Traversable wormholes via a double trace deformation involving $U(1)$ conserved current operators

We study the effects of conservation laws on wormholes that are made traversable by a double trace deformation. After coupling the two asymptotic boundaries of a maximally extended $(d+1)$ dimensional black brane geometry with $U(1)$ conserved current operators, we find that the quantum matter stress-energy tensor of the corresponding bulk gauge fields in the hydrodynamic limit violates the averaged null energy condition (ANEC), rendering the wormhole traversable. Applying our results to axionic two-sided black hole solutions, we discuss how the wormhole opening depends on the charge diffusion constant, how this affects the amount of information that can be sent through the wormhole, and possible implications for many-body quantum teleportation protocols involving conserved current operators.

C Vector field bulk-to-boundary propagators in the hydrodynamic limit 26 1 Introduction In holography [1][2][3], maximally extended two-sided black holes are dual to a pair of conformal field theories entangled in a thermofield double state [4].The wormhole connecting the two sides of the geometry can be made traversable by introducing a non-local coupling between the two asymptotic boundaries.This construction was first proposed by Gao, Jafferis and Wall [5], and it allows us to send a message from one side of the geometry to the other.From the point of view of the boundary theory, this can be viewed as a teleportation protocol [5,6].
In the simplest instance of quantum teleportation, two distant observers, often called Alice and Bob, share a pair of maximally entangled qubits.Alice has an additional qubit |ψ⟩ that she wants to teleport to Bob.To do so, Alice performs a measurement on the two particles in her possession in a particular basis (the Bell basis) and reports the result to Bob through a classical communication channel.Finally, Bob performs a unitary operation in his qubit to obtain the desired quantum state [7].The essential ingredients of this teleportation protocol are share entanglement, measurement, and classical communication, and it involves only three qubits.
Gao, Jafferis and Wall (GJW) traversable wormhole is related to a new type of teleportation protocol, the so-called traversable wormhole teleportation protocol.It involves two copies of a strongly interacting many-body system entangled in a thermofield double state.Let us call them the left and the right systems, and consider the teleportation of a qubit from the left system to the right system.In this setup, the quantum information to be teleported is initially scrambled among the degrees of freedom of the left system.Then, after a weak coupling between the left and the right systems, the quantum information reappears (unscrambles) in the right system after a time of the order of the scrambling time.In this protocol, the thermofield double state plays the same role that the maximally entangled pair plays in conventional teleportation protocols, while measurement and classical communication are used to implement the coupling between the two systems.See Fig. 1.The traversable wormhole protocol can be implemented in rather general chaotic many-body quantum systems, but the phenomenon of many-body quantum teleportation has distinct features in the case of systems that have an emergent gravitational description.This property makes the traversable wormhole protocol a powerful experimental tool to gain insights into the inner-working mechanisms of gauge-gravity duality [8][9][10].
Traversable wormholes violate the averaged Null Energy Condition (ANEC), which states that the integral of the stress energy tensor along complete achronal null geodesics is always non-negative where k µ is a tangent vector and λ is an affine parameter.In classical theories, the construction of traversable wormholes is prevented by the Null Energy Condition (NEC) T µν k µ k ν ≥ 0, which implies (1.1) and is valid in physically reasonable theories.GJW construction of a traversable wormholes overcomes this difficulty by considering quantum mechanical effects.They work in the context of the semi-classical approximation, in which the gravitational field is treated classically, but the matter fields are treated quantum mechanically.In this context, one writes Einstein's equations as follows where G µν is the Einstein tensor, and ⟨T µν ⟩ is the expectation value of the stress tensor in a given quantum state.Initially, before introducing the deformation, the wormhole connecting the two asymptotic boundaries is not traversable, which is consistent with the fact that the two boundary theories are not interacting.One then introduces a non-local deformation of the boundary theory which couples the two asymptotic boundaries.Here, S bdry denotes the action for the two copies of the boundary theory1 , and O L,R denotes a scalar operator that acts on the ddimensional left/right boundary theory.For certain choices of h(t, x), the deformation (1.3) gives rise to a stress energy tensor whose expectation value violates ANEC, rendering the wormhole traversable.The physical picture is that the deformation (1.3) introduces negative energy in the bulk, whose backreaction is given in terms of a negative energy shock wave that causes a time advance for the geodesics crossing it, as opposed to the usual time delay caused by positive-energy shock waves.In this way, a signal originated on the left boundary can cross the wormhole and reach the right boundary after interacting with the negative energy, as shown in Fig. 1.The ANEC can be violated in the GJW setup because the geodesics crossing the wormhole are not achronal -early and late times points along the horizon can be connected by a timelike curve passing through the directly coupled boundaries.schematic representation of the traversable wormhole teleportation protocol proposed in [10].The quantum information (shown in red) is introduced in the left system at early times, gets scrambled with the other degrees of freedom, and reappears (unscrambles) in the right system at late times after a weak coupling between the two systems.The coupling is implemented as follows.We measure some operator O L in the the left system, obtaining one of the possible values of o j .We then apply the operator V = e i hoj O R on the right system.For further details about this teleportation protocol, we refer to [10].
Despite the existence of a growing literature about wormholes that become traversable by a double trace deformation, most works only considered deformations involving scalar operators.It is then natural to question if such a construction is still possible for deformations involving other types of operators, such as vector and tensor operators.In particular, conserved current operators have different scrambling properties as compared to (nonconserved) scalar operators, and this is expected to affect the traversability properties of the wormhole.
In this work, we study the boundary deformations involving U (1) conserved current operators.We are interested in the corresponding bulk gauge fields in the hydrodynamic limit.In this case, the gauge field displays a diffusive behavior which is expected to affect the traversability properties of the wormhole.In general, the diffusive behavior of bulk gauge fields leads to a power-law behavior of two-point functions [36,37] and out-of-timeorder correlators2 at late times [39].Since the wormhole opening can be computed as an integral involving a product of two-point functions, we expect it to display a power-law behavior at late times.Moreover, we expect the wormhole opening and consequently the bound on information transfer to depend on the transport properties of the black hole horizon.In particular, we would like to understand the effects of conservation laws on many-body traversable wormhole protocols and obtain the expected behavior for systems that admit a dual gravitational description.Is teleportation favored in this case?Or do the different scrambling properties of conserved currents result in a less efficient teleportation protocol?The benchmark holographic behavior is well understood in the case where the double trace deformation involves scalar field operators but has not yet been explored for deformations involving vector and tensor operators, and this might be relevant in the experimental realization of traversable wormhole teleportation protocols, especially because of this possible interplay between traversability and hydrodynamic behavior.

