Reparameterization Dependence is Useful for Holographic Complexity
RReparameterization Dependence is Useful for HolographicComplexity
Ayoub Mounim ∗ and Wolfgang M¨uck † Dipartimento di Fisica “Ettore Pancini”, Universit`a degli Studi di Napoli “Federico II”Via Cintia, 80126 Napoli, Italy Istituto Nazionale di Fisica Nucleare, Sezione di NapoliVia Cintia, 80126 Napoli, ItalyFebruary 9, 2021
Abstract
Holographic complexity in the “complexity equals action” approach is reconsidered relacingthe requirement of reparameterization invariance of the action with the prescription that theaction vanish in any vacuum causal diamond. This implies that vacuum anti-de Sitter spaceplays the role of the reference state. Moreover, the complexity of an anti-de Sitter-Schwarzschildblack hole becomes intrinsically finite and saturates Lloyd’s bound after a critical time. It isalso argued that several artifacts, such as the unphysical negative-time interval, can be removedby truely considering the bulk dual of the thermofield double state.
In recent years, interest has been mounting in relating concepts from information theory to quantumfield theory and gravity. One aim of these efforts is to shed new light on our understanding oftopics such as the nature of space-time, black hole physics and, ultimately, quantum gravity [1–3]. ∗ E-mail: [email protected] † E-mail: [email protected] a r X i v : . [ h e p - t h ] F e b olographic complexity [4,5] is such a concept where, in the spirit of the AdS/CFT correspondence[6–8], a geometrical quantity related to a bulk space-time is identified as a measure of complexity of acertain boundary state. Two competing proposals are the ”complexity equals volume” ( C = V ) [4,9]and ”complexity equals action” ( C = A ) [10, 11] frameworks. In the C = A approach, which wewill use exclusively in this paper, complexity is identified with the action evaluated in a bulk regioncalled the Wheeler-de Witt (WdW) patch, C = S WdW π (cid:126) . (1.1)The WdW patch is defined as the region bounded by the null surfaces anchored at certain timeson the space-time boundary (left and right boundaries in the case of two-sided black holes) and,possibly, the black hole singularity. The action on the WdW patch is generically divergent, becauseof the contribution of the region close to the space-time boundary. Such a divergence is typicalin the AdS/CFT correspondence. Several regularization procedures have been studied, and it hasbeen proposed, just as in holographic renormalization [12–16], that the divergences can be removedby adding local, covariant counter terms on the boundaries [17–20]. Alternatively, one can subtractthe action of some reference space-time, which is also necessary to calculate the complexity offormation [21].Computational complexity is a concept from information theory measuring how difficult it is(or how many steps it takes) to compute (approximately) a desired target state starting froma given reference state using a certain set of elementary operations [22]. The precise definitionof computational, or cicruit, complexity depends on the system under consideration, the set ofelementary operations, the reference state and a parameter (cid:15) that specifies the tolerance withwhich the target state is reached. Typically, the complexity diverges when (cid:15) →
0. A geometricapproach to complexity, which can be applied to quantum field theory, was developed in [23]. Inthis approach, complexity is given by a weight function evaluated on a trajectory connecting thetarget and the reference state in some space of unitary operators. Several proposals for the weightfunction have been investigated in [24–29].In this paper, we will reconsider the complexity associated with global AdS space-time and AdS-Schwarzschild black holes. These are the simplest systems and have, of course, been investigatedalready in the very first papers on holographic complexity [10, 11, 30, 31] and more recently in [19], In the rest of the paper, we will work with the reduced action I = 16 πGS . Other settings have been considered, e.g., in [32–40]. the action in any vacuumcausal diamond vanishes . A causal diamond is defined as a region bounded only by null surfaces.Because global AdS can be seen as a causal diamond, this procedure achieves the goal by definition.Moreover, in the case of the AdS-Schwarzschild black hole, which is also a vacuum space-time, theregion that effectively contributes to the complexity reduces to a region bordering with the blackhole singularity. This region lies entirely behind the horizon, which directly incorporates the generalexpectation that complexity is a probe of the physics behind the horizon. Moreover, because theregion near the space-time boundary does not contribute, the resulting complexity is intrinsicallyfinite.The rest of the paper is organized as follows. In section 2, we briefly review the contributions tothe gravitational action on a WdW patch. In section 3 we show how, through a parameterizationchoice, we can set the complexity of empty AdS space-time to zero, effectively making it our pickfor the reference state. We then use this idea to compute the complexity of the AdS-Schwarzschildblack hole in section 4. We conclude in section 5.3
