JHEP02(2019)160 , , Springer and b,c January 18, 2019 February 10, 2019 February 25, 2019 : : : Received Accepted Published Leonel Queimada b Published for SISSA by https://doi.org/10.1007/JHEP02(2019)160 [email protected] , Robert C. Myers, [email protected] , b,c Astronomy, University of Waterloo, & . 3 1901.00014 Hugo Marrochio, The Authors. a c b AdS-CFT Correspondence, Black Holes, Black Holes in String Theory

We revisit the complexity = action proposal for charged black holes. We in- , [email protected] Department of Physics Waterloo, ON N2L 3G1, Canada E-mail: [email protected] [email protected] The University of Tokyo, Komaba, Meguro-ku, Tokyo 153-8902, Japan Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada b c a Open Access Article funded by SCOAP Keywords: ArXiv ePrint: theories. Finally, wefor connect the the two-dimensional previous Jackiw-Teitelboim theory. discussiontained Since to by the a two-dimensional the theory dimensional complexity=action is reductionnear-extremal ob- proposal from and Einstein-Maxwell near-horizon theory limit, in higher thesolution dimensions choices determine in of the a parent behaviour action of and holographic parent complexity background in two dimensions. pears for the switchback effect,show how e.g. the it inclusion ischarges of absent on for a purely surface an magnetic term equalrate. black to footing, holes. Next, the or we action investigate We how more can then thewithout generally causal put structure the change the influences surface the the electric late-time term and value growth for with magnetic of and charged the black late-time holes growth in a family of Einstein-Maxwell-Dilaton Abstract: vestigate the complexity forthe a late-time dyonic growth black is hole,particular, sensitive and the late-time to we growth find the ratecharge. the ratio vanishes surprising when If between the feature the electric black that dyonic and hole black magnetic carries hole only charges. is a perturbed magnetic In by a light shock wave, a similar feature ap- Beni Yoshida Holographic complexity equals which action? Kanato Goto, JHEP02(2019)160 ]. 17 ]. One strik- 12 – 1 47 ]. The concept of holographic 16 – 13 – 1 – 4 26 17 27 10 12 44 6 20 35 23 32 ]. For instance, holographic complexity is expected to be a useful diag- 1 17 38 4.2 JT-like model 4.3 Boundary terms? 3.1 Complexity growth 3.2 Boundary terms 4.1 Jackiw-Teitelboim model 2.1 Complexity growth 2.2 Maxwell boundary2.3 term Shock wave geometries correspondence [ nostic for late-time dynamics and into particular, increase the interior even of after a blackcomplexity the hole is boundary since it sensitive theory continues to hasi.e., perturbations reached even thermal of tiny equilibrium. perturbations the to system, In the namely addition, system the have a physics measurable of effect scrambling, on the complexity [ on deep conceptual puzzles into the study AdS/CFT the correspondence, dynamics and ofing, also strongly-coupled yet provide mysterious, quantum useful entry field tools to theories,circuit the e.g., gravity/information complexity: dictionary [ is the thegiven size concept reference of of quantum state the withcomplexity optimal naturally a circuit emerges set which from prepares of considerations a “simple” on the target gates bulk state [ causality from in a the AdS/CFT 1 Introduction Recent years have witnessed that quantum information theoretic notions shed new light B Bulk contribution from Maxwell boundary term 5 Discussion A More on shock waves 4 Black holes in two dimensions 3 Charged dilatonic black hole Contents 1 Introduction 2 Reissner-Nordstrom black hole JHEP02(2019)160 (1.1) (1.2) and the AdS N G ]. ) is the volume of B 19 ( V ,  ] Developing a proper understanding ) L . B 74 ( N – ~ V G 56 π WDW  I B ∂ – 2 – Σ= ]. A second direction of investigation has been to ]. In particular, we note that a number of new (Σ) = 55 A 55 , – C ] ] (Σ) = max 54 17 anchored to this boundary time slice. To produce a V 18 C , 21 B , 17 20 ]. Recently, there has been a great deal of interest in the JT model, ) produced the surprising result that the growth rate vanishes at late 76 , 1.2 ]. Furthermore, the JT gravity exhibits the late-time behavior of the ]. Further, more recently, a relation was conjectured between momentum 75 88 53 – 77 A second proposal, known as the complexity=action (CA) conjecture, relates anchored on Σ. ]. As such, the JT model should be an ideal platform to study the complexity 1 . B 90 , L 89 Our motivation for the present paper was to understand holographic complexity (and The study of holographic complexity is actively developing in two related directions. The original proposal for holographic complexity, known as the complexity=volume A more sophisticated approach to choosing the latter scale was described in [ 1 e.g. [ in various dynamical settings andplexity. investigate However, further our the initial implicationsthe calculation of CA of holographic proposal holographic com- ( complexity in the JT model using dilaton-gravity [ as it emerges ina particular the low holographic energy descriptioninvariance limit, [ of where the the Sachdev-Ye-Kitaevspectral system (SYK) form acquires model an factor in emergent which reparametrization are natural from the perspective of random matrix theory, for states in aof strongly complexity coupled in CFT, the e.g., boundaryproposals see and [ theory ultimately, is to essential produce to a properly derivation test of the onein various (or holographic particular, more) the of CA these conjectures. proposal) in the two-dimensional Jackiw-Teitelboim (JT) model of For example, one new proposal isconjecture, known which as suggests the that complexity=(spacetime volume) theWDW (or bulk CV2.0) patch dual [ of complexity isof the an spacetime infalling object volume of inevolved the the operator bulk radial on direction the and boundaryunderstand complexity [ of the the concept corresponding of time- circuit complexity for quantum field theory states, in particular The first is to explorein the the properties of CV the and newin gravitational CA the observables conjectures which boundary play and theory, aproposals the role e.g., have been implications see made of [ for these the conjectures holographic for dual of complexity complexity in the boundary theory. Here, the subscript WDW indicatesDeWitt that the patch, action which is calculatedsurfaces corresponds on to the so-called the Wheeler- causal development of any of the above bulk dimensionless quantity, thelength volume is dividedthe by complexity Newton’s of the constant boundaryregion state of to the the bulk gravitational spacetime action [ evaluated for a particular (CV) conjecture, asserts that [ where the boundarycodimension-one state bulk lives surfaces on the time slice Σ and then JHEP02(2019)160 (1.3) ]. In particular, JT 94 – 91 , 88 – 86 ]. Our first calculation in the following is ST, ] 91 , ∼ – 3 – 22 88 , C ) in this setting and in this case, we found the dt d 17 1.1 ]) when defining different thermodynamics ensemble, i.e., a 97 , 96 gives an account of the number of degrees of freedom while the . This result, of course, in tension with our common expectations 4 S ], had not shown any odd behaviour for the CA proposal. However, sets the scale for the rate at which new gates are introduced. Further 34 , T ] (see also [ 30 , 95 21 , 20 To better understand the vanishing of the complexity growth for the magnetic black However, there is a boundary term involving the Maxwell field, which one might add This apparent failure of the CA proposal motivated us to re-examine holographic tion in the presenceholographic of complexity the for Maxwell the surface JTthe term reduction model and of and the electrically for behaviour charged a black of related holes. the “JT-like” corresponding model,holes, derived from we might also ask whether this result is special to the Einstein-Maxwell theory. canonical ensemble with fixed charge,chemical as potential. opposed to We find a that grandsensitive with canonical to the ensemble with CA the fixed proposal, introductiongrowth the rate of holographic is complexity this is nonvanishing for Maxwell also the magnetic surface coefficient black term. of holes withcharged the In this case. surface surface particular, Given term term, the these while can results, late-time tuning yield we are a then vanishing lead growth to rate re-examine the in dimensional the reduc- electrically Of course, this vanishingdimensional matches reduction our of result these magnetic for black the holes. JT model,to which the gravitational would action. arise Thisdynamics in term [ the arises naturally in the context of black hole thermo- to examine holographic complexity forelectric a and dyonic magnetic black hole charges.that (in the four In dimensions) complexity this with growthcharge. both case, rate In even particular, is if if very thea the sensitive black vanishing hole geometry to growth is is the rate purely held ratio magnetic, at we between fixed, late find the we that times, CA two find and proposal types further, yields of that the switchback effect is absent. higher dimensions. Previouse.g. [ studies ofwith holographic hindsight, complexity we of note thatholes, charged whereas all the black of usual holes, these dimensionalholes investigations reduction made involved carrying to electrically a derive charged the magnetic black JT charge, model e.g., involves black [ Σ) continues to grow at a constant rate forcomplexity arbitrarily for late charged times. blackrived holes from in an higher appropriatedilaton-gravity dimensional dimensions describes reduction, since the e.g., the near-horizon JT physics [ model of certain can near-extremal be black de- holes in temperature since the JT modelmaximal is chaos, supposed to we capturepossibly would the can. certainly physics Rather expect of than thecan the considering SYK the also complexity model, CA examine should which prescription exhibits extremal the increase of volume holographic CV as (i.e., complexity, one proposal fast the length as ( of it the geodesic connecting the boundary points defining for complexity. It cancontinue be to argued grow quite with generally a that rate at given late by times, [ the complexity should where the entropy times — see section JHEP02(2019)160 ] to 21 , 20 ]. 101 , we comment – , we investigate B ], which has sig- 99 3 = 3 for the bound- 98 , we study the CA d 2 – 4 – , we describe in more detail the calculations of the A ]. In these theories, the Maxwell field is coupled to a 39 – 36 , 30 , we return to holographic complexity for two-dimensional black -2)-form potential field (i.e., the Hodge dual of the one-form Maxwell 4 d , as well as discussing some possible future directions. We leave certain technical 5 As this project was nearing its completion, we became aware of [ The remainder of our paper is organized as follows: in section focusing on the Einstein-Maxwell theoryary in theory. four These bulk dimensions, results i.e., aregravity easily theory extended to to a general ( potential). dimensions, if Our one main also objective couplesated is the with to the understand Maxwell the field. effect of As a mentioned new in boundary the term introduction, associ- we will find that although standing holographic complexity for the JT model recently appeared in2 [ Reissner-Nordstrom black hole In this section, we investigateevaluate applying the the holographic complexity=action (CA) complexity conjecture of [ the dyonic Reissner-Nordstrom black hole, while holographic complexity in the dyonicon shock subtleties wave concerning geometries. the evaluation In ofare appendix the present. Maxwell surface term when magnetic charges nificant overlap with the present paper. We add that an independent approach to under- holes. In particular, we showthat that this the situation late-time can growth rate befrom vanishes ameliorated four for if to the two the JT dimensions. Maxwell model,section surface We but summarize term our is findings included and indetails consider reduction to their the implications appendices. in In appendix of the Maxwell surface termgrowth to rate the for action the caneffect CA also by have injecting proposal. a small dramatic shockwaves In intoholographic effect the addition, on complexity dyonic we the black for also late-time hole. chargedtheories. briefly In black section investigate holes the In switchback in section a family of Einstein-Maxwell-Dilaton time growth rate with the CA proposal. proposal for dyonic blackdimensions. holes carrying We both firstsensitive electric to show and the ratio how magnetic between electric for charges and magnetic in a charges. four fixed We also bulk show geometry, how the the inclusion complexity rate of change is scalar field (the dilaton)With and the as dilaton a excited result,the in the these charged casual black solutions, structure holes theinvestigate of also nature to carry the of what “scalar extent the corresponding hair.” these spacetimeof black changes singularities the to holes and the holographic can spacetime complexity.spacetime geometry be geometry modify Our modified. the is conclusion behaviour the will Hence essential be feature we that leading can to the the causal unusual structure (i.e., of vanishing) the late- Nordstrom-AdS black holes are importantcomplexity for in the controlling CA the proposal. behaviourcomplexity As for of charged a black the step holes in holographic in this atheories. family direction, of Holographic we four-dimensional also complexity Einstein-Maxwell-Dilaton of investigate holographic Einstein-Maxwell-Dilatonstudied theories for has been several previously models [ Alternatively, the question can be phrased as which features of the corresponding Reissner- JHEP02(2019)160 (2.4) (2.5) (2.1) (2.2) (2.3) ] for a 26 ] to ensure 26 σa , √ , x ση , 2 ] for timelike and spacelike  √ d , 0 2 x Θ) Q 6 2 Σ L . µ ct 103 d Z I , ` . + Σ µν ν N + Z F 102 A R 1 ct N πG I µν  µν 8 1 g Θ log ( + F πG g F γ − ] for intersections of these segments, and + 8 µ − √ √ surf Σ θ I + γκ √ x d 2 105 4 ] for null boundary segments — see [ , x √ + K d 4 ] that this surface term must be included on M | θ 26 – 5 – d ∂ 2 h dλ d | 104 M Z Max 0 41 Z M I , 2 B p γ Z g dλ d Z x N + ) is a boundary term for the Maxwell field 40 0 2 3 B g N 1 = d 1 EH Z 4 πG 2.1 B 1 Q I µ Z πG N 16 − I 8 = 1 N = = πG = tot 1 8 I πG ct EH 8 Max I + I I . = 2.2 contains various surface terms needed to make the variational principle surf I surf I The final contribution in eq. ( We divide the action for four-dimensional Einstein-Maxwell theory in terms of the While introducing this boundarychange term the nature does of not theboundary change variational conditions principle the that of must equations the be ofwill Maxwell imposed motion, also field. for consistency it find That of is, does that thethis it it variational term changes principle. modifies for the We section the WDW action, but we reserve a complete discussion of properties of complexity. is not neededreparametrization invariance for on the the nullstudy boundaries. variational of Further, principle, it shock was wavethe but shown geometries null with it boundaries a in careful of was [ the introduced WDW patch in if [ the CA proposal is to reproduce the expected This contains the usualboundary Gibbons-Hawking-York term segments, [ the Hayward termsthe [ surface and joint termscomplete introduced discussion. in The [ null surface counterterm, The next term well-defined for the metric, where first two terms are integrated over the bulk of the manifold of interest action of the Wheeler-DeWittcoefficient (WDW) of patch. this Hence term)holographic we yield complexity. must a ask WDW which action choices which (for produces the the behavioursusual Einstein-Hilbert expected and of Maxwell actions, as well as various possible surface terms this boundary term does not modify the field equations, it has a strong influence on the JHEP02(2019)160 and (2.9) (2.6) (2.7) (2.8) + (2.10) (2.11) r , for ingoing u ) = 3 boundary 2 d and θdφ . v 2 , + r  = r

