# Holographic complexity in charged Vaidya black hole

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## Abstract

In this paper, we use the “complexity equals action” (CA) conjecture to discuss growth rate of the complexity in a charged AdS-Vaidya black hole formed by collapsing an uncharged spherically symmetric thin shell of null fluid. Using the approach proposed by Lehner et al., we evaluate the action growth rate and the slope of the complexity of formation. Then, we demonstrate that the behaviors of them are in agreement with the switchback effect for the light shock wave case. Moreover, we show that to obtain an expected property of the complexity, it is also necessary for the CA conjecture to add the particular counterterm on the null boundaries.

## 1 Introduction

In recent years, there has been a growing interest in the topic of “quantum complexity” which is defined as the minimum number of gates required to obtain a target state starting from a reference state [1, 2]. In the holographic viewpoint, Brown et al suggested that the quantum complexity of the state in the boundary theory corresponds to some bulk gravitational quantities which are called “holographic complexity”. Then, the two conjectures: “complexity equals volume” (CV) [1, 3] and “complexity equals action” (CA) [4, 5], were proposed. These conjectures have attracted many researchers to investigate the properties of both holographic complexity and circuit complexity in quantum field theory, e.g., [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43].

*S*is the on-shell action in the corresponding Wheeler–DeWitt (WDW) patch, which is enclosed by the past and future light sheets sent into the bulk spacetime from the timeslices \(t_L\) and \(t_R\). In particular, it was found that there is a bound of the complexity growth rate at the late time

Chapman et al. [45, 46] investigated the CA and CV conjectures for AdS-Vaidya spacetime which is sourced by the collapse of a spherically symmetric thin shell of null fluid [47, 48, 49]. They found that the standard definition of the WDW action is not appropriate for these dynamical spacetimes. In order to obtain an expected property of the complexity, we need to add a particular counterterm on the null boundaries. This counterterm also keeps the invariance under the reparametrization of the null generator on the null boundary. Moreover, they also demonstrated that the switchback effect for light shocks are imprinted in the complexity of formation and the full-time evolution of complexity when this counterterm is introduced.

In this paper, we follow the discussions in [45, 46] to investigate the holographic complexity for a charged AdS-Vaidya black hole which is sourced by an uncharged thin shell. This thin shell will generate a shape transition from a black hole with total mass \(M_1\) and charge *Q* to another one with mass \(M_2\) and same charge *Q*. With the approach proposed by Lehner et al. [11], we will evaluate the time evolution of complexity growth rate as well as the slope of the complexity of formation in the presence of the light and heavy shock wave. Using these results, we will argue that our results are also in agreement with the switchback effect for the light shock wave case.

The structure of this paper is as follows. In Sect. 2, we review the charged AdS-Vaidya background geometries. In Sect. 3, we first use the method proposed by Lehner et al. to calculate the complexity of formation as well as the action growth rate of the charged AdS-Vaidya black hole. Then, we investigate the action growth rate without the counterterm and compare our holographic results to the circuit behaviors. Concluding remarks are given in Sect. 4.

## 2 Charged AdS-Vaidya spacetime

*Q*to another one with mass \(M_2\) and same charge

*Q*, in which

## 3 Holographic complexity in charged AdS-Vaidya black hole

*u*and auxiliary time coordinate

*t*as

### 3.1 The change of the action

#### 3.1.1 \(\delta S_R\)

*v*,

*r*) and keeping the first order of \(\delta t_R\), one can obtain

#### 3.1.2 \(\delta S_L\)

#### 3.1.3 Counterterm contributions

### 3.2 Complexity of Formation

In the right panel of Fig. 3, we show the effect of heavier shock waves. In this regime, the slope starts at a finite value and suddenly drop to a minimal value, after that, it rapidly rises to the final constant value. It implies that the complexity of formation starts changing immediately and rapidly approach a regime of linear growth with increasing \(t_w\). This is very different with the light shock wave case.

#### 3.2.1 Large and small time behaviors

*q*, the shorter of time \(t_w\) dropping the minimal value, which means that this divergent peak only exists at \(t=0\) under the limit \(q\rightarrow 0\).

### 3.3 Time evolution of the complexity

*t*for the light and heavy shock wave in Figs. 5 and 6 separately. In these figures, we can see that the action growth rate develops a minimum or maximum at some finite time in very small charge case. These minimum or maximum becomes deeper and sharper for smaller charges. Therefore, the behaviors for the charged cases can smoothly approach that of the neutral cases. And the minimum or maximum is corresponding to the critical time in uncharged black hole [46].