Action on a WdW patch
The gravitational action on a WdW patch consists of contributions coming form the bulk of thepatch, its boundaries and the joints in which the boundaries meet. Given a bulk action, thevariational principle imposes a number of boundary and joint terms. The most familiar case forEinstein gravity is the Gibbons-Hawking-York term [43,44], which applies to space-like or time-likeboundaries. When the boundary is not smooth, additional joint terms are necessary [45, 46]. Inthe general case, which includes also null boundaries, the boundary and joint terms have beenconstructed, e.g., in [30, 42, 47, 48]. We refer to [42] for more references to the original work. Weshall now review the contributions one at a time.The bulk contribution is given by the Einstein-Hilbert action with a cosmological constant Λ I B = I EH = (cid:90) M d D X √− g ( R + Λ) . (2.1)The boundaries of the patch are hypersurfaces, which one can define in terms of a scalar functionΦ( X ) = 0. In this paper, we use the convention that Φ( X ) is negative inside the patch and positiveoutside of it. These conventions agree with those in [17,42] and ensure that the one-form dΦ alwayspoints outward. If the hypersurface is space-like or time-like, then the unit normal is taken to be n α = ∂ α Φ (cid:112) | g αβ ∂ α Φ ∂ β Φ | . In these cases, the boundary contribution to the action is given by the Gibbons-Hawking-York term I S = I GHY = 2 (cid:90) B d D − x (cid:112) | h | K , (2.2)where K = ∇ α n α is the extrinsic curvature of the boundary, and h is the determinant of theinduced metric.Instead, if the boundary is a null hypersurface, then ∂ α Φ must be proportional to the nulltangent vector, k α ≡ ∂X α ∂λ = e σ ( x ) ∂ α Φ . (2.3)Here and henceforth, λ denotes the hypersurface coordinate that parameterizes the null direction.The function σ ( x ) determines the parameterization of the null direction, i.e., the choice of λ . Thishas been discussed extensively in [30]. 4he analogue of the Gibbons-Hawking-York term is played by I N = 2 (cid:90) d λ d D − x √ γ κ , (2.4)where κ is the non-affinity parameter defined by k β ∇ β k α = κk α . (2.5)In (2.4), γ is the determinant of the induced metric on the codimension-two space-like part of theboundary that is orthogonal to the null direction.In addition to the codimension-one boundaries, the WdW patch has also codimension-twojoints in which two boundaries intersect. The contribution of a joint depends on the type of theintersecting boundaries. In the case of a joint formed by two null boundaries, it is given by I C = 2 (cid:15) J (cid:90) J d D − x √ γ ln | k · k | , (2.6)where k and k are the null tangent vectors of the two intersecting boundaries, and the sign (cid:15) J = ± σ ( x ). As we shall see below, performingan integration by parts in I N , one obtains σ -dependent contributions localized at the joints, whichmust cancel against the σ -dependent parts of the joint term I C . This determines the signs (cid:15) J . Werefrain from giving the joint terms in the other cases, because we will not need them.Adding up all these contributions gives an action with a well-defined action principle. Moreover,the full action is additive, i.e., one can safely divide a given space-time region into smaller subregions.However, in the presence of null boundaries, it is not invariant under a reparametrization of thenull directions, which can be seen from the fact that the arbitrary functions σ ( x ) remain explicit.A remedy is to add a counterterm such that its variation under a reparametrization will cancel thevariation of the action. We will use the term suggested in [30] I c . t . = 2 (cid:90) d λ d D − x √ γ Θ ln(˜ l | Θ | ) , (2.7)where Θ is the null geodesic expansion. It is defined by Θ = ∂ λ ln √ γ . The constant ˜ l is of dimensionlength and plays the role of a renormalization scale. The different sign with respect to the expression given in [30] derives from different conventions for k α and Φ.
5t has been pointed out in [42] that the choice (2.7) is not unique. There are, in fact, otherterms that would serve the same purpose, such as I (cid:48) c . t . = 2 (cid:90) d λ d D − x √ γ Θ ln d λ d t , (2.8)where t is an arbitrary affine parametrization, or I (cid:48)(cid:48) c . t . = 2 (cid:90) d λ d D − x √ γ Θ ln s ab s ab , (2.9)where s ab is the shear tensor of the null geodesic congruence. We will use the term (2.7) because,as it turns out, (2.8) is not compatible with the additive properties of the action, while (2.9) is notregular on the surfaces we need to study. In this section, we will compute the complexity of pure AdS spacetime, with the intention to finda systematic and well-defined way to set it to zero. The motivation behind our intention is that wewant to treat the CFT ground state dual to the AdS vacuum as the reference state, the complexityof which is null by definition. Then, we can interpret the holographic