dt + sin . 2  . e RNA  r ) dθ ∂r q r ( ( dθ ∂f 2 − ∗ r π ∧ r , 1 e (a), with an outer horizon 4 + ) q + r − , 1 r ) t = (˜  ˜ θ dφ r 2 r m . ( q 2 d = + 2 RNA sin r + dr f RNA dφ 2 e m = 0) f q ) q ∞ ,T θ r ) = 0). The mass, entropy and temperature γ = 0. That is, we examine the holographic + ( + + Z  2 + 2 r r tot γ – 6 – r cos − ( ω dt I dt N ) π − ∧ − , u r G = RNA ( ) = ) 0 f (1 r dr r I = ], we write the tortoise coordinates for the black hole ( ( is a parameter proportional to the mass. A Penrose m + 1 2 e ∗ RNA q q r 34 r 2 f 2 ω  ∗ RNA  r L − r + Reissner-Nordstrom-AdS black hole (with ) seems to be sensitive to the nature of the charge. In the N t N = ,S 1.2 ) = 2 = g g πG πG r N v ( 4 4 ds (defined by ω G dyonic √ ) = 0. The Eddington-Finkelstein coordinates, √ 2 − r RNA r ( f = = = F A ∗ RNA M r denote the electric and magnetic charges. m ), as ) by setting the parameter q →∞ r with 2.7 2.5 and is the AdS length, and ). This analysis reveals the puzzling feature that despite the fact that magnetic and e L q ] for further details. 2.7 Following the conventions of [ With this action, we apply the CA proposal to study the holographic complexity for For the calculations which are immediately following, we will drop the Maxwell bound- 34 hole ( electric charges are interchangeable atgrowth the in level the of the CA equations proposalfollowing, of ( we motion, provide the salient complexity to points [ in the calculation and we refer the interested reader and outgoing rays (from the right boundary), respectively, are given by 2.1 Complexity growth Next, we evaluate the growth rate of the holographic complexity for the dyonic black spacetime ( such that lim where As indicated above, thesponding black hole Maxwell field carries strength both and electric vector and potential magnetic can charges. be The written corre- as where diagram showing the causal structureinner is shown Cauchy in horizon figure are then given by a spherically symmetric dimensions). The spacetime geometry is described by the following metric, ary term ( complexity working with the action JHEP02(2019)160 = 0), (2.13) (2.12) t ] 34 ). In order . The shaded (and − 2.37 r 2 m r = ). The nonextremal = r r 2.7 . , ) — see eq. ( 2 m r w ( t 2 ) = 0 − 2 m r RNA ( = f − R t ∗ RNA r = + 2 m t dt 2 dr and an inner Cauchy horizon + – 7 – r = , , r ) 1 m 2. A typical WDW patch is illustrated in the Penrose = 0) while the past boundaries, at r ) = 0 ( t/ t 2 1 m r = ( RNA f R (and t ∗ RNA = r 1 m = r 1 m − L dt t t = 2 dr r (a). The position of these meeting points is determined by [ (a). The time evolution of the WDW patch can be encoded in the 1 1 ], we anchor the WDW patch symmetrically on the left and right asymp- 34 2. (b) Penrose-like diagram of an Reissner-Nordstrom-AdS black hole with a shock t/ . (a) Penrose diagram for the Reissner-Nordstrom-AdS black hole ( = R t Following [ = L and then the rate at which these positions change is simply given by time dependence of pointsboundaries where the meet null at boundariesas intersect shown in in the figure bulk, i.e., the future totic boundaries with diagram in figure black holes have an outer eventblue horizon region at corresponds to at typical WDW patch anchored towave symmetric inserted boundary on time the slicesto right with not boundary tilt at thethat the asymptotic the boundary boundary null after time geodesics the crossing shock the wave, collapsing we shock adopt wave the shifts. Dray-t’Hooft prescription Figure 1 JHEP02(2019)160 . ξ in  γ ], the √ (2.16) (2.17) (2.18) (2.14) (2.15) = 42 , log →∞ λ r 34 | ∂ ). With the t . ∂ , · , 2.2  . k  1 2 m m ) dependence of the  r r + 1 | 2 m ) ξ  q )  2 m r 2 4 r 2 ct ( − ) 2 ( r ` 2 ξ e 2 2 m q ξ r RNA RNA ( 4 f ), these terms yield f 2 ( . r . Notice that in the Maxwell |  2 ∂  1 2 2.9 m m 2 + 12 L r r r log 2  − 6  log L + ) becomes 2 m 2 = | 2 q ) − ) ) N r r 2 m  − R 2 m 2.16 ( G r 2 2 r 2 e 2 ) will eliminate the ( ξ q + ( RNA − 2.4 f +  | ) vanishes. This leaves only the joint terms  | 2 3 ) dr dt r r L 2.3 1 m – 8 – + 1  r ) log  WDW 2 ( r ξ  Z N ( 2 1 2 ct G + 1 ) RNA N ` 2 . The final result for these joint contributions is given f 2 1 1 m RNA |  G ξ 2 m r 4  ) and the Maxwell field ( = ( 4 2 ct r rf ` 2  2 = 2 max  2.7 ) requires evaluating the expansion scalar Θ = log ξ r 4 bulk and 2 dt approach the horizons and so the first term above vanishes. N log ) 2.4 Max  1  dI 1 m 2 G I 1 m , r 4 r 1 m 2 ( + r ) log N − =   1 m ), the time derivative of eq. ( G r EH r = 2 N (a), the WDW patch is cut off by a UV regulator surface at some I ( N 1 G 2 max 1 G = 2.12 r − 2 joint dt − = dI bulk . However, the boundary contributions coming from this time-like surface ct I = I max r joint = 0), the null surface term in eq. ( is the normalization constant appearing in the null normals, i.e., I = ξ κ ] r 34 the null boundaries of the WDW patch and the final result is given by Note that at lateHence times, only the second term contributes toCounterterm the contribution late-time growth rate. The boundary counterterm ( In a moment, the additionaction. of Using the eq. counterterm ( ( by [ where As shown in figure large segment and the correspondingto joints the yield time a derivative ofwhich fixed the constant, action. i.e., Further, they withat do the affinely-parametrized meeting not null points, normals contribute (for Joint contributions where we have used thecontribution trace of (i.e., Einstein the equations: secondappear term with opposite in signs! therate This integrand), of fact complexity the is for electrictime directly magnetic derivative and related black of magnetic to the holes, bulk the charges as actionand vanishing we reduces past of will to meeting the the see points, difference late below. of time terms Following evaluated [ at the future We start by evaluatingReissner-Nordstrom the geometry time ( derivative of the two bulk terms in eq. ( Bulk contribution JHEP02(2019)160 ) ) to 2.15 (2.20) (2.21) (2.22) (2.23) (2.19) (2.24) 2.1 . . 1 2 m m r r 1 2 m m  r r ) and (  2 ct ) ` 4 2 m | 2.17 into the magnetic q 1 2 m m ) r r 2 r T  r + ( r ) q 2 e r q ( ) has a compact form, RNA 2( f | RNA . −  f 2.18 ), this expression reveals a , r 2 1 2 m m 3  ∂ r r r e L r . 2 m = 0, the growth rate vanishes! log q r q − + ) e  ) = 0). This leaves the surprising r r q r + + 2

. 2 ct  ≡ ( N ` r   r 2 − + ( 2 χ r r r 2 ct 2 ct N ξ 2 T

` ` πG RNA q 4 4 4 r 2 ) is then given by the sum of eqs. ( | | RNA ) )  r f πG f 2 N e , the ratio of the electric and magnetic r r 2 2 q ( ( r r 1.2 2 χ − log χ πG 2 1 2 and m m RNA RNA ) χ r r f f

r = | | – 9 – ( 1 + r N   A N 2 e G C = dt q RNA 2 m 2 d ) as q ) log ) log A πG rf r r C dt ( ( +  d →∞ 2.22 lim 2 e t ), the electric and magnetic charges contribute with the − ) = q RNA RNA ct →∞ = lim I ≡ rf rf t 2.15 2 2 2 T ct +   q dt dI N N 1 1 dependence is eliminated, G G ), the rate approaches zero at late times when the black hole is joint 4 4 I ξ − − + 0, the late-time growth rate shrinks to zero. 2.22 = → ) = bulk ct I χ I ( , which fixes the spacetime geometry (e.g., + T illustrates the full time-dependence of the growth rate, as we change the d dt q 2 1 π ), which yields joint I = ( d 2.20 A dt Figure C dt d Again at late times, this contribution vanishes and so it only changes the transient be- monopole with ratio of the electric andexpected magnetic from charges eq. while keeping ( mostly the magnetic. spacetime geometry fixed. As Now fixing nontrivial dependence ofcharges. this growth In rate particular, on we see that as we put more of the charge Hence if we considerMore a generally, purely we magnetic might black introduce hole with which allows us to re-express eq. ( At late times, thespectively, past and so and the future second term meetingresult vanishes points (since meet the outer and inner horizons, re- The growth rate of theand holographic ( complexity ( Note that in contrastsame to sign eq. above. ( Total growth rate haviour in the growthexplicitly rate see at that early the times. It is useful to combine eqs. ( a fixed constant. Hencemeeting the points time in dependence the comes second only line. from The the time terms derivative evaluated of at eq. the ( The term in the first line comes from the UV regulator surface and again only contributes JHEP02(2019)160 0 → ] for (2.25) (2.26) m ), with 97 q , 2.7 96 1 curves are ) . 2.5 = 0 . ν χ ] — see [ ), when the charge is δA 95 are held fixed. Adding µν 2.22 µ F µ Σ 0 and the d → M 1.5 . e ∂ q Z ν ) changes the boundary conditions A 2 1 g µν 2.25 − F ν µ Σ δA d 1.0 µν are held fixed. This boundary term was also M ∂ F – 10 – Z Q µ 2 ∇ γ g = 10. Similarly the g = 1) to the Euclidean action produces the Legendre , − = χ µ √ Q γ µ . We fix the parameters that determine the geometry, but vary x δA I 0.5 4 L d = M ct Z ` would yield the Gibbs free energy, associated with the grand ) (with 2 1 0 g I and 2.25 = + r 0.0 5 Max . 0.6 0.4 0.2 0.0 -0.2 -0.4 δI ] — see further discussion below. = 0 L 108 , – + . The rate of change of complexity for the dyonic black hole given by eq. ( r ). Integrating by parts produces the equations of motion in the bulk but leaves a 3 106 . 2.2 As we noted above, adding this surface term ( = 0 − in the variational principleeq. of ( the Maxwell field.boundary term Consider proportional varying to the Maxwell action in temperature and total (electric)shown charge to play ain role four in resolving dimensions theholes and [ apparent the tension different between electric-magnetic partition duality functions of electric and magnetic black a discussion in theversion context of of the the AdS/CFT correspondence.canonical action ensemble In where the particular, temperature the andthe Euclidean chemical boundary potential term ( transform to the Helmholtz free energy, associated with the canonical ensemble where the an equal footing? Inmodify the the following, action we will with argue the that addition such of a the prescription Maxwell requires boundary that term we in eq. ( This surface term plays a natural role in black hole thermodynamics [ 2.2 Maxwell boundaryThe term discussion in theprescription previous for section the raises holographic the complexity question that of puts the whether electric there and is magnetic a charges consistent on Figure 2 r the ratio between electricmostly and magnetic, magnetic the charges. growthessentially rate matches As of the predicted complexity top approaches byindistinguishable zero curve eq. on at this for ( late scale. times. The limit JHEP02(2019)160 ]. 111 (2.30) (2.31) (2.32) (2.27) (2.28) (2.29) – . 109 B . = 0, then the ] . ν yields µν − + A r r F

. µ µν r Max ), i.e., including the I ∇ 1 2 m m . N 2 T γ r r ( q  µν γ δF πG tot 2 m F I q , + γ µν ) with r ν − µν + 2 2 e , 2 F g F δA q χ b 2.29 χ − µν ) µν 1) γ δA √ F 1 + b g F x (and the same choice of gauge), the − − ) a 4 ), the only change that has to be made is eliminated and the required bound- − γ γ d γ X ν (1 √ 2.1 ) (2 − M x γ = δA 4 Z − d is a unit vector orthogonal to the boundary [(1 − 1 − + 2 3 r r µ indicates that only the tangential components 2 µ r

L M − g Σ n a Z  2 m 4 – 11 – d γ = (1 2 2 N = 0, we recognize this as the Neumann boundary g γ γ q a M 1 2 G ∂ = r A + 2 Of course, the above expression takes the same form Z δA · = N 2 e µ 2 2 q ) is replaced by n 1 ∂ g ) ) = µ ), if we use the Maxwell equations πG γ ) and so we could just as well have re-expressed the bulk on-shell Q = 0, where

µ 2.15 on-shell γ n Q 2.1 − I

2.2 µ µa Q · · · − I + (1 µ δF I = + = will depend on details of the problem of interest, e.g., [ µ Q bulk n b µ I A Max a ( C I dt = 1, the dependence on the electric charge drops out of the numerator ) can be converted into a bulk term via Stokes’ theorem as = 0. With a general value of δI X d d dt γ a + = 1, the term proportional to 2.25 γ δA →∞ lim t Max µ ∂ δI µ ). As a result, eq. ( ), in which case the variation produces the boundary contribution n 2.14 2.25 = 0 on the boundary (where the index Subsequently, the final result for the late-time growth rate for the complexity becomes Returning to the action ( . If we choose a gauge where There is a subtlety here for the magnetic monopole contribution in that the boundary term must be a 2 M Therefore if we set and the late-time growth rate is primarily sensitive tointegrated the over magnetic the boundary charge. of In all patches particular where the potential is well-defined — see appendix Hence in evaluatingcontribution the of the WDW Maxwell boundary action term into for eq. the ( the previous general calculation isin action to eq. change ( the overall coefficient of the Maxwell contribution as the bulk Maxwellaction action as ( a boundary term. In any event, combining eq. ( boundary term ( which is explicitly gauge invariant. where the choice of Of course, with ary condition becomes ∂ condition potential would satisfy a mixed boundary condition, δA of the potential are fixed).term ( However, the latter can be modified by introducing the surface Hence, a well-posed variational principle requires a Dirichlet boundary condition setting JHEP02(2019)160 , 2 m q + , if at (2.34) (2.35) (2.33) , which 2 e ρσ q F , but both = 2 T ST q µνρσ ε ∼ 1 2 = ]. To examine this /dt C ) as indicated above. µν ) becomes d 41 e , switchback effect F 40 2.25 2.32 ↔ ), i.e., µν ] — see further discussion in , F 1.3 − + r r 32 ,

, r 2 m 18 ) q , this expression produces a puzzle in N γ + 5 −  , πG 2 e 2 q (1 ), we see that the combination of the bulk ). This conclusion can also be motivated = tot I 2 2.7 = 1 + 2.30 ). For example, eq. ( / – 12 – 1 2 ↔ , , =1 2.7 γ ) + + = 1.

r r γ 3 ( γ A C dt are governed by the spacetime geometry, as given in tot d I ] for T →∞ lim 108 t , the late-time growth rate drops to zero for an electrically γ and temperature 2 is singled out as the special choice which leaves the action unchanged in , as appears in the metric ( S / 2 m q . We will follow closely the analysis and notation of [ ). Of course, looking back at eq. ( = 1 + ). Hence it is natural to think that this rate should be controlled by 5 2 e γ q 2.33 2.8 Of course, the reader may wonder why we should expect that that magnetic and electric The above discussion shows us that the growth rate (or more generally the on-shell This equivalence was noted by [ = 3 2 T into a charged black hole.then If the the shell black only hole’s injects event a horizon small amount shifts of by energy a into the small system, amount, i.e., Another property that holographic complexity shouldis exhibit is related the to thesection complexity of precursorfeature, we operators consider a [ Vaidya geometry where a(n infinitely) thin shell of null fluid collapses by the shock waveelectric geometries, or which magnetic we black studythink holes in that exhibit the the the next holographic same section. complexitythe back-reaction should nature and In respond of hence this in the it context, the charge. is same both manner natural independent to 2.3 of Shock wave geometries black holes should computelate-time at growth the of same the rate. complexitythe should First, be entropy let given us by eq. recalleq. ( the ( expectation that the the combination appearing in the metric ( and as desired, the electrican equal and footing. magnetic charges However, influence asthe the we limit discuss complexity of in growth zero section rate charges. on still sensitive to theparticular, electromagnetic the field holographic complexity through onlyq its depends back-reaction on on the the duality geometry. invariant combination In i.e., we modify theThen coefficient of theeq. Maxwell ( boundary termand ( boundary terms for the Maxwell field vanishes on-shell. However, the complexity is charged black hole at late times. action) is symmetric underthe electric-magnetic same duality, time i.e., we exchange the action then, with this choice of JHEP02(2019)160 ], w t − 41 . Let , (2.37) (2.38) (2.36) w = t 40 s ), define − v = 0 and to = 2.11 t R t ) , the metric has 2 s = v , we find θdφ L = t 2 v → ∞ . r ) + sin s 2 v dθ − . Note that following [ ( . , v 2 w ( ) ) t r 2 , respectively. However, we must m b 2 q 2 m r H − 2 + r ( ω 2 ( r ∗ 2 + . = ∗ ω 1 r 2 e r  2 2 t q 2 , and − dr dv ) + − 1 )) + . Our goal here is to investigate to what , log s ) s ) r ) ) ω 1 1 v m s b ( ∗ scr v + 2 r ∗ r r 2 ( r 1 = ( ( ( r − 2 1 πT ∗ 2 ∗ ∗ 1 1 2 f 2 v r ω r r – 13 – dv ( − t > t ) − = = = 2 = 2 = 2 ) for each region and then following eq. ( − H r, v ∗ scr w w w w ( + 1 t t t t t ). Note that taking the limit ) with (1 2 F 2 r + + − − 2.10 ( 1 r L − 2.7 ∗ ω R L L R r t t t t = − ) = 2 ) = v v ds ( r, v s = ( f t F with , the meeting points of the future and past null boundaries, respectively; and ) denoting the usual Heaviside function). Before and after 2 m , the point where the null shell crosses the past right and future left boundaries, v (b) illustrates the spacetime geometry for a shock wave collapsing into a Reissner- b r ( but then the difference of complexities (for the perturbed and unperturbed states) r 1 H ∗ scr The geometry of the WDW patch is characterized by a number of dynamical points: and and after the perturbation is introduced, the complexity remains essentially unchanged for 1 s m In the following, itstudy is the behaviour sufficient resulting to from restrict pushing our the perturbation attention to to earlier the times case r respectively. These positions are determined by the boundary times with evaluate the tortoise coordinate ( the time coordinate as on the boundary. r where (with precisely the form given in eq. ( collapsing shock wave.carries For energy simplicity, but no we charges. assume The that corresponding the metric is thin shell is neutral, i.e., it Charged shock wave geometry Figure Nordstrom black hole fromwe the adopt right the boundary at Dray-‘t Hooft prescription that the null geodesics shift upon crossing the t t < t begins to grow linearlyextent afterwards, the i.e., CA proposal reproducesthe this previous behaviour sections. for the charged black holes discussed in scrambling time associated with this perturbation is then given by For the chaotic dual of the black hole, the switchback effect then predicts that for any time where the subscripts 1 and 2 indicate before and after the shock wave, respectively. The JHEP02(2019)160 , . A →∞ t | (2.40) (2.39) /dt A C 1, which is d . 2 , . is varied. We ) ' = 0 ∗ scr ∗ scr m w A , the complexity χ /q , there is a greater /dt e < t > t ), i.e., we again set χ ]. A q 2.1 C w w 41 d t t 2.5 = . 1 2 χ for for , , − − (a) that essentially they all r r (b). We note, however, that 4 = = 4 2 1 m m , we see that r r 5 ∗ scr but only the transition between the ]. ). The dependence on this ambiguity →∞ →∞ ) by simply multiplying the curves in > t but afterwards, the difference quickly w T 41 T lim lim 2.4 w ! w w )). Notice that the complexity remains t t t 2 2 , , /dt ∗ scr decreases to zero as more of the charge is 2.40 t − + A r r C

