Holographic Complexity of Einstein-Maxwell-Dilaton Gravity

We study the holographic complexity of Einstein-Maxwell-Dilaton gravity using the recently proposed"complexity = volume"and"complexity = action"dualities. The model we consider has a ground state that is represented in the bulk via a so-called hyperscaling violating geometry. We calculate the action growth of the Wheeler-DeWitt patch of the corresponding black hole solution at non-zero temperature and find that, in the presence of violations of hyperscaling, there is a parametric enhancement of the action growth rate. We partially match this behavior to simple tensor network models which can capture aspects of hyperscaling violation. We also exhibit the switchback effect in complexity growth using shockwave geometries and comment on a subtlety of our action calculations when the metric is discontinuous at a null surface.


Introduction
The physics of quantum information has played a growing role in our understanding of the emergence of spacetime and gravity from non-gravitational degrees of freedom in the context of the AdS/CFT correspondence. Entanglement [1][2][3][4], quantum error correcting codes [5], and even quantum state complexity [6] have all been used to illuminate various mysterious aspects of the emergent gravitational degrees of freedom.
Tensor networks have played an important role in these developments by providing a middle ground between quantum gravity and quantum information where many features of both sides can be cleanly identified and studied [4,[10][11][12][13][14][15]. In the context of complexity, the early discussions more-or-less identified the complexity of the field theory state with the number of tensors in the minimal tensor network needed to prepare the state (up to a constant) [6][7][8]. More sophisticated definitions are now being explored in various contexts [16][17][18].
On the gravity side, complexity has been conjectured to be dual to various geometric measures, including the volume of a certain maximal slice ("complexity = volume" or CV ) [7] and the action of a certain spacetime region ("complexity = action" or CA ) [8,19]. In particular, in the context of black hole geometries, it has been argued that the growth of the interior of the black hole is dual to the growth of the complexity of the field theory state under time evolution [6,13]. Moreover, using the action prescription it was observed that at late times a large class of black holes of equal mass have the same action growth rate and hence are conjectured to have the same complexity growth rate. This led to a conjecture that black holes complexify as rapidly as possible [8,20]; however, this proposal is known to be violated at least at early times [21] and in sufficiently exotic computational setups [8,22]. Very recently, while this paper was being prepared, some holographic examples showing late time violations of the proposed bound were exhibited [23,24]. There is a growing body of work extending these results to a wider class of gravity theories and exploring other proposals [25][26][27][28][29][30][31][32][33][34][35][36][37].
In this work, we study holographic complexity, meaning state complexity on the field theory side and its purported duals on the gravity side, in the context of a very broad class of gravitational theories known as Einstein-Maxwell-Dilaton (EMD) theories [38]. This is a class of models that have been considered in particular in the AdS/CMT (condensed matter theory) literature as possible starting points for describing aspects of strongly interacting physics at finite charge density, e.g., as occurs in the solid state [38][39][40][41]. For our purposes, the main interesting feature of these models is that they are dual to a broader class of scale invariant (but not conformally invariant) field theories and hence have a more general tensor network representation [42,43].
Our results are as follows. Focusing on so-called hyperscaling violating solutions of the EMD theory [38], we compute the rates of action growth and volume growth for finite temperature states dual to black holes as a function of the energy and the two scaling parameters describing the solutions. As we discuss in detail below, the "dynamical exponent" z controls the relative scaling of space and time while the "hyperscaling violation exponent" θ relates to the effective dimensionality of space. In terms of these parameters, we find that for z = 0, the previously obtained action growth for conformal field theories is obtained for all θ. However, for z > 1 we find that the action growth rate is enhanced by a multiplicative factor, Thus these black holes violate the conjectured action growth bound even at late times.
We are able to match these results to a simple tensor network model of complexity growth in hyperscaling violating field theories. We also compare our tensor network results to the rate of volume growth and find that we need a temperature dependent length scale on the gravity side to match tensor network expectations. Finally, we study shockwave solutions and verify the existence of the switchback effect; we also point out a subtlety concerning the proper definition of action when the metric is discontinuous.