Organization of this work
This work is organized as follows.In Sec. 2, we introduce our gravity setup.In Sec. 3, we review the boundary conditions for scalars fields associated with the GJW construction of a traversable wormhole.In Sec. 4, we discuss boundary conditions for vector fields.In Sec. 5, we consider a double trace deformation involving U (1) conserved current operators and show that it leads to a violation of the averaged null energy condition.In Sec. 6, we discuss our results.We relegate some technical details to the appendices A, B and C.

Gravity set-up
We consider a general (d + 1)-dimensional black brane background, with the line element of the form where (t, x i ) are the boundary theory coordinates, with i running from 1 to d−1, and z is the AdS radial coordinate.We take the boundary to be located at z = 0, where the geometry is assumed to asymptote AdS d+1 .We assume the horizon is located at z = z h , where G tt has a first order zero and G zz has a first order pole.All the other metric components are assumed to be finite and non-zero at the horizon.For simplicity, we take G ij = δ ij G xx , which corresponds to assuming full rotational symmetry in the x i directions.Near the horizon, we write the metric functions G tt and G zz as With the above assumptions and by requiring regularity of the Euclidean continuation of the above line element at the horizon, one obtains the inverse Hawking temperature as To analyze the equations of motion near the horizon, it is convenient to define the tortoise coordinate in terms of which the boundary is located at r = 0 and the horizon at r = −∞.In terms of the tortoise coordinate, the line element reads Finally, to describe the globally extended spacetime, we introduce Kruskal-Szekeres coordinates (U, V, x i ) as In terms of these coordinates, the line element takes the form where we used the fact that r is a function of U V .In these coordinates the horizon is located at U = 0 or at V = 0.The left and right boundaries are located at U V = −1 and the past and future singularities at U V = 1.The Kruskal and Penrose diagrams for this geometry are shown in Fig. 2.