complexity of other systems,e.g., black holes, obtained using the same prescription, as the complexity relative to the groundstate. Thus, what we are looking for is a way to set the action on the WdW patch in AdS vacuumto zero. First, we will see how this can be achieved making use of the parameterization dependenceof the null surface terms, if no counter term is included. Then, we show that the counter term(2.7), which renders the action parametrization independent, is generically divergent. Setting it tozero would require fine tuning the renormalization scale ˜ l .We consider n + 2-dimensional AdS space-time in global coordinates, with the metricd s = L cos ρ (cid:0) − d t + d ρ + sin ρ dΩ n (cid:1) . (3.1)Here, L is the AdS curvature radius, dΩ n the metric of a unit n -sphere, and the AdS boundary islocated at ρ = π/
2. Note that our metric tensor has dimension length , whereas the coordinatesare dimensionless. This requires a slight change with respect to the general construction in theprevious section, which we will mention in due course.The WdW patch is bounded by two null hypersurfaces, which meet at the AdS boundary ina joint. Using the time translation invariance, we are free to choose t = 0 at this joint. Because6f the divergences arising at the boundary, the WdW patch must be regulated. We shall use aregularization in which the radial coordiante of the joint is ρ ∗ = π/ − ε . Then, the following twoscalar functions define the null boundaries:Φ ± ( t, ρ ) = ρ − ρ ∗ ± t . (3.2)The contributions to the action to be calculated are the bulk term (2.1), the non-affinity termson the null hypersurfaces (2.5) and the joint term (2.6). Later, we will also consider the covariancecounter term (2.7). Let us consider them one at a time. The bulk contribution (2.1) is I B = (cid:90) d n +2 x √− g ( R − − n + 1) L (cid:90) d n +2 x √− g = − n + 1) L n Ω n ρ ∗ (cid:90) d ρ tan n ρ cos ρ ( ρ ∗ − ρ ) = − L n Ω n ρ ∗ (cid:90) d ρ tan n +1 ρ , (3.3)where we have integrated by parts in the last step. Ω n denotes the volume of a unit n -sphere.Let us now consider the null surface terms. Following the prescription outlined in Sec. 2, thenull tangents are, in coordinates ( t, ρ, (cid:126) Ω), k µ ± = e σ ± cos ρL (cid:16) ∓ , ,(cid:126) (cid:17) , (3.4)where σ ± are two auxiliary functions of λ incorporating the freedom of parameterization of the nulldirections. We remark that we have multiplied the expression resulting from (2.3) by a factor of L ,because our space-time coordinates, according to (3.1), are dimensionless, while the coordinates in(2.3) have dimension of length. Notice that (3.4) implies ∂ρ∂λ = cos ρL e σ ± . (3.5)Because this is positive, λ increases towards the corner on both boundaries.The non-affinity parameters κ ± turn out to be κ ± = ∂ λ σ ± , so that the action term I N is thesum of I N ± = 2 L n Ω n (cid:90) d λ tan n ρ ∂ λ σ ± = 2 L n Ω n σ ± ( ρ ∗ ) tan n ρ ∗ − ρ ∗ (cid:90) d ρ σ ± n tan n − ρ cos ρ . (3.6)7e have integrated by parts, and σ ± ( ρ ∗ ) denotes the values of σ ± at the joint.The corner contribution is simply I C = − L n Ω n tan n ρ ∗ ln k + · k − − L n Ω n tan n ρ ∗ [ σ + ( ρ ∗ ) + σ − ( ρ ∗ ) + 2 ln cos ρ ∗ ] . (3.7)We remark that the sign of the corner term follows easily from the fact that the terms with σ ± ( ρ ∗ )must cancel between I N ± and I C .Adding the contributions (3.3), (3.6) and (3.7) yields the action without the covariance term, I B + I N + I C = − L n Ω n ρ ∗ (cid:90) d ρ (cid:20) tan n +1 ρ + n σ + + σ − ) tan n − ρ cos ρ (cid:21) − L n Ω n tan n ρ ∗ ln cos ρ ∗ . (3.8)We can further rewrite the integral from the bulk contribution as ρ ∗ (cid:90) d ρ tan n +1 ρ = − ρ ∗ (cid:90) d ρ tan n ρ ∂ ρ ln cos ρ = − tan n ρ ∗ ln cos ρ ∗ + n ρ ∗ (cid:90) d ρ tan n − ρ cos ρ ln cos ρ , (3.9)so that (3.8) becomes I B + I N + I C = − nL n Ω n ρ ∗ (cid:90) d ρ tan n − ρ cos ρ ( σ + + σ − + 2 ln cos ρ ) . (3.10)We note that, written in this form, the integrand that gives the total action is proportional to aterm that has the same structure as the corner term. We also remark that the dependencies of σ + and σ − on ρ are to be intended implicitly along the null boundaries to the future and the past ofthe WdW patch, respectively.It is now obvious that there exists a class of parameterizations, for which (3.10) vanishes. Forexample, one may chooseon N +: σ + ( λ ) = a + ln cos ρ ( λ ) , on N − : σ − ( λ ) = a − ln cos ρ ( λ ) , (3.11)where a + and a − are two constants satisfying the constraint a + + a − = 2. We also see that thecorner term (3.7) vanishes separately with such a choice.8s we mentioned at the beginning of this section, a vanishing action on the WdW patch isexactly what we were looking for in the case of pure AdS. Nevertheless, let us also discuss thecovariance term (2.7), which, for general parameterization, is I c . t . = 2 L n Ω n ρ ∗ (cid:90) d ρ ( ∂ ρ tan n ρ ) (2 C − ρ + σ + + σ − ) . (3.12)Here, we have abbreviated C = ln n ˜ lL . (3.13)For general parameterizations, the terms with σ ± in (3.12) cancel those in (3.10) as expected.If we adopt the parameterization (3.11), for which the counter term represents the entireparameterization-invariant action, (3.12) becomes I c . t . = 4 L n Ω n tan n ρ ∗ ( C − ln sin ρ ∗ ) + ρ ∗ (cid:90) d ρ tan n − ρ . (3.14)The integral on the right hand side of (3.14) can be evaluated in closed form, ρ ∗ (cid:90) d ρ tan n − ρ = [ n − ] (cid:88) k =1 ( − k − tan n − k ρ ∗ n − k + ( − n ] ln cos ρ ∗ for even n , ρ ∗ for odd n . (3.15)Clearly, (3.14) is generically divergent for ρ ∗ → π/