Gravity model 2.1 Einstein-Maxwell-Dilaton theory
The Einstein-Maxwell-Dilaton (EMD) theory is (d + 2) dimensional Einstein gravity sourced by a U (1) gauge field (the Maxwell field) and by a scalar field Φ (the dilaton). The gauge field is coupled to the dilaton via a warping of the effective gauge coupling. The Lagrangian density can be written in terms of general functions V and Z via where 2κ 2 = 16πG is the gravitational constant. The action of the theory is where ... denotes additional boundary and corner terms which we specify later. Now consider the following ansatz for the metric, in which the length scale L reduces to the AdS radius in the conformal limit and r is the emergent holographic direction. We also assume only the time component of the gauge field is non-zero, Following Ref. [38], consider the situation in which the dilaton potential V (Φ) and the coupling Z(Φ) take the following asymptotic form as Φ → ∞, where α, β are two positive constants. Given these forms, one solution of the EMD equations of motion is given by Eq. (2. 3) with where the parameters areQ Q is an integration constant identified as the charge density in the dual field theory, and is defined as Given the solution in Eq. (2.7), we see that in order to have each term in Eq. (2.3) scale accordingly, the coordinates should scale as From these scaling relations we know that z is the "dynamical critical exponent". For example, in a weakly coupled theory Eqs. (2.12) and (2.14) would require a field theory dispersion relation relating energy ε to momentum k going like ε ∼ k z . From Eq. (2.15) we know that the proper length in the holographic theory transforms under the above scaling transformation. This implies that the hyperscaling invariance of the boundary field theory is violated as well. So θ is the so called "hyperscaling violation exponent".
Further, the coefficients g 0 and r 0 can be fixed and the combination of f 0 and h 0 given by f 0 h −2 0 can also be fixed.
More generally, there is an infinite class of solutions of the form Eq. (2.3) that correspond to a black hole geometry at non-zero temperature. The metric is modified to where r h is the parameter that labels the solution which is identified with the r coordinate of the horizon of the black hole. We see that the metric in Eq. (2.7) is the zero temperature limit of this class.
The Hawking temperature of the black hole is given by the surface gravity at the horizon The Bekenstein-Hawking entropy is related to the surface area of the black hole horizon and takes the form where Ω d is the regulated volumn of the d-dimensional hypersurface in the spatial x i directions. For later use, we can obtain the thermal energy E by integrating the thermodynamic equation dE = T dS with boundary condition E| r h =∞ = 0.

Action of the Wheeler-DeWitt patch
According to the CA conjecture [19], the complexity of a boundary state is proportional to the classical action of a region of spacetime called the Wheeler-DeWitt patch, which is the domain of the dependence of a Cauchy surface which intersects the boundary of the spacetime at a given time. A Wheeler-DeWitt (WDW) patch in the AdS-Schwarzschild black hole spacetime is shown in Figure 1.
As the WDW (Figure 1) patch is a manifold with boundary, we shall both consider spacelike surfaces, (such as CD, a segment of the future singularity) and null surfaces (segments AC, AE, BD, BD). The extrinsic curvature of a null surface is ill-defined, so the surface action of a null surface needs detailed consideration. Furthermore, the (d + 1)-dimensional hypersurfaces may have sharp boundaries when they intersect with each other. These joints are d-dimensional hypersurfaces shown as points (A, B, C etc) in the Penrose diagram 1.
The issue of the action for null surfaces and joints was considered in detail in Ref. [44]. We summarize the results that are of relevance to our calculation in Appendix A. Briefly, the total WDW patch action is given by in which we refer to Eq.