2.35 0. Hence the switchback effect vanishes (with = ) for ∗ scr d 0. r 1 t w is fixed) but the ratio → – 14 – t + ≥ 2.40 ) yields a simple result for the dynamical points → in eq. ( 1 1 χ 2 m  Several curves are shown in the figure where the , , w χ , the initial regime over which the complexity is constant q t w − + in eq. ( 2 χ 1 1 t , , r r 2.38 4 1 −

+ 6 + + (and the same charges). We begin by considering the r r − r and then we see in figure 1 2 10 e πT 1 2 q , as discussed in [ . The details of our calculations are given in appendix 2

= ω = w ∼ e = b s N /χ  2 = r r = 10 2 2.1 T ) 2 T /dt χ q 2 q 2  ω A vanishes as πG χ χ C →∞ →∞ in the null counterterm ( 2 d w T T 1 + lim lim χ ). w w ct 2 t t  (i.e., can be evaluated analytically to find (see appendix ` , namely, χ /dt w ) with 1 ) with the result in eq. ( A w , t 1 + 2.40 C + d 2.7 r ' 'O /dt 2.24 ) 6 A ) as in section ]. A A w w ) for a black hole with pure magnetic charge. This result might be expected C − 41 C C d γ ], the switchback effect is revealed (or not) in the ‘complexity of formation’ 2.1 dt dt d d 41 , we present the difference of complexities for a light shock wave producing 3 by the factor (1 + 3 = (1 + 10 We can confirm the scaling with The rate For heavier shock waves, e.g., Comparing eq. ( 2 4 5 , = 0 in eq. ( + this ambiguity does nottwo effect regimes the in final eq. rate ( essentially disappears, similar to the behaviour found foras neutral predicted black by holes [ in [ figure collapse onto a single curve.slightly The shifted only to exception the is right.sensitivity for to This the the behaviour smallest scale arises ratio, becausein for the smaller definition of the WDW action is illustrated in figure Hence as for therate growth after rate the of scrambling the timeand eternal depends in on black particular, the hole ratio case between in electric section and magnetic charges see that the rateput of into the the linear magnetic monopole, growth this i.e., choice as of since there is a closethe connection static between black the hole late-time and rate of growth of the complexity in γ In figure r unchanged by the perturbationmakes up a until transition togeometry linear is held growth. fixed (i.e., Following [ comparing the holographic complexitystatic of black the hole above ( shockwave geometryCA with proposal that for of the the action without the Maxwell boundary term ( in the limit of large Results for switchback effect us note that with these choices, eq. ( JHEP02(2019)160 , 2 , 4 + and r 2 , 5 . + = 10 to r 5 χ = 0 . 3 L = 0 1 . We essentially , 2 − r /χ , ) 2 2 2 , χ + r 5 . , as it is more sensitive to is the same after the shock χ T 4 and parameters q 1 = 0 1 1 and in dot-dashed , . L + r ) = 0 6 , both curves are essentially on top of − χ L 0 1 4 2 0 6 . 3 by the factor (1 + = 0 3 are more sensitive to this ambiguous scale. ct = (1 + 10 ` χ to be the same before and after the shock wave.) and parameters 2 , T 1 + , q 4 r – 15 – + r ) ). 6 2 . (b) The influence of the transient behaviour for the − ct ` 2.40 3 is then fixed by the condition that in units of L. For 2 = (1 + 10 , ct 2 − ` , 1 r 2 + ( r , the curves with small L L is determined by fixing = ∼ (a) (b) 2 , for a light shock wave , ct ct w ` − ` t 1 r 0 and 2 2.0 1.5 1.0 0.5 0.0 2.5 , + r . (a) The complexity for the shock wave geometry as a function of the insertion time . The difference of complexities of formation in the shock wave geometry as a function of 0 5 . 2 1 0 5 4 3 6 . (Further, L = 0 = 1 , , for a light shock wave − ct w complexity in the shockcontrast wave the geometry. effect of We varying each show other, in but solid for Figure 4 t ` We show the resultsee for that rescaling the the curves curvesthe lie in transient on figure behaviour top controlled of by each other, except the for smallest wave). The dashedparameters. vertical line We investigate is theafter the effect the scrambling of scrambling time time, varying the for themagnetic complexity ratio the charges, essentially as between shock remains predicted electric constant wave by for and with eq. the magnetic these ( solution charges: geometric with mostly Figure 3 the insertion time r JHEP02(2019)160 χ = 2 , + (2.41) r = 10), 015. For . χ . However, = 0 ) really only χ ∗ scr , . ) and therefore, 2.40 < t ct ∗ scr ∗ scr ` w t < t > t w w t t for for 2.5 . We show two examples, one for a L expression in eq. ( = ∗ scr ct ! l 2 2 , , , 2.0 + − 2 < t , r r

+ w r t r 1 5 . ) is replaced by + = 0) are modified if we include the Maxwell 1 1 = 0  , , 1.5 – 16 – . The exponentially growing mode only dominates at + − w 1 2 γ 2.40 , t r r . For the mostly electric black hole (with χ

1 − r r 1 5 πT , = 10 and one with mostly magnetic charge 2 . When this exponential mode is suppressed by small 2

, = 0), this exponentially growing mode is absent. χ e + A χ 1.0 , the amplitude of this initial mode is suppressed both by the r N 2 γ T 5 χ . q + πG 2 2 = 0 γ χ χ L ) + γ 0.5 2 2 1 + χ − χ ) suggests a regime of exponential growth for ) = 1, the roles of the magnetic and electric charges are reversed. For γ (1 until times of the order of the scrambling time (vertical black line), as in γ ). In particular, eq. ( 1 +  − 2.40 015), the amplitude of the exponential mode is suppressed by a factor = 0), the exponentially growing mode is absent. . 0.0 2.5 0 5 πT χ 1 — see appendix (1 -5 -20 -25 -30 -10 -15 = 0 & 'O ' , and also with = 2 1 , χ χ A A until times of the order of the scrambling time. For the mostly magnetic black , the dynamics is well approximated by an exponentially growing mode with Lyapunov w w L + C C χ λ r . The exponential growth for the complexity with a light shock wave, such that dt dt d d ) ). For the smaller value of 6 πT − , which shifts the corresponding curve down in the figure. In addition, we see that Of course, the above results (with Further, eq. ( 2 2.40 = 2 χ , it must compete with other transient dynamics (e.g., depending on L Hence if we choose example, with this choice, black holes with magnetic charges exhibit the desired switchback its role becomes lessmagnetic important (with in this regime. In particular,boundary term if ( the black hole is purely of the exponentially growing modereflects is the only fact that the the dominant analysisapplies producing contribution for the at earlier times.χ This this regime actually becomes smallerbecomes as small, the as black illustrated hole inwe becomes see figure mostly a magnetic, good i.e., agreement as λ with an exponentially growing modehole with the (with Lyapunov exponent energy of the shock wavevery and by early a times factor because of hole it has must only compete magnetic with charges other (i.e., transient effects. In the limit that the black Figure 5 (1+10 black hole with mostly electricthe charge larger exponent eq. ( JHEP02(2019)160 (2.42) (2.43) and the ]. Holo- 1 ]. However, ) and the , = 0. 0). Then the  23 , 122 γ r 0 – t 2.41 − ' , 119 0 2 ) for , t (  r 2.24 In fact, as discussed → t . ) 6 0 ). , t + ∆ 0 , ) and ( w t 2.32 2.40 → ) = (0 w w , t are essentially the same. Further, L , w , t t ) in a broader class of charged black R L A t C dt t , t d 1.2 − ]. The presence of these new couplings lead + ∆ dependence drops out of eq. ( A R ). L C t χ dt ) to shift ( d 113 – 17 – . Therefore, with this choice, both electric and , 2.42 → 2 m = 2.43 q L 112 A w + C 2, the dt d 2 e / q ] and they were extended to asymptotically AdS geome- = Therefore, we can anticipate that the switchback effect will t , t = 1 ) for the rate at large 2 T = 0. Of course, this is precisely the behaviour found in this 7 ∆ is related to the late-time growth rates of the complexity on γ 115 q , we use eq. ( , 0 γ − w t 2.41 R = 114 t /dt ) has a contribution corresponding to the growth rate on the right boundary at w A t → C d R 2.42 t , ]. Our investigation of the holographic complexity for these dilatonic ]. The AdS solutions were further explored in, e.g., [ w t 39 , , if we choose 118 – 38 , 2.2 116 36 , ], there is a more general relationship that extends to heavy shocks, i.e., In the following, we consider a simple extension of the Einstein-Maxwell theory, where As a final comment here, let us note that for a light shock wave, 30 Of course, there isThat a is, similar given relationship a between large the rates in eqs. ( 41 7 6 right-hand side of eq. ( very late times and anotherthe coming white from hole very early part times ofby on the the the Penrose time left diagram symmetry boundary. when of the Inup the complexity fact, to is unperturbed the an actually latter Penrose overall decreasing, is diagram, sign, e.g., probing this i.e., early-time [ the rate minus matches sign the in late-time eq. rate ( cally flat geometries intries [ in [ graphic complexity of dilatonicels black [ holes has been previously studied for several mod- singularities and the casuallike structure to understand of if the theseholographic corresponding changes complexity to black in the holes. an spacetime essential geometry Hence way. modify we the would behaviour ofMaxwell the field has andilaton. “exponential The coupling” corresponding to charged dilatonic an black additional holes scalar were introduced field, for the asymptoti- so-called holes. This investigation is motivatedresults by in the the question previous of understanding sectiontheory. to are In what special particular, extent in to our many theto string precise various theoretic couplings moduli settings, of or the the scalars, gaugeto Einstein-Maxwell e.g., field scalar will see hair also [ be on coupled the charged black holes, and may change the nature of the spacetime subsection. 3 Charged dilatonic blackIn hole this section, we investigate the CA proposal ( Hence for large either side of the shockbe wave. absent in exactly themagnetic same black situations holes where the with late-time growth rate vanishes, e.g., for this rate is essentiallyin twice [ the late-time growth rate in eq. ( because of the symmetry of the shock wave geometry under in section behaviour only depends on magnetic black holes exhibit the same switchback effect. two contributions in eq. ( effect while those with a purely electric charge would not. Further, similar to the discussion JHEP02(2019)160 . ),  1 3 2.4 − (3.3) (3.1) (3.2) − 2 is the 2 α α 3 L µν = 0, but  F . φ ) and ( i log µν φ 2 F 2.3 ) α α , where /α αφ 2 1 2 1+ − − = 0. In this case, α /L = ( ge 6 ) has a minimum at e φ φ 2 φ − − ( α √ ). The latter is tuned so V x . 4 + 8 3.2 d (0) = ∓∞ αφ M V 2 = 0, Z e ) 2 φ ) = 2 , g 1 φ α 4 ct ( . I ) has only a maximum at − . V − , are the same as in eqs. ( φ + (
ct /α ) I ∞ V 1 φ surf + (3 ( →∞ I = 0 and a minimum at → V ) = and αφ + φ/α φ φ α 2 − ( – 18 – surf − 2 V bulk I e ) I , namely, 1 and 3, and 1) α , ∂φ = . More generally, the potential is invariant with the φ 3 − 2( φ →∞ / 2 tot I αφ α ) given by → − = 1 ) has only the global maximum at (3 R − φ → − , we will also consider the effect of adding the analog of the 2 φ ( φ 2 ( . Moreover, lim α φ α g V h .  ˜ V − 3.2 2 ), as well as a new boundary term for the dilaton. Here, we are 2 2 α α √ L 3 3, 3, as well as the maximum at −∞ − 2 x 2.5 / − 3 ) controls the strength of the coupling of the dilaton to the Maxwell 4 1 1 ) has the maximum at 2 < d 2  φ α α ) = ( 2 < M φ V Z ( log 2 2 V (1 + 3, N < α α α − > 1 3 < α 1+ πG / 2 − →∞ α 16 ) = φ φ = = 0 is a critical point (i.e., a local maximum) with ( = For Asymptotically, we find lim For the special values it is symmetric under following substitutions: φ For 1 lim For 0 V φ The parameter As commented above, we will be studying holographic complexity in a theory where • • • • bulk I field, but it alsothat determines the shape ofcurvature the scale potential of in the eq.the corresponding ( potential AdS depends vacuum. on We the also value note of that the global shape of where the gravitational boundary terms, respectively. In subsection Maxwell surface term ( again focusing on the case of four bulk dimensions for simplicity. The total action takes the form where the dilaton potential the Reissner-Nordstrom black holes.the Hence spacetime our geometry conclusion is israte the in that essential the the feature previous causal leading section. structure to of the vanishing late-timegravity growth couples to a dilaton, as well as the Maxwell field (and cosmological constant), vious section (for certainfor choices charged of black charges holes.boundary and Rather, term boundary for modifies terms) the the is theoriesto complexity not studied reduce growth a rate here, the but the generic ratecomplished the analog to result in coefficient of zero the can the theories generally. not where Maxwell However, the be charged we chosen black will holes see have the that same the causal latter structure as can still be ac- black holes will show that the vanishing of the late-time growth rate found in the pre- JHEP02(2019)160 3, / 1 (3.6) (3.4) (3.5) ≥ positive 2 b α 3 where the 9 / . Hence in 1 2 but then reduce ≥ . Furthermore, c /L 2 6 6 α − Moreover, if we set ) from the previous ) vanishes with any , r = 8 ( ) 2.2 ). 2 , U , =0 ) α 2 r | ( r , ) θ dφ α → ∞ φ ) 2 U c b ( r where 2 α ( ) = 0. However, for between the event horizon and 1 + V = 2 L b  − r U + sin = r (  = αφ 3, the threshold corresponds to the point 2 b f 2 e + yields the usual coordinate system for the / r 2 2 1 dθ b 2 2 α α α α ) can be shown to be ] < + and the singularity is spacelike. Hence )( − − 2 1+ 1 r r ( 1 1 + 3.4 , q 116  2 → dt , e < α b 2 > b r + 2 U α r α ∧ c + 2 − + 1+ r  1 dr ) – 19 –  2 , as shown in the right panel of figure. r  N b r ( 2 + 1 αφ dr )  G f 2 r − c 2 r 0) and Schwarzschild ( e ( < r 1 + e , and hence the threshold solution contains a null singularity. U − = q 2 b →  − . The causal structure for these solutions is illustrated in 1 2 b N dt α → r  M ) + r  g πG r ( b < r 4 c ) = ) = f r r √ − ( ( ) becomes a naked singularity, e.g., if we begin with large ) reduces to the Einstein-Maxwell theory ( 2 f ), a class of static spherically-symmetric solutions describing elec- = = , i.e., 3.4 U = 0, the dilaton decouples from the Maxwell field and the poten- , and hence the solution becomes an extremal black hole (matching the behaviour 3.1 b 2 F + fixed. For the theories with 0 3.1 α r ds = b ], the mass of the black hole ( r . 3, there is an additional inner horizon at 121 α / are integration constants. We note that this solution interpolates between the 1 b < 2 ) reduces to a simple cosmological constant, i.e., coincides with and . The geometry has a curvature singularity at . In general, there are horizons determined by c − sufficiently large, e.g., α 6 r < α 3.2 c Following [ Implicitly, for the following, we will only consider nonextremal solutions, with For this theory ( Of course, just as for the Reissner-Nordstrom-AdS solution, there is a threshold beyond which the In the latter case, the coordinate transformation 8 9 = 0, the solution reduces to the (uncharged) Schwarzschild-AdS solution independently charged dilatonic solution ( its value while holding where of the Reissner-Nordstrom-AdS blacknonextremal holes). black holes However, only thehorizon have meets situation a the is single singularity, horizon. different i.e., for In this case, the threshold is reached when the event Schwarzschild-AdS metric. in examining the CA proposal,meet the we singularity will (at find late the times),for as future 0 illustrated null in boundaries the of leftthe the panel singularity of WDW at figure patch and figure finite one generally finds a single (real) solution where Reissner-Nordstrom black hole ( b of the value of with tial ( this limit, the theory ( section coupled to an additional massless scalar field. trically charged dilaton black holes is given by [ Of course, if we set JHEP02(2019)160 , 3, / ) 2 1 , we (3.7) (3.8) α 2 < (1+ 2 / α 1) − ≤ 2 α ( ). The left panel  + b 3.4 r − 1  b 4 ) . 2 ) ). − 2 α − ) α r 2 α = (1+ (1 + / r 2 in terms of the position of the event 1) c − (1 + πL 2 + 4 α ], however, we do not pursue this ques- r (3 3 97  , + + b ) r . The right panel corresponds to 0 96 2 . b α ) − – 20 – 2 = 1 α (1+ r /  ) 2 (1+ 2 3 + α / 2 r L − α 2 (1 +   + + + r b b r r = − − c 1 1   + ) = 0 to rewrite the parameter 2 + 3, for which the causal structure is similar to that of the Schwarzschild-AdS 1 + r / πr r 1 ( 4 N π f G ≥ = 2 + = α r = ) r +

