# Holographic complexity growth rate in Horndeski theory

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## Abstract

Based on the context of complexity = action (CA) conjecture, we calculate the holographic complexity of AdS black holes with planar and spherical topologies in Horndeski theory. We find that the rate of change of holographic complexity for neutral AdS black holes saturates the Lloyd’s bound. For charged black holes, we find that there exists only one horizon and thus the corresponding holographic complexity can’t be expressed as the difference of some thermodynamical potential between two horizons as that of Reissner–Nordstrom AdS black hole in Einstein–Maxwell theory. However, the Lloyd’s bound is not violated for charged AdS black hole in Horndeski theory.

## 1 Introduction

*etal*[11, 12], which states that the quantum complexity of ground state of CFT is given by the classical action evaluated on the ”Wheeler-DeWitt patch” (WDW), and the WDW patch is the spacetime region enclosed by future and past light rays started from a bulk Cauchy slice, reflected at the boundaries and then ended in another bulk Cauchy slice, e.g., see Fig. 1 in Sect. 2. The conjecture thus reads,

*M*[11, 12],

*F*is the free energy,

*T*,

*S*are the temperature and entropy of the black hole, and \({{\mathcal {H}}} = F + TS\) is the generalized enthalpy. It was pointed out that the result still holds for higher derivative theories. Recently, it was explicitly showed that the result is true for AdS black holes in the

*f*(

*R*) gravity, massive gravity theories [31, 32, 33, 34] and Lovelock gravity [35].

However, it was pointed out that the action growth expression is different for charged black hole with a single horizon in the Einstein–Maxwell–Dilaton and Born–Infeld theories [36]. It is thus worthwhile taking a further step to explore the pattern of the action growth of charged AdS black holes with only one horizon in higher derivative gravity theory. In this paper, we shall study the action growth of AdS black holes in Horndeski gravity theory.

Horndeski theory is a kind of higher derivative scalar-tensor theory which has the similar property of Lovelock gravity, that the Largrangian involves terms which are more than two derivatives, but the equations of motion are consisted of terms which have at most two derivatives acting on each field [37]. AdS black holes have been constructed in Horndeski gravities in [38, 39] and their thermodynamics were studied in [40, 41]. The stability and causality of these black holes were carried out in [42, 43, 44]. Holographic application of Horndeski theory were investigated in [45, 46, 47, 48, 49, 50, 51, 52], especially, it was shown in [51] that although there is no holographic *a*-theorem for general Horndeski gravity, there does exist a critical point in parameter space where the holographic *a*-theorem can be achieved, which suggests the Horndeski theory should have a holographic field theory dual. The AdS black holes we studied in this paper are in this critical point.

We shall study the action growth of unchanged black holes in \(D=4\) Horndeksi gravity in Sect. 2, and in Sect. 3, we compute the action growth of the charged black holes in the four dimensional Einstein–Maxwell–Horndeski gravity. We conclude our results in Sect. 4.

## 2 Neutral black holes in Horndeski theory

### 2.1 Planar black hole

*a*-theorem holds for this system as pointed out in [51]. When \(\mu = 0\), the solution turns out to be an AdS vacuum with the scalar \(\chi \) being logarithmic in terms of radial coordinate

*r*. The conformal symmetry of the AdS is broken down to the Poincare symmetry plus the scaling symmetry because of the logarithmic scalar \(\chi \), which means the dual field theory is scaling invariant. The Horndeski coupling \(\gamma \) doesn’t have a smooth zero limit and should not be treated as a perturbative parameter. It was also showed that the kinetic term of the scalar perturbation \(\delta \chi \) is non-negative as long as \(\gamma \) is great than zero [52].

