JHEP07(2017)082 Springer June 7, 2017 July 10, 2017 July 17, 2017 April 18, 2017 : : : : Revised Accepted Published Received d Published for SISSA by https://doi.org/10.1007/JHEP07(2017)082 [email protected] , and Hai-Qing Zhang a,c [email protected] , . 3 Xiao-Xiong Zeng 1704.03989 The Authors. a,b c AdS-CFT Correspondence, Black Holes, Classical Theories of Gravity, Ther-

We study the effect of inhomogeneity, which is induced by the graviton mass in , [email protected] Beihang University, Beijing 100191, China E-mail: School of Physical Sciences, UniversityBeijing of 100049, Chinese China Academy ofSchool Sciences, of Material ScienceChongqing and 400074, Engineering, Chongqing China Jiaotong University, Department of Space Science and International Research Institute of Multidisciplinary Science, CAS Key Laboratory of TheoreticalChinese Physics, Academy Institute of of Sciences, Theoretical Beijing Physics, 100190, China b c d a Open Access Article funded by SCOAP mal Field Theory ArXiv ePrint: the mutual information more rapidly,Besides, which the resembles greater the inhomogeneity reduces butterfly thejust effect dynamical as mutual in information in chaos more theory. the quickly static case. Keywords: field theory from themutual gauge/gravity information duality. increases as When thefrom graviton the mass homogeneity, system grows. the is However, mutual when near-homogeneous,adding information the the the decreases system perturbations is as of far theinformation energy graviton in into mass the the increases. system, shock we wave By investigate geometry. the We dynamical find mutual that the greater perturbations disrupt Abstract: massive gravity, on the mutual information and the chaotic behavior of a 2+1-dimensional Rong-Gen Cai, Influence of inhomogeneities oninformation holographic and mutual butterfly effect JHEP07(2017)082 ]. 4 9 (1.1) 3 11 7 7 , R i i ⊗ | L i i | i 13 ]. The typical example is the holographic E 3 2 β – 15 1 – 1 – − e 5 i X 16 i ≡ Ψ ]. Among these studies, one interesting thing is to adopt | 4 6 , 5 10 ], the authors defined the thermofield double states (TFD) to 20 ]. The holographic setup of mutual information can be obtained by 18 1 ]. In [ 21 – ]. 19 17 – 7 Mutual information measures the correlations between two subsystems in quantum 3.2 Effects of3.3 the graviton mass The and critical black width hole charge on the mutual information 4.1 Black hole4.2 charge and butterfly Graviton effect mass and butterfly effect 2.1 Shock wave2.2 geometry Holographic mutual information 3.1 Effect of the width of the strip on the mutual information calculating the length ofblack the hole wormhole [ whichdescribe connects the the states of two the sides entangled of two sides an of eternal the AdS black hole, i.e., views, please refer tothe refs. HEE [ to probesystems the [ thermalization process or phase transitions in theinformation strongly theory coupled [ entanglement entropy (HEE) proposed bycodimension-2 Ryu and bulk Takayanagi that surface relates toThe the the minimized Ryu-Takayanagi quantum conjecture information greatly inentropy simplifies the in the boundary the computations field of fieldbulk theory the geometry theory, [ and entanglement and the numerous quantum papers information in investigates the the boundary relation field between theory. the For recent re- 1 Introduction Over the past twotheory, decades, condensed matter there physics, havejecture quantum been information of a and gauge/gravity large geometry correspondence based overlap [ on studies the among con- high energy 5 Conclusions and discussions 4 Holographic mutual information in the dynamical background case 3 Holographic mutual information in the static background case Contents 1 Introduction 2 Reviews of shock wave geometry and holographic mutual information JHEP07(2017)082 B ) is and (1.2) B A and ∪ ) can be A A ( B ( ]. S S 29 – by Shenker and 24 ) and ) can be computed ) between , usually called the There have been a A ∗ ( B t S 1 ∪ A, B ] that the graviton mass ( A I ( 34 , S – ) respectively, while butterfly effect 32 B are the identical quantum states ∪ B R i A i ( | and S , which means there is no dependence A − ∗ and is the Bekenstein-Hawking entropy. The t . Holographically, ) L B S B i i ( | S ]. and – 2 – ) + 22 A [ A ( ]. As we know from [ B S , in which π 31 ≡ , 2 and ) / ) 30 A S A, B ( log( I β ∼ ]. ∗ 43 t – ) are the entanglement entropy of 35 B ( S ]. Specifically, as a small perturbation, such as a light-like energy perturbation, ), 23 on each side of the black hole, the mutual information A is the inverse of the temperature while ]. To be consistent with existing literatures, here we still take the term “inhomogeneity” to ( B S β 34 Firstly, we study the holographic mutual information of two identical strips in a static In this paper, we intend to study the static and dynamical mutual information in the The disruption of mutual information was related to the Strictly speaking, the term “inhomogeneity” used here represents the meaning of translational symmetry independently from the Ryu-Takayanagi conjecture, while 1 and the mutual information for the strips with shorter length willbreaking increase or with respect momentum topaper dissipation, the [ which wasrepresent used the same previously meaning. in the framework of