2. In order to get a vanishing result, onecould interpret the renormalization scale ˜ l or, equivalently, the renormalization constant C , as afunction of the cut-off ρ ∗ and fine tune it such that (3.14) vanishes. The choice C = 0, which wasadopted in [19], only removes the leading divergence. In this section, we will consider the case of AdS-Schwarzschild space-time with a spherical horizon.This case was discussed in [10, 11, 30, 31] and more recently in [19]. However, in most of thesepapers, only the time evolution of complexity, ∂ C /∂t , was discussed, because only the difference ofthe actions in two slightly different WdW patches is needed in such a calculation and the divergent9onstant drops out. Here, we will consider the entire WdW patch and compute the complexity ofthe black hole with respect to the pure AdS geometry.AdS-Schwarzschild space-time is described by the metricd s = − f ( r ) d t + f ( r ) − d r + r dΩ n , (4.1)with f ( r ) = r L + 1 − ω n − r n − . (4.2)The parameter ω is related to the total mass M = n Ω n πG ω n − . (4.3)The horizon radius is defined by f ( r H ) = 0. The black hole temperature is T BH = 14 π (cid:18) n + 1 L r H + n − r H (cid:19) , (4.4)and the entropy is given by the Bekenstein-Hawking formula S = A (cid:126) G , (4.5)where A = Ω n r nH is the spatial area of the black hole horizon.The AdS-Schwarzschild black hole has also a time scale relevant to the time evolution of thecomplexity know as the scrambling time [49, 50]. The scrambling time of a system is a measure ofhow fast said system can thermalize information by means of the interactions between its elementarydegrees of freedom. For the case of a black hole, the scrambling time is the time it takes the blackhole to completely and uniformly spread a local perturbation across its horizon. Interestinglyenough, black holes are believed to be the fastest scramblers in nature and their scrambling timeis of the order t ∗ ∼ β BH ln S . (4.6)The Penrose diagram of maximally extended AdS-Schwarzschild space-time is shown in Fig. 1.Each of the quadrants I–IV is covered by a set of coordinates ( t, r ) with metric (4.1), where r > r H in quadrants I and III and r < r H in II and IV and the future and past singularities are situated10 = 0 r = 0 r = ∞ r = ∞ r = r H r = r H IIIIII IV
Figure 1: Penrose diagram of the maximally extended eternal AdS-Schwarzschild space-time. Thearrows indicate the flow of the Killing vector field ∂ t .at r = 0. For the upcoming computations it will be useful to work with Eddington-Finkelsteincoordinates. We define the tortoise coordinate by r ∗ ( r ) = r (cid:90) R d rf ( r ) . (4.7)Note our choice of the lower integration limit, where R will be taken to agree with the cut-offinstead of the conventional ∞ . This will somewhat simplify the calculations.With ingoing Eddington-Finkelstein coordinates, the metric isd s = − f ( r ) d v + 2 d v d r + r dΩ n , (4.8)where v = t + r ∗ . The ingoing Eddington-Finkelstein coordinates cover two quadrants, I ∪ II orIII ∪ IV. Likewise, with outgoing coordinates, the metric isd s = − f ( r ) d u − u d r + r dΩ n , (4.9)where u = t − r ∗ . The outgoing Eddington-Finkelstein coordinates cover two quadrants, I ∪ IV orII ∪ III. 11he WdW patch is bounded by the null surfaces intersecting the left and right boundariesat the cut-off radius R and at times t L and t R , respectively, and possibly by parts of the futureand past singularities. The cut-off radius R will be taken to infinity at the end. Furthermore, tosimplify, we use time translational invariance to set t R = − t L = τ . The precise shape of the WdWpatch depends on the value of τ . When | τ | < τ = − r ∗ (0), the WdW patch touches both, thefuture and past singularities. For τ > τ , the WdW patch touches only the future singularity, andtwo null boundaries meet in the quadrant IV. Let us call r m the radius at which they meet. It isgiven implicitly by the relation r ∗ ( r m ) = − τ . Similarly, for τ < − τ , two boundaries meet in thequadrant II and the WdW patch touches only the past singularity. Because the setup is symmetricunder τ → − τ , this last situation does not need to be discussed separately. The WdW patch inthe other two cases are illustrated in Fig. 2. IIIIII IV t R t L u = t R u = t L v = t L v = t R IIIIII IV t R t L u = t R u = t L v = t L v = t R Figure 2: Left: WdW patch for the case − τ < τ < τ . The WdW patch touches both future andpast singularities. Right: WdW patch for τ > τ . Two null boundaries meet in the quadrant IV. We are interested in the action in the WdW patch, which represents the complexity of