# Holographic complexity of Born–Infeld black holes

## Abstract

In this paper, according to CA duality, we study complexity growth of Born–Infeld (BI) black holes. As a comparison, we study action growth of dyonic black holes in Einstein–Maxwell gravity at the beginning. We study action growth of electric BI black holes in dRGT massive gravity, and find BI black holes in massive gravity complexify faster than the Einstein gravity counterparts. We study action growth of the purely electric and magnetic Einstein–Born–Infeld (EBI) black holes in general dimensions and the dyonic EBI black holes in four-dimensions, and find the manners of action growth are different between electric and magnetic EBI black holes. In all the gravity systems we considered, we find action growth rates vanish for the purely magnetic black holes, which is unexpected. In order to ameliorate the situation, we add the boundary term of matter field to the action and discuss the outcomes of the addition.

## 1 Introduction

Holographic principle relates boundary CFT to bulk theory of gravity, through the correspondence one can study the problems of strong coupling CFT on the boundary through studying weak coupling gravity in the bulk. Remarkable progress has been made in applications of holographic principle in recent years, including applications of holography to study low energy QCD, hydrodynamics, condensed matter theory [1, 2, 3, 4, 5], etc.

Recently, the combination study of holography and quantum information shed light on understanding of quantum gravity. In the initial work [6], Maldacena and Susskind found any pair of entangled black holes are connected by some kind of Einstein–Rosen bridge, i.e., ER = EPR. However, the ER = EPR duality does not tell how it difficult to transmit information through Einstein–Rosen bridge. Therefore, the concept complexity was introduced. Complexity is the minimal number of simple gates needed to prepare a target state from a reference state. Complexity was originally conjectured to be proportional to the maximum volume of codimension-one surface bounded by the CFT slices, \(\mathcal {C}=\frac{V}{Gl}\), which is called CV duality [7, 8, 9, 10, 11, 12, 13, 14]. The length scale *l* is chosen according to situations. In order to eliminate the ambiguities in CV duality, CA duality was proposed [15, 16], which states that complexity is proportional to the action in Wheeler–DeWitt (WDW) patch, \(\mathcal {C}=\frac{I}{\pi \hbar }\). CA duality does not involve any ambiguities encountered in CV duality but preserves all the nice features of CV duality. CA duality have passed the tests of shock wave and tensor network.

It is natural to study complexity growth of black holes in different gravity systems to examine CA duality and Lloyd’s bound. For the progresses in this subject please refer to [16, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42]. In this paper, we intend to study complexity growth of BI black holes, since we are interested in the effects of nonlinearity of BI theory on complexity growth. The inner horizon of a BI black hole may turn into a curvature singularity due to perturbatively unstability [23], which implies a BI black hole may possess a single horizon [43, 44, 45]. It’s interesting to study the differences in complexity between BI black holes with single horizon and AdS-Schwarzschild black holes, although the casual structures of them are identical. Since the magnetic black holes have been studied rarely, in this paper we will pay much attention to the magnetic black holes, and make a comparison of the effects between electric and magnetic charges on action growth. As we will see in the following, action growth of magnetic BI black holes exhibit some specific properties that are not found in the electric ones. We are also interested in studying action growth of BI black holes in massive gravity and study the effects of graviton mass.

Recently, the authors of Refs. [46, 47, 48] found that, action growth rates vanish for purely magnetic black holes in four dimensions. Which is unexpected since the expected late-time result \(\frac{dI}{dt}\sim TS\) and electric-magnetic duality cannot be restored. Similar result was also found in this paper. In order to ameliorate the situation, a boundary term of matter field was proposed to be included to the action. In this paper, we add the boundary term of matter field proper to the gravity systems we considered and discuss the outcomes of the addition of the boundary term.

The paper is organized as, we study action growth of dyonic black holes in Einstein–Maxwell gravity in Sect. 2, BI black holes in massive gravity in Sect. 3, and EBI black holes in Sect. 4. In all the gravity systems we considered, we add the boundary term of matter field and discuss the outcomes of the addition. We summarize our calculations in the last section.

## 2 Dyonic black holes of Einstein–Maxwell gravity

*K*is the Gibbons-Hawking term. \(\kappa \) measures the failure of \(\lambda \) to be an affine parameter on the null generators. \(\eta _{j_i}\) is the joint term between non-null hypersurfaces. \(a_{m_i}\) is the joint term between null and other types of surfaces. The signatures \(sign(N_i), sign(j_i), sign(m_i)\) are determined through the requirement that the gravitational action is additive.