holography in the background of a 3+1-dimensionalmutual massive information gravity. between them For decreases thehowever, monotonically the strips as the with mutual graviton information larger masscreases length, between increases; with two the respect shorter to strips(or the equivalently first spatial graviton increases inhomogeneity) mass. would and havelength. greater then A When effects de- the plausible on system the reason is strips is near-homogeneous with (or that larger equivalently the with graviton small graviton mass mass), the graviton mass plays the rolegraviton of mass inhomogeneity in is the boundarynumber dual of field to papers theory, i.e., investigating greater the greater inhomogeneoussee inhomogeneity effects for caused in example by [ the massive gravity boundary. so far, background of massive gravity [ in the bulk breaksnon-conserved the in diffeomorphism the dual invariance,causes which field the momentum makes theory. dissipation the in The the stress-energy non-conservation boundary tensor of field theory. the Therefore, stress-energy from tensor this aspect scrambling time in blackthe hole black systems, hole, is e.g., proportionaldisruption to of the the logarithm mutual information ofto is the the a entropy piece initial of of conditions,effect. evidence which of Recent the literatures reminds system’s relevant us to high the of sensitivity butterfly the effect terminology can in be found chaos in theory-butterfly [ Stanford [ is added to onesides side may be of disrupted theor after eternal entanglement an black between amount the hole, of two the time sides mutual of information the between black the hole. two The time the entanglement entropy of the unioncalculated of by the areas ofB the minimal surfacesfrom in the the bulk minimal geometry surfaceconnecting associated which the to crosses two subregions the horizon and stretches through the wormhole on the two-sided AdS blackA holes. Supposing there arecan two be identical computed space-like as subregions where where JHEP07(2017)082 ]. 24 we study the dy- . Through out this 4 5 and discussions in the 4 . In section 3 we briefly review the dynamical holo- 2 – 3 – = 1) for simplicity. ]. However, as the graviton mass grows big enough, ~ 24 = c = G . 3 ]. We find that as the shift grows (more added perturbed energy at the 44 This paper is organized as follows: in section In order to study the dynamical behavior of mutual information in the boundary field In all the parameter regimes we considered in this paper, we find that the critical width 2 Reviews of shockThe wave butterfly geometry effect and of a holographic blackwe mutual hole are is information going usually studied to inFor briefly the simplicity, introduce shock we wave the will geometry, therefore, key adopt ingredients a of planar the symmetric shock black wave hole geometry in at 3 first. + 1 dimensions with the graphic mutual information in theinformation shock in wave the geometry. massive We gravitynamical investigate background the mutual in static information section mutual bygravity. adding energy Finally perturbations we into drawpaper the our we bulk conclusions use of natural and the units discussions massive ( in section of the phenomenon inwe chaos also theory find — that the butterflyindicates greater effect. that the the mass By spatial is, turning inhomogeneity thestatic will on smaller reduce case the the the mentioned mutual graviton mutual information above. mass, information will just be, like which in the will lead toinformation a on shift the on two sidesperturbations the of can horizon the be black of hole. convenientlycoordinates the investigated The in [ black dynamics the of hole. shock theinitial bulk wave time), This geometry after the adding with shift mutual the Kruskal will information will affect be the reduced mutual more significantly, which reminds us know, in the massive gravityincreases. the temperature Therefore, of we the cantemperature black of infer hole the will that decrease black the decreases, as critical which the width is charge consistent of with thetheory, the strip one results increases of obtained when the in the [ approaches is to add extra energy perturbations into the bulk, which case is given in section of the strip, whichto renders the the graviton mutual mass. informationthat Moreover, vanishing, the as always critical decreases the according width charge of which the disrupts boundary the strip mutual increases, information it increases is as found well. As we role in decreasing thehand, mutual if information the compared strips are withthe longer, the graviton the mass temperature. mutual regardless information of On decreases thealy monotonically the system suggests according being other to that near or the far spatiallarger from length inhomogeneity homogeneity. by would It destroying have intuition- the greater mutualthe impacts information relation to monotonically. between the A mutual strips detailed information discussion with of and the graviton mass in the static background main context), which is similarture in to the the pure relation homogeneousi.e., between case far mutual from [ information homogeneity, the and mutualgraviton tempera- information mass, will instead which decrease behaves withthat distinctly respect the from to greater