Boundary conditions for scalar fields
In this section, we review the boundary conditions for scalar fields that are associated with the GJW traversable wormhole.Let's consider a minimally coupled scalar field with mass m propagating on the background (2.1).Near the boundary, the field behaves as where y = (t, x) denotes a boundary point, and For m 2 > −d 2 /4 + 1, the first term in (3.1) is non-normalizable and is associated to a deformation of the boundary theory of the form where the single trace operator O α (y) has scaling dimension ∆ + = d/2 + ν.The second term in (3.1) is normalizable, and is related to the expectation value of O α For −d 2 /4 < m 2 < −d 2 /4 + 1, both terms in (3.1) are normalizable.In this regime, we are free to impose boundary conditions on either α or β.Each choice corresponds to a different boundary theory.In the boundary theory in which α(β) is fixed, the bulk field ϕ is dual to an operator of dimension ∆ + (∆ − ).We refer to these boundary theories as CFT ∆ + and CFT ∆ − .In particular, the so-called alternative boundary condition is associated to deformations of the form in which the boundary operator In this case, the leading order term is now associated to the expectation value of O β (y): The above equations allow us to symbolically write the deformation as W β ∼ d d y β α.A linear boundary condition relating the faster and the slower falloff parts corresponds to a double trace deformation of the form [40] which is a relevant deformation, because ∆ − < d/2.Starting from the CFT ∆ − , the deformation (3.9) produces a renormalization group flow which is expected to end at the CFT ∆ + in the infrared.We now explain how the linear boundary condition (3.8) can be used to construct a traversable wormhole.GJW construction of a traversable wormhole relies on the use of a relevant deformation, because in this case the near boundary geometry is not modified by backreaction in an uncontrolled way [5].In this way, their construction can be shown to be embeddable in a UV complete theory of gravity.To have a relevant deformation, they consider a massive scalar field with mass in the range −d 2 /4 < m 2 < −d 2 /4 + 1 with the alternative boundary condition, in which the boundary operator has dimension ∆ − = d/2 − ν.For the fields propagating in the left and right exterior regions, we expect the following near boundary behaviors In order to have a deformation coupling the two asymptotic boundaries, GJW impose the following boundary conditions which correspond to a non-local double trace deformation of the form The deformation (3.13) modifies the 1-loop expectation value of the scalar field stressenergy tensor.For some choices of h(y), this leads to a violation of the ANEC that renders the wormhole traversable.

Boundary conditions for vector fields
In this section, we briefly review possible boundary conditions for vector fields in AdS/CFT.
A general discussion about this topic was first presented in [41].We consider a massless vector field A µ propagating on the background (2.1) with action where r = (z, t, x) ∈ M denotes a bulk point and y = (t, x) ∈ ∂M denotes a boundary point.Here, γ denotes the determinant of the induced metric on the boundary, and n µ is the outward pointing unit vector normal to ∂M.The equation of motion for A µ reads

Dirichlet boundary conditions for vector fields
Near the boundary (z → 0), the gauge field behaves as In general, when d ≥ 4, the leading term a µ is non-normalizable and must be fixed.This corresponds to imposing a Dirichlet boundary condition on A µ .From the point of view of the boundary theory, such boundary condition correspond to a deformation by a global U (1) conserved current operator with a µ acting as a source for J µ .Note that, since A µ = a µ at the boundary, we can write The expectation value of the conserved current is given by This allow us to write W A ∼ ∂M d d y a µ b µ .Similarly to the scalar field case, a double trace deformation can be introduced by imposing a linear relation between a µ and b µ , i.e., a µ = Q µν b ν , which leads to which is an irrelevant deformation, with dimension 2d − 2, because the scaling dimension of J µ is d − 1.Here Q µν is a set of parameters that we will specify later.Just like in the scalar field case, one can construct a traversable wormhole by coupling the two asymptotic boundaries of an asymptotically AdS two-sided black hole with conserved current operators.This can be done as follows.First, we write the near boundary behavior of the gauge field as follows where the superscripts R and L denote the right anf left exterior regions, respectively.Then, an irrelevant double trace deformation can be introduced by imposing the following boundary condition which leads to a deformation of the form We show in the next section that for some choices of Q µν the deformation (4.10) leads to a violation of the ANEC, rendering the wormhole traversable.
In this work, we consider the deformation (4.10) because we are interested in the hydrodynamic limit of the gauge fields, associated with long-distance physics.We can think about our setup as the IR fixed point of a renormalization group (RG) flow.As explained in the next section, a relevant deformation can be introduced if one considers gauge fields satisfying Neumann boundary conditions.In this case, the initial UV theory flows under the RG flow to an IR fixed point in which the gauge fields respect Dirichlet boundary conditions, which corresponds to our case.By that reasoning, we expect that our construction has a well-defined UV fixed point.
Our analysis should be contrasted with the one conducted by performed by Gao, Jafferis, and Wall [5].In their case, from the dual CFT viewpoint, the double trace deformation is relevant in the context of the renormalization group, leading to a well-defined UV fixed point.The deformation we employ in our analysis (4.10), however, is irrelevant, and thus, there is no guarantee that it leads to a well-defined fixed point.Nevertheless, as discussed in [41], there is evidence supporting that the deformation (4.10) results in a well-defined UV fixed point in four, five, and six bulk spacetime dimensions.