the dualstate. Before considering the entire WdW patch, let us focus on a causal diamond, which we mayplace, without loss of generality, in the quadrant I. The causal diamond is bounded by four null12 N N N N Figure 3: A generic causal diamond. The labels of the null boundaries and the corners used in thetext are shown. The arrows on the null boundaries indicate the orientation of the λ -integrals inthe corresponding boundary terms of the action.surfaces, which we label N , . . . , N , counting them clockwise starting from the north east. Thefour intersection points are counted clockwise starting from the north. Obviously, their coordinatessatisfy v = v , u = u , v = v and u = u . The setup is illustrated in Fig. 3. In what follows,we work in outgoing Eddington-Finkelstein coordinates (4.9).The four scalar functions defining the null surfaces are given byΦ ( u, r ) = u + 2 r ∗ ( r ) − v , (4.10a)Φ ( u, r ) = u − u , (4.10b)Φ ( u, r ) = v − u − r ∗ ( r ) , (4.10c)Φ ( u, r ) = u − u . (4.10d)13rom these, we obtain the following expressions for the null tangent vectors, k α = e σ (cid:18) − f , ,(cid:126) (cid:19) , (4.11a) k α = e σ (cid:16) , ,(cid:126) (cid:17) , (4.11b) k α = e σ (cid:18) f , − ,(cid:126) (cid:19) , (4.11c) k α = e σ (cid:16) , − ,(cid:126) (cid:17) . (4.11d)The four functions σ , . . . , σ implement the parameterization dependence.Let us start with the surface terms (2.4). For all four surfaces, we have κ = d σ d λ . The orientationof the λ -integrals can be read off from (4.11a)–(4.11d), because k α = d x α d λ . For example, for N weget I N n = (cid:90) d λ r n κ = r (cid:90) r d r r n d σ d r = − n r (cid:90) r d r r n − σ + r n σ ( r ) − r n σ ( r ) . (4.12)Proceeding similarly for the other three boundaries and summing up all the terms, we find I N n = − n r (cid:90) r d r r n − σ − n r (cid:90) r d r r n − σ − n r (cid:90) r d r r n − σ − n r (cid:90) r d r r n − σ (4.13) − r n [ σ ( r ) + σ ( r )] + r n [ σ ( r ) + σ ( r )] − r n [ σ ( r ) + σ ( r )] + r n [ σ ( r ) + σ ( r )] . The bulk action (2.1) gives I B n = − L u (cid:90) u d u (cid:2) ρ ( u ) n +1 − ρ ( u ) n +1 (cid:3) , (4.14)where the functions ρ ( u ) and ρ ( u ) are defined implicitly by Φ ( u, ρ ) = 0 and Φ ( u, ρ ) = 0,respectively. Using ρL = 12 (cid:20) d f ( ρ )d ρ − ( n − ω n − ρ n (cid:21) = − dd u ln | f ( ρ ) | −
12 ( n − ω n − ρ n , we can rewrite (4.14) as I B n = u (cid:90) u d u (cid:20) ρ n dd u ln | f ( ρ ) | − ρ n dd u ln | f ( ρ ) | (cid:21) . (4.15)14fter integrating by parts and changing the integration variable, this becomes I B n = n r (cid:90) r d r r n − ln | f ( r ) | + n r (cid:90) r d r r n − ln | f ( r ) | (4.16)+ r n ln | f ( r ) | − r n ln | f ( r ) | + r n ln | f ( r ) | − r n ln | f ( r ) | . This is identical to I B n = na u r (cid:90) r d r r n − ln | f ( r ) | + na u r (cid:90) r d r r n − ln | f ( r ) | (4.17)+ na v r (cid:90) r d r r n − ln | f ( r ) | + na v r (cid:90) r d r r n − ln | f ( r ) | + r n ln | f ( r ) | − r n ln | f ( r ) | + r n ln | f ( r ) | − r n ln | f ( r ) | , where a u and a v are two real constants that are constrained by a u + a v = 1.The corner terms (2.6) contribute I C n = r n [ σ ( r ) + σ ( r ) − ln | f ( r ) | ] − r n [ σ ( r ) + σ ( r ) − ln | f ( r ) | ] (4.18)+ r n [ σ ( r ) + σ ( r ) − ln | f ( r ) | ] − r n [ σ ( r ) + σ ( r ) − ln | f ( r ) | ] . It is apparent that we have manipulated the surface and bulk actions in such a way that theboundary terms arising from the integrations by parts precisely cancel the corner contribution.Thus, after summing (4.13), (4.17) and (4.18), the action of a causal diamond is obtained as I B + I N + I C n = n r (cid:90) r d r r n − [ a u ln | f ( r ) | − σ ] + n r (cid:90) r d r r n − [ a v ln | f ( r ) | − σ ] (4.19)+ n r (cid:90) r d r r n − [ a u ln | f ( r ) | − σ ] + n r (cid:90) r d r r n − [ a v ln | f ( r ) | − σ ] . Here, we note that the integral involving σ must proceed along the null surface N , because σ isdefined only on N , and similarly for the others.Now, we can require that the action in any causal diamond (of a vacuum region) vanishes. Thismeans that we must choose a parametrization such thaton N : σ ( λ ) = a u ln | f ( r ( λ )) | , on N : σ ( λ ) = a u ln | f ( r ( λ )) | , on N : σ ( λ ) = a v ln | f ( r ( λ )) | , on N : σ ( λ ) = a v ln | f ( r ( λ )) | . (4.20)15nterestingly, because a u + a v = 1, this choice implies that all corner terms in (4.18) vanish sepa-rately.Our choice (4.20) has an important consequence for the complexity calculation. Using theadditivity of the action, we now know that portions of the WdW patch that are bounded only bynull segments do not contribute to the action. Non-zero contributions may come only from theremaining regions, which border with the singularity. These are the dark shaded areas in Fig. 4.Notably, these areas lie entirely behind the horizon. We will now turn to the action on theseremaining regions. IIIIII IV IIIIII IV
Figure 4: The only portions of the WdW patch that contribute to the action in our computationare those shaded in dark gray.