the the graviton mass homogenous (or case. stronger inhomogeneous Therefore, effects) we would argue play a dominant temperature of the black hole (referring to the right panel of figure JHEP07(2017)082 2 µ E /  h r r (2.2) (2.3) (2.4) (2.1) | . ) ν ]. The r ( 0 44 f = 0 κ  approaches the r r ) we know that the h 0 ,  2.4 , κt r 2 , in which , κV − e e represents the radial direc- ) π 1, respectively. The light is 0 in the left exterior region , 2 2 ∓ − r ) − 2 dy κ/ , where a dot represents a two = = = Ν dy + 1 , ν < = 0 2 + ]. Supposing that at the boundary µν ). Therefore, as T  2 dx r 44 r ( ( Μ 2 µ > dx ( r 2 r ]. Thus from eq. ( + dr/f and 0 respectively. The Penrose diagram ) = 0 and + 24 2 R r ( dr µν f  = −∞ – 4 – r dµdν + ? r ) 2 r ( dt µν ) f are the Eddington-Finkelstein coordinates, which can r 2 ( 1 ? κ , ν tends to , µ/ν f r = constant. ? the horizon radius. From the AdS/CFT dictionary, the ? − κU = + ν κr r h − 2 2 t = r e e 2 ds  − = ) directions. We suppose 0 ds = = 0 V  x, y  µ r r µν ) is shown on the left panel of figure and 0 in the right region as in [ 2.2 ) can be rewritten as ? is the shift on the horizon between the left and right Kruskal coordinate r can be regarded as the temperature of the dual field theory. h 2.1 = constant and − ) are the transverse directions in the bulk while T µ , ν > t 0 x, y = . Penrose diagrams for an eternal black hole without (left panel) and with (right panel) we add a light-like perturbation of the energy, which goes along a constant U w µ < t The shock wave geometry can be obtained by adding a small perturbation of energy  r event horizon and boundary aregoing located along at into one side, for instancetime left side of the black hole [ where be defined byevent the horizon tortoise and coordinate boundary, of the geometry ( dimensional space in ( while in which The shock wave geometrymetric is in conveniently eq. discussed ( in the Kruskal coordinates [ tion. The Hawking temperatureis of the this surface blacktemperature gravity hole with is 2.1 Shock wave geometry line element, in which ( Figure 1 a perturbation. JHEP07(2017)082 ) ) y h is B c x ( ], in A δ (2.5) (2.6) (2.7) ) S ) + ) µ h ( h r δ r ( w − t c r π β 2 log[ Ee . Therefore, we π β (log( c 2 e κ 1 ∼ 2 (1) for in this case ). The difference ∼ ]. so that the left and = O µµ ? ≈ of the subregion t 45 2.7 B C ? ∼ δT ). We will consider the A r )[ ], one often employs the = S . h h ) and the first law of the r ) and extends along the 23 x, y ( ?R A h , 0 r c 2.7 R trajectory of the perturbation + , x κ 2 µ ), where when R (0 e h ν r w − , in which t ∈ ≡ ]. Our objective is to compute the − . of the perturbation is much smaller µµ = . ) x r w 1 20 ( R δT E hR κt ν C r − R = e , we employ the relation − = R µν ) is still finite in eq. ( = ? ν hL G κr R r hR – 5 – 2 ( ) is a step function. The shock wave geometry is r µ , µ e , an eternal black hole has two asymptotic AdS w µ and has been used. It should be stressed that even − ?L = κt 1 r L on the left asymptotic boundary and its partner L L ν C ]. On the basis of eq. ( hL . On the other hand, we are interested in the case κ 1 in the numerics. Ce µ r 2 for the energy A ( e 23 ≡ = M + w κ − Y R κt R ν e . In this case, we can approximate = = , respectively. The constant C h ), where Θ( is defined as the value of R r = L R µ ν = κ Therefore, the entanglement entropy ? L , ν t L → ν R µ 2 L )Θ( = . r µ θ C ( L Y h κ is the shift close to the horizon. The Penrose diagram of the shock and as a strip, which has the width + R ν L ν B 0, the formula , ν → L or ν µ → and ) ) is a function of the black hole horizon. In various gravity models, this form is A L h ν r hR , which implies ( r a constant of integration. Hence, c c − We are interested in the 3+1-dimensional planar black holes, thus the AdS bound- The scrambling time To get the standard shock wave geometry as showed in [ Without loss of generality, we set → ∞ 2 hL r w ary has a 2-dimensionalsubregion space parameterized by coordinatesdirection ( with length regions, which can beidentical, holographically non-interacting described conformal field bymutual the theories information so-called [ of a TFDon subregion states the of right the asymptoticright boundary. two boundaries For are simplicity, identical. we will let which universal and the only difference is embodied in2.2 the function Holographic mutual information As depicted on the left panel of figure strictly a solution tois the the Einstein boost equation energy arising from the null perturbationthe at mutual the information initial vanishes time. black [ hole thermodynamics, the scrambling time can be written as ( between wave geometry is shown on the right panel ofreplacement figure with have the identification in which the relation Generally speaking, than the mass oft the black hole right side as implies In order to find the relation between trajectory, into the left boundary. We label the Kruskal coordinates on the left side and JHEP07(2017)082 , B B B ∪ ∪ A A (2.8) (2.9) (2.12) (2.13) (2.10) (2.11) (2.15) (2.14) S − , one can B L S (left) and + A . . A  f S r . f √ r . As stressed in the  √ 4 A ) = , ) dr 0 2 dr x , /r r ∞ ( h ], the total area Area r . 