r N ( . Causal structure for the charged dilatonic black hole given by eq. ( 2 ∂r ∂f G π 1 πU 4 = = S T 3.1 Complexity growth We will nowdilatonic study black the holes time-dependence presentedthe of above previous discussion the using of holographic the the CA dyonic complexity Reissner-Nordstrom-AdS proposal. of black holes the Of in charged section course, in contrast to It would be interestingdynamically stable to (e.g., study in which analogytion black to here). refs. hole [ solutions are thermodynamically and Then the temperature and entropy of the black hole can be expressed as It is useful to use horizon of the black hole, Figure 6 corresponds to black hole, with afor spacelike which the singularity causal at structuretimelike is singularity similar lies to behind that an of inner the Cauchy Reissner-Nordstrom-AdS horizon black (at hole and the JHEP02(2019)160 (3.9) 2 +1 (3.10) (3.12) (3.11) α 2 4 α .  6 b r . − 3, the WDW . Similarly, at / 0 . 1  1 6 +  2 1 m m b ) r r ≥ = 2 .   r 2 0

α 1  2 α − α  + − 2 b ) α 1+ r 3 = ( r 3, the past meeting point

instead. Hence the GHY ∗  / r + (3 b r 1 2  — see figure ) 1 ) − r r − − < +1 ( ( 2 − 2 1 ∗ ∞ α 2 r f α U 3 r  ) corresponds to the position of the 2 
+ → ) 2 b 1 b m r < α ) r t r 2 ( r 1 m 0. The trace of the extrinsic curvature − ( − r U ) 3, ). In contrast for ( U 1 2 → / r   and so as usual, we introduce a regulator + 1  , α ) ∂ 0 r b  2 # ( ≥ α 2 2 e < r f = 2 8 r q ) (1 + ) + 2 α r 2 1 m  r – 21 – ) r + 2 2 ( ) r ( L ) r f  < r 2 ( 2 r U 1 ) ∂ α ( U 2 +1 − − 2 r 2 2  r α r 2 ∂ α α r ) 2 r 3, the future tip of the WDW patch is the joint where the (1 + (  ) yields 2 / f b r (1 + 1 ) + 2 L 1 . − r 3.1 − − ( < α + ) here and defer the discussion of additional boundary terms to p f 1 . On the other hand, for 0 " 2 2 2 r e b r  q ∂ 3.3 − =    1) < α 3, we will assume that the WDW patch has already lifted off of the dtdr = 2 for all / 1 m . We must evaluate the Gibbons-Hawking-York (GHY) term, given in N ) − N 1 r 0 1 r + 2 G  K ( G r 2 ≥ WDW α 2 U + Z 2 (3 b → = − α 2 N α 2 m 1 = = G r bulk 2 r + dt dI = GHY ), on this surface and consider the limit I , e.g., as in the Schwarzschild-AdS solution, but ], which is straightforward to adapt to these solutions. Further, we will only be 2 2.3 r bulk 34 I However, notice that innot integrating this termcontribution over from the the regulator surface, surface the becomes spherical measure is eq. ( of the regulator surface is given by As noted above, for 0 future null boundaries meetpatch (with ends on the spacelikesurface singularity at at singularity, that is approaches the Cauchy horizonlate at times, late times, i.e., GHY contribution For black holes with just one horizon, i.e., The time derivative then becomes Bulk contribution Evaluating the bulk action ( in [ considering the action ( the next sections. Sincetheories with we are primarilypast interested singularity in in the the late-time following growth calculations, rate, as illustrated for in the left panel of figure only have the solutions carrying purely electric charges here. We will follow the discussion JHEP02(2019)160 . ) 2 here ) (3.16) (3.17) (3.19) (3.13) (3.14) (3.15) (3.18) 3 r . The 2 b ( ) 6 U r ( ( r 2 1 m m U r r ∂ ct .  ξ` | | ) ) which vanishes ) r 2 0 2 ( m  ξ 2 r f ξ ) log ( | . + 2 — see figure f ) | b 2 1 m m r r r 2 m (  r 1 3 1 3 = ) log . ) are added together, the log U ) r ξ ( 2 r ( ) r = > ) ( f 2 2 r m 3.19 2 2 N and 2 ( . r )) ) U α α ( G dr ∂ ) r 2 U m r 2 1 m 2 ( ( ( r r r r 2 ( U max for U 2 r ∂ 2 m = U r ct = ( − U ) and ( r Z . r |  . However, the time dependence has a ξ` ) for r ∂ ) ) 2 b N 3 2 α 1 b 1 m L ) 3.15 2 G 2 r − r ξ , ) ( ) + ( 0 yields − c ) log r r f + r ξ ( ( r | U ( b ( ( ) would absent at late times if we consider the ) → f U 2 r 2 f = ) r − 0 ) α ) 2 1 m log ∂ – 22 –  c r 2 ξ∂ λ α r 2 ( ) )   ), we begin by choosing the affine parameter along ) (1+ r 1 U m r ) ( N N N ( ( r r N 3 (1+3 2.4 r ( U G G ( G Θ = U ( 8 4 2 ∂ 2 G 2 r 0 limit produce the extra term proportional to 4 ct U U ∂      ξ` − → 3. 3, the future boundary of the WDW patch is the regulator / = N 0 = /  1 1 G
) = ) ) log 2 = 1 2 2 m ≥ GHY 2 m dt ) r 2 r 2 = ( r 3, the only joints which contribute to the time dependence are ( α dI ( / α ct 1 U 3. As expected, when eqs. ( ( dt joint joint dI r / I < I 1 0. Hence the corresponding joint contributions vanish. Therefore in this 2 ≥ ) + dr ∂ α ) + 1 → 2 m 1 m max r α r 0 r 1 m (  r ( Z joint N joint I 1 I G d = dt ct I Implicitly, the contribution evaluated at solutions for combined contribution to the time derivative is independent of This integration is nontrivial forsimple general form, values of which then yields The sum of the counterterm contributions on the four null boundaries then reads precisely given by the expression above evaluatedCounterterm at contribution To evaluate the surface counterterm ( the null boundaries as As discussed above for surface just above theboundaries spacelike meet singularity. this While surface,in there their the are size limit joints is where proportionalcase, the the to future contribution null to the time derivative comes from the past meeting point and it is The time derivative then yields Joint contributions If we focus on those at the futurecorresponding joint and contributions past are meeting given by points, i.e., Notice that subtleties inwhen the we have precisely Now taking time derivative and the limit JHEP02(2019)160 ) 2, ct / ` (3.21) (3.20) = 1 2 α ) and consider , this boundary 3 for which the / 1 3 1 3 1 3 ) for the charged 1 3.19 2.2 < = > 1.2 < 2 2 2 2 α α α ), α ]. We also note that for ) and ( ) for the latter solutions. 3.1 for for for ≤ 55 , as discussed above, and at , 3.15 2 m , i 2.22 0. We also note that in the r 34 2 , where the black holes become b ), ( L αφ = 3 2 N → b i r − → G e N 4 3.13 α ν + G ) ), this rate will vanish as we approach r − A 2 b ), ( α N 3.7 b µν 3 G 4 (1+ F 3.10 µ 3, which again parallels the behaviour of the − − Σ / i d + + 1 r r + – 23 – 2 2 1 e e r q q N N M < ∂ G G the full time evolution of complexity for 2 − Z 7 α − − − 1 2 α r g γ ≤ h M M 2 2 = N 2 e h h ) reduces to precisely the growth rate of the (electrically q Q 1 1 πG π π µ for 0 ]. Up to a certain critical time, the WDW patch ends on both I          3.20 − 34 r = ) is effected by the addition of two boundary terms, involving the → A 3.4 C dt + d r →∞ lim . Further, we observe that using ( t , we recover the late-time growth rate of the Schwarzschild-AdS solution, i.e., 3 solutions, the rate vanishes in the limit . In particular, we note that in the theories with 0 M/π / ). The behaviour is very similar to that of the latter neutral black holes, as shown is a free parameter. Following the same reasoning as in section 6 1 6 α = 2 → ∞ ≥ γ α 2 As an example, we show in figure /dt α A C We begin with the Maxwell boundary term for the new theory ( where term changes the boundary condition imposed on the Maxwell field in the variational Next we examine how thedilatonic growth black rate holes of ( theMaxwell holographic and complexity dilaton ( fields. Maxwell boundary term the complexity exhibits a transient behaviourand (which then depends by on a thelimit counterterm time scale which of it the subsequently order approaches of from the above. 3.2 inverse temperature, it has Boundary overshot terms the late-time in figure in the detailed analysis ofthe [ past and futureAfter singularities, this and critical time, during the this pastlate time, null times, boundaries this the meet joint complexity at approaches remains the event constant. horizon. In this period of time, rate of change of (electrically charged) Reissner-Nordstrom-AdS black holesthe [ null singularities. for which the causal structure resembles that of an Schwarzschild-AdS black hole (left panel late-time rate above hasIn precisely fact, the same the form resultcharged) as in Reissner-Nordstrom-AdS in eq. black eq. ( holeslimit ( when d extremality, i.e., as Of course, the late-timesee growth figure rate depends oncausal the structure causal matches structure that of of the the black Reissner-Nordstrom-AdS hole black holes, — the form of the Now we combine all of the contributionsthe in late-time eqs. ( limit, to find Total growth rate JHEP02(2019)160 2, 5. . √ / ) are (3.23) (3.24) (3.22) = 2 ), if the = 1 c 3.20 α 2.29 2.5 ) on the time 1 3 1 3 1 3 75 and . < = > 3.21 2 2 2 α α α = 0 b for for for 2.0 . 9, . µν i 2 , at times of the order of the F L ct = 0 ) = 0, this boundary term is 3 ` µν N b L G µν F 4  0 discussed below eq. ( , F 1.5 i αφ 1 m L 1 − 2 r αφ N → ) 2 − ) are now replaced by c c G = − ) α − α α 2 N e γ γ g e ct ( α G + 2 m + ` 3.20 1 µ − b 4 b r = 0. Further in analogy with ( 3( (1+ ∇ √  and x 2 e 1.0 4 M − − q i N ∂ d | 2 2 e e α + 1 G q q – 24 – r γ ) ) + + M δφ r r α α → ∞ Z − γ γ − N N 2 − − G G α α − = 1 g (1 (1 r γ 2 h Q 0.5 − − µ 2 e = dt q ) dI N M M α γ 2 2 πG h h − on shell 1 1 (1 π π

, the limits Q            µ α 0.0 I γ = 1.0 0.5 0.0 -2.0 -2.5 -0.5 -1.0 -1.5 A C dt d →∞ lim t . The time dependence of complexity for an electrically charged Maxwell-Dilaton black Notice that, if weunchanged, i.e., fix they yield the late-time growth rates of the Schwarzschild-AdS and elec- Of course, all ofthe the above contributions expression, calculated the previously late are time limits unchanged. in Hence eq. adding ( in Hence, it is straightforward todependence evaluate the of effect the of WDW this action boundary and term one ( finds principle. However, we shouldboundary add condition that for implicitly the we dilaton,Maxwell would i.e., field also satisfies be the assumingequivalent equation to a of Dirichlet motion which corresponds to a blackAdS hole black with a hole. causalIn structure analogy The that to resembles parameters that thecritical of are time, Schwarzschild-AdS an chosen black where Schwarzschild- the hole, tothe WDW the be patch late complexity leaves time the does limit pastinverse not temperature. singularity. from change Then, above, until the with a complexity a approaches certain transient dependence on Figure 7 hole (without the addition of the Maxwell boundary term). We evaluate for concreteness JHEP02(2019)160 ). 3.21 (3.28) (3.26) (3.27) (3.25) . On the other = 1, for which ]. However, one 1 3 1 3 1 3 α . In this case, the 2.2 γ 26 < = > 6 2 2 2 3, and as we approach α α α / 1 for for < 2 = 0 corresponds to a Dirichlet α i ) in the previous section), this i φ N , ≤ φ . 3, in which the causal structure 2 γ G 2 µ / L ) 3 2 1 b N 2.25 b α σφ G φ ∂ 2 < . √ µ (1+ x 2 Σ − 2 = 1 yields a Neumann boundary condition d d − = 0 N 0 b φ 3 G Σ M φ γ < α M 4 ∂ ) does not yield the expected growth rate of Z dt  dI Z ) (or eq. ( − – 25 – 2 N are nonvanishing, the dilaton will have a more 2 3. N α φ 3.25 α / φ γ α 2 M →∞ lim 1 3.21 πG 1+ t γ γ h πG 8 h ) is actually tangent to the boundary. Therefore the 4 ≥ π 1 1 2 π 0 for = 2 = and α            φ 3.26 = 0), setting I φ 0), eq. ( φ I = γ M → ∂ =1 | α b γ δφ in which case the causal structure matches that of the Reissner-

A 1 3 for further discussion. C dt d 5 < in which case the causal structure is similar to the Schwarzschild-AdS 2 = 0). More general choices of this parameter lead to mixed boundary α →∞ 1 3 lim t M ≤ > ∂ | 2 δφ α µ ) becomes ∂ µ — see section n 3.24 Let us first consider black holes for 0 One interesting choice to consider for the boundary coefficient is M/π where each joint termfinds carries in a this case sign according to the conventions of [ resembles the Reissner-Nordstrom-AdS blackboundary hole as term shown lives inthe only figure derivative on appearing the inboundary eq. null term ( boundaries reduces of tonamely an the integral WDW over patch the joints but where in the this null boundaries case, intersect, boundary condition (i.e., (i.e., conditions. Further, ifcomplicated boundary both condition involving terms proportional to the integrand in eq. ( As for theterm Maxwell modifies boundary the termdilaton character ( in of the the variational principle. boundary For condition example, which while must be imposed on the 2 Dilaton boundary term Next we consider the following boundary term for the dilaton Nordstrom-AdS black holes, the (electrically) chargedify dilatonic at black late holes times. fail tohand, This complex- for precisely matches theblack behaviour holes, found the in late-time section growththe rate uncharged remains limit nonvanishing. (i.e., However, we observe that in That is, for 0 above rate will vanishthe as limit we of approach a extremality null for singularity 0 for eq. ( trically charged Reissner-Nordstrom-AdS black holes, respectively. Further, as before, the JHEP02(2019)160 ) ) 2.1 3.26 (3.29) ]. More 94 – = 1 sets the 91 α , γ 88 – ) for the electrically 86 . Evaluating eq. ( 3.24 . 6 0  + b ). We return to this point in = r

