How to Make Traversable Wormholes: Eternal AdS_4 Wormholes from Coupled CFT's
Suzanne Bintanja, Ricardo Espíndola, Ben Freivogel, Dora Nikolakopoulou
HHow to Make Traversable Wormholes:Eternal AdS Wormholes from Coupled CFT’s
Suzanne Bintanja, a Ricardo Esp´ındola, a Ben Freivogel a,b and Dora Nikolakopoulou a a Institute for Theoretical Physics, University of Amsterdam b GRAPPA, University of Amsterdam
E-mail: [email protected] , [email protected] , [email protected] , [email protected] Abstract:
We construct an eternal traversable wormhole connecting two asymptoticallyAdS regions. The wormhole is dual to the ground state of a system of two identical holo-graphic CFT’s coupled via a single low-dimension operator. The coupling between the twoCFT’s leads to negative null energy in the bulk, which supports a static traversable wormhole.As the ground state of a simple Hamiltonian, it may be possible to make these wormholes inthe lab or on a quantum computer. a r X i v : . [ h e p - t h ] F e b ontents A Propagators 25B Stress tensor 26C Matching 28D Correlators in (cid:104) H int (cid:105) Wormholes have been a puzzling topic for physicists for a century. Many efforts have beenmade to build traversable wormholes using different kinds of fields and techniques, most ofwhich require either the insertion of exotic matter [1–7] or higher derivative theories [8–11]which lack UV completions [12].Recent work has shown how to build traversable wormholes in physically sensible theories.Gao, Jafferis, and Wall (GJW) [13] showed how to make asymptotically AdS black holestraversable for a short time by coupling the boundaries to each other. This approach hasbeen extended in a number of other works since then [14–29]. The first eternal traversablewormhole was constructed by Maldacena and Qi [30] in asymptotically nearly-AdS spacetime.– 1 –ore recently, Maldacena, Milekhin and Popov [31] found a long-lived 4 D asymptotically flattraversable wormhole solution in the Standard Model (see also [32]).In this paper, we make use of the ingredients developed by GJW and MMP in order toconstruct an eternal traversable wormhole in asymptotically AdS spacetime. Our motivationis twofold. First, by constructing wormholes in asymptotically AdS spacetime, we can useAdS/CFT to learn more about them. Second, our wormhole solution can be used to learnmore about CFT’s. To this end, we identify a family of Hamiltonians consisting of two copiesof a CFT coupled by simple, local interactions whose ground state is dual to the traversablewormhole.This last point is significant for constructing traversable wormholes in a lab or on aquantum computer. Some very interesting ideas on how to do this are described in [33–35].Given access to a holographic CFT, one simply needs to implement the coupling and allowthe system to cool to its ground state, which is dual to a traversable wormhole.Concretely, the bulk theory we consider is described in Section 2 and consists of Einstein-Maxwell theory with negative cosmological constant, a U (1) gauge field and massless Diracfermions coupled to the gauge field. A particular solution is the magnetically chargedReissner-Nordstr¨om (RN) black hole. Due to the magnetic field, the charged fermions developLandau levels. The lowest Landau level has exactly zero energy on the sphere, so we can thinkof them as effectively 2 D fermionic degrees of freedom once we dimensionally reduce on thesphere.The classical solution consists of two magnetically charged RN black holes connectedthrough an Einstein-Rosen bridge which is non-traversable. The traversability of the worm-hole is achieved by introducing a coupling between the two CFT’s (labelled L, R ) of theform S int = i (cid:90) d x h (cid:0) ¯Ψ R − Ψ L + + ¯Ψ L + Ψ R − (cid:1) . (1.1)Here Ψ R is the bulk field at the right boundary that is dual to the charged fermions in theright CFT, and Ψ L is defined analogously. Note that this is a local coupling involving asingle, low dimension operator in each CFT; this contrasts with the beautiful constructionof Maldacena and Qi [30] in the AdS context, where a large number of operators must becoupled.In Sections 2.3 and 2.4, we describe how this interaction has the effect of modifying theboundary conditions and the vacuum state. The stress tensor receives a quantum correctionof the form (cid:104) T ++ ( x ) (cid:105) = − π qλ ( h ) R , (1.2)where R is the sphere radius, q is the charge of the black hole, and λ is given by (2.33). Forsmall coupling h , λ ( h ) ≈ h , but our analysis remains valid for finite h . A priori it is notclear whether a self-consistent solution exists in which the negative null energy supports atraversable wormhole. Since it is only the quantum correction that has a chance of makingthe wormhole traversable, the quantum effects have a large backreaction on the metric.– 2 –ypically, this would constitute an intractable problem: we cannot calculate the quantumstate, and hence the stress tensor, until we know the geometry, but on the other hand wecannot solve the Einstein equations to determine the geometry until we know the stress tensor.In this case, we are able to self-consistently solve the system because the stress tensor takesa particularly simple form, depending locally on the metric (up to an overall factor).In Section 3, we discuss properties of both the linearized and non-linear solutions. Thewormhole geometry has the following two regimes. The middle of the wormhole is nearlyAdS × S . As we move away from the middle of the wormhole, the geometry smoothlyinterpolates to the near-extremal region of two RN black holes. Far away, the quantumcontribution (1.2) becomes negligible and the geometry is that of two magnetically chargedRN black holes (see Fig. 2).As a consequence of the boundary perturbation, the mass of the wormhole is slightly decreased by a term proportional to the coupling M = M ext + ∆ M , with ∆ M ∼ − λ ( h ) . (1.3)An infalling observer will experience that she is approaching a naked singularity from infinity.All of a sudden, deep in the throat region, the wormhole opens up and she comes out to theother side safely.In Section 4.1, we identify a simple Hamiltonian whose ground state is dual to the worm-hole. The procedure is to begin with two identical holographic CFT’s, each with a global U (1) symmetry, so that they are dual to Einstein-Maxwell theory at low energies. We thenturn on a chemical potential for each CFT separately, and turn on a coupling of the form ¯Ψ R Ψ L where the Ψ operators are dual to a bulk massless charged fermion.Concretely, the Hamiltonian we analyze is H = H L + H R + µ ( Q L − Q R ) − ih(cid:96) (cid:90) d Ω (cid:0) ¯Ψ R − Ψ L + + ¯Ψ L + Ψ R − (cid:1) , (1.4)This Hamiltonian is similar to the construction of Cottrell et al [36]. The authors showedthat the Hamiltonian in their case has the thermofield double state as its ground state. Thatconstruction, however, did not have a semiclassical gravity dual.We show that the ground state of this theory is dual to our eternal traversable worm-hole geometry for some range of the coupling h and chemical potential µ . We comparethe wormhole to other geometries with the same boundary conditions, which may dominatethe ensemble. In particular, we consider two disconnected black holes and empty AdS. Wecompute the ground state for different values of the parameters h and µ , and find that thewormhole is the ground state for h > h c and µ > µ c , with the critical values given by h c = ¯ r G N q (cid:115) π C (cid:18) r (cid:96) (cid:19) and µ c = √ πm p , (1.5)with m p the Planck mass. Interestingly, as the non-local coupling vanishes, there is a triplepoint located at h = 0, µ = µ c where the three phases meet. For values h <
0, the groundstate is dominated by either empty AdS or the black hole phase.– 3 –he challenge of building a traversable wormhole is to have enough negative energy toallow defocusing of null geodesics, allowing the sphere to contract and re-expand. Here wehave added two ingredients so that the bulk dual remains semiclassical. First, the chemicalpotential makes the decoupled system closer to being traversable, since the near-horizongeometry for an extremal black hole is
AdS × S . Because the size of the sphere is constantnear the horizon, a small amount of negative energy will allow the sphere to re-expand, andrender the wormhole traversable . Second, by using bulk charged fermions in combinationwith a magnetically charged black hole, as was done in MMP [31], we enhance the negativeenergy due to the quantum effects. The key point is that a single 4d charged fermion actslike a large number q of 2d light charged fields due to the large degeneracy of lowest Landaulevels. Note:
We understand that overlapping results will appear in [37]. We thank S. Banerjeefor discussions. Also, [38] appeared very shortly before this work. There, asymptoticallyAdS wormholes are also constructed, but with rather different ingredients. In addition, thesolutions of [38] have different symmetries than our solution: they preserve the full Poincareinvariance in the boundary directions. It would be interesting to understand the relationshipbetween the two constructions better. We thank M. van Raamsdonk for discussions. We start this section by describing the particular theory of interest, as well as setting upthe notation and conventions of spinors in curved space. Afterwards, we describe how theboundary conditions change once we couple the asymptotic boundaries. Finally, we computethe resulting stress tensor.