1 ∞ I h + r Z 4 min 24 − Z ) 2 r 0 Y ( 4 r where the mutual information Y /r ) 1 . − 4 c − ) as a ‘Lagrangian’ − 1 0 1 4 -direction, that is 1 min min f − x − ) x r 1 = 0. According to the symmetry f 2.8 , ( p 4 p /r 0 ) 2. With the help of the conserved r/r f p r / − ( r 2 min . Therefore, our next step is to find 0 /r √ min 1 r affects the mutual information. Substi- dx r x p r B ( dr r 0 0 p min dr = x f x r 1 = 0 − ∞ f h ( 2 √ dr r r 0 Z ∞ 1 min x and r √ r Z r ) together, we have − 1 Y Z p 1 Y − A – 6 – dr ∞ min f = f − r 3 r p 2.12 Z r √ γ ∞ = 4 min + f f r r r √ 2 √ Z B √ dr r = 2 ∪ thus takes the same form as Area Y A dr p dr ∞ min ) and ( B r dx dydx ) can be rewritten as ∞ h Z 0 = 2 ∞ r min x ), we obtain r Z Area Z 2.11 2.8 Y 0 is the area of the minimal surface in the bulk, i.e., A Z Z  = A + , Area Y , which is the minimal surface connecting regions ), ( 2.13 = A 4 A 2 B 2 min Area 0  r ∪ 2 min 1.2 x r 1 4 A 0 = Area c ) = 0, which leads to 0 can be written as Y x c . If regarding the integrand in eq. ( ) = 0 x 0 1 2 0 x x ) into eq. ( x ( ( I I 4, where Area ), ) = dr/dx / is the turning point of the surface with 0 2.10 A x 2.9 ( = is identical with I 0 min r r or the area Area B B = Area ∪ A A It is of great interest tovanishes, find i.e., the critical value of the width We are interested in howtuting the eq. width ( of the strip Combining all the eqs.( to study the correlationS between regions (right) by passing through the horizonincluding of both the sides black of hole. the From [ horizons can be expressed as Since introduction, we will employ the mutual information, defined by equation ( and the minimal area in eq. ( where of the surface, the turning point locates at where define a conserved quantity associated to translation in S JHEP07(2017)082 2 ). is c µν (3.2) (3.3) (3.4) (3.1) 2.10 f and 1 ]. Diffeo- . In 3+1- c , αµ 31 0 , , f c  ) 30 µα from eq. ( g ρσ p , f min ) one finds that as . r = µν ] g 4 µ , ( µ i 2.10 2 K U K i m 6[ . c 2 is the field strength, 2 c ) can have the gravitational − 2 0 µ i ] = 4 2 m c π ] X A 1 2.1 K 2 4 ν c 2 + 0 ∂ K is also correct, since as we know r c m 2 − 2 and [ + + ν m . From eq. ( λ 1 2 2 ] + 3[ A µ µν c 2 µ 0 m K A F π ∂ c 2 , ][ 4 c ] 3 = µν 2 0 = 3 + c K F λ are symmetric polynomials of the eigenvalues 2 K ν 2 1 4 ) l µν r i + diverges, which can be seen as well from the – 7 – F U A − + 8[ 0 π 2 + √ 2 2 x 6 ] + 2[ ] Q 2 ( l 2 16 2 r ν K µ Q 4 K + ) ][ − , ][ 2 ] : A + R 2 π 3 K K 2 in order to render the background thermodynamically  4 √ αν K / g r [ 3[ 6[ f 1 M ]. Therefore, from this sense the graviton mass plays the = − − − − − µα − 34 , √ g 2 3 4 ] ] ] ] = x MG √ 4 K K K K 2 ) = T d c r ≡ ( are constants, and = [ = [ = [ = [ ν Z f i stands for ( µ 1 2 3 4 c are the mass and charge of the black hole respectively; K U U U U = 1, K . Intuitively the left panel of figure πG 1 1 Q 2 c 16 = ] = ]. The temperature of the black hole is readily obtained as 0 and c S 35 , the integral for the width 4 matrix , 39 h , is the graviton mass parameter, M r × 32 35 m → min of the strip inbetween the the width background of ofThe the massive strip relation gravity. and is One ther shown position can of on readily the read the turning offleft left point plot the panel of relation of figure figure 3.1 Effect ofFirstly, the we width are of going the to study strip the on relation the between mutual the information mutual information and the width in which are constant parameters associatedparameters to the gravitonstable mass. [ In this paper, we will fix the The square root in dimensions, a black hole solutionpotential with as line element in eq. ( of the 4 where the reference metric, role of inhomogeneity onthe the effect boundary of field inhomogeneitygoing theory. on to the explore It the holographic is effects mutualout of of information, energy the great perturbations in graviton interest mass in particular, to onfollows the we the investigate [ following are mutual context. information with The and action with- of the massive gravity is as Massive gravity is a deformation ofmorphism the invariance Einstein is gravity broken with in graviton massivethe mass gravity [ stress because of energy theconservation tensor graviton mass. of is Therefore, the not stressdual conserved energy boundary any tensor field more corresponds theory in to [ the a dual momentum field dissipation theory. on the The non- 3 Holographic mutual information in the static background case JHEP07(2017)082 c 6, 0 min . r min x r = 0 1.4 m for 1.2 0 x ] we know that . For both cases, 1.0 24 min r . Right: the relation ), one can see that as 0.8 min r 0 x 2.14 ). From [ 2.0 0.6 2 and ) or ( 0 x 1.8 0.4 2.13 ) and width of the strip 0 0.2 1.6 x ( I  0 25, which indicates that there is a critical x 0.0  . 1.4 I 2.0 1.5 1.0 0.5 0.0 1 ≈ 1.2 – 8 – . In addition, from this subgraph we observe that min r min 2 r 5 1.0 , from which we can clearly see that the critical width ) and the position of the turning point 3 0 0.8 x ( I 13. In addition, we find that the mutual information always 4 . ) rendering the mutual information vanishing. Combining the 1 = 1. 0.6 2 h  ≈ r 0 x  c I 0 0.5 0.4 0.3 0.2 0.1 0.0 x , we plot the relation between the mutual information and the width 3 2 = 2, and explicitly in figure Q = 1. 