Neumann boundary conditions for vector fields
In the previous section, we adopted the most common boundary condition for vector fields in AdS/CFT, which corresponds to imposing Dirichlet boundary conditions on A ν .As pointed out in [41], for d = 3, 4 and 5, it is possible to consider an alternative boundary condition for vector fields, in which one imposes Neumann boundary condition on the gauge field by fixing at the boundary.This boundary condition fixes the value b µ , but leaves a µ unconstrained.For d = 3, this corresponds to a deformation of the boundary theory by an operator O F ν , with dimension one.One can show that this operator is a U (1) gauge field [41][42][43].
Imposing the condition b µ = Q µν a ν corresponds to introducing a relevant double trace deformation of the form [41] which has dimension 2. Under the deformation (4.12), the boundary theory associated to Neumann boundary conditions flows in the infrared to the boundary theory associated to Dirichlet boundary conditions [41].
Once again, an wormhole connecting the two sides of an asymptotically AdS two-sided black hole can be made traversable by introducing the following relevant deformation which is associated to boundary conditions of the form 5 Opening the wormhole with conserved current operators In this section, we propose a generalization of the GJW traversable wormhole that involves a double trace deformation constructed out of conserved currents For some choices of Q µν , the deformation (5.1) introduces negative null energy in the bulk, which makes the wormhole traversable.For simplicity, we consider the coupling between the two conformal field theories as involving only J (R) t and J (L) t , i.e., we set Q µν as Here, h(t 1 , x 1 ) controls the coupling between the two boundary theories.We choose a perturbation that is instantaneous in time t 0 and uniform in the transverse space where h is a constant.By dimensional analysis, h has dimensions of [E] 2−d .Later, we will factor out this dimensional dependence and write this constant as h = h T 2−d .The traversability of the wormhole can be measured near the event horizon (U = 0) by the averaged null energy where T V V is the V V component of the Maxwell stress energy tensor.To study the effect of the deformation on T V V , we compute the 1-loop Maxwell stress energy tensor using a point splitting technique.The Maxwell stress energy tensor is given by To simplify our calculation, we consider that only A V and A U are non-zero, and we consider a metric of the form ds 2 = G U V dU dV + G ij dx i dx j .With these assumptions, the V V component of the stress energy tensor reads Now, using and the point splitting method, we write where ⟨A V (r)A V (r ′ )⟩ is the bulk two-point function between two points r = (z, t, x), and r ′ = (z ′ , t ′ , x ′ )3 .

Modified bulk two-point function
To compute stress energy tensor involving the deformation δH in (5.1), we first compute the modified bulk two-point function.Due to the deformation, the gauge field operators in (5.7) evolves in time with the operator U (t, t 0 ) = T e in the interaction picture.Then, the modified bulk two-point function in (5.7) is given by where we used the superscripts H and I to distinguish between the Heisenberg and interaction picture, respectively.For notational simplicity, we will omit the superscript I in what follows.
Working at first order in perturbation theory, we expand where The first correction to the two-point function of A V is obtained as follows, In the second line, we put (5.1) into the expression.In the third line, we assume that the fields A V (r) and A V (r ′ ) act on the right exterior region.Then, we can use large N factorization and the fact that the operator on the right exterior region commutes with the operators on the left boundary.
(5.13) for t > t 1 .Then, the expression of the first order correction to the bulk two-point function of A V can be rewritten as follows, G In the first line, we used the KMS condition [44] which gives the relation between J (L) µ and J (R) µ , and in the second line, we included the perturbation (5.2).
In this work, we consider the bulk-to-boundary propagators (5.12) and (5.13) under the hydrodynamic approximation because in that case the gauge field reveals a diffusive behavior, controlled by a charge diffusion constant.In this approximation, we focus on large times and large distances.This translates to solving our equations of motion for low-energy modes with large wavelengths.The bulk-to-boundary propagators in the hydrodynamic limit can be written as (see Appendix C or [39]) , (5.14) where D c is the charge diffusion constant: Using the above expressions, G V V is obtained as where D k = Dc 2πT k 2 , and V 1 = e 2πT t 1 .