3. No such choice was possible ]. One perspective of JT gravity 3.1 / φ ) that choosing 1 r 83 – < φ ∂ 3.25 78 ) 2 N r α ( G f 2 2 ) ) does not change the complexity growth rate r = 0. However, when the Maxwell boundary ( ) are summarized eq. ( α U γ – 26 – 3.26 3, in which the causal structure resembles the φ / γ 3.21 1 0 ], which has a simple action linear in the dynamical → ≥ 0 77  – 2 = lim α 75 φ dt dI ) when we set →∞ lim t 3.20 6= 0, this expression is divergent. Therefore adding the dilaton ) spoils the good behaviour of the regularization procedure at the φ γ and therefore it appears that the causal structure of the black hole was 3.26 2 ) was included, we also showed in eq. ( . 3. The former behaviour was analogous to that found in the Einstein-Maxwell / 5 3.21 1 Hence, our general results for the late-time growth rate of the holographic complexity Next, we turn to the case ≥ 2 dual of the Sachdev-Ye-Kitaevacquires (SYK) an emergent model reparametrization in invarianceis [ the that it low describes energy physicsof limit, (of near-extremal the where spherically charged the symmetric blackspecifically, sector) system in we holes the focus in near-horizon on region higher deriving dimensions, the e.g., action for [ JT gravity by reducing the action ( 4 Black holes in twoIn dimensions this section, we willmensions. focus Our on main studying motivation dilaton isJackiw-Teitelboim gravity evaluating (JT) models the model growth in of [ two holographicdilaton bulk complexity field. spacetime for di- This the theory has received great deal of attention recently as the gravitational consider dyonic or magnetically chargednot black holes. yet Unfortunately known the for lattersection the solutions Einstein-Maxwell-Dilaton are theory ( the Reissner-Nordstrom-AdS black holes,when i.e., the for causalα structure had thetheory form in of section theone Schwarzschild-AdS of black the holes, essentialnote i.e., features that for producing our analysis the here unusual focused behaviour only found on electrically there. charged black However, holes we and we did not including the Maxwell boundary termcharged ( black holes. Ofboundary course, these term results in match eq. theterm ( growth ( rates without thelate-time Maxwell growth rate to zero for the cases where the causal structure was like that of Unfortunately, for boundary term ( singularity. Therefore, we do not consider these boundary terms further here. to the late-time growth rate.(spacelike) However, regulator there surface is at an the additionalon future contribution this coming singularity boundary from — the and see figure considering the late time limit, we find at late times for thesecomplexity black holes. at early Nevertheless, the times transient will behaviour be of modified the holographic by thisSchwarzschild-AdS black term, hole. but In we this will case,WDW not the patch contribution explore still from this the reduce here. null to boundaries contributions of the on various joints, which again do not contribute Hence, adding the dilaton boundary term ( JHEP02(2019)160 ) d ), ), 2.9 2.4 2.6 (4.1) (4.2) (4.3) ]. ) illustrates 125 in eq. ( ) in the four- 0 4.2 I 2.2  . We will discuss these . (Ψ) -form field strength in 2
d U 2 ). Hence while we begin by − 2 θdφ ) or the null counterterm ( 2.5 2 ) but without the addition of the . Ψ) 2.3 2 ∇ , and the result is precisely that , 2 m ) and this magnetic field into the 2.1 q Ψ 2 F + sin = 0. The two-dimensional Maxwell Ψ 4.1 2 ]. We decompose the four-dimensional µ + 2 ( + 2 m dθ q ∇ ) results from integrating by parts in the 2 2 R 129 µ 2 2 – L Ψ n 4.2 | 6 Ψ γ  126 | − + Ψ g ) with – 27 – b 2 p − − dx ], where the effective cosmological constant is dy- 2.9 √ = 0 [ dx a x γ 2 M d dx ∂ 124 ) , Z (Ψ) = M x ( Z U N 123 1 ab G N g 2 1 = 0. Substituting eq. ( G = 4 + e 2 q = ds 2D mag I ) with ), we integrate out the spherical directions to produce the following two- 2.9 2.2 = 0. In addition in this section, we will analyze an analogous two-dimensional e q dimensional reduction. We emphasizedimensional that action it and arises is from unrelated to thewhose the dimensional bulk surface reduction terms terms we ( ( willthe explicitly fact examine below. that The restricteda action two-dimensional to ( gravity spherically model symmetric with solutions, a our dilaton theory field. can However, be no recast approximations have as with the potential given by The boundary term in the second line of eq. ( dimensional action If we assume thatcharge, the we Maxwell can fieldgiven use in by four this eq. dimensions metric ( bulk corresponds ansatz action to to ( a solve pure for magnetic 4.1 Jackiw-Teitelboim model We begin with the dimensional reductionMaxwell of boundary the term, action i.e., ( setting metric as the holographic complexity forwe both will the find JT thethe and holographic corresponding JT-like complexity four-dimensional models. for black boththeories holes As and models discussed might the section behaves be holographic in complexity expected, the in same more detail way in as an for upcoming work [ the Brown-Teitelboim model [ namically controlled by the energydimensions. density Further, of our an analysisshown antisymmetric of the holographic important complexity roleexamining in of the the the dimensional Maxwell previous boundary reduction sectionswe term has without also ( this consider term, the dimensional i.e., reduction by of reducing this boundary term and its contribution to cally charged in fourwith dimensions, i.e., the four-dimensionaltheory gauge that field can has describe theing form the a ( near-horizon purely physics electricfield of is charge, four-dimensional an i.e., black essential eq. holes ingredient ( carry- for this JT-like theory and so it has a form reminiscent of to two dimensions while assuming the background is spherically symmetric and magneti- JHEP02(2019)160 , 8 (4.7) (4.8) (4.9) (4.4) (4.5) (4.6) has a (4.10) 1. The ab g  . 0 ) (within the . ) . Φ 2 2 2 / 2 2.7 b µ 2Λ µ 2 2 πL ) can be recovered ) with a description − 2 L 2 2.7 r = 4.9 R − ≡ JT Φ( . As depicted in figure ) c g r r ( − ) which allows us to express f h √ r . For latter purposes, we define . ( x 2 h ,T 2 r  . d ) 2 0 RNA µ with = , 6 f L  M 2 2 = Z 2 h ]. The boundary value of the dilaton + r L ) . The black hole is characterized by the 2 . N 85 r JT + N 2 h µ ( + Φ) 1 2 – h r r dr G f  1 0 πG 4 79  πr 1 + 3 4 = + ) = 0 = (Φ 16 −  h + Φ( 2 r ≡ JT 2  π h 0 – 28 – 1 + ( = dt r r 4 0 Φ ) 2 2 r R Φ 1 = 1. In particular, applying this expansion (to linear = ( RNA L = g ). That is, we write f f 2 ]. That is, the near-horizon region of the extremal are taken as energies conjugate to the coordinate time − −  − JT Ψ 88 2 T,ext 4.4 √ 0 [ = JT q = x T Φ 2 2 h 2 / r d Λ ) yields the Jackiw-Teitelboim action, M determines the position of the physical boundary of our system. ,S Z 4.2 c c r r N 2 2 , ds 0 = 2 L c r πG Φ µ r r N b b 16 Φ πG = and temperature and the linear approximation remains valid as long as Φ 16 Φ = Φ b ). JT bulk JT = I M 4.2 JT M Now, in considering small deviations from the extremal throat, we expand the dilaton Next, we are interested in describing the near-horizon region of the near-extremal (which will be taken as the time in the boundary theory). Of course, one can treat the JT The mass t model as an independent theory, orof one the can near-extremal match the throats JTlinear of solutions approximation the ( applied Reissner-Nordstrom-AdS above). black holes ( is denoted Φ metric has an outer andfollowing inner parameters horizon at In the dilaton solution,this we time-like have surface introduced theThe cut-off dynamics radius of thehas boundary extensively position been reproduces studied the in IR recent physics years of [ the SYK model, as order in Φ) to the action ( The solutions derived from this action can be written as around the extremal value in eq. ( with the understanding that Φ Further, in theconstant extremal negative throat, curvature, which the is two-dimensional related to geometry the described higher dimensional by parameters by the extremal horizon area as For the extremal solutions,the we extremal have charge in terms of the horizon radius, from eq. ( black holes. Recalllong that throat in of extremal limit,solutions a is the fixed described charged by radius a black constant holes dilation profile develop Ψ an infinitely been made at this point, and so the full four-dimensional solution ( JHEP02(2019)160 (4.12) (4.11) ) at late 4.16 . 9 . ) as part of the WDW . 2 m c r r

2 m c 4.12 r r N .  | µ ) Φ  r πG ( 4 = f | + JT  r , respectively. 1 m c log r r 2 m

= r N r N 0 ) in this surface term, we find a term = Φ Φ πG r πG 8 4.7 4  ], we anchor this region at the boundary + and 34 ) in its own right, without any reference to higher 1 m c Φ = 1 m r r µ ) included a surface term which was not incor- 4.8 r  | ∇ – 29 – ) = 4.2 r µ ( r 1 would be slightly modified, e.g., in figure f | . dλ k 0 following [ JT log B , we consider the WDW patch anchored on the physical Substituting eq. ( tT 2 Z N 8 0 10 ) on the WDW patch, which yields N Φ ). πG 1 8 4.8 πG  4.8 8 = = JT bulk 2. Further, as illustrated in the figure, we denote the meeting points of I JT totder . As in section I t/ solution of the JT model and the WDW patch. Physical boundary is depicted c r 2 = = R t r . AdS = L t First, we evaluate eq. ( One might also wish to consider JT gravity ( 10 dimensions. In this case,action. we would not However, include dropping thetimes, total this but derivative the contribution contribution ( transient would behaviour not for change the vanishing growth rate ( However, recall that the reducedporated action in ( the JT actionthat ( is linear in Φsurface and term when yields it is evaluated on the null boundaries of the WDW patch, this the future and past null boundaries as Complexity growth Next, we consider theproposal. growth As of depicted holographic in complexityboundary figure for the JTtimes model using the CA Figure 8 with a blue curve. The outer and inner horizons appear at JHEP02(2019)160 ). ). → 2.3 1 m 4.10 2 and r (4.15) (4.16) (4.13) (4.17) (4.14) / ) 1 m r ( f . = 2 c m r r + Φ). Combining 1 m c /dt 0 r r  . 1 | m ) is vanishing in both ) , ) have a simple dimen-  1 2 m m r dr ) , r r | ( . 2 0 d ) ) then yields the relevant
f 2 4.15 2.4 Φ r
log (Φ | ) (  ) 2 b 2 0 c Θ 2 c λ c | , of the future and past null 4.7 f r ct ) Φ ∂ Φ | ` 2 r m 2 b 2 c 2 ct ( µ/r r 2 0 ` r µ/r Φ ( = f , the complexity growth rate is ( ) and ( Φ 1 | 2 ct  1 2 b log ( d 2 − c ` ), we find ) in eq. ( − 2 r and d 2.3 Φ ) and the null surface counterterm. r  2.1 2 ( a 2 ct , 1 m f ` ). The growth rate of the holographic 2.12 . 0 i tanh r Σ 

− 4.9 + tanh a + Φ) log = 0 t t 0 ) yields the holographic complexity for the + Φ) Θ ) log + Φ) log JT 0 JT JT r A (Φ 0 1.2 ( dt C + Φ) πT (Φ f d πT − 0 – 30 – (Φ  (Φ dλ − N 2Φ 0 →∞ lim ) into an expression involving the boundary or phys-  t G i B tanh 2Φ c X Z b N ) in eq. ( r  − , ) yields the total of the boundary contribution for the Φ 4.15 2 1 N N 8 N . = = tanh πG π λ 1 1 8 4.11 1 ) is defined in eq. ( 4.12 ∂ 1 2 16 πG πG m m r µ µ πG r r 8 8 ( + 8 . Hence the prefactor of − = f µ approach the inner and outer horizons, respectively, i.e., = = = ≡ ≡ µ = ∂ = = 0 and we can ignore the corresponding surface term in eq. ( 2 m JT µ JT 1 m 2 m ct ct r JT A JT I κ I joint k x x JT + dt C I r + d 2 where and / → ) JT joint 1 m 2 I m 2 m r r r ( + f − and JT totder µ = I − It is interesting to recast eq. ( As the higher dimensional calculations in section The remaining boundary terms introduced in eqs. ( /dt = 2 m JT − First, we define dimensionless coordinates for the meeting points. r contributions and thus we have ical parameters of the JT model, e.g., the mass, temperature and entropy in eq. ( complexity is then given by At late times, CA proposal. determined by the dynamics ofboundaries the of meeting the points, WDWdr patch. In analogy to eq. ( Adding this expression to eq. ( where the two-dimensional ‘scalarthese surface expansion’ terms reads withWDW Θ eq. patch illustrated ( in figure Dimensionally reducing these two termsboundary and terms substituting for eq. the ( JT model, normal normalized as sional reduction. Of course,WDW patch, with we affine have parametrizationThis along leaves the only null the boundaries null joint of terms the (proportional to Here, we are assuming that the null boundaries are affinely parametrized, with the null JHEP02(2019)160 ]. 1. 40 ,  (4.18) (4.20) (4.21) (4.19) c µ 29 r , , 26 b 0 Φ Φ .
t JT . πT 2 . 1 m 2 m − ] — see also [ x x 1 e 41 −  | O 2 . ) , which is an ambiguity that x 2.0 2 2 2 x c + ( r ). Different colours correspond to L t ˜ ≡ f /L | ) JT ≡ ct 2 2 x c 4.19 ` T c ( r µ t ˜ µ f 2 b 2 0 JT Φ Φ . As we can see, the curves are qualitatively 1.5 2 πT 2 2 2 2 ct L ` L − e =  where 2 ct ) ` – 31 – JT JT ) log T ) where x c 1.0 and πT µ ( , with a negative transient behaviour, and the vanishing c r ˜ 3 f (2 2 −  r/µ = ), the late-time limit of the complexity growth is zero. π µ ( − ˜ 2 f c JT 2 c JT π ) as 2 2 µ = 10 2 µ 4.16 T M µ L b 0 . π Φ Φ − JT and 0.5 9 2 4.15 0 ) also depends of the ratio M = ) = Φ r ( / 32 JT A b f 4.19 dt C − d = 0.0 JT A 0 dt C -2 -4 -6 -8 -10 d is the conformal breaking scale in the boundary theory. Both of these ratios ), in contrast to the expectation from higher dimensional black holes. The c . Complexity growth in the JT model from eq. ( µ 4.19 As we already saw in eq. ( Further, we note that thisin limit is eq. generically ( approached fromleading below, contribution given at the late expression times is given by Note that the resultarises ( in defining holographic complexityWe with show the examples CA of proposal the [ of time the evolution temperature of in the figure holographic complexity for different values That is, should be small as the near-extremal and near-horizon limit requires both Hence apart from thedimensionless overall factor ratios, Φ of the mass, the above expression is a function of the Then we write the blackening factor as Finally we can rewrite eq. ( Figure 9 different temperatures. We set similar to the purelylate magnetic time curve rate in of change. JHEP02(2019)160 ) ), 2 2.24 ) but (4.22) (4.23) 4.1 2 F 2 Ψ g − ] it was argued to ) was derived from √ x 130 2 4.8 d M Z ]. In [ 2 π g 130 )-plane. In this case, the bulk − = 0. Recall that in section ( ], with a dynamical cosmological and hence the dimensionally re- 2  e ) is identical for black holes with ) , x rt q . 2 1 124 F , 2 2 x 2.7 (Ψ L Ψ . e 6 U 123 ) leads to the same equations of motion ]. However, we will show that holographic − − 2.1 2 2 86 = 0, assuming the metric ansatz ( 4.22 − Ψ) γ , – 32 – ∇ 2 ) = = 0, i.e., we did not include the Maxwell boundary 2 Ψ γ µ + 2 ( (Ψ 0). In particular, the throat of an electrically charged ) was our main motivation to revisit the holographic ] and more recently in [ ∇ ) with , e µ U T R n 2 q 2.1 | 4.16 129 ) with Ψ , γ is supported on the ( |  2.1 g ) = ( p F ) has a single component 127 − m ]. When we evaluate the complexity growth for the new dimen- , dx √ , q 2.9 e x 126 q M 2 132 , ∂ d Z M 131 N Z 1 G ) and ( N 2 T 1 G , q + 4 ) reduces to the following two-dimensional action = ), and with a purely magnetic solution, i.e., 2.2 = 0. The result in eq. ( ) after putting the gauge field on-shell [ ) = (0 2.5 χ m We recall that the four-dimensional geometry ( Consider again the action ( The vanishing of the complexity growth rate at late times may seem puzzling for the JT 2D electric 4.2 I , q e q the JT model studied above. ( extremal black hole iscorresponding identical to two-dimensional that geometry of and a the magnetic (constant) extremal dilaton black are hole. identical Therefore in the the with the potential We note that itas ( is possible tocomplexity derived show with that this ( action using the CA proposal yields a different result from sionally reduced theory, we find nonvanishingfrom complexity the growth higher at late dimensional times, analysis as in expected section a purely electric field,action ( i.e., form reminiscent of the Brown-Teitelboim modelconstant [ controlled by a fieldwas strength also of studied maximal in rank. indescribe A [ the similar physics two-dimensional of action charge the studied extended in [ SYK models with complex fermions with conserved We now turn our attention toa another purely possible two-dimensional electrically theory charged thatMaxwell black is field derived hole from strength in ( fourduced dimensions. theory In incorporates the aform latter Maxwell of black potential maximal holes, and rank the in the two corresponding dimensions. field In strength this is sense, a the two-dimensional theory has a term ( we found that thewith late time growthcomplexity of rate charged also black holes. vanished in this4.2 case, e.g., JT-like see model eq. ( gravity when we consider thatof this the model SYK is supposed model,nontrivial to which complexity capture is growth the low maximally fordimensional energy chaotic very reduction, dynamics and long this feature hence times. isthe would not four-dimensional However, be action so from ( surprising. expected the The to perspective action exhibit ( of the JHEP02(2019)160 . ˜ f with (4.28) (4.24) (4.25) (4.27) (4.29) (4.30) 2 ) with ). When 4.7  4.24 ). Of course, cd ˜ f . ) reduces to the 4.6 cd i ) 0