We have seen in the previous subsection that the calculation of the action can be reduced to theregions that border with the past and future singularities. We first recall that the time translationsymmetry is fixed by setting t R = − t L = τ . Let us start with the case τ > τ = − r ∗ (0). Inthis case, there is only one contributing region, because the WdW patch touches only the futuresingularity. It is bounded by the two null surfaces u = t R + 2 τ and v = t L − τ , as well as the part16f the future singularity with t L − τ ≤ t ≤ t R + τ . In outgoing Eddington-Finkelstein coordinates,the bulk region is given by t L ≤ u ≤ t R + 2 τ and 0 < r ≤ ρ ( u ), where ρ ( u ) is determined implicitlyby u + 2 r ∗ ( ρ ) = t L − τ . (4.21)The bulk term is easily computed, I B n = − L t R +2 τ (cid:90) t L d u [ ρ ( u )] n +1 . (4.22)Now consider the null surface contribution. Without loss of generality, let us pick the values a u = 1, a v = 0 in (4.20). Then, only the null surface with constant v contributes, with σ = ln( − f ). Itstangent vector is k α = e σ (cid:18) f , − ,(cid:126) (cid:19) = (cid:16) − , f,(cid:126) (cid:17) . (4.23)This gives I N n = (cid:90) d λ r n dd λ ln( − f ) = − t R +2 τ (cid:90) t L d u [ ρ ( u )] n dd u ln( − f ) = 12 t R +2 τ (cid:90) t L d u ρ n dd ρ f ( ρ )= 1 L t R +2 τ (cid:90) t L d u [ ρ ( u )] n +1 + 12 ( t R − t L + 2 τ )( n − ω n − . (4.24)As we mentioned at the end of the previous subsection, the corner term from the joint betweenthe two null surfaces vanishes in our choice of parameterization. The two corners at the futuresingularity do not contribute either, because their volumes are zero. The remaining contributionis the Gibbons-Hawking-York term (2.2) at the singularity. Consider a surface with small, butconstant r , which will be sent to zero at the end. In Schwarzschild coordinates, the normalized,outward-pointing normal vector on this surface is n α = (0 , √− f ,(cid:126) K = 1 r n ∂ r (cid:16) r n (cid:112) − f (cid:17) = nr (cid:112) − f − √− f (cid:20) rL + ( n − ω n − r n (cid:21) . (4.25)Therefore, the Gibbons-Hawking-York term is I S n = t R + τ (cid:90) t L − τ d t lim r → (cid:16) r n (cid:112) − f K (cid:17) = 12 ( t R − t L + 2 τ )( n + 1) ω n − . (4.26)17dding up all contributions yields I WdW = I B + I N + I C + I S = 4 n Ω n ω n − ( τ + τ ) , (4.27)where we have set t R = − t L = τ in the final result.Eqn. (4.27) holds for τ > τ only. The other cases can be obtained using the t → − t symmetry.In particular, for | τ | < τ , we need to add the contribution from the region bordering with the pastsingularity. Its contribuition is simply (4.27) with τ replaced by − τ . The sum of the contributionsfrom the two regions results in a constant, I WdW , = 8 n Ω n ω n − τ .We are now in a position to translate our result into a complexity. Using (1.1) and I = 16 πGS ,we obtain C = 4 Mπ (cid:126) | τ | + τ for | τ | > τ ,2 τ for | τ | ≤ τ . (4.28)To conclude this subsection, let us discuss what we have found. First, the complexity is manifestlyfinite. This is ensured by the fact that the corners at the cut-off boundary, where divergences mayoccur, belong to causal diamonds, which do not contribute following our prescription. No furthercounter term is needed. We will see in the next subsection that including the covariance counterterm re-introduces the generic divergence. The limit R → ∞ has not been taken explicitly. In fact,the only R -dependence is hidden in τ = − r ∗ (0), which includes R as the lower integration limit in(4.7). τ remains finite in the limit R → ∞ . For example, for n = 2 we have τ = L r H + L r H (cid:18) L r H (cid:19) + 3 r H + 2 L (cid:113) r H + 4 L arctan (cid:113) r H + 4 L r H = L α α (cid:20)
12 ln (cid:0) α − (cid:1) + 2 + 3 α √ α arctan (cid:112) α − (cid:21) . (4.29)On the second line we have introduced the adimensional ratio α = r H L . We will always assume α > α < τ > τ and saturates theLloyd bound [10, 11, 51] . Compare this to the reparameterization-invariant approach, which wewill review in the next subsection. There, linear growth holds only at late times. Third, (4.28) is To compare with the standard expression for Lloyd’s bound, one should set t = 2 τ . − τ ≤ τ ≤ τ . Using (4.29), the constant value of the complexity is(for n = 2) C = 4 L α (1 + α ) π (cid:126) G (1 + 3 α ) (cid:20)
12 ln (cid:0) α − (cid:1) + 2 + 3 α √ α arctan (cid:112) α − (cid:21) . (4.30)If we perform an expansion in the large black hole limit, α (cid:29)
1, we find C = 4 L √ (cid:126) G (cid:34) α + √ π + 23 − α − + O (cid:0) α − (cid:1)(cid:35) = 43 √ π S + 4 π C T
81 (9 + 2 √ π ) − π C T √ S + O (cid:0) S − (cid:1) . (4.31)The final expression has been written in terms of physical expressions, in particular the black holeentropy (4.5) and the central charge of the boundary CFT [52], C T = L π (cid:126) G . We can interpret thisconstant value as the complexity of formation of the black hole from the empty AdS space-time [21].The leading order of (4.31) shows the same linear relation between complexity of formation andblack hole entropy highlighted in [19, 21], at least up to a numerical factor. The sub-leading termsare different, and in particular we do not have the logarithmic divergence term that shows upin [19, 21]. These differences are due to the fact that a different parametrization was used in thecomputation. In [21] an affine parametrization is used, so that the boundary contribution of thenull segments can be discarded. Incidentally, such a parametrization gives the same result as withthe parametrization invariant action, which was used in [19].Last, for τ < τ , the complexity decreases linearly. This behaviour does not appear to bephysical, and we interpret it as an artifact of the eternal, two-sided black hole. In subsection 4.5we will show that treating the black hole as a thermofield double state removes this artifact as wellas the plateau at small τ . As we mentioned above, the complexity (4.28) is not invariant under a change of parameterizationof the null boundaries. Because parameterization independece may be considered as a necessaryfeature, we will calculate here the counter term (2.7) that would render the action invariant.The quantity we need to compute the counter