The theory consists of Einstein-Maxwell gravity with matter described by the action S = (cid:90) d x √ g (cid:18) πG N ( R − − g F + i ¯Ψ /D Ψ (cid:19) . (2.1)In particular, we are considering a single massless Dirac fermion of charge one. In this section,we follow the approach and conventions of [31].We consider g to be small, so that loop corrections are suppressed. A general classof spherically symmetric solutions with magnetic charge, denoted by the integer q , can beparametrized as follows ds = e σ ( x,t ) ( − dt + dx ) + R ( x ) d Ω , A = q θdφ . (2.2)Note that in this metric the range of x is compact and fixing this range can be seen as agauge choice. For now we use x ∈ [0 , π ]. To have a well-defined representation of the Clifford We thank Daniel Jafferis for suggesting this approach. – 4 –lgebra at each point of the spacetime we introduce the vierbein e = e σ dt, e = e σ dx, e = Rdθ, e = R sin θdφ . (2.3)and by solving de a + ω ab ∧ e b = 0 , ω ab = − ω ba , (2.4)we compute the spin connection components ω = σ (cid:48) dt + ˙ σdx, ω = R (cid:48) e − σ dθ, ω = R (cid:48) sin θe − σ dφ, ω = cos θdφ . (2.5)Here a prime denotes a derivative with respect to x , while a dot denotes a derivative takenwith respect to t . We use the following basis for the gamma matrices in flat space γ = iσ x ⊗ , γ = σ y ⊗ , γ = σ z ⊗ σ x , γ = σ z ⊗ σ y . (2.6)In this basis the Dirac operator has the form /D = e − σ (cid:104) iσ x (cid:16) ∂ t + ˙ σ (cid:17) + σ y (cid:16) ∂ x + σ (cid:48) R (cid:48) R (cid:17)(cid:105) ⊗ σ z R ⊗ (cid:104) σ y ∂ φ − iA φ sin θ + σ x (cid:16) ∂ θ + 12 cot θ (cid:17)(cid:105) . (2.7)In the static case, the metric (2.2) has two Killing vectors ∂ t and ∂ φ . Introducing the followingansatz will allow us to decompose in Fourier modes on the sphere S ,Ψ = e − σ R (cid:88) m ψ m ( t, x ) ⊗ η m ( θ, φ ) . (2.8)Here ψ m and η m are bi-spinors. In the rest of the paper we will suppress the indices on ψ .In this ansatz the Dirac equation is given by e − σ R ( iσ x ∂ t + σ y ∂ x ) ψ ⊗ η = − λ ,e − σ R σ z ψ ⊗ (cid:18) σ y ∂ φ − iA φ sin( θ ) + σ x (cid:18) ∂ θ + 12 cot( θ ) (cid:19)(cid:19) η = λ . (2.9)Restricting to the lowest Landau level decouples the equations and admits solutions of theform ψ ± = (cid:88) k α ± k e ik ( t ∓ x ) , η m ± = (cid:16) sin θ (cid:17) j ± ± m (cid:16) cos θ (cid:17) j ± ∓ m e imφ , j ± = 12 ( − ∓ q ) , (2.10)where ψ ± are the components of ψ , and we choose σ z η ± = ± η ± as the basis for η . If we take q >
0, the solution is η + = 0 , η − = (cid:88) m C jm η m − , − j ≤ m ≤ j , (2.11)– 5 –here we define the quantum number j := j − , so that in the lowest Landau level the degen-eracy of the two-dimensional fields is q . The normalization constant is given by( C jm ) = 12 Γ(2 + 2 j )Γ(1 + j − m )Γ(1 + j + m ) , (2.12)so that (cid:90) d Ω ¯ η m η m = δ m m . (2.13) According to the AdS/CFT dictionary, a bulk Dirac spinor of mass m is dual to a spin 1 / O of conformal dimension∆ ± = 32 ± √ m (cid:96) , (2.14)where (cid:96) is the AdS radius [39, 40]. The stability bound requires m ≥ γ ) = 1 and ( γ ) † = γ . We can then decomposethe bulk fermions onto the eigenspace of γ Ψ ± := P ± Ψ , P ± = 12 (cid:0) ± γ (cid:1) , (2.15)and similarly for the Dirac conjugate. The orthogonal projection operator satisfies the twoconditions P = P and P † = P . More explicitlyΨ + := 12 e − σ R (cid:32) ψ + − iψ − i ( ψ + − iψ − ) (cid:33) ⊗ (cid:32) η + η − (cid:33) , Ψ − := 12 e − σ R (cid:32) ψ + + iψ − − i ( ψ + + iψ − ) (cid:33) ⊗ (cid:32) η + η − (cid:33) . (2.16)The variation of the Dirac part of the action (2.1) with respect to Ψ ± after integration byparts becomes ∆ S D = bulk terms + i (cid:90) ∂ d x √ γ (cid:0) ¯Ψ − δ Ψ + − ¯Ψ + δ Ψ − (cid:1) , (2.17)where γ is the determinant of the induced metric at the boundary.The bulk terms are propor-tional to the equations of motion. In order to have a well-defined boundary value problem,we should include a boundary term of the form S ∂ = i (cid:90) ∂ d x √ γ (cid:0) a ¯Ψ − Ψ + + a ¯Ψ + Ψ − (cid:1) . (2.18)– 6 –e can then either fix Ψ + = 0 or Ψ − = 0 (and thus ¯Ψ + = 0 or ¯Ψ − = 0) at the boundarydepending on whether we set ( a = − , a = 0) or ( a = 0 , a = 1) respectively in the totalvariation of the action δS D + δS ∂ .In the massless case, both modes Ψ ± are normalizable. We can then identify the asymp-totic values Ψ ± := lim x → π R ( x ) − Ψ ± , (2.19)with the normalizable part of the dual operator O . After reducing on the S sphere, theeffective 2 D fermions obey reflective boundary conditions in both types of quantizations ψ + = e iα ψ − , with α = (cid:40) π , standard π , alternate , (2.20)which correspond to taking Ψ = 0 or Ψ − = 0 respectively. Intuitively, they would notallow the charge and energy to leak out at the boundary. In fact, by using the conservationequations it is easy to see that at the boundary˙ E = T (cid:12)(cid:12)(cid:12) ∂ = 0 and ˙ Q = J (cid:12)(cid:12)(cid:12) ∂ = 0 , (2.21)where J is the x component of the U (1) current J = ψ † σ z ψ ⊗ η † η = (cid:16) ψ †− ψ + − ψ † + ψ − (cid:17) ⊗ η † η , (2.22)and T is the energy flux, which is given by the tx -component of the stress tensor T = i R (cid:16) ψ † + ( ∂ x − ∂ t ) ψ + + ψ †− ( ∂ x + ∂ t ) ψ − − ( ∂ x − ∂ t ) ψ † + ψ + − ( ∂ x + ∂ t ) ψ †− ψ − (cid:17) ⊗ η † η . (2.23)Now consider two decoupled and identical conformal theories with fermionic degrees of free-dom. In principle, each one has