Advances in High Energy Physics

Volume 2018, Article ID 7183608, 8 pages

https://doi.org/10.1155/2018/7183608

## Mutual Correlation in the Shock Wave Geometry

^{1}School of Material Science and Engineering, Chongqing Jiaotong University, Chongqing 400074, China^{2}College of Science, Agricultural University of Hebei, Baoding 071000, China^{3}School of Mathematics, Sichuan University of Arts and Science, Dazhou 635000, China^{4}School of Science, Hubei University for Nationalities, Enshi, 445000, China

Correspondence should be addressed to Xiao-Xiong Zeng; moc.361@scisyhpgnezxx

Received 11 March 2018; Revised 13 June 2018; Accepted 25 June 2018; Published 13 August 2018

Academic Editor: Elias C. Vagenas

Copyright © 2018 Xiao-Xiong Zeng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

We probe the shock wave geometry with the mutual correlation in a spherically symmetric Reissner-Nordström AdS black hole on the basis of the gauge/gravity duality. In the static background, we find that the regions living on the boundary of the AdS black holes are correlated provided the considered regions on the boundary are large enough. We also investigate the effect of the charge on the mutual correlation and find that the bigger the value of the charge is, the smaller the value of the mutual correlation will be. As a small perturbation is added at the AdS boundary, the horizon shifts and a dynamical shock wave geometry form after long time enough. In this dynamic background, we find that the greater the shift of the horizon is, the smaller the mutual correlation will be. Especially for the case that the shift is large enough, the mutual correlation vanishes, which implies that the considered regions on the boundary are uncorrelated. The effect of the charge on the mutual correlation in this dynamic background is found to be the same as that in the static background.

#### 1. Introduction

Butterfly effect is ubiquitous phenomenon in physical systems. One progress on this topic recent years is that it also can be addressed in the context of gravity theory [1–15] with the help of the AdS/CFT correspondence [16–18]. In this framework, one can define the so-called thermofield double state on the boundary of an eternal AdS black hole [19]. As a small perturbation with energy is added along the constant trajectory in the Kruskal coordinate to one of the boundary at early time , one finds that a bound of infinite energy accumulates near the horizon and a shock wave geometry forms at , which is the so-called butterfly effect in the AdS black holes [20]. The evolution of the shock wave is dual to the evolution of the thermofield double state according to the intercalation of the AdS/CFT correspondence. The mutual information, defined by is often used to probe the effect of the shock wave on the entanglement of the subsystems and living on the boundary [20], where , are the entanglement entropy of the space-like regions on and , which can be calculated by the area of the minimal surface proposed by Ryu and Takayanagi [21], while is the entanglement entropy of a region which cross the horizon and connects and .

There are two important quantities characterizing the butterfly effect. One is the scrambling time, which takes the universal form [20]: where is the black hole entropy and is the inverse temperature. The scrambling time is the time when the mutual information between the two sides on and vanishes. The other is the Lyapunov exponent , which has the following bound [22]: and the saturation of this bound has been suggested as the criterion on whether a many-body system has a holographic dual with a bulk theory [22]. A remarkable example that saturates this bound is the Sachdev-Ye-Kitaev model [22].

In the initial investigation, the dual black hole geometry is the nonrotating BTZ black hole [20]. The area of the minimal surface equals the length of the geodesic on the boundary. The mutual information thus is defined by the geodesic length. In this paper, we intend to study butterfly effect in the 4-dimensional Reissner-Nordström AdS black holes. Though the area of the minimal surface does not equal the length of the geodesic, we want to explore whether there is a quantity defined by the length of the geodesic that can still probe the butterfly effect. We define this quantity as mutual correlation: in which and are two points on the left and right boundaries, , are the space-like geodesic that go through points and , respectively, and is the geodesic length cross the horizon and connects and . The results are not expectable since we cannot view simply the mutual correlation as the spatial section of the mutual information by fixing some of the transverse coordinates. The metric components of the transverse coordinates are not one but the functions of the radial coordinate so that they have contributions to the area of the minimal surface.

In the 4-dimensional space-time, though the geodesic length does not equal the area of the minimal surface, it has been shown that both the geodesic length and area of the minimal surface, which are dual to the two point correlation function and entanglement entropy respectively, are nonlocal probes and have the same effect as they are used to probe the thermalization behavior and phase transition process [23–40]. Thus it is interesting to explore whether the mutual correlation can probe the butterfly effect as the mutual information for both of them are defined by the nonlocal probes.

In [1], the author has probed the shock wave geometry with mutual information in the 4-dimensional plane symmetric Reissner-Nordström AdS black branes. They have obtained some analytical results approximately and found that for large regions the mutual information is positive in the static black hole, and the mutual information will be disrupted as a small perturbation is added in dynamic background. In this paper, we will employ the mutual correlation to probe the shock wave geometry in the 4-dimensional spherically symmetric Reissner-Nordström AdS black holes. Our motivation is twofold. On one hand, we intend to give the exact numeric result between the size of the boundary region and mutual correlation as well as the perturbation and mutual correlation. One the other hand, we intend to explore how the charge affects the mutual correlation in cases without and with a perturbation. Both cases have not been reported previously in [1].

Our paper is outlined as follows. In Section 1, we will construct the shock wave geometry in the Reissner-Nordström AdS black holes. In Section 2, we will study the mutual correlation in the static background. We concentrate on the effect of the boundary separation and charge on the mutual correlation. In Section 3, we will probe the butterfly effect with the mutual correlation in the dynamical background. We concentrate on studying the effect of the perturbation and charge on the mutual correlation. The conclusion and discussion are presented in Section 4. Hereafter in this paper we use natural units () for simplicity.

#### 2. Shock Wave Geometry in the Reissner-Nordström AdS Black Holes

Starting from the action,one can get the Reissner-Nordström AdS black holes solution. For the case , we havein which , where is the mass and is the charge of the black hole.

In order to discuss the butterfly effect of a black hole, one should construct the shock wave geometry in the Kruskal coordinate firstly. We will review the key procedures and give the main results as done in [20] for the consistency of this paper though there have been some discussions on this topic.

The event horizon, , of the black hole is determined by . With the definition of the surface gravity, , we also can get the Hawking temperature , which is regarded as the temperature of the dual conformal field theory according to the AdS/CFT correspondence. In the Kruskal coordinate system, the metric in (6) can be rewritten asin whichwhere , , are the Eddington coordinate, which are defined by the tortoise coordinate . We will suppose at the right exterior as in [20]. As approaches to the event horizon and boundary, we know approaches to and 0, respectively. Thus from (9), we know that the event horizon and boundary locate at and , respectively.

Next we will check how the space-time changes as a small perturbation with asymptotic energy is added on the left boundary at time following a constant trajectory. We label the Kruskal coordinate on the left side and right side as and . The constant trajectory propagation of the perturbations implies To find the relation between and , we will employ the following relation: Generally speaking, for the energy of the perturbation is much smaller than that of the black hole mass . On the other hand, we are interested in the case , which implies . In this case, we can approximate for there is a relation . In this case, , here . So we have the identification where we have used the relation . From (12), we know that there is a shift in the Kruskal coordinate as the small perturbation is across the horizon of the black hole. For computations, the shift in is often written as , where is a step function. In this case, (7) changes into a standard shock wave: in which we have used the relation and the replacement The Kruskal diagram for the perturbed space-time is shown in Figure 1.