*k*being the null normal to the hypersurface \(v=const\) and \(\bar{k}\) being the null normal to the hypersurface \(u=const\). For the affinely parametrized expressions \(k_\alpha =-c\partial _\alpha v\) and \(\bar{k}_\alpha =\bar{c}\partial _\alpha u\), we have \(a=-\ln \left( -f/(c\bar{c})\right) \), therefore \(h(r)=-r^{d-2}\ln \left( -f/(c\bar{c})\right) \). Using \(dr=-\frac{1}{2}f\delta t\), we obtain

## 3 BI black holes in massive gravity

*f*is a fixed symmetric rank-2 tensor. \(c_i\) are constants and \(\mathcal {U}_i\) are symmetric polynomials of the eigenvalues of matrix \(\mathcal {K}^\mu _{\;\nu }\equiv \sqrt{g^{\mu \alpha }f_{\alpha \nu }}\)

It is easy to note that, when \(q\rightarrow 0\), action growth rate (42) reduces to the one of AdS-Schwarzschild black holes, for which Lloyd’s bound is saturated (shown by the blue line of the upper plot in Fig. 2). Although the causal structure of single-horizoned BI black holes is identical with the one of AdS-Schwarzschild black holes, the manners of action growth differ between the two types of black holes due to the presence of electromagnetic field. Therefore, BI electromagnetic field slows down complexification of the black holes.

The lower left plot in Fig. 2 shows the case that there exists an extremal black hole for the selected parameters, the left most point *A* represents the extremal black hole for which the inner and outer horizons merge. The difference between the line above *A* and the line under *A* is action growth rate of the black hole with double horizons. The lower right plot in Fig. 2 shows the case that no extremal black hole exists for the selected parameters. It is easy to see that action growth rates increase as *m* increases.

After addition of the BI boundary term, action growths versus *m* are presented in Fig. 3 for different values of \(\gamma \). The straight blue line on the upper plot in Fig. 3 still corresponds to the Lloyd’s bound. The straight red line shows action growth of the black hole with single horizon for \(\gamma =1\). The difference between the straight blue line and the straight red line is just \(C_1\). The three points *A*, *B*, *C* on the lower left plot in Fig. 3 still represent the extremal black holes. From the three plots in Fig. 3, one sees that, action growth rates decrease as \(\gamma \) increases, and that action growth rates increase as *m* increases.

## 4 EBI black holes

*r*. The Einstein equations (55) imply the dyonic black hole solution in general dimensions [53]

*n*, let’s consider the following special cases.

### 4.1 Purely electric EBI black holes

*G*(

*r*) in (60) with the magnetic charge parameter \(p=0\). Action of the \(r=0\) surface is

*q*. Since it is difficult to work out

*q*as \(q(r_{+})\), and subsequently \(\frac{\delta I}{\delta t}\) as \(\frac{\delta I}{\delta t}(r_{+})\), from (59) for fixed \(\mu \), we fix \(r_{+}\), then

*M*and \(\frac{\delta I}{\delta t}\) are functions of

*q*. The black line on each plot in Fig. 4 is the 2

*M*line. From Fig. 4, one sees that action growth rates decrease as

*q*or \(\gamma \) increases, Lloyd’s bound is satisfied. Note that \(\frac{\delta I}{\delta t}\) will not be zero as

*q*increases, since for certain value of the black hole mass

*M*,

*q*can’t be arbitrarily large to ensure the existence of the black hole horizon.

### 4.2 Pure magnetic EBI black holes

In this subsection, we calculate action growth of purely magnetic EBI black hole in general dimensions and make a comparison between the effects of electric and magnetic charges on action growth.

*p*, it does not depend on the product of magnetic charge and potential \(Q_m\Phi _m\). This differs from the purely electric case (64), where the action growth depends on mass, some constant and the product \(Q_e\Phi _e\). By comparison of the first line of (77) and the first line of (64), we see that, for electric EBI black holes, the term corresponding to \(\bar{H}(r_{+})\) disappears since

*p*vanishes, therefore the term \(Q_e\Phi _e\) in action growth remains. The calculation details of simplifying Eq. (77) from the first line to the second line can be found in the appendix.

*n*to be odd, while \(\bar{C}\) vanishes for

*n*to be even. For odd

*n*, according to CA duality we have \(\frac{d\mathcal {C}}{dt}<\frac{2M}{\pi \hbar }\), Lloyd’s bound is satisfied. For even

*n*, we have \(\frac{d\mathcal {C}}{dt}=\frac{2M}{\pi \hbar }\), Lloyd’s bound is saturated. Therefore, if we don’t consider the boundary term of electromagnetic field, action growth of magnetic EBI black holes is in the same manner as the one of AdS-Schwarzschild black holes in some dimensions. In this case, magnetic charge affects action growth through back-reaction on the geometry.