its own bulk gravity dual. The boundary action then acquiresthe form S ∂ = i (cid:90) ∂ d x √ γ (cid:0) a ¯Ψ R − Ψ R + + a ¯Ψ R + Ψ R − + b ¯Ψ L − Ψ L + + b ¯Ψ L + Ψ L − (cid:1) . (2.24)There are various options depending on what type of boundary sources we would like to keepturned-on. The guiding principle we will follow is CPT invariance. CPT-related boundaryconditions imply a vanishing T ++ component consistent with the fact that vacuum AdS cannot support finite energy excitations [42]. For the purpose of this work, we choose thefollowing CPT conjugate boundary conditionsΨ R + = 0 CPT −−−→ ¯Ψ L − = 0 , (2.25)which correspond to the coefficients a = − , b = 1, and a = b = 0. The vanishing energycan also be understood as due to the conformal anomaly contribution present in the mappingbetween the energy of a CFT on the strip to AdS [30]. Note that since we consider two copies of the theory we now have x ∈ [ − π , π ], and we denote the left(right) boundary at x = ∓ π with L ( R ). – 7 – .3 Modified boundary conditions We are interested in the case where the two bulk geometries are two magnetically charged RNblack holes. Intuitively, we can think on them as being connected through an Einstein-Rosenbridge. A priori, however, it is not obvious how to connect both bulk geometries throughthe horizon. Moreover, in order to render the wormhole traversable, we need to establish aconnection between the two asymptotic boundaries. We achieve that by using a non-localcoupling of the form S int = − i (cid:90) d x √ γ (cid:0) h ¯Ψ R + Ψ L − + h ¯Ψ L − Ψ R + + h ¯Ψ R − Ψ L + + h ¯Ψ L + Ψ R − (cid:1) . (2.26)This term will provide us with the negative energy we need and will open up the wormhole.It is important to mention that if instead of fermions, we considered interacting scalar fields,similar to [13], the lowest Landau levels would have positive energy on the S sphere, makingthe problem of finding a traversable geometry much harder.We are looking for an eternal traversable wormhole, so we let the coupling constants beturned on for all times. For the purposes of this work, we focus on the case where the couplingconstants are real and h = h = 0 . The boundary conditions turn out to beΨ R + + h Ψ L + = 0 , and Ψ L − + h Ψ R − = 0 , (2.27)where h = h = − h . Notice that the sources at the boundary are vanishing. In terms of thespinor components this implies the following boundary conditions ψ R + − iψ R − + hψ L + − ihψ L − = 0 , and ψ L + + iψ L − + hψ R + + ihψ R − = 0 . (2.28)See Fig. 1 for an illustrations of the modified boundary conditions.In order to obtain a solution to the equations of motion (2.9) with the boundary conditions(2.28), for the lowest Landau level, we use the following ansatz: ψ + = (cid:88) k α k √ π e iω k ( t − x ) and ψ − = (cid:88) k β k √ π e iω k ( t + x ) . (2.29)Filling in this ansatz in to the boundary conditions (2.28) leads to the following constraintequations ( i ω k + hi ω k ) α k + ( i ω k +3 + hi ω k +3 ) β k =0 , ( i ω k + hi ω k ) α k + ( i ω k +1 + hi ω k +1 ) β k =0 , (2.30)with solution ω k = 2 k − iπ log (cid:18) − h ± i | − h | h (cid:19) , β k = ( − k +1 α k , (2.31) In general, the coupling constants can be complex. However, they must satisfy h = h ∗ , h = h ∗ , in orderfor (2.26) to be real. It would be interesting to understand other combinations of the non-local couplings. Note that the equations are invariant under h (cid:55)→ h and β k (cid:55)→ − β k , so the theory exhibits S-duality. – 8 –here k ∈ Z . The solution can be written in the following form ω k = 2 k + 12 + ( − k π λ ( h ) , (2.32)where λ is a function of h given by λ ( h ) = 12 arctan (cid:18) h | − h | (cid:19) . (2.33) Figure 1 : A right moving massless fermion, with amplitude | ψ R + | = 1,traveling on the strip hits the right boundary. The probability of theresulting left mover is equal to ( h − ( h +1) , and the right mover emerging fromthe left boundary has amplitude h ( h +1) . Using the solution (2.32), we write the fermionic fields as ψ + = (cid:88) k √ π α k e iω k ( t − x ) , and ψ − = (cid:88) k ( − k +1 √ π α k e iω k ( t + x ) . (2.34) In the remainder of this work, we will use light-cone coordinates defined by x ± = t ± x , whenever theyare more convenient. – 9 –he modes α k obey the following anti-commutation relations { α k , α † j } = δ k,j , { α k , α j } = 0 and { α † k , α † j } = 0 , (2.35)and the vacuum is defined as α k | (cid:105) = 0 ∀ k ∈ Z < , and α † k | (cid:105) = 0 ∀ k ∈ Z ≥ . (2.36)Using equations (2.34)-(2.36) we calculate the propagators. We present one of them here andthe rest can be found in the Appendix A (cid:104) ψ † + ( x − ) ψ + ( x (cid:48)− ) (cid:105) = 1 π e i ( x (cid:48)− − x − ) − e i ( x (cid:48)− − x − ) + e i ( x (cid:48)− − x − ) e i ( x (cid:48)− − x − ) iλ ( h ) π ( x (cid:48)− − x − ) + · · · . (2.37)We proceed by stating the relevant components of the stress tensor. Since (2.2) is spher-ically symmetric and does not depend on time, the only off-diagonal component of the stresstensor that could be nonzero is T . However, in Appendix B we show explicitly that T vanishes for our setup. Therefore, we only need the diagonal components of the stress tensor,which are given by T = i R (cid:16) ψ † + ∂ t ψ + + ψ †− ∂ t ψ − − ∂ t ψ † + ψ + − ∂ t ψ †− ψ − (cid:17) η † η ,T = − i R (cid:16) ψ † + ∂ x ψ + − ψ †− ∂ x ψ − − ∂ x ψ † + ψ + + ∂ x ψ †− ψ − (cid:17) η † η ,T = − ie − σ R (cid:48) R ψ † σ z ψη † η ,T = − i sin( θ )2 e − σ R R (cid:48) sin( θ ) e − σ ψ † σ z ψη † η . (2.38)In order to compute the quantum contribution to the components of the stress tensor due tothe non-local coupling, we apply the point-splitting formula (cid:104) T µν (cid:105) = lim x (cid:48) → x iη ab (cid:16) e a ( µ γ b ∇ (cid:48) ν ) − ∇ ( µ e aν ) γ b (cid:17) (cid:104) ¯Ψ( x )Ψ( x (cid:48) ) (cid:105) . (2.39)By using the propagators and after subtracting the vacuum contribution, we end up with thefollowing finite result (cid:104) T hµν (cid:105) = − π qλ ( h ) R diag (1 , , , , (2.40)where the factor q comes from the fact that in the lowest Landau level the degeneracy of thetwo-dimensional fields is q . The range for the compact radial coordinate (∆ x = π ) is presentin the prefactor in the above expression. Picking a different gauge would result in a rescalingof the stress tensor. One can easily check that the stress tensor is conserved and traceless dueto conformal symmetry. Details of the stress tensor calculation can be found in Appendix B.– 10 – Wormhole geometry
We start this section by describing the two different regimes of the wormhole geometry, afterwhich we analytically solve the (linearized) Einstein equations in both regimes. We continueby showing that the solutions in the two regimes can be consistently patched together througha coordinate transformation in the overlapping region of validity. We end the section bysolving the full, nonlinear Einstein equations numerically.
The next task is to solve the semi-classical Einstein equations to find a magnetically chargedgeometry sourced by (2.40) G µν + Λ g µν = 8 πG N (cid:104) T µν (cid:105) . (3.1)As we approach the AdS boundaries located at r → ±∞ , the electromagnetic contributionof the stress tensor dominates over the Casimir energy. Then, far away from the wormholethroat the solution should look like Reissner-Nordstr¨om AdS ds = − f ( r ) dτ + dr f ( r ) + r d Ω , (3.2)with emblackening factor f ( r ) = 1 − G N Mr + r e r + r (cid:96) , and r e = πq G N g . (3.3)Here M denotes the mass of the black hole, q is an integer and r e denotes the magnetic chargeof the black hole. Close to extremality, the geometry developes an infinitely long throat. Thevalue of the extremal radius has the form ∂ r f ( r ) (cid:12)(cid:12)(cid:12) r =¯ r ! = 0 ⇒ ¯ r = (cid:96) (cid:32) − (cid:114) r e (cid:96) (cid:33) . (3.4)Inverting this relation gives the charge of the black hole in terms of the extremal horizonradius and the AdS length r e = ¯ r (cid:18) r (cid:96) (cid:19) . (3.5)In the range of masses that we are interested in, the quartic polynomial f ( r ) = 0 admitscomplex conjugate roots . We choose to parametrize them by r , = ˆ r (1 ± i(cid:15) ) with (cid:15) > r >
0. We can analytically solve for the other two roots, r and r , and the parameterˆ r by matching the quadratic, cubic, quartic and constant contributions to r f ( r ). Thisparametrization is symmetric with respect to (cid:15) (cid:55)→ − (cid:15) . Therefore, the expressions for ( r , r , ˆ r ) As a function of r e , the disciminant interpolates between ∆( r e = 0) = − G N M (cid:96) ( (cid:96) + 27 G N M ) toinfinity. In particular, ∆ ≈ (cid:96) r e when r e (cid:29) (cid:96) and ∆ ≈ − G N (cid:96) M when r e (cid:28) (cid:96) . In both cases, there isat least one pair of complex conjugate roots. – 11 –ill involve only even powers of (cid:15) . In the near extremal limit ( (cid:15) (cid:28) f to order O ( (cid:15) ) by f ( r ) = 1 (cid:96) (cid:32)(cid:18) r − ˆ rr (cid:19) + (cid:18) ˆ r(cid:15)r (cid:19) (cid:33) ( r − r )( r − r ) , (3.6)with ( r − r )( r − r ) = (cid:96) + r + 2 r ¯ r + 3¯ r − (cid:96) ( r + 4¯ r ) + 2¯ r ( r + 6¯ r ) (cid:96) + 6¯ r ¯ r(cid:15) + O ( (cid:15) ) , (3.7)and ˆ r = ¯ r + (cid:15) ¯ r C (¯ r ) (cid:18) r (cid:96) (cid:19) + O ( (cid:15) ) . (3.8)Here C ( r ) is defined by C ( r ) = 6 (cid:16) r(cid:96) (cid:17) + 1 . (3.9)In the region where r − ¯ r (cid:28) ¯ r and (cid:15) is small, we can approximate the metric (3.2) as ds = −C (¯ r ) (cid:32)(cid:18) r − ¯ r ¯ r (cid:19) + (cid:15) (cid:33) dτ + dr C (¯ r ) (cid:16)(cid:0) r − ¯ r ¯ r (cid:1) + (cid:15) (cid:17) + ¯ r d Ω . (3.10)By making the following identifications, ρ = r − ¯ r(cid:15) ¯ r and t = C (¯ r ) τ (cid:15) ¯ r , (3.11)the metric can be brought to global AdS × S form ds = ¯ r C (¯ r ) (cid:18) − ( ρ + 1) dt + dρ ρ + 1 (cid:19) + ¯ r d Ω . (3.12)Following [31], we expect that in the wormhole region the solution is a slight perturbationof the near extremal RN black hole. We make the following gauge choice for our ansatzgeometry in the throat ds = ¯ r C (¯ r ) (cid:18) − (1 + ρ + γ ) dt + dρ ρ + γ (cid:19) + ¯ r (1 + ψ ) d Ω , (3.13)where the functions ψ ( ρ ) and γ ( ρ ) are small fluctuations and ¯ r is given by (3.4). In thesecoordinates, the stress tensor contribution has the approximate form (cid:104) T hµν (cid:105) ≈ − π qλ ( h )¯ r diag (cid:18) , ρ ) , , (cid:19) . (3.14)The linearized Esintein’s equations in this geometry are given by tt : ζ ρ + ψ ( ρ ) − ρψ (cid:48) ( ρ ) − (1 + ρ ) ψ (cid:48)(cid:48) ( ρ ) = 0 , (3.15)– 12 – ρ : ζ ρ − ψ ( ρ ) + ρψ (cid:48) ( ρ ) = 0 , (3.16) θθ : 4 C (¯ r ) (cid:18) r (cid:96) (cid:19) ψ ( ρ ) + γ (cid:48)(cid:48) ( ρ ) + 2 ρψ (cid:48) ( ρ ) + (1 + ρ ) ψ (cid:48)(cid:48) ( ρ ) = 0 , (3.17) φφ : sin ( θ ) (cid:18) C (¯ r ) (cid:18) r (cid:96) (cid:19) ψ ( ρ ) + γ (cid:48)(cid:48) ( ρ ) + 2 