ab )–( ˜ 2

F 4.28 f ) 2 ( 0 ) ab 4.4 0 ab ) F g 0 ( F 2 , F − ( ] b + ( − 0 2 0 ˜ a 2 0 and eq. ( , defined in eq. ( . a bc E [ 0 ) . When we expand the bulk 2 E 2 , as a dynamical field, however, ∂ 0 ab ) 0 2 E  0
0 + 2Φ F F 2 0 ( 0 ab E 2 A c + 2 ab (4.26) g ) E E a ]
g 0 2( ˜ b f Φ] + ) F Φ ≡ − and again, our prescription is that we − 0 2 + 2 , the resulting action takes the form bc − + 4 ab ˜ A = 2 ) bc ). We will call this theory the “JT-like” ab f 0  ( ) 0 )  2 a 0 bc 0 [ F ) ˜ , + Φ)( [Φ f 4.8 F ( 0 F 0 ∂ ∝ c relative to ( c Φ + Λ ( 0 g ab F a Φ T,ext c 2 a ) ) 0 ab q ) a − 0 0 (Φ ) ∇ 0 ) = 2 N Φ 0 h F 0 √ F – 33 – F 2 / ( N F x g F ab π g g 4( ab 2 ˜ 4( πG G f 4 0 Φ − g 4( d  4 a √ Φ √ 0 0 √ , i.e., + , ∇ M − x Φ Φ Φ ab Z 2 ab = Φ + 2  ab ). As a consequence, we also expand the field strength as ) d b + N N N ) 2 2 0 0 0 ab ). In contrast, we treat 2 2 2 g 2Λ ∇ F ) ab E M g g g 2 F a ˜ 0 πG πG πG 4.24 ( f Z 2 2 2 F a a + 4.24 2 = ( ( − − ∇ R − ∇ −∇ − g 1 4 ab JT bulk F I − . That is, the two-dimensional geometry becomes locally AdS is simply treated as a constant parameter defining the theory, as is the 2 = ). Again, the deviations from this extremal form are captured by 0 ) yields the solution ( 2Λ ) to linear order in both Φ and : 0 = : 0 = : 0 = 4.24 a a JT-like 4.27 bulk Φ : 0 = ab ) ˜ a I 4.22 δ R − 0 δ defined in eq. ( δg A is precisely the action given in eq. ( ( 0 δ to its extremal value, we note that ( E captures corrections of order Φ 1. However, in the present case, we also wish to capture small corrections to the 0 JT bulk ˜ f F I  0 Now, as in the previous subsection, we wish to construct a theory which captures small Φ / Of course, eq. ( should choose the prefactor to bewe set the extremal electric field expected 0 = constant our prescription is thatstrength the in solution eq. is ( alwaysIt chosen is to useful yield to precisely write the out extremal the full field equations of motion where model. As before, Φ where action in eq. ( deviations from the extremalΦ throat. Hence,extremal we field expand strength the in eq. dilaton ( as in eq. ( the difference is thatthe the two-dimensional volume electric form throat is supported by a Maxwell field proportional to present case as in the previous subsection, i.e., as described by eqs. ( JHEP02(2019)160 T ). δq ) as 4.10 (4.33) (4.34) (4.32) (4.35) (4.31) ). The 4.29 since this 4.26 ab . , ) 0 0 ) reduces to  F ) (i.e., compared which is allowed ( 0 . a which capture the Φ . The second term Φ δE/E 4.30 ) for the JT model. 11 0 ˜ ∇ ab f 4.34 Φ − ). We note that the 4.9 / T,ext 0 b q δE g and E Φ δE δE , 4.30 = 0 ), eq. ( 0 0 . T in eq. ( E F 2 0 Φ 0 1 + δq T ), we can express the on-shell 0 4.31 Λ Φ  δq E g 4.8 . N . 2 − c N δE/E r r g 2 ab √ πG g b  8 x πG 2  8 0 d + , this collection of terms vanishes. Upon + Φ E ≡
M ab 0 q  Φ  Z Φ 2 1 0 Φ ). We can write the solution of eq. ( 2 0 E − E 2 − c g = 0 – 34 – r r 4.27 Φ Φ + Λ δE ab  ) is shifted by a small amount proportional to 2 )  again corresponds to the value of the dilaton at the ) to the JT action ( q + 0 where Φ ∇ b = F 4.10 4.26 ab ab ˜ g f q on-shell Φ = Φ | Φ + JT bulk still describes the solution for the JT-like model ( b I 8 ∇ = a and hence we may write the solution as in eq. ( ˜ ), we have dropped two terms proportional to Φ = Φ + Φ −∇ 2 on-shell ) for the JT model), and the field strengths | 4.29 0 = . However, as a result, the dilaton has a new value when evaluated at the 4.9 c and so the entropy ( r JT-like bulk µ I = r = r ˜ Φ satisfies the dilaton equation appearing for the JT model. Hence our final dilaton As described above the geometry is precisely the same as in the JT model and so the Lastly, we turn to the dilaton equation of motion in eq. ( Now in eq. ( . However, when we substitute ( One might also think of this as a shift in the extremal charge, with ˜ f 11 Given the close connection ofbulk eq. action ( as new features are a small shiftto of the the solution dilaton ( proportional to extremal Maxwell field and the leading correction toComplexity this growth extremal two-form. boundary horizon (relative to the JT model). Penrose diagram in figure solution for the JT-like model becomes The parameters are chosen so that Φ We can absorb the last term with a simple constant shift of the dilaton, i.e., Then or substituting this extremal field as well as the perturbation ( The first term representsby a the small dynamical shift Maxwell in fieldrepresents — the the note background leading that electric correction we to field, assume the field strength created by thesecond running line of is the dilaton. actually the leading contribution since it is not suppressed by a factor of Φ Further, the mass, entropy and temperature all take thefactor same already form vanishes as according given to in eq. eq. ( ( a curvature set by Λ JHEP02(2019)160 ) in (4.37) 11 (4.36) (4.38) for the corre- ). In fact for large from footnote 2.1 T . 4.38 δq 2 1 m m r r  . ) can be combined with 2 ) to leading order in the r π  b q 0 c 0) (with 2.22 , r Φ Φ 0 T + Φ  + δq ) terms agree in that they vanish, but q 2Φ 0 T 0 , we also found that the Maxwell + Φ Φ Φ 0 δq E δE ( E − δE and therefore the complexity growth − O 2.2 T,ext r  q 0 µ ). In fact, we found that the late-time ) (divided by 1 + 2 2 h h  E δE r r  q 0 → 1.2 . Then the leading terms for the late-time growth ) = ( Φ Φ µ 4.36 T + 3 + 6 2 m m 2 0 1 + 2 2 δq r 2 + , q L L E  e 0 q 0 – 35 – πg  g E and δE 2 and − 2Φ − µ √ µ = ) terms agree, the x = 1 + µ 2 q 0 ( d and and ( Φ Φ  O → − µ M 2 0 dt JT-like A Z 1 m  E 2 ), we note that r C 2 g 0 h 0 d r E 2 Φ 4.33 g = 0 ) to the CA proposal ( = Φ →∞ lim t 4D  The expression in eq. (  r 1, these two corrections will cancel one another. d 4.35 dt 13 ) agree, i.e., the  ) can have a dramatic effect on the growth rate and so in the following, we 4.38 /L h r 2.5 ) matches the higher dimensional result in eq. ( 12 ). Hence we can easily extend the analysis of the holographic complexity for ) and ( ) terms disagree. 4.38 T . ) to give a full description of the growth rate for the holographic complexity in the 4.13 ) and expand to linear order in both 2.22 µ δq 10 ( 4.15 2.22 That is, we substitute Given the prefactor in eq. ( O 13 12 and hence both ofblack the holes, i.e., corrections are the same ordereq. in ( the second factorin of eqs. eq. ( ( the investigate the dimensional reduction ofcomplexity this in surface the term two-dimensional and gravity its theories. effect on the holographic For both the JTcomplexity matched and (at the leastsponding JT-like qualitatively) the black model, results holes we found insurface found in four term section that ( dimensions. the In growth section of the holographic near extremal limit. eq. ( JT-like model. Some examplesfigure of the full time profile of the growth4.3 rate are illustrated Boundary in terms? for the JT-like model.the Again, corresponding the nonvanishing (electrically resultconstant charged) here growth may black rate not holes at be inthat late so eq. times. surprising four ( since dimensions In fact, also a exhibited careful a translation of the parameters shows Again at late times,rate we becomes have growth rate of the holographicmodel, complexity vanished the for late-time the growth JT rate model, and will so come in entirely the from JT-like the time derivative of this term, terms for the null boundariessame of way the as WDW patch before,by are i.e., eq. dimensionally the reduced ( in null exactly jointthe the JT terms model and from the thesecond null previous term subsection surface in by counterterm simply eq. are investigating ( the given contribution of the Further, while this expression only refers to the bulk action, it is clear that the surface JHEP02(2019)160 1 2 = (4.42) (4.39) (4.40) (4.41) γ . ) to express .   ab 2.29 b ˜ f ) 0 ab ) A ( 0 F ab ( ˜ f , 0 + ab b F ˜ 2.0 a ab . ). We fix the dimensionless ratios ab 2 F ) 2 Ψ 0 b F Ψ 4.26 + Φ) + 2Φ ( A ) and in the Maxwell perturbation . The solid curves are the complexity g 0 2  ab − 0 L 4.7 F (Φ 1.5 √ a = ab x Σ ) 2 = 0, the dashed curves correspond to ct d 0 l d F γ ( M M ∂ + Φ) + Φ ab Z and Z ) – 36 – 0 decreases the late time growth of complexity for the 0 0 ), the two-dimensional surface term becomes 2 2 1.0 F πγ E γ g πγ (Φ g ( 2 b 4 4.1 01  ) . 0 g = = A − = 1 . As in the electrically charged black holes discussed in ( = 0 √ γ ab ) Q x 0 δE on-shell 2D,elec µ 2 0.5

I F , d ( 0  Φ M 4 Q a Z 2D,elec µ − Σ I 2 d g γ 2 M = 10 0.0 ∂ = 2 2 Z 0.6 0.4 0.2 0.0 1.0 0.8 2 πL γ g ), we find , 4 on-shell = 0

4.25 Φ . Complexity growth in the dimensionally reduced model derived from the RN black holes 3 , a higher value of the parameter = 0, the “JT-like” model given by the action in eq. ( − ) reduces in a straightforward way to a boundary term for the WDW patch in two Q Q JT-like 2 µ JT-like µ m I I q 2.5 Alternatively, when the Maxwell field is on-shell, we can also use eq. ( = 10 in eq. ( b a ˜ or after the usual linear expansion, we arrive at the Maxwell surface term inbulk terms integral of becomes a bulk integral. The two-dimensional version of this Expanding to linear ordera in the dilaton with eq. ( case, the Maxwelleq. field ( is stilldimensions. a Substituting dynamical the field ansatz ( in the two-dimensional theory and so section “JT-like model”. JT-like model We start by analyzing the role of the Maxwell surface term in the JT-like model. In this Figure 10 with Φ growth without the additionand of the the dot-dashed correspond counterterm to JHEP02(2019)160 ) = ) is m 4.45 ) and q (4.45) (4.46) (4.44) (4.43) 4.46 4.24 . . 2 1 m m , the evaluation r r  ). Alternatively,

B 0  ) E δE B.6 q ) γ 2 + Φ b − , (Φ  c ) to leading order in the near- 0 . 2 r + (1 Φ r 2 Φ 2 )) on the WDW patch, with the 1  Ψ 4.44 − q 0 − g 1 Φ Φ 4.42  − ). Expressing the former in terms of q  √ ) for the on-shell Maxwell field, which ), the extremal field strength ( g 1 + x B.4 ) and ( 2 − + Φ  d ) √ 2.29 0 4.34 γ x 2.32 M E δE 2 Z d − 0 – 37 – 2 m N M (1  Z G γq + 2Φ µ 2 m N ] where the holographic complexity is equated with the 0 = 2 0 2 G Φ 53 E 0 πγq  . 0 πg ) for the magnetically charged black holes is more subtle. Φ 4 ), the dilaton ( r Q ) (or equivalently eq. ( , the JT model arises from the dimensional reduction of a  2D,mag µ 2Φ 13 2.5 = I 4.9 2 0 2 Q = 4.40 4.1 JT µ g ) and expand to first order in Φ, we see that the Maxwell surface I γ E − 4.7 ) to the Maxwell field. The time derivative then yields dt JT-like A ). Note that since the magnetic Maxwell field and the relevant surfaces = C d 4.31 4.5 ) then yields ) and so combining this result with the contribution from the Maxwell Q dt JT-like µ →∞ lim t 4.40 4.38 dI , we find agreement between eqs. ( ) as a surface term in the two-dimensional theory. Rather, here we are modifying 2.2 given by eq. ( Next, we evaluate eq. ( 4.46 T,ext the standard CA prescriptionholographic in complexity. the In JT particular, we modelsimply observe by proportional that to adding the the first a spacetime contribution newreminiscent volume in of of eq. bulk the the ( CV2.0 contribution WDW proposal patch to [ andspacetime the so volume this to contribution the is WDW patch. That is, when the Maxwell surface term is included in the JT model. Recallin that the in these near-extremal formulas throat theq two-dimensional with model the describes gravity magnetic chargeare equal integrated to out the extremal inor charge the ( dimensional reduction, there is no way to think of eqs. ( Then substituting eq. ( term contributes as the patches where thewe gauge can potential again is useyields well the precisely defined, bulk the e.g., expression samethe see result in two-dimensional as eq. eq. variables, shown the ( ( in surface term eq. becomes ( As we described inmagnetically section charged black hole and in thisout” case, in the the Maxwell field dimensional is reduction.of completely “integrated Further, the as Maxwell we surface explained termThe in ( appendix interesting contributions to the surface term actually live on the surface(s) dividing Comparing this expressionsection to the late-timeextremal growth limit — for see four-dimensional footnote black holesJT in model Without the Maxwell surfacegiven term, by the eq. late-time ( surface growth term rate ( for the JT-like model was solution given by the metric ( the perturbation ( JHEP02(2019)160 . ). ). µ , we  2.5 2.2 ) and h (4.47) (4.48) r 2.3 4.9 = is given in 4D  r γ ) through the ) on the holo- 2.7 1.2 ). In section ). The general result 1.3 2.24 ), as well as . T,ext 2 1 m m , q r r