term is the null expansion along the boundariesΘ = n dd λ ln r ( λ ) = nr k r . (4.32)19irst, we consider the causal diamond of subsection 4.2. For the segment N we readily find I c . t .,N n = n r (cid:90) r d r r n − ln n ˜ lr + n r (cid:90) r d r r n − σ = r (cid:32) ln n ˜ lr + 1 n (cid:33) − r (cid:32) ln n ˜ lr + 1 n (cid:33) + n r (cid:90) r d r r n − σ . (4.33)It is evident that the dependence on the parametrization function σ in (4.33) cancels the corre-sponding integral term in (4.12). The contributions of the other boundaries take essentially thesame form, and the total counter term reads I c . t . n = 2 r (cid:32) ln n ˜ lr + 1 n (cid:33) − r (cid:32) ln n ˜ lr + 1 n (cid:33) − r (cid:32) ln n ˜ lr + 1 n (cid:33) + 2 r (cid:32) ln n ˜ lr + 1 n (cid:33) (4.34)+ n r (cid:90) r d r r n − σ + n r (cid:90) r d r r n − σ + n r (cid:90) r d r r n − σ + n r (cid:90) r d r r n − σ . The action in the causal diamond, including the counter term, is obtained by adding (4.34) to(4.19), which gives I tot n = 2 n r (cid:90) r d r r n − ln n ˜ l (cid:112) | f | r + 2 n r (cid:90) r d r r n − ln n ˜ l (cid:112) | f | r . (4.35)Here, we have chosen to leave the counter term in integral form. As expected, the result is inde-pendent of the parametrization of the null boundaries. However, there is no way to set this actiongenerically to zero.To see what are the consequences of this and the differences with the result (4.28), we computethe action of the whole WdW patch with the contribution of the counter term. Let us focus on themore interesting case τ > τ . The counter term contribution of the four null boundaries reads I c . t . n = n R (cid:90) d r r n − (cid:32) n ˜ lr + σ + σ (cid:33) + n R (cid:90) r m d r r n − (cid:32) n ˜ lr + σ + σ (cid:33) . (4.36)In computing (4.27) we have set a v = 0 and a u = 1, which by (4.20) implies σ = σ = ln | f | and20 = σ = 0. Using this parametrization in (4.36) we find I c . t . = 4 n Ω n R (cid:90) d r r n − ln n ˜ l (cid:112) | f | r + 4 n Ω n R (cid:90) r m d r r n − ln n ˜ l (cid:112) | f | r = I . t . − n Ω n r m (cid:90) d r r n − ln n ˜ l (cid:112) | f | r , (4.37)where we have defined the constant contribution I . t . = 8 n Ω n R (cid:90) d r r n − ln n ˜ l (cid:112) | f | r . (4.38) I . t . is just the counter term of the case | τ | < τ , when the WdW patch touches both singularities.We note that it diverges in the R → ∞ limit.We can now sum (4.37) and (4.27) to find the action on the WdW patch with the counter term, I WdW,tot = 4 n Ω n ω n − ( τ + τ ) − r m (cid:90) d r r n − ln n ˜ l (cid:112) | f | r + I . t . . (4.39)The second term in the brackets, which stems from the counter term, is also time-dependent,because τ = − r ∗ ( r m ). The time derivative of the action is˙ I WdW , tot = 4 n Ω n (cid:34) ω n − + r n − m f ( r m ) ln n ˜ l (cid:112) | f ( r m ) | r m (cid:35) . (4.40)Let us briefly discuss the second term in the brackets. At late times, when r m → r H , it vanishes,reproducing the known linear growth of the complexity. For τ just above τ , i.e., when the WdWpatch detouches from the past singularity just after the time interval with constant action, it tendsto −∞ . This behaviour was observed already in [31], where it was proposed to smooth out thespike in the complexity by averaging over a time interval shorter than the thermal time.In Fig. 5 we plot (4.40) for some values of ˜ l (we use the n = 2 case for simplicity). We can seethat at late times (4.40) agrees with the expected behaviour, namely linear growth, for any valueof ˜ l . We also see however, that at early times the actions is decreasing. The time interval for whichwe have a decrease in the action is dependent on the arbitrary scale ˜ l .The growth in complexity of the black hole is supposed to reflect the growth of the bulk volumebehind the horizon, and as such it is expected to be at least non decreasing. Moreover, we find21he fact that the time interval for which the action is decreasing depends on the arbitrary scale ˜ l iseven more unphysical. ττ -50-40-30-20-10010 ˙ I Ω ω Figure 5: Time evolution of the action as a function of r m for different values of ˜ l . Solid line:˜ l = 0 . r h , dashed line: ˜ l = r h , dash-dotted line: ˜ l = 2 r h . ( n = 2, α = 5). As we have seen in the previous subsections, the action on a WdW patch in the eternal (two-sided)Schwarzschild-AdS black hole spacetime, with or without the counter term, is symmetric undertime reflection, τ → − τ . Moreover, it is constant in the time interval | τ | < τ . These two featuresare difficult to interpret when the action is taken as a measure for complexity. Now, we will showthat taking literally the interpretation of the two-sided Schwarzschild-AdS black hole spacetime asthe dual of a thermofield double state [53–55] resolves both issues.Consider a quantum system with Hilbert space H , Hamiltonian H and the energy spectrum H | n (cid:105) = E n | n (cid:105) . A thermofield double state is the following temperature-dependent entangled statein the doubled Hilbert space H ⊗ H , | ψ (cid:105) = 1 (cid:112) Z ( β ) (cid:88) n e − β En | n L (cid:105) ⊗ | n R (cid:105) . (4.41)The subscripts L and R refer to the two copies of the Hilbert space, Z ( β ) is the partition functionof the theory, and β the inverse temperature. The state (4.41) is prepared by an Euclidean path22ntegral over the interval τ E ∈ (0 , β ), which prepares the state at some initial Lorentzian time τ = 0. For τ >