ρψ (cid:48) ( ρ ) + (1 + ρ ) ψ (cid:48)(cid:48) ( ρ ) (cid:19) = 0 , (3.18)where ζ is a constant given by ζ = G N qλ ( h ) π ¯ r . Note that the first two equations do not dependon γ . Therefore, we can find an expression for ψ ( ρ ) by solving the first order equation (3.16).This results in ψ ( ρ ) = ζ (1 + ρ arctan( ρ )) + cρ , (3.19)with c an integration constant. A simple check shows that (3.19) also solves the tt componentof the Einstein equations (3.15). By using the solution for ψ ( ρ ), we can now use the angularcomponents of the Einstein equations to solve for γ ( ρ ). It turns out that γ ( ρ ) = − ζ (cid:16) ¯ r (cid:96) (cid:17) C (¯ r ) (cid:0) ρ + ρ (3 + ρ ) arctan( ρ ) − log(1 + ρ ) (cid:1) + c + ρc , (3.20)solves (3.17) and (3.18). Integration constants can be set to zero by requiring that thegeometry is invariant under ρ (cid:55)→ − ρ and by a redefinition of ρ and t .In the next subsection, we show that there is an overlapping region between the twosolutions deep in the RN-AdS throat and construct the full wormhole geometry. Intuitively, once the non-local coupling h is turned on, the wormhole is formed and the throatacquires a certain finite length L which we will determine below. Outside this range, thelinearized solution found in the previous section will not be valid anymore. In fact, bothperturbations ψ and γ increase with the value of ρ , as we approach the wormhole mouth.More precisely, we expect the slightly deformed solution to be valid up to values of ρ forwhich the term ζρ is no longer subleading (since this is the leading order behaviour of γ ).In the following, we consider ρ to be large, but ζρ small and fixed, and ζ small. We take thenear-horizon limit of (3.6) f ( r ) = C (¯ r ) (cid:15) + C (¯ r ) (cid:18) r − ¯ r ¯ r (cid:19) − C (¯ r ) (cid:18) r − ¯ r ¯ r (cid:19) (cid:15) − (cid:18) r (cid:96) (cid:19) (cid:18) r − ¯ r ¯ r (cid:19) + · · · , (3.21)where we have expanded up to third order in (cid:15) and r − ¯ r ¯ r combined. As a first approximationlet us set ρ = L ¯ r r − ¯ r ¯ r , t = C (¯ r ) τL . (3.22) Note that we can write q in terms of ¯ r . This results in ζ = gλ ( h ) π ¯ r (cid:114) G N (cid:16) ¯ r (cid:96) (cid:17) π . From this we see thatwe can let ζ be small at finite h and independent of the ratio between ¯ r and (cid:96) . – 13 –n the limit ρ (cid:29) , L ¯ r (cid:29) , and r − ¯ r ¯ r (cid:28) , (3.23)equation (3.22) matches the order O (2) of the unperturbed ansatz geometry. Here L is anintegration constant that denotes the rescaling between the t and τ coordinates. Furthermore,by considering the relation between ρ and r , one can see that L is a measure up to which wecan trust the ansatz; so that ρ has a cutoff at ρ ∼ L ¯ r . By comparing to the matching of thenear-extremal Reissner-Nordstr¨om black hole given in (3.11), we see that L is connected to (cid:15) through L = ¯ r(cid:15) . (3.24)By matching the angular coordinates we see that r = ¯ r (1 + ψ ( ρ )) → r − ¯ r ¯ r = ψ ( ρ )2 + O (cid:0) ψ (cid:1) = πζ ρ + O (cid:0) ζ (cid:1) , (3.25)where we have expanded (cid:112) ψ ( ρ ), and in the third equality we used the expansion of ψ ( ρ )at large ρ . Using (3.22) and (3.25) we can find the value for L by examining ρdt = C (¯ r ) r − ¯ r ¯ r dτ = ⇒ L = dτdt C (¯ r ) = ρ ¯ r ¯ rr − ¯ r = 4¯ rπζ . (3.26)One can easily see that with this value for L , (3.22) and (3.25) are consistent with one another.This also gives a relation between the non-local coupling constant h and (cid:15) . With (3.24) and(3.26) we see that (cid:15) = π ζ
16 = G N g λ ( h ) π ¯ r (cid:18) r (cid:96) (cid:19) = G N q λ ( h ) π ¯ r . (3.27)The matching of the time component of the geometry is discussed in Appendix C. In thisappendix we show that the equations above give a consistent matching between the Reissner-Nordstr¨om geometry and the deformed AdS × S . A final comment we make concerning thematching is that the deformation γ gives a correction to the range of the radial coordinate,which in turn leads to a correction to the stress tensor. However, this correction is of order ζ , and therefore will not influence the matching . The full wormhole geometry with the tworegimes is schematically shown in Fig. 2.An important fact to notice about the wormhole solution we find is that there are threeindependent parameters: the charge, the non-local coupling and the AdS length by whichthe solution is determined. As soon as these three parameters are fixed, there is a unique,static and spherically symmetric wormhole geometry that solves Einstein’s equations. At radii Recall that (cid:15) encodes how “far” from extremality the near-extremal black hole metric is. The correction can be calculated by considering ∆ x = (cid:82) ∞−∞ dyg yy , with y the holographic coordinate,resulting in ∆ x = π (1 + ζf (¯ r )), for some function f . Since we consider ζ to be small, the matching isconsistent. If we had taken this correction into account the stress tensor would have been given by (cid:104) T hµν (cid:105) = x qλ ( h )2 π R diag (1 , , , – 14 –elow the cutoff the geometry is that of deformed AdS × S . As ρ increases, the geometrysmoothly interpolates to a near-extremal Reissner-Nordstr¨om black hole in AdS . This blackhole is characterized by its charge r e , while its mass is given by M W H = M ext + ∆ M, (3.28)with M ext = ¯ rG N + 2¯ r G N (cid:96) , and ∆ M = − ¯ r(cid:15) G N C (¯ r ) = − g λ ( h )2 π ¯ r (cid:18) r (cid:96) (cid:19) C (¯ r ) , (3.29)where M ext is the mass of an extremal black hole.Since it is not very pleasant to have a factor of g in this formula, we use the definitionsto rewrite this as ∆ M = − G N q λ ( h )2 π ¯ r C (¯ r ) ∼ − qλ ( h ) C (¯ r )¯ r ζ . (3.30)Therefore, the black hole is indeed near-extremal, with mass just below the extremal mass.Coming from infinity, as an observer approaches the wormhole mouth, the observer wouldexperience getting closer and closer to a naked singularity. All of a sudden, the wormholethroat opens up and she traverses through the wormhole reaching the other side safely.In the limit (cid:96) (cid:29) r e , where the AdS radius is larger than the radii of the throats. Thechange in the mass due to the non-local coupling has the form∆ M = − G N q λ ( h )2 π r e , (3.31)which has the same scaling as the binding energy, relative to the the energy of two disconnectedextremal black holes, coming from the wormhole throat in the asymptotically flat case [31]. One might be concerned that the solution presented in the previous subsection only existsin the linearized analysis. We will proceed to find a similar solution to the full Einstein’sequations. The geometry ansatz we will consider is the following ds = ¯ r C (¯ r ) (cid:18) − f ( ρ ) dt + dρ f ( ρ ) (cid:19) + R ( ρ ) d Ω , (3.32) ρ ∈ [0 , ±∞ ), t ∈ ( −∞ , ∞ ) and we have assumed the extremal value for the radius in theoverall factor. The non-zero components of the Einstein equations can be written as tt : 3¯ r f ( ρ ) C (¯ r ) (cid:96) − πG N q ¯ r f ( ρ ) g C (¯ r ) R ( ρ ) + 4 G N qλ ( h ) π R ( ρ ) + ¯ r f ( ρ ) C (¯ r ) R ( ρ ) − f ( ρ ) f (cid:48) ( ρ ) R (cid:48) ( ρ ) R ( ρ ) − f ( ρ ) R (cid:48) ( ρ ) R ( ρ ) − f ( ρ ) R (cid:48)(cid:48) ( ρ ) R ( ρ ) = 0 , (3.33) ρρ : − r C (¯ r ) (cid:96) f ( ρ ) + πG N q ¯ r g C (¯ r ) f ( ρ ) R ( ρ ) + 4 G N qλ ( h ) π f ( ρ ) R ( ρ )– 15 –dS × S ∂ AdS ∂ AdS θρ cutoff ρ cutoff Figure 2 : Wormhole geometry: In the throat region the metric has the AdS ×S form(3.12) up to the cutoff located at ρ ∼ L/ ¯ r . Around this point, where the limits (3.23)are satisfied, the geometry smoothly interpolates to near-extremal Reissner-Nordstr¨om blackholes in AdS . − ¯ r C (¯ r ) f ( ρ ) R ( ρ ) + f (cid:48) ( ρ ) R (cid:48) ( ρ ) f ( ρ ) R ( ρ ) + R (cid:48) ( ρ ) R ( ρ ) = 0 , (3.34) θθ : − πG N q g R ( ρ ) − R ( ρ ) (cid:96) + C (¯ r ) R ( ρ ) f (cid:48) ( ρ ) R (cid:48) ( ρ )¯ r + C (¯ r ) R ( ρ ) f (cid:48)(cid:48) ( ρ )2¯ r + C (¯ r ) f ( ρ ) R ( ρ ) R (cid:48)(cid:48) ( ρ )¯ r = 0 , (3.35) φφ : sin ( θ ) (cid:18) − πG N q g R ( ρ ) − R ( ρ ) (cid:96) + C (¯ r ) R ( ρ ) f (cid:48) ( ρ ) R (cid:48) ( ρ )¯ r (cid:19) + sin ( θ ) (cid:18) C (¯ r ) R ( ρ ) f (cid:48)(cid:48) ( ρ )2¯ r + C (¯ r ) f ( ρ ) R ( ρ ) R (cid:48)(cid:48) ( ρ )¯ r (cid:19) = 0 . (3.36)These differential equations depend on three independent physical parameters of the form G N q g (cid:96) , G N qλ ( h ) (cid:96) , and (cid:96) . (3.37)Since both functions f and R appear in the differential equations with two derivatives, therewill be four integration constants. By requiring the solution to be symmetric around ρ = 0we fix two of those. Requiring this Z symmetry is equivalent to setting f (cid:48) (0) = R (cid:48) (0) = 0.Furthermore we have the freedom to rescale the time coordinate. This allows us to pick f (0) = 1. Now the constraint equation (3.34) fixes R (0) in terms of f (0). By these choicesall integration constants are then fixed. Also note that due to spherical symmetry wheneverthe θθ equation is solved, the φφ equation is automatically satisfied. With the integrationconstants as mentioned, we can now solve the tt and θθ equation numerically. The results ofsolving the non-linear Einstein equations are shown in Figures 3 and 4. In order to compare– 16 –ith the linearized results, we pick the integration constants so that the non-linear and linearsolutions agree at ρ = 0. We should note however that the non-linear solution makes sense forother parameter values and integration constants as well. We expect the linear and non-linearresults to agree up to | ρ |∼ ρ cutoff = L ¯ r . As a final comment note that, as can be seen fromFigure 3, for large ρ the numerical solution behaves as R ( ρ ) ∼ (cid:0) L ¯ r (cid:1) − | ρ | , which is preciselywhat we expect in light of equation (3.22) and by the fact that away from the wormhole weexpect the S radius to be equal to r . Figure 3 : Solutions of R ( ρ ) with parameters G N q g (cid:96) = 0 . G N qλ ( h ) (cid:96) =0 . (cid:96) = 100. The initial condition is R (0) = 17. For these parame-ters we expect agreement up to ρ cutoff = 7 .
2. We see that for larger ρ thelinear solution starts to deviate. Figure 4 : Solutions of f ( ρ ) with parameters G N q g (cid:96) = 0 . G N qλ ( h ) (cid:96) =0 . (cid:96) = 100. The initial condition is R (0) = 17. For these parame-ters we expect agreement up to ρ cutoff = 7 .
2. We see that for larger ρ thelinear solution starts to deviate, and even becomes negative. Of course, inthis region the RN AdS black hole dominates.– 17 – Thermodynamics
This section contains a calculation of the on-shell Hamiltonian of the wormhole solution. Wepropose a Hamiltonian and show that the wormhole solution is the ground state for a region ofparameter space. Furthermore we give a qualitative discussion of the thermodynamic stabilityof the wormhole solution in the (grand) canonical ensemble.