Action growth in the EMD theory
In this part we focus on the rate of change of the action as a function of time rather than on the absolute value of the action. Note that the WDW patch depends on the Cauchy surface through its intersection with the boundary at r → 0, (e.g. A, B in Figure (1)) rather than on the specific details of the Cauchy surface. Our primary interest is thus the dependence of the action of the WDW patch on the combination (t A + t B ). The combination (t A + t B ) appears because the time in the left and right spatial regions flow in opposite directions. Without loss of generality, we study the time evolution as a function of the left time t A = t L . An illustration of the change of the WDW patch when we evolve the t L by δt is shown in Figure 2. Note that the deviation of EMD black hole metric from the AdS-Schwarzschild metric does not affect the qualitative structure of the Penrose diagram.
After analyzing the different parts that may contribute to the action growth (see Appendix B for the details of the calculation), it turns out that only the bulk regions V 1 and V 2 , the spacelike surface section δΣ, and the null-null surface joints E, E will contribute to the change of the action: In the late time limit, t A → ∞, the null surface BE lies close to the horizon and the action growth rate takes a remarkably simple form in terms of E, θ, z and d.
This indicates a significant violation of the complexity growth bound conjectured in Ref. [8] and inspired by the Lloyd's conjecture [20]. However, we remind the reader that early time violations of the conjecture were already known and that some models of computation have been exhibited which also violate the conjectured bound. Nevertheless, within the confines of the CA duality conjecture, it seems that hyperscaling violating black holes complexify much more rapidly than their conformal cousins. v u Figure 2: Illustration of the change of the WDW patch after evolving t L by δt. The Penrose diagram for an EMD black hole shares a similar structure to that of the AdS-Schwartzschild black hole, and the deviation does not affect the analysis hereafter.

Maximal volume slice and CV duality
In the CV conjecture [7], the complexity of a state |ψ(t L , t R ) is taken to be proportional to the volume of a maximal volume slice which intersects with the two r → 0 boundaries at t L and t R .
At late times, the maximal volume slice asymptotes to a fixed slice as t L → ∞ and t R → ∞ [7]. When t L → ∞ and t R → ∞, the configuration has time translation invariance so the shape of the maximal volume slice is independent of t L , t R . It turns out that in this case the slice is a constant r surface whose value r m is obtained by maximizing the measure, which amounts to finding critical points: The solution r = r m is proportional to r h , Then the change rate of the volume with respect to t L or t R is given by the spatial The proportionality is obvious from the relation in Eq. (2.28). It is worth pointing out that in the case of non-vanishing θ, V d has different dependence on r h as compared to the energy E in Eq. (2.23) and an extra dimensional scaleQ is introduced to get the right dependence on dimensionful parameters. Thus we find that while the volume and the action both grow linearly with time at late time, the rate of growth has a qualitatively different dependence on temperature in the two cases.

Tensor network model
In this section we present two tensor network models that partially capture the complexity growth of the thermofield double state corresponding to the EMD black hole geometry. When carrying out the calculations, the following assumptions are made. Since the boundary theory has a scaling symmetry, we assume that the time evolution can be "renormalized" by passing it through the renormalization group (RG) circuit so that it acts only on low energy degrees of freedom [8]. This renormalization already vastly reduces the naive complexity of time evolution. Rather than using a detailed model for the thermal state at a given temperature, we instead approximate the thermal state by taking the ground state RG circuit and truncating it once the renormalized correlation length is equal to the lattice spacing. At that final thermal scale where the low energy degrees of freedom reside, we assume that the complexity growth is proportional to the energy scale of the Hamiltonian and the number of degrees of freedom.

(d−θ)-dimensional tensor networks embedded in d-dimensions
By studying the boundary behaviour of the metric Eq. (2.3) with solution Eqs. (2.7), (2.19) we know that the boundary field theory should live in d-dimensional space (the number of x i s). Here we show that the scaling of the temperature, entropy, and complexity growth can be captured by a system composed of a direct sum of (d − θ) dimensional tensor networks. We also incorporate the dynamical critical exponent z into the analysis.
Suppose each copy of the (d − θ) dimensional tensor network has lattice length a and overall size L. The remaining θ dimensions are regularized to have length L 0 , and each copy separated by a displacement of length l 0 along each of the θ directions. An illustration of this set up is shown in Figure 3. In total we have As a simple example of this kind of physics, non-interacting Weyl fermions in d = 3 spatial dimensions in the presence of a magnetic field organize at low energy into a set of d = 1 dimensional chiral modes propagating along the magnetic field direction. There is also a similar phenomenon in holographic models [45].
We perform the RG transformation until the lattice spacing reaches some temperature dependent correlation length ξ so that each site is uncorrelated with the rest. We denote the complexity generated from performing the series of RG transformation as C RG . Now the total number of the sites in each (d − θ) dimensional network can be written as Combined with Eq. (3.1), we have the total number of sites in the whole system as We now compute how RG transformation acts on an infinitesimal time evolution step, Here V is an isometry that transforms the operators defined with lattice spacing a to operators defined with lattice spacing 2a. The superscripts denote the lattice size the operators act on. ∆ H is the scaling dimension of the Hamiltonian operator. Since the dynamical critical exponent relates scaling of time and space, we take ∆ H = z. In order to renormalize to the low energy degrees of freedom at scale ξ, we must perform the isometry n times where n is n = log 2 ξ a . (3.5) The result of applying the RG isometry n times is To sum up, the complexity of the unrenormalized state evolving for time t can be identified with the complexity of the RG transformation to an uncorrelated state plus the complexity of this renormalized state evolving for time T Λ 0 t. Let c be the complexity generated by the Hamiltonian acting on each site per infinitesimal time step. Then the total complexity of the state is Refering to Eqs. (2.23) and (2.21), we see that Eq. (3.7) indeed captures the temperature dependence of the complexity growth rate in Eq. (2.26) up to a multiplicative factor which depends on the details of the Hamiltonian.

d-dimensional branching tensor tetwork, s = 2 θ fixed point
In this section we present another tensor network model which has a similar character to the first model without the explicit decomposition into non-interacting copies. We still assume that the effective dispersion relation is ε ∼ k z , and we require the system to be at s = 2 θ fixed point. (citation required) The structure of the tensor network is now more elaborate, similar to a so-called branching MERA tensor network in which at each stage of the RG, spatial lengths are reduced but multiple copies of the system at the longer scale are produced. If a non-branching MERA can be understood as an isometry relating lattice space a and 2a, |ψ (a) = V |ψ (2a) , then a branching MERA with s branches gives We set up the model as before: d dimensional tensor networks with lattice spacing a and overall size L. We denote the momentum scale corresponding to the inverse lattice as Λ 0 .
At an s = 2 θ fixed point, the thermal density matrix ρ(H) ∝ exp(−H/T ) splits into a direct product of s copies after one RG transformation step An illustration of the density matrix splitting for θ = 2 is shown in Figure 4.
We assume that after performing n = log 2 ξ a iterations of the isometry V , the Hamiltonian and the state decompose into 2 nθ disjoint copies: Here each H (ξ) is a distinct decoupled Hamiltonian acting on one of the 2 nθ copies of The assumption of complete decoupling of the copies may be unrealistic, but we expect corrections to it will not modify the complexity growth estimate dramatically.
The counting of degrees of freedom is now similar to the previous model. For the renormalized state with lattice spacing ξ, the number of sites in each copy is The density matrix splitting effect gives With all the same arguments as for the previous model, our complexity estimate is which has the same temperature dependence as Eq. (3.7) and thus also captures the growth rate Eq. (2.26) .