In the left panel of Fig. 5, we show the action growth rate for a very light shock wave with \(\delta \omega =\omega _2-\omega _1=10^{-4}\) at \(t_w=5\). These figures show the same pictures with that of the internal RN black hole, which can be understood by the switch back effect since \(t_w=5<t_\text {scr}^*\) in this case. In the right panel, we show the growth rate at \(t_w=14\) such that \(t_w>t^*_\text {scr}\). After the scrambling time, the action of the light shock wave will be clearly illustrated. Therefore, in this case, it will share the similar behaviours with the case of heavy shock wave as shown in Fig. 6. Moreover, for the small charge case, a minimum value of the action growth rate appears at a finite time. Under the uncharged limit, this minimum point will reduce to the critical time in the neutral case as shown in Fig. 2 of Ref. [46].

In Fig. 6, we show the action growth rate for a heavier shock wave with \(\delta \omega =1\) at \(t_w=5\). We can see that there might exist two critical times under the uncharged limit, which will coincide with the neutral case for the heavier shock wave in Fig.3 of Ref. [46]. In addition, as shown in Fig. 6, for the non-extremal case, there exists at least two horizontal periods, in which the rate can be regarded as constant. However, for the extremal case, there only exists one horizontal period, i.e., the late time period.

#### 3.3.1 Early and late time behaviors

### 3.4 Complexity without counterterm

### 3.5 Circuit analogy

*I*when \(t<t^*_\text {scr}\). This feature is connected to the switchback effect [3, 7] and can provide a deeper explanation of our holographic results.

We denote the rate of the complexity to \(c_1\) before the operator \(\mathcal {O}_R\) is inserted and \(c_2\) after it [45]. Under the limit of light shock, we have \(c_1\approx c_2\approx c\).

## 4 Conclusion and discussion

The action of AdS black hole within the WDW patch has been related to the quantum complexity of a holographic state. Following the procedure in [50], we calculated the action growth rate of the charged AdS-Vaidya black hole in \((d+1)\)-dimensional Einstein–Maxwell gravity. We first introduced a charged AdS-Vaidya geometry which is source by the collapse of an uncharged thin shell of null fluid. And this thin shell generates a shape transition from a black hole with total mass \(M_1\) and charge *Q* to another one with mass \(M_2\) and the same charge *Q*.

Using the approach proposed by Lehner et al. [11], we studied the complexity of the formation and discussed its small and large time behaviors in Sect. 3.2. We found that the slope of the complexity of formation shares the similar behaviors with the uncharged case. Meanwhile, these results are also in agreement with the switchback effect. After that, the growth rate of the complexity was evaluated in Sect. 3.3. By comparing it to the uncharged case, we found that the behaviors for the charged cases can smoothly approach that of the neutral cases. Furthermore, we also found that when \(t_w<t^*_\text {scr}\), the action growth rate is the same as the unperturbed case, and when \(t_w>t^*_\text {scr}\), it shares the similar behaviors with the heavy shock wave case. And these behaviors can be explained by the switchback effect. In addition, we show that the late time growth rate is given by the average value of the two RN rate without shockwave, which is consistent with the uncharged case. In Sect. 3.4, we investigated the early and late time behaviors of the complexity without the counterterm. We demonstrated that, in order to obtain an expected property of the complexity, it is also necessary to introduce the counterterm on the null boundaries for the charged Vaidya black hole. Finally, by analysing the circuit model, we showed our results our holographic results are in agreement with that of the circuit model.

In this paper, we only considered the CA conjecture in charged RN black hole sourced by the collapse of an uncharged thin shell of null fluid. It would also be interesting to further investigate the CV conjecture in the charged Vaidya black hole. As discussed in the uncharge case [50], the CV conjecture also shares the similar results with the CA conjecture, such as the late time behaviors and the switch back effect. Therefore, we have good reason to believe that the CV conjecture have same behaviors with the CA conjecture in the charged Vaidya black hole, such as the late time action growth rate can also be expressed as the sum of the average value of the two RN rate without shockwave. In addition, it would be interesting to investigate the charged Vaidya black hole with a charged shock wave, in which we might possible to study the one-side charged Vaidya spacetimes which formed by the collapse of an charged spherically symmetric shell to the AdS vacuum spacetime, and consider the process from the finite temperature black hole to extremal black hole.

## Notes

### Acknowledgements

This research was supported by NSFC Grants nos. 11775022 and 11375026. The author is grateful to the anonymous referees for their useful comments which have significantly improved the quality of our paper.

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