1-loop stress tensor
The V V component of the 1-loop expectation value can then be computed as where we used the fact that G (0) V V does not contribute to the opening of the wormhole.Using (5.3) and performing the integral in V 1 , the expectation value of the stress-energy tensor can be written as (5. 19) By writing k in spherical coordinates, the expectation value of the stress-energy tensor can then be written as and k max is a momentum cutoff.Since we derived the bulk-to-boundary propagators using a hydrodynamic approximation, in which k 2πT ≪ 1, it is natural to set k max ∼ 2πT .In this work, however, we choose to introduce a double trace deformation that only involves low energy modes, and we write the momentum cutoff as k max = 2πT √ f , where the parameter f controls the maximum energy of our deformation.This is conceptually similar to what was done in [6,45], in which the authors consider a signal that only contains low energy modes by changing the corresponding wave functions as ψ(p) → ψ(p)e −p 2 /σ 2 .Here we do a similar procedure for the deformation, instead of the signal, and introduce a hard cutoff, instead of a smooth (Gaussian) cutoff.Our motivation to consider such type of deformation is because we are interested in the connection between traversability and diffusion, and diffusion only takes place for low energy modes.Before evaluating the integral in (5.22), let us first introduce a new variable, u = D k = Dc 2πT k 2 , and factor out the temperature dependence of h, D c and G xx (z h ) by writing them as follows h = hT 2−d , D c = D c /T and G xx = Gxx /T 2 .In terms of these new variables, the expectation value of the stress-energy tensor can be written as where the overall factor contains information about the geometry, while the remaining part only depends on u max .Note that k max = 2πT √ f implies u max = 2πD c f , which is typically a O(1) number4 .Fig. 3 shows the behavior of N −1 ⟨T V V ⟩ as a function of V for several values of V 0 with d = 4 and u max = 1/2.This figure shows that the stress-energy tensor diverges at the insertion time V = V 0 , and quickly decreases to zero for larger values of V .Similar behavior was also observed in [5].

Averaged null energy
In this subsection, we compute the averaged null energy in the presence of double trace deformation involving U (1) conserved current operators.By integrating (5.22) along complete achronal null geodesics, the averaged null energy can be computed as sin(πu)P (V, V 0 ) u , (5.24) where N ( h, z h , d, D c ) is given by (5.23).The above integrals can be computed numerically.In Fig. 4, we plot N −1 T V V dV versus V 0 for several spacetime dimensions using (5.24).
The averaged null energy becomes less negative as we increase d, suggesting that it is more difficult to open the wormhole in higher dimensional setups, in accordance with previous results in the literature [16].
The relation between the averaged null energy and the wormhole opening ∆U reads (see Appendix A) (5.25) Using (5.24), we write the wormhole opening as follows where sin(πu)P (V, V 0 ) u . (5.27) Here we write the UV cutoff as k max = 2πT √ f , which leads to u max = 2πD c f .The factor f controls the size of the UV cutoff.