Shockwaves and the switchback effect
In this section we study CA duality for hyperscaling violating black holes in the presence of a bulk shockwave. This shockwave is dual to the insertion of a perturbation in the past of the thermofield double state. The complexity added by this perturbation can be understood in terms of the minimal quantum circuit needed to apply the Heisenberg operator W (t w ) = e iHtw W e −iHtw to the thermofield double state. As discussed in Ref. [7], we expect a partial cancellation of the forward and backward time evolutions generating W (t w ), so that the total additional complexity for large t w is proportional to 2(t w − t * ) with t * the scrambling time.
Here we verify that this "switchback effect" is also present for hyperscaling violation black holes. We do this in the context of CA duality by explicitly evaluating the action ρ(H (a) ) one RG step ρ(H (2a) ) ⊗2 θ n RG steps ρ(H (ξ) ) ⊗2 nθ 2 nθ copies Figure 4: An illustration of a branching tensor network with θ = 2, s = 4. The circles at each layer represent copies of the thermal density matrix of the corresponding Hamiltonian. After n = log 2 ξ a RG steps, we get 2 nθ copies of the density matrices ρ(H (ξ) ), corresponding to the Hamiltonian on a lattice length ξ, where ξ is the correlation length. growth of an eternal black hole perturbed by a shockwave. We find that the switchback effect is indeed reproduced and roughly matches our tensor network expectations. One subtlety which arises is how to correctly evaluate the action when the spacetime has discontinuities along null surfaces as in the shockwave geometry. We show that one way to consistently calculate the action using the "Perimeter-style" method of Ref. [44] (so that it agrees with the "Stanford-style" method of Ref. [8] and reproduces our physical expectations) is to open the geometry up along the null discontinuity and take a twosided limit.
It is convenient to carry out the shockwave calculation in Kruskal-Szekeres coordinates. These coordinates can be defined throughout the eternal black hole spacetime as where β here is again the inverse temperature, We set up the configuration such that the null shell is injected from the left boundary at time t w → −∞ with infinitesimal energy δε. When the shell arrives at the horizon, the U U component of the energy-momentum tensor is exponentially blue shifted to As stated above, the the shochwave leads to a discontinuity of the metric at the U = 0 horizon. The discontinuity turns out to be a finite shift in the Kruskal-Szekeres variable An illustration of the Penrose diagram for the shockwave geometry is shown in Figure 5. The calculation of the action of the WDW patch proceeds as before, except that we find that the discontinuity at U = 0 must be treated specially. We show in Appendix C the details of the action calculation and discuss how the discontinuity can be dealt with (see Section. (C.1)).

The result of the calculation is
where ... denotes other terms that are not time-dependent. We see two principle features. First, the extra action contributed by the perturbation is proportional twice t w times the previously computed action growth rate without the shock. Second, the switchback effect is present as expected. As shown in Ref. [46], the scrambling time in the hyperscaling violating black hole is still proportional to β 2π , i.e., they continue to maximally scramble [47]. These features are also present in the tensor network models discussed above.

Discussion
In this work we studied the CA and CV conjectures in the context of a general class of scaling solutions to the EMD theory. We found that CA and CV differ in their temperature dependence for these EMD black holes. In particular, when the dynamical exponent z is larger than 1 , we found that the rate of complexity growth was enhanced relative to the z = 1 result. We were able to match the results of the CA calculation using simple tensor network models.
There are several interesting directions to pursue. Of course, the EMD theory considered here is not expected to be a UV complete theory of quantum gravity, so it would also be interesting to study action growth and its analogs in a more complete theory, perhaps a string theory, to at least gain some insight into the physics of the singularity. Other directions include the formulation of a more defined tensor network model and comparisons to the recent free field theory complexity calculations, perhaps in the context of the branching tensor network in Ref. [48]. Finally, given that the conjectured complexity growth rate bound in Ref. [8] is now thoroughly falsified [22,23], it is interesting to consider other possible bounds that might illuminate in which senses black holes are the fastest computers.