 . c 2 r µ ) = (0 r N 0 b 2 m m G Φ 0 2Φ , q γq e Φ 8 q − r =  JT – 38 – A ) on the WDW patch with the solution ( 2 m N dt C = 0), the late-time growth rate vanished for black ), the late-time growth rate is governed entirely by ) with ( d G 0 γ 4.46 πγq 2. We might recall that when we are evaluating the Φ 4 / 4.46 2.32 →∞ lim t = = 1 Q JT µ γ ). Hence once we reconsider the holographic complexity with dt , i.e., with the same form as the bulk Maxwell action ( dI µν 4.16 , the complexity only depends on the same combination with this F 2 m µν q , by investigating the complexity=action proposal ( F + = 1, the roles of the electric and magnetic charges described above are 2 the rate in eq. ( 2 e 2, these boundary and bulk contributions precisely cancel, as is evident γ q µ / ) to define the “gravitational action” (i.e., we only keep the geometric = 2 T = 1 2.1 q γ ). A possible alternative then would be to define complexity=(gravitational ). We found that the results were very sensitive both to the type of charge ) which yields ). In particular, we found that the electric and magnetic charges contribute on an 2.2 2.30 4.47 2.32 However, this picture changes dramatically when the Maxwell boundary is included. Evaluating the above expression ( field for eq.contributions ( to the action)the and WDW then patch. define Of thecombination course, complexity because by the evaluating chargesdefinition. this only It action enter might on the be metric interesting ( to investigate this proposal in other settings. equal footing with theMaxwell choice boundary term foras an a on-shell bulk gauge integralHence field of with that we canin express eq. the ( contribution action). That is, we could drop both the bulk and boundary terms involving the Maxwell For example, with reversed, i.e., the late-timewith growth rate magnetic vanishes charge. witheq. electric The ( charge behaviour and of is nonvanishing the late-time growth for general Without the latter surface termholes (i.e., carrying purely magneticvanishing charge, constant while in for accord the withfor electrically the dyonic charged general black holes case, expectations carrying of it bothalso eq. is kinds noted ( a of that charge non- the is given switchback in effect eq. exhibited ( a similar sensitivity to the type of charge. We began in section graphic complexity of charged four-dimensionaltheory black ( holes in the(i.e., usual electric Einstein-Maxwell versus magnetic) and to the inclusion of the Maxwell boundary term ( Comparing this result to thoseto in linear four dimensions, order we in see that this new expression matches 5 Discussion Without the Maxwell surface term, the late-timevanished, as growth rate shown for in the eq. holographic ( complexity the addition of the “surface”eq. term ( ( produces a complexity=(action + spacetime volume) prescription for theconsidering JT the model. time derivative then yields in the usual complexity=action prescription in four dimensions, the dimensional reduction JHEP02(2019)160 . ) ) , 6 3.1 5.1 ). In (5.1) (5.2) ) via ) was µν 1 and e F , 3.1 4.8 3 3.21 µν / e F ˜ φ α = 1 2 2 (controlling the − α α ge − √ x φ . 4 d − M = ). Generally then, with the Z ˜ ) and hence for which eq. ( φ ˜ ). However, as we noted above φ ) of the original theory would 2 3. The former behaviour was φ g 1 / − ( → ( 4 5.2 1 3.4 did not consider dyonic or mag- V V − 3 ≥ ) for charged black holes in a family  2 ) = ) 1.2 φ ˜ α φ , φ = 1 sets the late-time growth rate to zero − − ( ( α ρσ γ V V F − 2 – 39 – ) µνρσ ). As the value of the parameter ˜ ε φ ∂ 3.1 αφ 2 2( − e ), it is straightforward to produce magnetic solutions given = R − ) that choosing  µν 2.2 g e . F − 3.25 2 √ → x ) except for the appearance of 4 µν 3. No such choice was possible when the causal structure appeared d / F 3.1 1 ), the electrically charged solutions ( M Z < 5.1 , we turned to holographic complexity for black hole in two dimensions. , we investigated the CA proposal ( 2 N α 4 3 1 πG 16 ). The latter mirrored the behaviour for the magnetic Reissner-Nordstrom-AdS ), there are three special cases for which = 4.16 3.4 In section Let us note, however, that the analysis in section In section bulk I In particular, weeq. showed ( that theblack late-time holes growth in rate fourdimensional dimensions, vanishes reduction. which for This play the situation asional can JT role be reduction model in ameliorated describing by constructing the in instead the near-horizon considering JT physics a action of dimen- ( near-extremal electric black holes hole. Hence at leastgrowth of of these the three magnetic cases,construct black it holes magnetic is is or straightforward nonvanishing. dyonic tomore It black verify generally would, holes that and of for the to course, late-time the fully be investigate Einstein-Maxwell-Dilaton interesting holographic theories to complexity ( in these theories. which matches eq. ( transformation ( become magnetically charged solutions of theeq. new ( theory ( leaves the theory invariant. Note3, that lie all in three the of regime where these the special causal cases, structure i.e., matches that of the Schwarzschild-AdS black The action of the “dual” theory is then the electrically charged blackas holes straightforward using for electric-magnetic the duality. Einstein-Maxwell-Dilatonreplace theories, This in operation which is case it not is natural to behaviour found in section netically charged black holes forsolutions the have simple not reason yet that, to beenthe the Einstein-Maxwell constructed theory best for ( of the our Einstein-Maxwell-Dilaton knowledge, theory such ( included, we showed in eq. ( for the cases where theholes, causal structure i.e., was for like thatas of the in Reissner-Nordstrom-AdS the black Schwarzschild-AdSanalogous black to holes, that found i.e.,causal in for structure the of Einstein-Maxwell theory the and black therefore hole it was appears one of that the the essential features producing the unusual of Einstein-Maxwell-Dilaton theories ( coupling between the dilaton andsingularities the and Maxwell field) the isGenerally, causal varied, the the structure late-time nature of of growththe the rate the electrically curvature of charged black black the holes. holes holographic However, change, when complexity the was as Maxwell nonvanishing boundary shown term for ( in figure JHEP02(2019)160 ], 133 ) would 1.2 ]. Hence 97 Preliminary – 95 14 2 seems to be special as / ]. In particular, in shock = 1 41 γ , ), the late-time growth rate is 40 4.26 . However, it is not immediately obvious ). The slightly unusual feature is that as a γ 4.48 – 40 – ). As we noted previously, in this way, this surface ). Hence it may be that electric-magnetic duality (or ). We also considered the dimensional reduction of 2.28 ) in both cases. Again, the effect of this surface term 2.32 4.38 2.5 ) in the gravitational action [ 2.4 , we found that the Maxwell boundary term played an essential role in order 2 ]. In fact, it is this principle that fixes the overall coefficient with which the null 27 , While Maxwell boundary term does not effect the equations of motion, it does play Implicitly, we are suggesting that the action used to evaluate the holographic complexity ( 26 14 magnetic charges would make distinctinvestigations with contributions simple in qubit different models ensembles. but seem of to course, indicate it that would this be is interesting indeed to the study case this [ questionbe further. the same as that used to evaluate the Euclidean action for the thermodynamic ensemble. a role in thegauge field, variational as principle described around byterm eq. changing ( plays the a role boundary inEuclidean conditions black action imposed hole depending thermodynamics, on on i.e.,one the it the might becomes thermodynamic ask an ensemble important if of part different of interest ensembles the [ will “compute” differently, i.e., if the electric and it allows both magneticfooting, and as electric can charges be to seenS-duality) participate from provides in“computations” eq. the on ( guiding an principle(to equal us) to that fix this is the correct choice — see further comments below. guiding principle that suggeststhat that the action this is boundary invariant underin term reparametrizations [ of should the be null boundaries,counterterm included as is is emphasized added to to ensure coefficient the of action. the Maxwell In boundary the term. present We did case, find we that have not definitely fixed the not be surprising thatbe holographic sensitive complexity, or to morethe surface specifically null terms the counterterm CA sincewave ( proposal, an geometries, can this analogous surface behaviour termcomplexity plays was exhibits an both essential already role late-time observed in growth ensuring with and that the the switchback holographic effect. Of course, another In section for the CA proposalgrowth to and produce the switchback the effect) for expected dyoniccharges. black properties holes A of carrying priori, both the this electric surface complexity andMaxwell term magnetic (e.g., would action, not late-time appear to e.g., be obviously, an essential it part of does the Einstein- not affect the equations of motion. It should prescription becomes a complexity=(actionmodel, + i.e., spacetime we volume) have a prescriptiondiscuss mixture for the of two-dimensional the the results JT CA further and below. CV2.0 proposals in two dimensions.Maxwell We will boundary term, revisited nonvanishing, as shown inthe eq. Maxwell ( boundary termmimicked that ( found in fourmodel dimensions. is In now particular, nonvanishing, as the shown late-timeresult in growth of eq. rate including ( of the the Maxwell JT boundary term in four dimensions, the complexity=action (in four-dimensions). In the resulting JT-like theory ( JHEP02(2019)160 ], We 21 (5.4) (5.3) , in the 15 20 γ , and also 0) for either ≥ 3 2 χ , i.e., there are no 0. If we consider this → → ∞ T . ··· q γ max + r + 2 ) 2 . ]. On the other hand, we might χ r χ ( ) γ dr 21 ··· , 1 + RNA − f + ], independent of the choice of 20 (1 29 2 e max max r q r M π 2 2 Z + r + – 41 – 2 γ L e = . For example, for an electrically charged black r N 4 γ 0 ), we find γ q N G → ]. 4 G T , there will only be one class of black holes, i.e., with q 2.32 γ

) corrections to this saddle point evaluation of the holo- 134 ∼ ∼ − A N C dt ), we find d 2.9 (UV) →∞ lim t µQ I ) and ( 2.7 0. On the other hand, from the bulk perspective, there is a nontrivial , for which the expected rate is recovered in the zero-charge limit. → T /γ q 0. 1) ≤ − γ γ However, to close our discussion here, we observe that the limit of zero charge is subtle. In passing, we note that while the holographic complexity defined by the CA proposal Even though the Maxwell term is associated with modifying the boundary conditions 1 or Notice that we can only produce the desired limit with a physical charge ratio (i.e., = ( 15 ≥ 2 , if we are evaluating the holographic complexity with CA prescription for a boundary do not have anyzero-charge insight into limit this for issue, thewith but Einstein-Maxwell-Dilaton higher let curvature theory us corrections studied add [ in that section similar subtleties arise inγ the The first factor correspondsand to hence the we only late recover timeis, this growth for expected rate each rate choice of when of theχ a the second neutral coefficient factor black is hole equal [ to one. That in the limit and abrupt change inlimit the for causal the structure late-time of growth spacetime rate ( with UV divergent contributions coming fromgraphic the complexity Maxwell has boundary the term.boundary same Therefore UV term. the structure holo- [ Naively, from perspective of the boundary CFT, nothing particularly strange should happen Hence the leading UV contribution vanishes in the limit which exhibit this sensitivity. Thependent asymptotic of contributions the to choice actionhole are of given essentially the in inde- coefficient eqs. ( into or out ofwonder if the “quantum” WDW (i.e., finite- patch,graphic as complexity discussed would involve in fluctuationsthe [ of boundary fields conditions on determined the by our WDW choice patch of which surface respect is terms. sensitive to the presence of the Maxwell boundary term, it is only the infrared properties for the field equations,applied we on are the not boundaryγ suggesting of that the these WDWstate patch. boundary dual conditions For to should example, athe be irrespective solution of bulk in the solution which choice should of various not charges be are modified circulating in and the the bulk charges spacetime, would appear to freely flow JHEP02(2019)160 ) 2.21 ]. This ) matches 136 4.16 deformation of the ], it is not surprising ¯ T T 135 boundary, while the two- . Of course, both of these 4 2 , 1 m r = r . For example, the vanishing of the throat of the near-extremal black holes. introduced for the two-dimensional black 2 2 c ]. r 125 = r – 42 – ). surfaces in the four-dimensional geometry [ ) inherit a causal structure with an outer and inner 2.22 r 4.9 ), and this explains the close match between the results in . That is, the growth rate is determined “infrared” part of the 4.36 + 16 r → = 0) in eq. ( 2 m γ ) and ( r black holes ( is implicitly a constant radius surface deep in the interior of the corre- 2 4.15 8 and , the causal structure was identified as an essential feature in determining − r 3 ), we see that all of the contributions to the four-dimensional growth rate corre- → 1 m 2.31 r In fact, one finds that the same late-time growth rate will be derived for WDW Given these differences, one must examine the results more closely to understand the On the other hand, the close parallels between the two- and four-dimensional results We should add that the dimensional reduction does not require any modification of the time coordinate 16 let us addinterpretation that of there moving the have AdS been boundary significant into developments theand concerning hence bulk another the as importantmeasured a holographic with ingredient respect in to this the match same time is coordinate that [ the rates in two and four dimensions are patches anchored to anyagain fixed points to thethe holographic importance complexity of for the theor CA causal anchor proposals. radius structure However, is in we varied, note determining the that the details when the behaviour of cut-off the of early-time transients are modified. Further, Further, in these geometries with twoby horizons, quantities the evaluated late-time at rate issame the completely behaviour corresponding determined for bifurcation the surfaces. contributionse.g., to Of see the course, eqs. growth we rate ( two find in and the the four two-dimensional dimensions. geometry, similarities in the complexity growthand rates in ( two and fourspond dimensions. to First, in terms eqs. evaluated ( junctions at are the in meeting the junctions throati.e., and at at late times,four-dimensional they geometry, i.e., approach by the the inner near-AdS and outer horizons, regions of the spacetime inpatch the is two different anchored contexts. to That adimensional is, cut-off WDW the patch surface four-dimensional is near WDW anchored thefour-dimensional to asymptotic a black constant AdS hole. radius surface deep in the throat of the may seem unexpected whenand one JT-like recalls models that focuses the ondimensions. dimensional the That near-horizon reduction is, region producing the of the cut-offholes near-extremal surface JT in black at holes figure in four sponding four-dimensional solution. Hence the WDW patches correspond to very different four-dimensional charged blacklate-time holes growth in rate section forwith the the CA vanishing proposal rate found foundcharge for (and for the using the four-dimensional JT black model holes carrying in only eq. magnetic ( In section the unusual behaviour ofsince the holographic the complexity AdS forhorizon charged from black the holes. near-extremal blackthat Therefore, holes the in higher holographic dimensions [ complexity exhibits behaviour analogous to that found for the Back to two dimensions JHEP02(2019)160 ]. ge- 2 ] (see ) and 72 , ] between 1.1 71 147 , 21 , 20 146 ]. ) comes with a very 101 4.46 ] are easily generalized 34 ). This also reminds one of 1.3 . Of course, these approaches to ]. Of course, the latter is a natu- ST 145 was that when the Maxwell surface ∼ 4 /dt however, it remains an interesting future C ) or to the two-dimensional near-AdS d 17 ], 2.7 – 43 – 99 ] indicating that the entire entropy of the extremal ), i.e., 55 ]. While we remind the reader that this relation is known to 1.3 ]). In particular, this approach relies on defining a new cut-off 144 98 ] suggest a very different understanding of holographic complexity for JT ]. Some recent progress in connecting these developments with 101 143 – – 99 ]. 137 Note that with the latter prescription, the purely “topological” sector 53 18 ), it is straightforward to examine the behaviour of these other proposals to ). For the CV proposal, the techniques developed in [ 1.2 4.9 ], it no longer seems to be an essential ingredient for the new approach [ ]) and hence another interesting extension of the present work is to consider holo- 37 – 148 The JT model provides a simple setup to study traversable wormholes [ One of our most interesting results in section 34 We observe that in this construction, the spacetime volume contribution ( We note that refs. [ 18 17 gravity than developed heresurface (and behind in the horizon, [ the CA which proposal was originally andfail the defined [ Lloyd using bound the [ conjectured relation [ specific prefactor. InCV2.0 prescription general, [ the normalization of the holographic complexity is an ambiguity for the also [ graphic complexity growth for traversable wormholes.which In has order fallen to retrieve deep a quantum into state the bulk, one would need to perform some operation which would not be sensitivewould to be an the interesting type questionextension of to where examine thermodynamic the either ensemble ensemble of played in these acomplexity. role proposals question. For in had instance, determining a one However, the well-motivated might behaviour it of explore of the the the holographic role recent of understanding boundary of conditions the in holographic the dual context of the bulk symplectic form [ ometries ( to these new metricsresults and for for the the bulk CV2.0with action. proposal, the one general In expectations need either of simplydescribing case, eq. adapt ( holographic the the late-time complexity appropriate are growth is only found sensitive to to be the spacetime in geometry keeping and they a few more observations.other prescriptions The of preceding holographicthe discussion CV2.0 complexity, of proposal. in the particular, Whileproposal JT the our ( model analysis CV in reminds proposalthe this us ( four-dimensional paper of charged focused the black almost holes entirely on ( the CA ral perspective here where thecomplexity two-dimensional were model both and derived the with prescription a for holographic dimensional reductionOther from future four directions dimensions. We have already commented on various future directions above, but let us close with the JT model. (representing extremal black holes with Φis in = keeping 0) with exhibits the complexity recent growth.black results of hole Of [ course, contributes this to thethe suggestion late-time that complexity the growth JT ( modela should quantum be gravity theory interpreted as with a a low-energy larger sector Hilbert embedded space in [ holographic complexity was reporteddirection in to [ fully develop these connections. term is included inmensional the reduction usual produces a complexity=action complexity=(action prescription + in spacetime four volume) prescription dimensions, for the di- boundary theory [ JHEP02(2019)160 . 2 m . r ) ) b b (A.1) r r ( ( , 1 2 and ) f f , 2 m 1 ) m ) r r 2 1 m b are given by ( , r r 1 2 . This calcu- s 2 ( ( w f r , t 2 1 1 f f , − b r = = 0 = = and . The shock wave b s L L L L 1 2 R m m dr dr dt dt dt dt 2.3 , t dr dr L t , ) ) s s r r ( ( 2 1 f f , ) , ) ]. Research at Perimeter Institute is 2 s m ) r r 2 2 1 m ( ( 98 r 2 1 2 ( f f , 2 f − − = = = = 0 b s R R 1 2 R R m m – 44 – dr dr dt dt dt dt dr dr can be determined by studying the time evolution ], so we refer the reader there for more details (as w , , 41 t   ) ) ) ) s s b b r r r r ( ( ( ( 2 1 1 2 f f f f − − 1 1 , ,   ) ) b s ) ) r r 2 2 2 1 m m ( ( r r 1 2 2 2 ( ( f f ] may also prove fruitful. 1 2 f f − − ), we find that the time derivatives with respect to 150 = = = = , b s w w 2 1 w w m m ]). 2.38 149 dr dr dt dt dt dt dr dr 40 : : : : The complexity dependence on b s 1 2 m m r r r r From eq. ( geometry is represented inlation the below Penrose-like follows diagram the inwell analysis as the in [ right [ of figure of the special positions on the boundary of the WDW patch labeled by A More on shockIn waves this appendix,Reissner-Nordstrom-AdS we background, discuss as in we more presented detail in the section calculation of the switchback effect in Perimeter Visiting Graduate Fellows programsof for KG support was during supported this research.thanks in Perimeter The part Institute work by for the theirwould JSPS hospitality also during like Research to various Fellowship thank stages for the ofstages KITP Young this at Scientists. of UC project. Santa this KG Barbara RCM project. forScience their hospitality At Foundation during under the the Grant final KITP, No. this NSF research PHY-1748958. was supported in part by the National supported by the Governmentof of Ontario Canada through through the Industryin Ministry Canada part of and by Research Discovery by & Grantsof the from Innovation. Canada. Province the RCM RCM Natural and also SciencesQubit” BY received and funding are collaboration. Engineering from supported Research the LQ Council Simons would Foundation like through the to “It thank from the Perimeter Scholars International and We would like toRuan, Tokiro thank Numasawa, William Shira Donnelly, Zachary Fisher, Chapman,for Ronak useful Soni Juan and comments Alex Hernandez, and Streicher discussions. RobieSusskind We for Hennigar, would sharing also Shan-Ming with like to us thank a Adam draft Brown of and their Lenny paper [ would be interesting totion investigate the that relation can between the becomplexity. amount transmitted Examining of through these quantum ideas informa- theprotocol from [ wormhole the and perspective the of corresponding the holographic Hayden-PreskillAcknowledgments recovery would prevent or reverse the natural tendency of the system to complexify. Perhaps it JHEP02(2019)160 = )) s L r t ( (A.7) (A.5) (A.6) (A.2) (A.3) (A.4) ∗ 1 r . 1 b m . r r ) + 2