Application for linear axion models
In this subsection, we would like to investigate how the averaged null energy depends on the charge diffusion constant.In order to do that, we consider a linear axion model where the charge diffusion constant depends on a parameter that controls the momentum relaxation of the dual field theory, and becomes arbitrarily small as one increases the momentum relaxation parameter.
More specifically, we consider a simplified version of the linear axion model proposed in [46], with action of the form where L is the AdS length scale.For convenience, we set L = 1.We consider the background solution found in [46], which takes the form where the boundary is located at R → ∞, and the black hole horizon is located at R = R 0 .
Here, the index a runs from 1 to d − 1, labeling the d − 1 spatial directions, I is an internal index labeling d − 1 scalar fields, and α Ia are real arbitrary constants.The emblackening factor is given by (5.30 where α is a parameter that controls the momentum relaxation of the system, and it is given by where ⃗ α a • ⃗ α b = d−1 I=1 α Ia α Ib , and we made the assumption that ⃗ α a • ⃗ α b = α 2 δ ab for all values of a and b.The black hole temperature is given by Since there is no background gauge field in the above solution, the charge diffusion constant associated to fluctuations of a probe gauge field can be computed using Eq.(B.27) derived in Appendix B: The dimensionless charge diffusion constant takes the form (5.34) , and it becomes arbitrarily small in the zero temperature limit, in which the ratio α/T becomes arbitrarily large. 5e would like to understand how the presence of momentum relaxation affects the traversability properties of the wormhole.In our hydrodynamic limit, our result is almost universal, and the dependence on the system comes basically from the dimensionless charge diffusion constant, which naively appears only as an overall factor of . In the regime of strong momentum relaxation, D c becomes arbitrarily small, and the overall factor of becomes arbitrarily large, causing ∆U to diverge.However, once we introduce an UV cutoff of the form k max = 2πT √ f , this implies that the wormhole opening takes the schematic form and the limit of strong momentum relaxation (in which D c → 0) does not lead to any divergence of ∆U , due to the fact that the upper limit of integration is also proportional to the dimensionless charge diffusion constant, i.e., u max = 2πD c f .In Figure 5 we show the result for the averaged energy condition for several spacetimes dimensions and for several values of α/T , which is the parameter that controls the momentum relaxation in the system.From this figure, we can see that the wormhole opening decreases as we increase the momentum relaxation parameter, and approaches zero as α/T ≫ 1.That suggests that, at least in our hydrodynamic approximation, the presence of momentum relaxation does not favor traversability.In fact, a small diffusion constant leads to a very small wormhole opening 6 .

Fitting the data
In this subsection, we show that our numerical results for the averaged null energy can be described reasonably well by the following function: where the fitting parameters a and b are functions of d and α/T .Figure 6 shows our numerical results for the averaged null energy, obtained with Eq. (5.24) as well as the

Bound on information transfer
A traversable wormhole allows us to send a signal from one side of the geometry to the other side.In this section, we derive a bound on the amount of information that can be down roughly at Finally, plugging (5.41) and (5.35) into (5.40), and we obtain The fact that the bound is proportional to is consistent with the analysis of [20] for higher dimensional traversable wormholes.The functional dependence of the result on D c implies that N bits → 0 as D c → 0. From the point of view of the boundary theory, that suggests that the presence of momentum relaxation makes the teleportation protocol less efficient.It would be interesting to check if this also happens in many-body quantum teleportation protocols in which the classical communication involves the measurement of conserved current operators.