0 6, is closer to the infinite boundary, therefore, it is obvious that the greater x . h 2 r the mutual information diverges, since in this case the widths of the strips on the . Left: the relation between the width of the strip min = 0 h . The relation between the mutual information r r m → = 2, and 0 1 x min 2.0 1.5 1.0 0.5 0.0 of the strip of the strip is roughly grows as the widthsubsystems of on the the strip two increases, asymptoticgreater which boundaries as is are well. easy larger to and their understand entanglements for becomes in this case the the mutual information foralso divergent be strips found on will the be rightthe panel divergent mutual of information too. figure vanishes where Thisvalue for phenomenon the can position offrom the the left turning panel point of (ortwo figure equivalently panels a in critical figure width of the strip greater corresponds to the smaller widthr of the strip. From eq.two ( boundaries are nearly divergent (cf. the left panel of figure Figure 3 Q we set Figure 2 between the mutual information JHEP07(2017)082 c c m Q m 20 = 1. = 1. h h r r 6, . grows the = 2, ) we see that m = 0 15 Q m 3.4 . In particular, when by fixing 4 10 m while fixing increases when fixing other Q Q we see that for each curve, the 5 4 ). Therefore, from the left panel 3.4  0 x  I 0.08 0.06 0.04 0.02 of the black hole on the mutual information . As we know from the preceding subsection, increasing from 1.21 to 1.27 with steps 0.02; right: ) and the graviton mass 0 Q Q – 9 – x min ( increasing from 1.21 to 1.27 with steps 0.02. r min I r ) and black hole charge 0 3.0 min x r ( I 2.5 and charge also indicates that smaller subregions have smaller mutual m 4 . From the left panel of figure 2.0 4 1.5 is smaller) the mutual information first increases to a maximum value and ] we learn that when the temperature of the black hole grows, the critical 0 24 x 1.0 actually corresponds to smaller width of the strip on the boundary. There- we see that as charge grows (temperature decreases) the mutual information . Left: the relation between 4 min 0.5 r mation grows the temperature of the black hole grows as well when fixing other parameters. is bigger ( ) are shown in figure However, it is interesting to see that the mutual information does not have monotonic From ref. [  0 m 0 x x  I ( min 0.10 0.08 0.06 0.04 0.02 then decreases with respect to thewhich corresponds graviton mass. to the Let’s maximum call valueas of the mutual critical information. graviton mass From as eq. ( Therefore, from the conclusionsmutual in information the preceding should subsection increase it as seems well, that since as the mutual information increases with decreases monotonically, which matches the conclusionsalso mentioned checked before. other Incidentally, cases we of the graviton mass, similardecreasing results behavior were to obtained. the gravitonr mass on the right panel of figure width of the strip decreases.grows It also the means that mutual for amassive information fixed gravity, width the of of temperature the the strip, will asparameters decrease two temperature as of strips the the charge will background,of grow figure please as refer well. to eq. In ( the case of the between the paired subregionsmutual we information considered. is smaller For forbigger a greater small fixed charge,fore, we the find left that panel the information of between figure them, which is consistent with the preceding subsection. The effects of graviton mass I mutual information decreases as thethat charge makes increases. the Besides, mutual there information is vanishing, a which critical means charge that there is no entanglement the relation between the mutual information Curves from top to down correspond to 3.2 Effects of the graviton mass and black hole charge on the mutual infor- Figure 4 The curves from top to down correspond to JHEP07(2017)082 Q 6. . 2.5 = 0 m = 2, we 2.0 Q , the mutual the inhomo- 4 c by fixing charge m 1.5 m < m grows for various black the inhomogeneous effects 1.0 m c , which contradicts the above c that when 0.5 , by fixing the charge . In fact the top curve corresponds m > m while fixing the graviton mass 4 and graviton mass 5 m c Q 0 m > m x decreases with respect to the increasing c c 0 0.0 x 0 0.9 0.8 0.7 0.6 0.5 0.4 x increasing from 1 to 5 with steps 2, respectively. also indicates that the higher temperature h 5 r m – 10 – 20 increasing from 1 to 2 with steps 0.4, respectively. and the charge h r c of the strip). Therefore, it makes us speculate that 0 x 0 . x c 0 decreases as the graviton mass x 15 c grows in the regime 0 increases); however, when x , the critical width m m m of the strip is the width that renders the mutual information van- c 10 0 x (bigger width ) = 0. On the left panel of figure c . This means that the greater inhomogeneity will reduce the critical width since now the inhomogeneous effects dominate. 