0, we will use the convention that τ = t R = − t L , so that no time reflection isneeded in the bulk description. This can be acieved by taking H L = − H R , or by considering a brastate (cid:104) n L | instead of | n L (cid:105) .The bulk geometry of the Schwarzschild-AdS thermofield double state is obtained by gluingthe half of the Lorentzian space-time with τ > τ E ∈ (0 , β ) [53].Therefore, this construction gives us a precise definition of an initial time. The resulting geometryis illustrated in Fig. 6, where we also show a WdW patch. t L = − τ t R = τ Figure 6: WdW patch in the thermofield double state. The shaded area indicates a WdW patch,the darker part is the region that effectively gives the action.When we compute the action on the WdW patch using the prescription that causal diamondsdo not contribute, then the region that effectively gives the action is the dark shaded area in Fig. 6,which borders with the singularity. We note that the triangular regions bordering with the τ = 0spacelike hypersurface, which remain after removing causal diamonds, do not contribute. This isbecause they effectively add up to a causal diamond and the τ = 0 hypersurface has zero extrinsiccurvature, so that the corresponding surface term vanishes. Therefore, the result is simply givenby (4.27), C = 4 Mπ (cid:126) ( τ + τ ) . (4.42)It is evident that the complexity growth is now linear at all times τ > τ = 0, the complexity is half the complexity of the double sided black hole, as we would expect23ince the WdW patch touches only one singularity. Consequently, the complexity of formation ofthe black hole is also halved.We can quite easily see how the action would look like if we included again the counter termcontribution. Such contribution takes the same form as (4.37), we only need to exchange r m withthe radius r in which the wdw patch intersects the boundary at τ = 0. The τ -dependence of r isgiven implicitly by: r ∗ ( r ) = − τ . When τ = 0 we have r = R , and as τ grows to ∞ r approaches r H . The action with the counter term is then: I WdW,tot = 4 n Ω n ω n − ( τ + τ ) − r (cid:90) d r r n − ln n ˜ l (cid:112) | f | r + I . t . . (4.43)This expression suffers the same problems that affect (4.39). The term I . t . introduces a divergencein the complexity, and the time derivative reads˙ I WdW , tot = 4 n Ω n (cid:34) ω n − + r n − f ( r ) ln n ˜ l (cid:112) | f ( r ) | r (cid:35) . (4.44)Now r goes from R to r H as time increases. As show in Fig.7, in the limit R → ∞ , when τ → l .However as an improvement with respect to (4.40), the action is now always increasing. ττ ˙ I Ω ω Figure 7: Time dervative of the action for different values of ˜ l . Solid line: ˜ l = 0 . r h , dashed line:˜ l = r h , dash-dotted line: ˜ l = 2 r h . ( n = 2, α = 5).24 Conclusions
In this paper, we have reconsidered the holographic complexity of pure AdS space-time and theAdS-Schwarzschild black hole in the C = A approach. The novelty of our treatment lies in thedeparture from the requirement that the on-shell action be invariant under reparametrizations ofthe null components of the boundary of the WdW patch. This requirement would mandate acounter term in addition to the minimal action determined by the variational principle and wouldleave the renormalization scale ˜ l as the only parameter. Instead, we regard the parameterizationdependence as a feature that allows to describe physically different situations. On the bulk side,the action terms on the null boundaries describe the heat content on these boundaries [41]. Wethink that the interpretation in the dual CFT is related to the details of the definition of circuitcomplexity, i.e., the reference state and the set of elementary gates, but we do not have anythingprecise to say on this point. Having disposed of reparameterization invariance, we have introduceda criterion that selects physically sensible parameterizations. Our criterion is that the action inany vacuum causal diamond vanishes . This immediately sets the action on the WdW patch inpure global AdS space-time to zero, fixing this state as the reference state. This was not possiblein approaches similar to holographic renormalization. For the AdS-Schwarzschild black hole, ourcriterion renders the action intrinsically finite, because the regions near the space-time boundarycan now be discarded. The region that effectively gives the complexity lies entirely behind thehorizon and borders with the singularity.In the case of the eternal (two-sided) black hole, our calculation not only confirms the lineargrowth of complexity at late times, but makes the growth linear at all times τ > τ , where τ isa critical time, saturating the Lloyd bound. For 0 < τ < τ , the complexity is constant. Thisconstant value is interpreted as a complexity of formation and agrees up to a numerical factor andto leading order with previous results for that quantity [19, 21]. We also considered the thermofielddouble state interpretation of the AdS-Schwarzschild black hole replacing the unphysical region ofnegative times τ (where complexity decreases) with an appropriate Euclidean space-time regionthat creates the thermofield double state. In this case, the complexity turns out to be linear at alltimes τ >
0, starting at a positive value that can again be interpreted as a complexity of formation.One might object that the divergence of holographic complexity in the reparameterization-invariant approach, when the cut-off is sent to infinity, reflects the generic divergence of circuitcomplexity when the precision parameter (cid:15) is sent to zero. Although it is true, this argument25as a loop hole. Physically, one can reformulate this expectation by saying that constructing,with infinite precision, a generic target state from a given reference state using a given set ofelementary operations requires, generically, infinitely many operations. However, this is only thegeneric situation, and holographic complexity still is divergent when generic parameterizationsof the null boundaries are considered. It all depends on what happens to the set of elementaryoperations as the limit (cid:15) → all unitary operations areallowed as elementary operations, only a single operation would be needed to obtain the targetstate trivializing complexity. Therefore, one should think that for parameterizations satisfying ourcriterion the set of elementary operations is suitably enlarged in the limit (cid:15) →
0, such that thecomplexity remains finite. It would be very interesting to investigate the generalization of our criterion for non-vacuumsituations, such as Vaidya space-time or charged black holes. Another interesting direction would bethe study of the complexity behaviour under conformal transformations [56, 57], since when usingthe ”Complexity=Action” conjecture, the counter term gives rise to effects of difficult physicalinterpretation . Acknowledgements
This work was supported partly by the INFN, research initiative STEFI.
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