). Since −1 < ω < −1/3 so this term is a negative term. Horizon radius r h as we know is a solution of f (r h ) = 0. In a charged black hole solution with no dark energy around it, corresponds to a = 0, the butterfly velocityṽ B is the same as (3.14) without the last term and different horizon radiusr h for f (r h ) = 0, in which From (2.5) and (3.15) it is simple to conclude that for fixed mass and fixed charge we have f (r) <f (r). This equation is situated properly for any horizon radius such as the horizon of quintessence solution, namely f (r h ) < f (r h ). Since f (r h ) = 0 thenf (r h ) > 0 and becausef (r h ) = 0 so it leads tõ f (r h ) >f (r h ). This means the horizon radius of charged black hole solution must be greater when it is surrounded by dark energy, r h >r h . If we consider non-charge solution for simplicity then it could be found easily: To compare the butterfly velocities with and without the effects of quintessence dark energy we attend to the last term of the equation (3.14) which tell us v B <ṽ B . In other words we can result.
It means that the butterfly velocity is smaller when the black hole is surrounded by quintessence dark energy. Slower butterfly velocity means slower complexity growth in this regime which is caused by the effects of dark energy. It could be deduced easily from (3.17) the smaller ω leads to the smaller v B .

Concluding remarks
We studied the complexity growth rate by using "CA" conjecture [5,6] for a simple model of gravity when its black hole solution is surrounded by quintessence dark energy [25]. The effects of this kind of dark energy is investigated earlier in various works [25][26][27][28][29][30] and it seems challenging to see how it effects the holographic characteristics. We found an extra term related to the quintessence dark energy is added to the total action growth. By attention to the conjugated potential for the quintessence parameter it is also proved that the Lloyd bound [7] is satisfied. All these considerations lead to add an extra term to the average amount of black hole energy and the Lloyd bound satisfies with a correction related to the conjugated potential of dark energy. We also investigate the action growth of this model for shock wave geometry [9]. Actually when the boundary perturbs by a small amount of energy, the geometry in the bulk is affected. The local shock wave spreads near the horizon with the "butterfly velocity" which could be obtained by the equation of motions for the new stress-energy tensor. The new stress-energy tensor is the same of the old tensor plus an extra term related to the shock wave which has "uu" component due to a massless particle moving with the speed of light at u = 0. We showed that the effect of dark energy makes the spread of shock wave slower than usual case, so the complexity growth would be slower as well.