Discussion
We constructed a traversable wormhole by coupling the two asymptotic boundaries of a general (d + 1)-dimensional black brane with U (1) conserved current operators.The nonlocal coupling introduces a quantum correction to the expectation value of the stress-energy tensor that violates ANEC, rendering the wormhole traversable.
The double trace deformation involving U (1) conserved current operators is dual to a gauge field fluctuation in the bulk.In the limit where the frequency is small, the gauge field in the bulk displays a diffusive behavior.We found that the diffusive properties of the gauge field affect the behavior of the wormhole opening ∆U (see Fig. 8), causing it to have a power-law behavior as a function of the insertion time t 0 .In fact, by using that V 0 = e 2πT t 0 , and taking the limit where 2πT t 0 ≫ 1 in (5.36), we can write the averaged null energy as The power law part is reminiscent of the power law behavior observed in two-point functions [36,37] and out-of-time-order correlators involving conserved current operators [39], and it is related to the diffusive behavior of the bulk gauge field.The exponential part also takes place in the large time limit when the double trace involves scalar operators.For scalar operators of dimension ∆, one finds [16,20] (6.2) In the limit t 0 ≫ 1/T , one obtains A scalar ∼ e −2πT (2∆+1)t 0 .Therefore, while the exponential decrease with t 0 is present in both cases, the power law part only appears in the case involving conserved current operators.
We studied how the wormhole opening depends on the charge diffusion constant.In order to do that, we consider two-sided axionic black hole solutions (see Sec. 5.3.1) and study the behavior of the averaged null energy as a function of the momentum relaxation parameter.In these backgrounds, the charge diffusion constant becomes arbitrarily small as one increases the momentum relaxation parameter α/T .We computed the averaged null energy in the same limit and checked that it becomes arbitrarily small as α/T ≫ 1. See Fig. 5.This suggests that the presence of momentum relaxation makes it harder to open the wormholes using U (1) conserved current operators.
To compute the charge diffusion constant, we introduced a bulk gauge field in the probe approximation.If the bulk gauge field backreacts in the geometry, then our formula for the charge diffusion constant Eq.(B.27) is not valid.It would be interesting to generalize our results for systems with non-zero chemical potential.Another interesting future direction would be to consider double trace deformations involving other types of conserved currents and study how the wormhole opening depends on the corresponding transport coefficients.(2.1), these metric components are written as g tt = −g rr = −G tt , and g xx = G xx .For simplicity, we follow [39] and we choose a configuration in which A i = 0.This corresponds to analyzing the longitudinal sector of the gauge field fluctuations, which are known to be controlled by a diffusion pole [47,48].
We first decompose the gauge field in Fourier modes and assume the momentum to be parallel to the x axis.With these assumptions, Eq. (B.1) for ν = r becomes Solving (B.4) for A r , we find Now we turn our attention to Eq. (B.1) for ν = t: Substituting (B.5) into (B.6),we find an equation for A t : The above equation has two independent solutions, which near the horizon (r → ∞) take the form e ±iωr , corresponding to out-going and in-falling boundary conditions.We focus on the solution that satisfies in-falling boundary conditions at the horizon, and look for a solution of the form With this ansatz, Eq. (B.7) becomes To find a solution in the hydrodynamic limit, we set (ω, k) → λ(ω, k) and take λ ≪ 1.We then expand the function F as follows and solve the equations of motion at each order in the parameter λ.At zero order in λ, we have the following equation whose solution reads To find a regular solution it is convenient to write F 0 in terms of the coordinates (t, x i , z) defined in Sec. 2. In these coordinates, we find In the near horizon region, the solution takes the approximate form For this solution to be regular at the horizon we need to set C 1 = 0.At linear order in λ, we find Using that F 0 = C 0 , we find In terms of the coordinates (t, x i , z), the solution reads In the near horizon region, the solution takes the approximate form Requiring regularity at the horizon fixes the integration constant C 2 as Then, we can use C 3 to write the solution in terms of a definite integral where we wrote the argument of the metric functions explicitly to avoid confusion.To first C Vector field bulk-to-boundary propagators in the hydrodynamic limit In this section, we derive bulk-to-bulk propagators for a gauge field propagating in the background (2.1).We start by writing Maxwell-Einstein equations with a source term where the source satisfies the equation The gauge field can then be written as where r = (r, t, x) and G µν (r − r ′ ) solves (C.1) with a delta function as the source.
Following [39], we use the gauge A i = 0.By writing the propagators and the sources in momentum space we can find a solution for G µν (r, ω, k) in the hydrodynamic limit.To find G tt , we require J r bulk = 0, and J t bulk (r, ω, k) = δ(r − r ′ ).Then (C.2) implies J x bulk (r, ω, k) = ω k δ(r − r ′ ).The t components of (C.1) reads while the r component of (C.1) reads By solving (C.6) for G rr and substituting the result in (C.5), we find which, after some simplifications, can be written as (C.8)A solution to the above equation satisfying in-falling boundary conditions at the horizon and Neumann boundary conditions on the boundary corresponds to the retarded bulk-to-bulk propagator in momentum space: G ret tt (r, r ′ , ω, k) = −i dt d d−1 xe iωt−ik•x θ(t)⟨[A t (r, t, x), A t (r ′ , 0, 0)]⟩ .(C.9) For small ω and small k, one can show that the solution in the near-horizon region reads [39] G ret tt (r, r ′ , ω, k) ∼ In position space, the above bulk-to-bulk propagators can be written as G ret tt (r, t, x; r ′ , t ′ , x ′ ) = −i e ik•(x−x ′ ) θ(V − V ′ ) .(C.12) The Wightman function can be obtained as The corresponding bulk-to-boundary propagators can be obtained from the above expressions by taking one of the bulk points to the boundary.Denoting the bulk point as (U = 0, V, x) and the boundary point as (t 1 , x 1 ), we can write the retarded bulk-toboundary propagator as (V e −2πT t 1 ) where we used that r(z → 0) = 0 and consequently V → e 2πT t 1 as z → 0.
In the calculation of ∆U , we need K w V t and K ret V t , which can easily be obtained from K w tt and K ret tt .Since A V = At+Ar 2πT V , we can write ⟨A V J t ⟩ = 1 2πT V ⟨(A t + A r )J t ⟩.This can be simplified even more in the hydrodynamic limit, in which A r = A t .Using this last equality, we find ⟨A V J t ⟩ = 1 πT V ⟨A t J t ⟩.With the above considerations, we find , (C.16) . (C.17) where V 1 = e −2πT t 1 , and an overall factor of T was included for dimensional reasons.These formulas were derived in the hydrodynamic limit, which means they are only well-defined near the black hole horizon.Since we consider the expectation value of stress energy tensor in the near horizon region, the large time separation modes (V /V 1 ≫ 1) give a dominant contribution to it.Also, the stress tensor ⟨T V V ⟩ computed from the above propagators is independent on the insertion time t 0 .This happens because ⟨T V V ⟩ is proportional to7 K ret (V, t 0 )K w (V, −t 0 ) and the t 0 dependence cancels out.Consequently, the expression for the averaged null energy ∞ V 0 dV ⟨T V V ⟩ derived using (C.16) and (C.17) is only valid at late times.To fix this problem and obtain a result for ANE that depends on the insertion time, we used the propagators proposed by Swingle and Cheng in [39]: (V e −2πT t 1 − 1) According to [39], the above propagators have qualitatively the same behavior as (C.16) and (C.17) in the late time limit and have the desired analytic properties connecting K ret and K w .They allow us to study the dependence of ANE for any value of the insertion time, which is more physical, since there is nothing special about t 0 = 0.