2 . 2 , , r , we set ) ) ( + − b b  r r ∗ 1 r r ))

) ) r ( ( r 2 2.3 1 2 2  ( 2 m f f ∗ 1 q 2 m − r  q 4 2 θ dφ − r w r 2 m 2 t − 2 e − q 2 b m q 2 e r r . + r L

q − ) 2( t L s  2 e t + sin v + + q + 2 m 2 − ) 2 q 3 − w 2 + r w t 6 , L r )( dθ L v t − ( r 2  ( − 3 can be written as 2 2 m ( 2 e − r + L b q r H q 1 )( N  2 I  2 r R 1 . r + + G t 2 ( ω + N b N 2 2 e 2 )) 1 I )( 1 3 q G G b r r 2 − L dr r 4 r 2 1 ( , )) + ( + b drdv 1  1 , , + s ∗ 2 I r = + 2 ) m + − v r dr r N r r r 2 v 2

+ 2 . 2 1 s m 1 ( Z r − , G r r 1  w F 2

+ 2 t v f r – 45 – b ) ) dr r ( 2 2 ) + 2 m π r s s dv − s q r − g )) + 4 ) r r − Z , which is given by r b ( r ( ( r r ∗ 2 w 1 ( = s m − 2 1 ). We will assume a dyonic black hole in four bulk − t r r Z r − H ∗ 1 f f r, v s

2 e w 2 r ( + 1 v t q )) +   2 (1 2 r F − 2 1 2Λ) ( 2 2 m m + r )) + − L − ω ∗ 1 q q w r ) t 2 r − 3 ( r r s 2 = − − ∗ r 2 L r R + r ( ) = ) = 2 2 2 e e −  ( 2 ∗ 1 q q v R r ( t )( ds N − N 1 r, v r + + 1 ( f ( G )( π )( )(2 b 2 2 2 r 3 3 r F 4 πG r I ( ( r r ( L L b − b b 2 16   I I I 2 = 2 2 N N dr r 1 1 ) = and G G with r dr r 2 2 dr r dr r →∞ 1 1 ( , , max b s b w r r r 1 + + m t + − , the derivative becomes I r r r

max w Z Z Z r s = t r w + + + Z s bulk dt w = s dI bulk dt We will show next how to obtain the complexity of formation as a function of how early dI = 0 and probe the dependence on s bulk R I For large t We are interested in investigating the switchback effect, so as in section The bulk action in the Wheeler-DeWitt patch in figure Bulk contribution The integrand of thewritten as bulk action (after integrating over the angular directions) can be The shock wave is then inserted at the shock wave isdimensions inserted and (i.e., with of aneutral spherical shock horizon, waves, as i.e., inequal. the the For black convenience, main hole we text. rewrite charges the before Further, metric we and here only after as consider the shock wave remain JHEP02(2019)160 . (A.8) (A.9) , and 2 2 , , (A.12) (A.10) (A.11) ξ do not + − r r  s

r  = ) ) 2 b b m . r r q ( ( and  r 2 1 | + →∞ f f b ) r 2 e r | 0 2 m q t r ∂     log ( ξ ξ − 1 · 2 1 2 1 2 1 2 1 2 b f r 2 3 | . k + + + + r L N   1      G , log 2 ct ct ct ct N ) ) 2 + 1 + 1 b b 2 1 2 1 m m m m 1 ) G + r r ξ ` ξ ` ξ ` ξ ` r r r r   2 m 2 ( ( 2 2 2 2 ) ) r 2 1 2 ct 2 ct s s f f ` ` ,     , + r r 4 4 ( ( ) ) 2 2 ) ) ξ 1 1 | | + ( 1 2 , , s s b b ) ) ) ) | f f log log log log + − r r r r = 1 2 ) m m r r 1 2 m m ( ( ( (    

r r r r 1 2 2 1 1 m 00 00 2 2 2 2 ( ( ( ( f f f f  r ) ) ) ) log 2 1 ( 2 s f f 2 ξ ξ 1 2 1 2 m m m m m 2 | | r q r r r r f log log = 0 for the null geodesic across the shock ( ( ( ( |   r N

+ N N N N 2 2 κ 1 G 2 e ) ) 1 1 1 1 G G G G 2 log log q log – 46 – 2 1 m m read 2 2 2 2   2 r r ) − ( ( 2 2 , ξ 2 m − − − − ) ) 1 m 2 r 3 ) ) − − r 1 2 m m s s r L ( 2 2 r r b s r r   ( ( ( ( r r 1 2 and N f f N N N 1 1 1 1 N N 1 G m ξ G G G ] 1 1 r 2 are to ensure 2 2 2 G G = 2 2 00 − 41 0 − − = , ξ ξ + + = “UV terms” + 40 = “UV terms” = “UV terms” = “UV terms” = “UV terms” →∞ ct and I w (I) as [ ct t (II) 0 ct

I (IV) (III) ct ct I ξ ξ ) I I , the derivative simply reads X ct , w I t + ξ ]. The joints at P ] for more details), where (I) refers to the the past null boundary that extends = “UV terms” w 41 joint + 41 I dt is fixed at the boundaries with the usual prescription joint joint ξ I I ( If we add the joint and counterterm contributions, we find that the overall answer is The counterterms associated to each of the null boundaries of the Wheeler-DeWitt d Then, for large independent of future one that touches the right AdS boundary. patch (see [ to the right asymptoticboundary, AdS (III) to boundary, the (II) past to one that the touches future the one left AdS that boundary touches and the finally left (IV) to AdS the where the conditions for wave. Because ofcontribute [ the affine parametrization condition, the joints at We now evaluate the jointorder and to counterterm impose contributions affine to parametrization thenull across shock normalization the wave shock spacetime. wave, we In have the condition on the Joint and counterterm contributions JHEP02(2019)160 , ) w ) T t 2.29 ]. In (B.1) (A.14) (A.15) (A.13) 2.9 34 for small ), and this 5 . = 0, such that 2.29 ], because now becomes), these . t for times smaller 41 = 0 in eq. ( χ w t e ) on the boundaries q ) with 2.25 + transient  2.12 , such that if the black was . w + transient w . t 2 2 t 1 , , N  − + r r πT w A ), a similar expression holds

t 2 1 2 e r ≡ e q πT 2.25 2 N  dφ b 1 e for large 1 ) r θ πG 2 e  q b − 1 , at very early insertion times, r + ), we obtain the simple result for the time . There is one big difference with respect s cos w 1 1 1 r , , t ∗ scr − − − + r r s −

A.12 1 r – 47 – (1 2 < t e 1, has a simple expression r m 2 q − r N w & t N  m m 2 1 χ πG r ) and ( N gq 4 πG 2 T  q √ = πG A.7 N 2 T = ) could be written as the bulk contribution ( q γ A w C πG A dt + d 2 2 2 2.25 χ χ 2 χ ) χ γ 1 + 1 + − , in analogy to the time evolution of the eternal black hole [  ct ` (1  the solution for meeting point equation in eq. ( . If we include the surface term in eq. ( 'O , we argued that by using Stokes’ theorem and the equations of motion, the m m A r w r C 'O 2.2 dt d = A w 2 m C r dt d Consider the gauge potential for the magnetic charge by setting The dominant contribution, for Combining equations eqs. ( = terminates long before the scrambling time, as these effects compete with each other. 1 m is nonvanishing — seethe details following. below. We elucidate the resolution of this inconsistency in Maxwell boundary term ( identity was later used toHowever, simplify there some is of clearly our acharge, calculations subtlety: it of if is the one not holographicof were hard complexity. to the to consider WDW see patch the that yields case evaluating zero of the while a boundary the purely bulk term magnetic integral ( on the right-hand side of eq. ( B Bulk contribution from MaxwellIn section boundary term We denote r transient effects will become important,χ which is the reason the curve in figure to the switchback effectboth for the Schwarzschild past black andof holes future at as boundaries a discussed of spacelike in thethat singularity. Wheeler-DeWitt [ depends patch Therefore, on end thereaddition, at is the joints, always instead more a magnetic transient charge term that for is small present (i.e., the smaller purely magnetic, the derivative would vanish. More on the switchback effect Let us evaluate more carefullythan the the rate scrambling of time, complexity with such respect that to derivative of the holographic with respect to Hence this derivative is directly proportional to JHEP02(2019)160 2 in π/ N (B.4) (B.5) (B.6) (B.2) (B.3) A . Now = and so ). For 2 π θ S dθ )) are well- = ∧ θ N × B.2 = θdφ . As a result, this ν M . sin ) π ) S 2 S N A = m . πG − θ ∂H 4 g q , N 2 √ µ θ r A δ component. However, as we dt dr ( 2, we use the potential = N × ) θ ( θ N dφ . µν ). Certainly if we only integrate . , then the result is zero since as F Z ) which covers the south pole with F 2 ∪ θ cos µ S 2 S 2 2 m ) 2.25 N cancel. θ < π/ r Σ 2 − N H 2 N dt dr d G γ q ≤ ) yields where the sign indicates the two bound- (1 ∂H N N × ∂H not 2 N = θ Z ∂ (1 + cos only has a 2.25 N × do 2 m m µν ∂H = Z 2 sin ν m q q N N × S F 4 2 ( − A G γ r A N g γ q µν ) yields – 48 – M µν = ∪ ∂ ). N covering 0 g = = F 2 πG S 2 g F ) and 4 2.29 2 N g ν 2 − πG 2.29 √ N S ∂H A H 4 √ A − x µν = 4 = d F ν N × S µ A ∂ M = 0) and the other covering the south pole ( A Σ Z µν 0 d ) ), the field strength becomes θ = ( 2 F 0 g 1 M ) 2 2.9 ∂ ( M Z . Hence we can not properly apply Stokes’ theorem in the way that ∂ 2 ), the combination ( γ π but in the case, where the charge is purely magnetic. Given the form g ), the WDW patch has the form of a direct product 2 = B.2 2.7 θ , we define π = 0 in eq. ( ) is integrated on the boundaries of all of the patches used to define the gauge e ≤ ), while on the complementary hemisphere q 2.25 B.1 < θ Let us explicitly illustrate this point for the Reissner-Nordstrom-AdS black holes dis- We can evade this problem by describing the gauge potential on two patches, one 2 where in the first line,aries we have have used opposite orientations.term Hence ( with the prescription that the Maxwell boundary Hence if we integrate over this boundary, eq. ( we saw in eq.discussed ( above, we shouldwhere the actually gauge integrate potential overboundary is the integral well-defined, runs boundaries e.g., over using of the all prescription of outlined the above, patches the Next we turn toover the the boundary boundary term of as given the in WDW eq. patch ( where the contributions involving cussed in section of the metric ( setting evaluating the right-hand side of eq. ( Since the two gauge potentialsdefined and on the their corresponding respective integrands patches, (asbut we in now can eq. apply the ( Stoke’s resulting theorem boundary in each terms patch now separately include an integral over the surface covering the northexample, pole on ( a(n open)eq. hemisphere ( π/ problem is inherited by the integrand appearing in the boundary term, i.e., is singular at was implicitly done in deriving eq. ( Of course, we recognize that this choice is not well-defined at JHEP02(2019)160 ) , B.4 ]. , (2011) (2006) JHEP , 01 08 SPIRE IN (2012) 065007 ][ JHEP Gravitation from JHEP , , Lect. Notes Phys. D 86 , , Springer ]. (2015) 149 06 ]. ]. SPIRE Phys. Rev. IN arXiv:1312.7856 , 19 ][ [ JHEP SPIRE , SPIRE IN Relative Entropy and Holography IN Holographic quantum error-correcting ][ ][ ]. = 0). A Covariant holographic entanglement γ (2014) 051 ]. SPIRE 03 ]. IN (when – 49 – SPIRE ][ 2 arXiv:0705.0016 N IN [ Bulk Locality and Quantum Error Correction in SPIRE ∂H JHEP ][ Holographic Entanglement Entropy Holographic Entanglement Entropy hep-th/0603001 , IN − [ arXiv:1411.7041 ][ ]. = [ ]. 2 S ]. ), which permits any use, distribution and reproduction in ]. 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