0 h min x r ( r m I 5 . We know that the greater horizon corresponds to higher temperature of the h . Left: relation between the critical width r Finally, let’s come to the top curve on the right panel which shows that the mutual The possible way coming to rescue is that the conclusion in the previous subsection is . For a fixed value of is big enough. Therefore, we can infer that greater inhomogeneity will spoil the mutual c = 2. Curves from top to down correspond to c 0 0 0 x 0.8 0.6 0.4 0.2 0.0 hole horizons x horizon black hole, therefore, thealso left reduces plot the critical of width figure according to 3.3 The criticalThe width critical width ishing, i.e., see that the critical width information decreases monotonically with respectto to a smaller the inhomogeneity will havethe greater mutual information. effects Therefore, on for longer a strips long than strip the shorter mutual ones information to only decreases reduce geneous effects aretemperature tiny. grows Hence, (i.e., inare significant, this which case will disrupt them the mutual mutual information information and finally stillinformation. render grows it vanishing as as the conclusions. mainly valid in theinhomogeneous (near-)homogeneous effects case. induced by However,Therefore, the when we graviton taking can mass, into infer the account conclusions from of should the the be right different. panel of figure Right: relation between critical width Curves from top to down correspond to respect to the temperature. However,information as we decreases see as from the right panel of figure Figure 5 Q JHEP07(2017)082 c 0 is x and 6 (4.1) (4.2) (4.3) c 0 x increases = 0. From Q r and to study , the induced x B ∪ A is small the critical width , direction for a long enough . 2 h , we can define a conserved 2 r ν ˙ , r dy L 1 0 2 f − r f − + decreases. Therefore, the right plot increases more mildly. As we know + 2 p 0 c f dt r MG 0  − T x 2 sits in the left asymptotic boundary and = ˙ r p 2 ) A ˙ r r is affected by the shock wave since it passes 1 are unaffected by the shock wave since they 1 ( ]. At a surface with constant dt r ) for a fixed boundary separation. f − B is big, B x f ∪ – 11 – Z ( 26 rf , h A h + ) + r increases, which states that the higher temperature tends to 0, which corresponds to the case that the − r 24 ) as a ‘Lagrangian’ f ( ) = h [ . For a fixed horizon, we find that when r f − h H 4.2 ( h 6 , r − ) near the horizon in the p B h , we plot the relation between the critical width and Area x ∪ r  ( 5 A = A h = → . This statement matches the conclusions above. H 2 c . Our main goal thus is to calculate the Area 0 0 is the radial position behind the horizon satisfying ˙ r Area , when a small perturbation is added from the left boundary to 6 ds x 0 2 r decreases as increases the temperature c in the right boundary. After adding the light-like perturbations, the increases as well. In particular, when 0 Q x c B 0 . The area of minimal surface for the regions 1, 2 and 3 in figure , we suppose that the strip ) and x 0 3 as r ( H f for various horizons dr/dt ) that as = Q = 0 indicates that as temperature of the black hole decreases the critical width 3.4 is shown in figure r f 5 . A shock wave geometry thus forms and the wormhole connecting the left and ), we know that as B w ∪ t A 4.3 Because of the symmetry of the minimal surface, we should only calculate the areas As in section On the right panel of figure , the critical width ‘Hamiltonian’ in which eq. ( in which ˙ then given by If regarding the integrand in eq. ( for the regions 1,metric 2 can and be 3 written in as figure areas of minimal surfacesdo Area not cross theacross horizon, the while horizon the Area andArea stretches through thehow wormhole. it changes The with sketchy respect plot to of the the shift surface holographic mutual information in the staticwe background are of going massive gravity; to in investigatein this the section, the dynamical shock behavior wave of background the of holographic massive mutual gravity. information its identical partner As we know from section the bulk, there willtime be a shift right regions may be destroyedbetween in the some circumstances. two Therefore, subregions the will mutual information be disrupted. In the previous section we studied the will increase, which isQ consistent with the previousreduces analysis. the Moreover, critical for width a fixed charge 4 Holographic mutual information in the dynamical background case the charge the critical width increases more rapidly; however,from when eq. ( of figure JHEP07(2017)082 .  2 (4.8) (4.5) (4.6) (4.7) (4.4) fr 2 + ), we can r ) depends 2 0 4.2 H p h, x coordinate can ( . t . I 2 dr 2 ), h r 0 fr fr r 4.3 Z 2 + 2 + 8 r ) into eq. ( 2 r 2 − -independent term which H 4.4 H separates the line 2 and 3. 2 h p 0 p fr r . dr 2 2 + = dr , . We see that r h 2 r h r 2 0 fr h r 0 r H r ) which connects the left and right Z 2 + fr Z h r 2 p 2 ( − B H + 4 dr ) and + 2 H ∪ 0 2 dr p A ∞ 2 h can be rewritten as r fr 6 Z 1 + fr dr h, x 4 ( 2 + p 2 +

I r Z − 2 r f 2 – 12 – 4 H ) H /2

Z ) = p 3 /r p h  ( , the mutual information in the shock wave geometry dr

B B min dr . Thus in order to proceed, we should find the relation r 1 ∪

∞ ( ) = h h ) coordinates). The left half of the surface is divided into three 2 A ∞ h r r r r ( 0 respectively. Substituting eq. ( Z − t Z x, y 1

0. From the conservation equation eq. ( Area 1 r p r < ) = → and Area ) = 2 f h r h h ( A ( √ B B ∪ for a fixed ∪ 0 and ˙ dr A 0 A r ∞ min r r > Z Area Area 4  4 Y denote ˙ . The Penrose diagram of the shock wave geometry. The horizontal colorful line is the ) =  0