A Rules for calculating the action: null surfaces and joints
We list the results that are of relevance to our calculation below. They are discussed in detail in Ref. [44].
• For a spacelike / timelike (d+1) dimensional hypersurface, the action is given by the York-Gibbons-Hawking surface action where Σ is the surface of interest, h = det h ab is the determinent of induced metric h ab , and K = h ab K ab is the extrinsic curvature. If Σ is spacelike, sign(Σ) = 1(−1) if Σ lies to the past (future) of the bulk of interest. If Σ is timelike, then sign(Σ) = 1.
• For a null surface N , the boundary action is given by where √ γd d x i is the volumn element for the d dimensional spacelike hypersurface and λ is the parameter of the geodesic that generates the null surface N . sign(N ) = 1(−1) for N in the past (future) of the bulk of interest. If we denote the vector along the null geodesic as k α = ∂x α ∂λ , then the surface gravity κ is given by Note that if λ is chosen to be an affine parameter of the null surface, then by definition κ = 0. So when we affinely parametrize the null surface N , the corresponding boundary action I N = 0.
• For joints that come from intersection of at least one null surface, the boundary action is given by If a null surface N intersects with a spacelike surface S, and if k is the null tangent vector of N and n is the normal vector of S, then the function a j is given by a j = log |k · n|.
If a null surface N intersects with another null surfaceN , and if k and k are the null tangent vectors of N andN respectively, the function a j is given by The rule of the sign is that sign(j) = 1 if the null segment N lies to the past (future) of the bulk of interest and the joint is at the past (future) end of the segement, and sign(j) = −1 otherwise.

B Action growth in the EMD theory: calculational details
The incoming and outgoing Eddington-Finkelstein coordiantes are convenient for the calculation. By defining where ρ * is given by we get the variations of the metric Eq. (2.3) The change of the action arising from evolving t L for δt can be written The meaning of the terms are explained in Figure 2.
We first show that the action given by the joints at A and C are independent of time. The future directed unit normal vector of CD, at a constant ρ slice, is The null surface AC is determined by a scalar function Φ(u, ρ, x i ) = u−const.
By the same argument as before, I A depends only on the combination of u − v, hence it is invariant under time translation. So (I A − I A ) = 0.
For the null surface action I CAE with the null normal vectors chosen to be k u = 1 for CA andk v = 1 for AE, we can calculate explicitly that Thus I CAE = 0 since κ = 0 on both AE and CA. So also is I C A E = 0.
To summarize so far, the action change in Eq. (B.6) has been reduced to Eq. (2.25) as claimed. We now turn to the different parts in Eq. (2.25).
• The bulk action (I V 1 − I V 2 ) and changing variable u = u 0 + v 0 + δt − v, it turns out that 14) The result is • The spacelike surface action I δΣ is where the extrinsic curvature K is given by and the future directed normal vector n α is defined in (B.7).
The result is We note when ρ(u 1 , v 0 ) → ρ h , this term approaches

C Action of the shockwave geometry in EMD theory: calculational details
In this appendix we present the calculation of the action in a geometry perturbed by a spherically symmetric null shell falling into the blackhole. The null shell sets of a shockwave whose physical manifestation is a null shift along the shockwave. For the convenience of the calculation hereafter, we introduce the metric after changing the u coordinate in Eq. (B.3) to the Kruskal-Szekeres variable U , Recall that β is chosen such that the metric is non-singular at the horizon, We set up the configuration such that the null shell is injected from the left boundary at time t w → −∞ with infinitesimal energy δε. When the shell arrives at the horizon, the U U component of the energy-momentum tensor is exponentially blue shifted to The shochwave leads to a discontinuity of the metric at the U = 0 horizon if it is injected from the left boundary. The discontinuity turns out to be a finite shift in the Kruskal-Szekeres variable V δV = h ∼ e 2π(|tw|−t * )/β (C. 4) where t * = β 2π log L d G N . An illustration of the Penrose diagram for the shockwave geometry is shown in Figure 5 . We will show in Section (C.1) that the discontinuity across U = 0 crucially contributes to the time dependence of action.
A very distinct feature of the shockwave metric is that the WDW patch can intersect with past and future singularities simultaneously, while in the unperturbed metric the patch can only touch both or intersect either past or future singularity. The condition that the patch intersects with the past singularity is where ρ J is the radius of the point J where null boundaries of the past domain of dependence intersect. We can relate the U 0 , V 0 to the time on the left and right boundary via We would like to calculate the time-dependent (boundary condition dependent) part of the WDW patch action. The time-dependent bulk parts are colored in blue while the independent parts are colored in green in Figure 5. Besides the action contributed by the discontinuity Appendix C.1), the time dependence of the boundary and joint action are the same as discussed in Appendix B. To be specific, in Figure 5(a), only the action of the boundary CD and the joint J are considered. In Figure 5(b), only the action of the boundaries CD and M N are considered.