*t*,

*r*), (

*u*,

*r*) and (

*v*,

*r*), we have,

*K*is the Gibbons-Hawking term and

*a*will be illustrated later. Due to the equation of motion the Horndeski term in the action vanishes, the bulk contribution has a simpler form,

*B*approaches black hole event horizon, it turns out to be,

*K*, the normal vector is \(n_\alpha = {\frac{1}{\sqrt{f}}} \partial _\alpha r \), and

*K*is given by,

*a*is defined by,

### 2.2 Spherical black hole

*h*is not equal to

*f*any more. The mass of the black hole is,

## 3 Charged black holes in Horndeski theory

### 3.1 Planar black hole

*f*diverges,

*h*in terms of \(r_*\),

*r*”. We find that local extremes of profile

*h*, where \(h'(r_e)=0\), are equal to,

*h*are always positive. Since

*h*approaches \(\infty \) as

*r*approaches \(\infty \) and

*h*approaches \(-\infty \) as

*r*approaches 0, there should be at least one zero point. And as analysed before all the extremes are positive, thus there exists one and only one zero point for the profile

*h*, which means that there is one and only one event horizon for the black hole solution, which is quite different from RN-AdS black hole.

*M*, satisfying the Lloyd’s bound.

### 3.2 Spherical black hole

*a*vanishes on the black hole event horizon. Again, there is an additional singularity \(r_*\) where

*f*diverges,

*h*, where \(h'=0\), are equal to,

*h*are always positive. Since

*h*approaches \(\infty \) as

*r*approaches \(\infty \) and

*h*approaches \(-\infty \) as

*r*approaches 0, there should be at least one zero point. And as analysed before all the extremes are positive, thus there exists one and only one zero point for the profile

*h*, which means there is one and only one event horizon for the black hole solution.

*q*the combination of \(Q \Phi + {{\mathcal {C}}}_1\) is given by,

*q*is small, the black hole radus \(r_0\) should be great than \({\frac{q}{2 \sqrt{2}}}\), in this limit we have that,

*q*. For general parameter range we can not prove the combination \(Q \Phi + {{\mathcal {C}}}_1\) is always greater than zero, however we plot \(Q \Phi + {{\mathcal {C}}}_1\) as a function of

*q*and \(r_0\) for large number of parameter choices which imply that \(Q \Phi + {{\mathcal {C}}}_1\) is greater than zero, we present several of them in Fig. 2. It seems that the combination of \(Q \Phi + {{\mathcal {C}}}_1\) is always greater than zero, and the holographic complexity satisfies the Lloyd’s bound.

## 4 Conclusions

In this paper, we studied the holographic complexity in Horndeski gravity theories through the “Complexity = Action” conjecture. In particular, we calculated the gravitational action growth of neutral and charged AdS black holes in Horndeski gravities. We found that the rate of change of action for neutral black holes with planar and spherical topologies is 2*M* , which is the same as the universal result [11, 12] and saturates the Lloyd’s bound [28].

The charged black holes are more complicated. We analysed the metric profiles carefully and found that the charged black holes with planar and spherical topology both have only one event horizon which are quite different from that of the RN black holes. We computed the gravitational action growth for the charged black holes with planar and spherical topologies. It turns out that the action growth for planar topology is less then 2 *M* thus satisfies the Lloyd’s bound. Whilst, for spherical case, we showed that the action growth is less than 2*M* when *q* is small. Though we did not prove analytically that the result holds for the whole range of parameters, we did numerically studies a substantial parameter choices and found that the action growth is less than 2*M*, which leads us to believe that the action growth is always less than 2*M* and satisfies the Lloyd’s bound.

Here, we just studied the late time rate of change of holographic complexity, it is worthwhile going a step further to see the effect of the higher-derivative non-minimally coupled Horndeski term to the complexity of formation [54], subregion complexity [16, 55, 56, 57] and also to explore full time dependence of the holographic complexity [58, 59, 60].

## Notes

### Acknowledgements

We are grateful to Luis Lehner and Hong Lu for useful discussion. H.-S.L. is grateful for hospitality at Yangzhou University, during the course of this work. X.-H.F. is supported in part by NSFC Grants No. 11475024 and No. 11875200. H.-S.L. is supported in part by NSFC Grants No. 11475148 and No. 11675144.

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