2 10 5 6 20 A ANEC violation 22 B
Neumann boundary conditions for vector fields Opening the wormhole with conserved current operators 10 5.1 Modified bulk two-point function 11 5.2 1-loop stress tensor 13 5.3 Averaged null energy 15 5.3.1 Application for linear axion models 16 5.4 Bound on information transfer 18 Discussion Vector field fluctuation on a generic black brane background 22

Figure 1 .
Figure 1.Left panel: Penrose diagram for the GJW traversable wormhole.The negative energy shock wave (shown in blue) causes a time advance ∆U for geodesics crossing it.The geodesics are actually continuous, but they appear to be discontinuous because the diagram is drawn using discontinuous coordinates.Using the equations of motion one can see that the time advance is proportional to the averaged null energy, i.e., ∆U ∼ ⟨T V V ⟩dV .The signal (shown in red) originated on the left boundary and propagating along the horizon (U = 0) can cross to the other side of the geometry if ∆U < 0, which happens when ANEC is violated, i.e, ⟨T V V ⟩dV < 0. Right panel: schematic representation of the traversable wormhole teleportation protocol proposed in[10].The quantum information (shown in red) is introduced in the left system at early times, gets scrambled with the other degrees of freedom, and reappears (unscrambles) in the right system at late times after a weak coupling between the two systems.The coupling is implemented as follows.We measure some operator O L in the the left system, obtaining one of the possible values of o j .We then apply the operator V = e i hoj O R on the right system.For further details about this teleportation protocol, we refer to[10].

Figure 2 .
Figure 2. Kruskal diagram (Left) and Penrose diagram (Right) for a two-sided asymptotically AdS black brane.

Figure 4 .
Figure 4. averaged null energy versus V 0 in asymptotically AdS d+1 spacetimes for several values of d.Here we fix u max = 1/2 and h = −1.

Figure 5 .
Figure 5. T V V dV as a function of V 0 for several values of the momentum relaxation parameter α/T and d for the asymptotically AdS d+1 background solution in (5.29).Here we fix u max = 1/2 for α = 0 by setting f = (d − 2)/d.

Figure 6 .
Figure 6.T V V dV as a function of V 0 for several values of the momentum relaxation parameter α/T .The dots represent numerical results obtained with Eq. (5.24) and the continuous curves represent the fitted curves of the form (5.36)

Figure 8 .
Figure 8.The boundary double trace deformation with U (1) conserved current operators corresponds to gauge field fluctuations in the bulk.One can view the red lines as photons that are delocalized with different longitudinal energies, and it indicates the diffusive behavior of the bulk gauge fields.The right and left past (future) horizons intersect at the point P 1 (P 2 ).The size of the wormhole opening ∆U can be obtained by integrating along an achronal null geodesic, which is represented by the blue line.