h, x ( I We are going toon find the the location relation of between It should be stressed thatmust the be first subtracted segment by contains a athe pure divergent contribution AdS of contribution Area as wecan compute be it numerically. expressed Considering as The second term contains aregions prefactor have 2 the stemming same from area.boundaries the thus The fact is total that area the Area second and third Therefore, the area of regions 1+2+3 in figure where get a time independent integral segments, i.e., black line, red line and yellow line. Theshock surface wave is absent for be written as a function of Figure 6 minimal surface connecting onedimensional of surface the spanned two by ( ends of the strips (one dot in the line represents a two JHEP07(2017)082 for ) = r (4.9) (4.14) (4.15) (4.10) (4.11) (4.13) (4.16) (4.12) ) with = ¯ µ, ν ) at the 2 r vanishes, 4.8 , ν h 2 µ means there is h , 1 2 0) to ( µ µ , 1 . , µ = 2). Therefore, with the   ,  2 . Combining eq. ( 2 h/ . 2 h  r fr fr  2 = fr 2 2 2 2 − − 3 fr . − . Next, we will study the relation , 2 1 fr H 1 H 1 h ) 2  H − , ν 3 − 1 f H dr 1 1 + . The vanishing of H 1 + 1 + 0 = 0 + Ξ r 7 p for a fixed p ) and = 1 in numerics), the shift ¯ r 1 + 2 p 3 0 1 + Z 0 µ h − − p r − r κ p 1 1 + Ξ 2 h, x 1 − (   1  −  I 1 f f ) to ( 1 – 13 – dr , dr  f 2 dr  h exp f r f h 0 ∞ dr , ν h dr (we set r 2 r 0 f r 2 dr 1 r 0 h µ Z Z µ r r h 0 Z ¯ r r κ r ∞ κ h can be determined by the relation κ = = 2 exp(Ξ r Z Z , the first segment goes from the boundary at (  − Z 2 − 2 κ κ h → 6  ν µ  κ can be determined by choosing a reference surface 0 r = 2 = 2 = 2 ). The second segment stretches from ( ν 1 2 3 = exp Ξ Ξ Ξ = exp = exp 2 2.3 numerically in the background of the shock wave geometry in the ν h 1 2 1 0), in which 2 , µ µ µ h 1 = depends on the location of µ 3 2 h . From figure as ν ν 0 r ) and h ) = ( 0 . The coordinate 0 r µ, ν and h, x ( = = 0 in the black hole interior. Thus, we reach h I ), one can get the relation between r ? r 1) to ( 4.13 − , behavior of mutual informationprevious in subsection the that following as which context. can also As be weno already added found perturbation knew in into from the theunperturbed the bulk, plot or therefore, (a) the the static of shock case figure wave which geometry we goes studied back in to the the previous section. between massive gravity. 4.1 Black hole chargeWe are and going butterfly to effect study how the charge affects the shock wave geometry and the dynamical It is obvious that eq. ( where relation we can express which The third segment stretches from ( surface In contrast, the coordinate (1 where we have used eq. ( between JHEP07(2017)082 . 0 r is 7 1.00 h ) for = 50, (4.17) 4.17 min r ; panel (c): 0.95 Q 2, . 6275 which is . , greater values 0 = 0 0.90 0 for various charges r r ≈ 8 respectively. 0 . r 0 , crit 7 . for different 0.85 r (b) 0 , h 0 , r 6 12 . ) is is a conserved quantity, the ) = 0 0.80 = 0 H divergent. For instance, in the ) and 4.17 0 Q that renders mutual information crit h 10 r 0 ( that for a fixed r h, x . As we know that the shift f ( 0.75 7 h I 8 + 2  0 x and the minimal radius crit 0.70 h,  h r I 0.5 0.4 0.3 0.2 0.1 0.0 . In numerics, we have set ¯ ) 6 diverges. Since Q (c) crit h r 0 – 14 – r ( 0 increases, which is shown on panel (b) of figure 1.0 f 0 ), we can see that if the denominator in the integrand r 4 = . We have also checked the correctness of eq. ( 4.16 crit for various 0.9 the shift r 7 , h = 2 r

) 6, a physical solution to eq. ( = 0 . 2 in eq. ( 0.8 2  0 ) and 3 x fr dr 0 ( fr = 0 h,  I 0.5 0.4 0.3 0.2 0.1 0.0 2 d − h, x m ( 0.7 H (a) I 1 + 6 and is the critical position that makes the shift 0.6 . p = 1. The red, yellow and green lines correspond to crit = 0 r h . Panel (a): the relation between the shift r 0.5 Q that the more the added energy perturbation is, the smaller the mutual information 6, . 7 The mutual information grows as From the formula of Ξ = 0 h ; panel (b): the relation between the mutual information 0.4 0 400 300 200 100 From this plot we see that therevanishing. are also In critical values fact, of oneof can the see from charge plotproportional correspond (a) to to of the energy figure smaller offigure the values added of perturbation, thus shift we can deduce from panel (b) of in which, case of consistent with plot (a) ofother figure parameters. vanishes, i.e., above equation can be readily transformed to Figure 7 Q the relation between m JHEP07(2017)082 ; 0 0 r r m 1.0 = 50, min according r ; panel (c): h 2, m , the mutual . 0.9 for various 0 crit r = 0 h vanishes; besides, r 4 respectively. . h 1), i.e., there is no can be more clearly 0 , for different graviton h 3 . 0.8 → for various 0 (b) 0 h r = 1, , 0 0 r 2 r h . r and 50 = 0 → h ) and 0 m 0.7 0 r diverge. The critical values of corresponds to larger h h, x m ( 40 I and the minimal radius  0 h x 0.6 h,  . In numerics we have set ¯ I 0.8 0.6 0.4 0.2 0.0 30 , larger m 0 r (c) 0 – 15 – r 1.0 20 . For a fixed value of shift, the mutual information . As the shift grows to a critical value that makes the shift for different 0.9 7 0 10 h r ). For a fixed 0.8  0 ) and 4.17 x 0 h,  I 1.0 0.8 0.6 0.4 0.2 0.0 butterfly effect h, x ( 0.7 (a) I shows the relation between the shift 8 0.6 = 1. The red, yellow and green lines correspond to h . Panel (a): the relation between the shift r 0.5 5, . = 0 h 0.4 0 400 300 200 100 4.2 Graviton massPlot and butterfly (a) effect of figure masses. Just like the casethere in is the also preceding a subsection,match critical as the value ones of from eq. ( in chaos