C.1 Contribution from the shockwave discontinuity
We divide the total bulk into four subregions which are colored in Figure 5, thus introducing three segements as internal boundaries: F O, O H, EG. Since the metric around F O and O H is continuous, the induced null surfaces and joints will have identical action up to a sign on each side and cancel with each other. However, the metric is discontinuous along EG, so we should not expect the action on both sides to add up to zero. We calculate the effect of the discontinuity by comparing the two null surfaces U = and U = − which approach to EG in the same way when → 0 (as in Cauchy principal value integration). Figure 6 illustrates the treatment.
Using the Kruskal coordinate U, V and Eq. (A.2), it is a straightforward calculation to obtain Using Eq. (A.4), we have where H ± (ρ) ≡ ρ d log(±L 2 f (1/ρ)) as a short-hand notation. The (+) sign applies for joints outside the future or past horizon while (−) sign applies for those inside the horizon.
As → 0, the radius ρ → ρ h for all of the four joints. Expanding (ρ − ρ h ) for H ± (ρ) we get where c 1 is a constant depending only on d, z, θ and is same for both (+) and (−) case.
On the other hand, recalling Eq. (B.2), we can perform the integral and expand in (ρ − ρ h ): where ± sign has the same structure as H ± (ρ) and c 2 is another constant that depends only on d, z, θ and is same for both (+) and (−) case.
Using the definition of the Kruskal coordinates, and similar expressions for E , G , G .
Combining Eq. (C.12) with Eqs. (C.10), (C.9), and using Eq. (2.23) for the energy, we can get an expression for Eq. (C.8) in terms of a power series of (ρ − ρ h ) 13) in which the log terms precisely cancel with each other provided the four joints approach to the U = 0 horizon in the same way and the higher order terms in the last line will vanish in the limit → 0.
So we conclude that the discontinuity of the metric will contribute to the total action by (C.14) C.2 Contribution of Figure 5(a) The two bulk regions behind the future and past horizon also contribute to the time dependence of the action: where the function F (ρ) is defined in Eq. (B.13). Note that the last line of both calculations are of no interest here since they are time independent.
We refer to Eq. (A.1) for the calculation of the boundary action The joint action is obtained from Eq. (A.4) It is cumbersome to sum up all the parts directly, but it is easy to show that in the limit h → 0 and ρ J → ρ h , summing up all the parts gives I total = 2E(t L + t R ) 1 + z − 1 d − θ , (C. 19) which recovers Eq. (2.26) when performing either a t L or t R derivative.

C.3 Contribution of Figure 5(b)
In this case, the bulk regions behind the future and past horizon still contribute to the time-dependent action. The one behind the future horizon has action identical to Eq. (C.15), while the one behind the past horizon has action (C.20) There is no longer a joint action that is time dependent. However, we should take the boundary action of the spacelike segment M N into account: This result is identical to the result obtained by Ref. [8] when θ = 0, z = 1. The calculation in Ref. [8] considers an extra boundary term for the past and future horizon F OG and EO H, which is given by the null limit of the spacelike boundary action In the calculation method of this paper, these terms are not present, but a similar contribution is given by considering the effect of the discontinuity of the metric along U = 0 horizon.  Figure 5: Illustration of the Penrose diagram for the shockwave geometry. Left (5(a)): the patch only intersects with future singularity; Right (5(b)): the patch intersects with both singularities. The bulk regions whose actions are time-independent are colored in green while those whose actions are time-dependent are colored in blue. Figure 6: Illustration of the treatment of the discontinuity along null segement EG.
We take the limit → 0 so the null segement E G and E G approach to EG in the same way.