*n*. Therefore, in the dimensions with even

*n*, action growth of magnetic EBI black holes takes the specific form \(\frac{\delta I}{\delta t}=2M-\gamma Q_m\Phi _m\).

Figure 5 shows us \(\frac{\delta I}{\delta t}\) versus *p*. Similar to the electric case, we fix \(r_{+}\), the black line on each plot in Fig. 5 is the 2*M* line. On the left two plots, the black lines coincide with the red lines, this is because \(\bar{C}_1\) in (84) vanishes for even *n*. One sees from the figure that, Lloyd’s bound may be violated for some values of \(\gamma \). Therefore, the addition of matter field boundary term to action may lead to violation of Lloyd’s bound. The late-time violations of Lloyd’s bound have also been found in Einstein-dilaton system, in Einstein–Maxwell–Dilaton system [26, 36], etc. Note also that, \(\frac{\delta I}{\delta t}\) will not be zero as *p* increases, since *p* can’t be arbitrarily large for certain *M* to ensure the existence of black hole horizon.

### 4.3 Four-dimensional dyonic EBI black hole

Since the integral (60) can not be integrated out for general *n*, for simplicity we only consider the \(n=1\) case, i.e., the dyonic black hole in four dimensions.

*q*and \(\frac{\delta I}{\delta t}\) versus

*p*. It is easy to see that, \(\frac{\delta I}{\delta t}\) decreases with

*q*(or

*p*) for fixed

*p*(or

*q*), and Lloyd’s bound is satisfied. Note that, on the left plot in Fig. 6, the lines with larger \(\gamma \) are lower than the lines with smaller \(\gamma \), while, on the right plot in Fig. 6, the lines with larger \(\gamma \) are upper than the lines with smaller \(\gamma \). This is because the coefficient before \(Q_e\Phi _e\) in Eq. (96) is \((1-\gamma )\) while the coefficient before \(Q_m\Phi _m\) in Eq. (96) is \(\gamma \).

## 5 Summary and discussion

In this paper, we study action growth of BI black holes. As a comparison, we first review action growth of dyonic black holes in Einstein–Maxwell gravity in general dimensions, and notice that similar to the four-dimensional case, action growth rates vanish for purely magnetic black holes if we don’t consider the Maxwell boundary term. After the inclusion of Maxwell boundary term, if we set \(\gamma =\frac{1}{2}\), then electric and magnetic charges contribute to action growth on equal footing, which is in accord with electric-magnetic duality. If we set \(\gamma =1\), then action growth rates vanish for purely electric black holes.

We study action growth of electric BI black holes in massive gravity, and find that BI black holes in massive gravity always complexify faster than their Einstein gravity counterparts due to the back-reaction of graviton mass on geometry. If the BI boundary term is included, we find that action growth rates decrease as \(\gamma \) increases.

For the EBI black holes, we first study action growths of the purely electric and magnetic black holes in general dimensions. Before considering the BI boundary term, action growths of electric and magnetic EBI black holes with single horizon are in different manners, which are \(\frac{\delta I}{\delta t}=2M-Q_e\Phi _e-\hat{C}\) and \(\frac{\delta I}{\delta t}=2M-\bar{C}\) respectively. We notice that, for the magnetic black holes, the constant \(\bar{C}\) vanishes for even *n*. Therefore, in the dimensions with even *n*, action growth of magnetic EBI black holes saturates Lloyd’s bound, i.e., it takes the identical form with that of the AdS-Schwarzschild black holes. In this case, magnetic charge affects action growth through back-reaction on the geometry. For EBI black holes with double hirizons, action growth of the electric ones takes the identical form with that of AdS-RN black holes, while action growth vanishes for the magnetic ones, this agrees with the result obtained from the dyonic black holes in Einstein–Maxwell gravity.

If we include the BI boundary term to the action, action growth of the electric (magnetic) EBI black holes with \(\gamma =1\) takes similar form with that of the magnetic (electric) ones with \(\gamma =0\). We find that, for electric EBI black holes with single horizon Lloyd’s bound is satisfied, while, for the magnetic ones Lloyd’s bound may be violated for some values of \(\gamma \). Therefore, the inclusion of BI boundary term to action may lead to the violation of Lloyd’s bound.

For the four-dimensional dyonic EBI black hole, calculation shows the manner of action growth of the black hole agrees with that of the purely electric and magnetic EBI black holes in general dimensions. When we set \(\gamma =\frac{1}{2}\), electric and magnetic charges contribute to action growth on equal footing. Lloyd’s bound is satisfied for the four-dimensional dyonic EBI black hole.

## Notes

### Acknowledgements

KM would like to thank Profs. Peng Wang, Haitang Yang, Liu Zhao and Haishan Liu for valuable discussions.

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