theory, i.e., decreases as thesection. charge grows, In which particular, isadded perturbation when consistent into the the with bulk, shift the itsection. is will statements go zero back in to (or the the equivalently static previous case discussed in the previous will be. The relationshipseen between from mutual the information panel andinformation (c) the of shift will figure vanish, i.e.,disrupt the when entanglement between the the two addeddepending strips. perturbation on The disruption the is of the added big mutual perturbation information enough, in it the will initial finally time reminds us of the phenomenon Figure 8 panel (b): thethe relation relation between between the mutualQ information JHEP07(2017)082 , which ] or for h 46 corresponds corresponds m m is shown in the h for these relations and found that m – 16 – . 8 ; meanwhile we find from the panel (b) that larger 8 , the mutual information decreases according to the shift 7 , which can be obtained from the rest two plots (a) and (b). Just like in 8 . We have also checked other values of c h It would be interesting to study the holographic mutual information in the background By adding the light-like perturbations into the bulk, we studied the dynamical mutual The relationship between the mutual information and the shift that render the mutual information vanishing, and it is found that larger c We found that the largercritical the values values of of thewould the shift reduce charge the are, and mutual graviton which information. mass indicates are, that the both smallerof the the real charge and spatial inhomogeneity inhomogeneity, such as adding lattice structures in the bulk [ produce a shift on theshock horizon wave geometry, in we the know Kruskal that coordinates. thethat shift the From is the more proportional existing the to studies the addedsuggests of added energy that the energy. was, the We the found added smallertwo perturbations sides the would of mutual the reduce information black the wouldon hole. be, mutual the We which information also critical between investigated values the the of effect the of shift, the charge where and makes graviton the mass dynamical mutual information vanish. investigated the effect of theinformation charge decreases on the as mutual the informationstrip. and charge found increases, that which the mutual is independent ofinformation the in width the of shock the wave geometry of the massive gravity. The added perturbations mass. For longer strips theto mutual the information graviton would mass, decrease which monotonicallythe implies with mutual information respect that of the strips spatial withwhen inhomogeneity larger the length. has system larger With is influence the farrole above to from results, in the we affecting argued homogeneity, that the the spatial mutual inhomogeneity information plays than a the dominant temperature of the black hole. We also gravity. In the staticinformation case, would we increase found as that thewith for temperature shorter the of strips the conclusions the black in holelarger near-homogeneous grows, graviton mutual the which mass, is pure consistent which homogeneoustheory, we corresponds case found to that studied greater the previously. mutual inhomogeneity information in However, would the decrease for with boundary a respect field to the graviton they have similar behaviors as in figure 5 Conclusions and discussions In this paper we studied the holographic mutual information in the background of massive the plot (c) ofmeans figure the added perturbationof will the destroy black the hole. mutual Andrespect information for between to a the the fixed two graviton value sides h of mass the as shift, we the already mutualto information discussed smaller decreases above. with There are also critical shifts to smaller mutual information, whichmutual information. indicates that This is larger consistent inhomogeneity withsection. the will statements disrupt in the the static case in the previous plot (c) of figure to the panel (a) of figure JHEP07(2017)082 ] , 47 ]. ]. , ]. 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This work isNos. supported 11375247, 11435006, by 11447601, the 11405016,of 11575270, National CAS, and China Natural 11675140), Postdoctoral by Scienceence Science a Foundation Foundation of key Foundation Education project (Grant of Committee No. ofResearch China Chongqing 2016M590138), Project (Grant (Grant of Natural No. Science KJ1500530), Sci- and and Technology Basic Committee of Chongqing (Grant No. cstc2016- the parameters of theholographic scalar sources mutual really information play in thestudy. the same massive role as gravity. the We graviton will mass leave to this the Acknowledgments as our future the latter is thatthe the sources bulk spacetime of ismeanings the homogeneous of massless and the isotropic; scalar momentum moreover,that relaxation fields turning the in on on parameters the related the boundary tosectors field boundary the of theory. would scalar the It sources provide was massive are more shown gravity. similar in to physical Therefore, [ the it graviton mass would in be some very interesting to check whether simplicity with the spatially dependent sources on the boundary [ JHEP07(2017)082 , ]. ] , ]. ]. 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