We expect the wormhole geometry presented in the previous section to be dual to the asymp-totic field theories in some particular entangled state. In particular, it should be dual tothe ground state of a certain local Hamiltonian whose ground state is approximately thethermofield double state with chemical potential [30, 36].From the gravity point of view, given the set of boundary conditions, Einstein’s equationsfill in the bulk geometry smoothly. We will consider three solutions with the same boundaryconditions at zero temperature: the wormhole, two disconnected black holes and empty AdS.Depending on the values of λ ( h ) and µ , there is a dominant saddle. For concreteness we willfocus on the symmetric case where the total magnetic charge is Q := Q R = − Q L and themass is M := M R = M L . Of course, less symmetric cases can also be considered but webelieve they will not dramatically modify the presented results.In a general covariant theory, the on-shell Hamiltonian can be computed as a boundaryintegral as follows H [ ζ ] = (cid:90) ∂ Σ d x √ σu α ζ β T αβ , T αβ := 2 √− γ δSδγ αβ , (4.1)where T αβ is the Brown-York stress tensor , u α is the unit normal to a constant time hyper-surface, ζ β is the flow vector and √ σ is the volume element of the boundary at fixed time.The energy of a gravitational solution is associated to time-translation symmetry, i.e. , to theKilling vector ζ = ∂ τ .The wormhole solution presented in the last section is a solution to the action withinteracting term S int = ih (cid:90) d x √− γ (cid:0) ¯Ψ R − Ψ L + + ¯Ψ L + Ψ R − (cid:1) . (4.2)The interacting part of the boundary stress tensor then has the form T αβ := 2 √− γ δS int δγ αβ = ihγ αβ (cid:0) ¯Ψ R − Ψ L + + ¯Ψ L + Ψ R − (cid:1) . (4.3)The metric close to the AdS boundary at r → ∞ and the time-like unit vector are of theform ds = − f ( r ) dτ + dr f ( r ) + r d Ω and u = (cid:112) f dτ . (4.4) In order to avoid IR divergences in AdS, we need to include counterterms in the purely gravitational partof the action (2.1). In d=3, they result in a modified stress tensor T αβ = K αβ − Kγ αβ − (cid:96) γ αβ − (cid:96)G αβ , where G αβ is the Einstein tensor computed on the boundary induced metric γ αβ [43]. – 18 –he Z symmetry along the radial direction allows us to define a notion of gravitationalenergy by applying the formula (4.1) at the asymptotic AdS boundaries H int := H [ ζ τ ] = − ihr (cid:112) f (cid:90) d Ω (cid:0) ¯Ψ R − Ψ L + + ¯Ψ L + Ψ R − (cid:1) . (4.5)At this point we need to evaluate the bulk spinors close to the asymptotic boundary. We canachieve this by evaluating the scaling factor close to the boundary ¯Ψ R − Ψ L + = e − σ R (cid:104) ( ψ R ⊗ η ) † P − γ P + (cid:0) ψ L ⊗ η (cid:1)(cid:105) (4.6)= C (¯ r ) Lr √ f (cid:104) ( ψ R ⊗ η ) † P − γ P + (cid:0) ψ L ⊗ η (cid:1)(cid:105) , and a similar expression for ¯Ψ L + Ψ R − . We then find H int = − ih C (¯ r ) (cid:15) ¯ r (cid:90) d Ω (cid:104) ( ψ R ⊗ η ) † P − γ P + (cid:0) ψ L ⊗ η (cid:1) + ( ψ L ⊗ η ) † P + γ P − (cid:0) ψ R ⊗ η (cid:1)(cid:105) . (4.7)We can compute the semi-classical interacting Hamiltonian in the state defined in (2.36) bycomputing the following expectation value (cid:104) H int (cid:105) = − ih C (¯ r ) (cid:15) ¯ r (cid:90) d Ω (cid:104) (cid:104) ( ψ R ⊗ η ) † P − γ P + (cid:0) ψ L ⊗ η (cid:1) (cid:105) + (cid:104) ( ψ L ⊗ η ) † P + γ P − (cid:0) ψ R ⊗ η (cid:1) (cid:105) (cid:105) . (4.8)We can evaluate the boundary integral by taking first the angular spinors on-shell. Thecorrelators involved in (4.8) can be computed perturbatively in the limit where λ ( h ) ∼ h .They are explicitly given in Appendix D. Finally, we obtain the result (cid:104) H int (cid:105) = C (¯ r ) (cid:15) ¯ r qhπ (cid:18) − hπ (cid:19) = 4∆ M + O (cid:0) h (cid:1) , (4.9)with ∆ M given by (3.29). In order to get the total wormhole energy, we need to add theenergy asssociated to the non-interacting parts, i.e. , of two near-extremal RN black holes (cid:104) H (cid:105) W H = 2 M W H + 4∆ M − µQ = 2 M ext + 6∆ M − µQ , (4.10)where M ext is the black hole extremal mass and Q the extremal charge. In this equation,we have taken into account the change in energy due to the chemical potential µ for theasymptotic charges. This is similar to the electric case [44].Now that we understand the energy of the wormhole geometry, we would like to inves-tigate whether it is the ground state of some Hamiltonian. We propose the following localboundary Hamiltonian In the RN background (4.4), far away from the horizon the metric is conformaly flat. The relation betweenthe coordinates is t = τL C (¯ r ), and x = (cid:82) dr L f ( r ) C (¯ r ), and the conformal factor equals e σ = (cid:16) L C (¯ r ) (cid:17) f . Note that in the second equal sign we only take into account the terms up to order h , even though theexpression in the middle contains a third order term. This is done to compare to the results of the previoussection, which included terms up to second order. – 19 – = H L + H R − ih(cid:96) (cid:90) d Ω (cid:0) ¯Ψ R − Ψ L + + ¯Ψ L + Ψ R − (cid:1) + µ ( Q L − Q R ) , (4.11)where H L and H R are the Hamiltonians associated to the boundary dual of the two identicaloriginal systems, and again we take into account the change in energy due to the chemicalpotential µ for the asymptotic charges. It is important to notice that equation (4.11) is anexpression written purely in terms of boundary data. In particular, we have defined theboundary spinors, denoted as Ψ , by removing the scaling factor defined in (2.19), so thatΨ ± = R − Ψ ± . (4.12)Note that the interacting term is inspired by (4.2), which in terms of the boundary data canbe written as S int = ih(cid:96) (cid:90) dτ d Ω (cid:0) ¯Ψ R − Ψ L + + ¯Ψ L + Ψ R − (cid:1) . (4.13)The Hamiltonian determines the time-evolution with respect to the asymptotic time definedin (4.4). Note that the total charge of the field theories is conserved as a consequence of aglobal symmetry.Next, we consider the expectation value of the Hamiltonian for the different phases. Firstof all note that the expectation value of (4.11) is precisely equal to (4.10) for the wormholesolution, since that is the primary reason for the definition of (4.11). Secondly, note that theempty AdS geometry has a vanishing Hamiltonian. Finally, we note that for the disconnectedblack holes, the interaction term of the Hamiltonian does not contribute to the energy. Thiscan be seen from the fact that we can Wick rotate the RN black hole solution, after whichthe geometry is conformal to the disk. However, the conformal factor vanishes if the blackhole is extremal. Since the correlators on the disk must be finite, the total contribution ofthe interacting part of the Hamiltonian must indeed be equal to zero.The difference between the wormhole and the extremal black holes phases is given by (cid:104) H (cid:105) − (cid:104) H (cid:105) WH > . (4.14)It is easy to see that (cid:104) H (cid:105) has a minimum at the point¯ r = (cid:96) √ π (cid:115) µ m p − π , for µ m p > π , (4.15)for which (cid:104) H (cid:105) < (cid:104) H (cid:105) Vacuum = 0 (see Figure 5). Then, the wormhole phase (where itexists) dominates the ground state for values of the chemical potential µ > µ c . The completephase diagram is shown in Figure 6. We see that the point ( h = 0 , µ = µ c ) is actually a triplepoint where the three different phases meet. Intuitively, empty AdS is the dominant saddlefor very small values of h , for which the wormhole has not been formed yet, and µ so thatthe charge contribution is negligible. For h < µ > µ c , the black holes phase is thedominant saddle. Alternatively, for positive values of the coupling and µ > µ c , the wormholephase will be the ground state of (4.11). – 20 – igure 5 : Expectation value of the Hamiltonian (4.11) in the two-disconnected black holes phase for different values of ¯ r . The minimumis located at r min , which is given in (4.15). We briefly discuss possible instabilities of the solution. When considering scalar fields in aReissner-N¨ordstrom AdS background there are instabilities that lead to hairy black holes.These instabilities can be understood, for near-extremal black holes, as originating from thedifference between the Breitenl¨ohner Freedman bounds for AdS and AdS ; fields that areallowed tachyons in the asymptotic AdS spacetime lead to instabilities in the AdS near-horizon region. Even though intuitively similar arguments would lead to fermionic instabili-ties, no evidence for the existence of fermionic hairy black holes has been found [45]. Sincethe argument crucially depends on the fact that there is an asymptotic AdS geometry, weexpect the same result to hold for the wormhole phase.Besides investigating whether the wormhole solution is the ground state of the Hamil-tonian (4.11), one could also wonder whether it is the thermodynamically favored phase inone of the standard thermodynamic ensembles. We must couple the two CFT’s in order fora wormhole solution to exist; if signals can cross from one boundary to the other in the bulk,it must be possible to transfer information between the CFT’s [24].Before coupling the CFT’s, each CFT has a global U (1) symmetry with an associatedcharge conservation. After coupling the theories, charge can flow from one to the other,so only a single U (1) survives. In our conventions, the conserved charge us Q L + Q R . Inour conventions, the wormhole solution has Q L = − Q R . One can picture magnetic field linesthreading the wormhole, so this convention is natural. Therefore, the conserved charge for thewormhole is Q L + Q R = 0. In the standard construction of thermodynamic ensembles, one canonly turn on a chemical potential for this conserved charge. The term µ ( Q L − Q R ) appearingin our Hamiltonian looks like a chemical potential, but it is not really, because Q L − Q R isnot conserved: it does not commute with the interaction term in the Hamiltonian.– 21 – µ AdS BHWH µ c h c Figure 6 : Diagram that shows the ground state of the Hamiltonian (4.11) for different valuesof h and µ . Empty AdS is the dominant contribution at the origin up to the critical values h c = ¯ r G N q (cid:114) π C (cid:16) r (cid:96) (cid:17) and µ c = m p √ π where the wormhole phase becomes the groundstate. The point ( h = 0, µ = µ c ) is a triple point where the three phases meet. For negativevalues of h , there is a competition between the empty AdS and the black holes phases. Notethat depending on the mass of the monopoles in the theory there could be a region in thediagram where the ground state is AdS with monopoles.One can ask whether the wormhole dominates one of the standard thermodynamic en-sembles at zero temperature; for example, consider the canonical ensemble. The differentphases that should be compared are the following: empty AdS, two black holes, the worm-hole solution.The free energy is F = E − T S , (4.16)in the canonical ensemble. At zero temperature, the wormhole phase will have free energyequal to F = 2 M W H , while empty AdS has a vanishing free energy. Therefore, empty AdSdominates the ensemble.Finally, one could consider other ensembles in the hope of finding an ensemble in whichthe wormhole solution dominates. One candidate is the grand canonical ensemble, withpotential given by Φ = E − T S − µQ. (4.17)– 22 –owever, the wormhole phase has a conserved charge Q = Q L + Q R = 0. Therefore, addinga term proportional to Q will not change the potential of the wormhole phase. Because ofthis it can never dominate the ensemble. The only thing that we have achieved by changingensembles is that there can be even more phases with a lower potential than the wormhole.Note that it is not very clear how to interpret magnetic charges (and how to fix them)in the different ensembles. The electromagnetic duality suggests they should be treatedin the same way as electric charges, but in the standard AdS/CFT context black holes withdifferent electric charges are different states in the same theory, while black holes with differentmagnetic charges live in different theories. Presumably one can make a choice when imposingboundary conditions for the gauge field analogous to the standard vs alternate quantizationfor other light fields, and this choice determines whether electric or magnetic states live inthe same theory. It would be interesting to understand this better, since our setup dependson the magnetic charges, but this subtlety is mostly orthogonal to our work here.To summarize: the wormhole appears to be a stable solution that corresponds to theground state of our Hamiltonian. However, it does not seem to arise as the dominant phaseof one of the standard thermodynamic ensembles with chemical potential. We have found an eternal traversable wormhole in a four-dimensional AdS background. Thisgeometry is a solution to the Einstein-Hilbert gravity action with negative cosmological con-stant, a U (1) gauge field and massless fermions charged under the gauge field. To open upthe wormhole we need negative energy, which we acquire by coupling the CFT’s living on thetwo boundaries of the spacetime.By calculating the backreaction of the negative energy on the geometry, we find a statictraversable wormhole geometry with no horizons or singularities. This wormhole is dual tothe ground state of a simple Hamiltonian for two coupled holographic CFT’s. The parametersin the Hamiltonian are the chemical potential, the coupling strength, and the central charge.The wormhole dominates in some region of parameter space, while disconnected geometriesdominate other regions.Working in the semi-classical approximation, the authors of [24] proved that there areno traversable wormholes that preserve Poincar´e invariance along the boundary field theorydirections in more than two spacetime dimensions. The geometry found in this work evadesthis result because our solution is not Poincar´e invariant.There are a number of interesting future directions. Traversable Wormholes in the lab; energy gap.
One possible application of our resultsis to build traversable wormholes in the lab by implementing our interacting Hamiltonian andallowing the system to cool to the ground state. For this process to be efficient, it is importantthe energy gap between the ground state and the first excited state is not too small.It would be interesting to carefully calculate the gap in this system. A rough estimate canbe obtained by calculating the maximum redshift. Black holes have infinite redshift near the– 23 –orizon and therefore support excitations with arbitrarily small energy in the semi-classicallimit.Looking back at our ‘matching’ section, we see that the black hole geometry is valid downto r − ¯ r ∼ (cid:15)L ∼ ¯ r . (5.1)At this location, the redshift is f ( r ) ∼ C (¯ r ). Inside this matching radius, the geometry isAdS × S . The relative redshift between the middle of the wormhole and the matchingsurface is f match f ∼ ρ ∼ (cid:15) . (5.2)Combining these results, and restoring units using the AdS radius, our guess isGap ∼ C (¯ r ) (cid:15) (cid:96) . (5.3)This result looks concerningly small due to the (cid:15) ; however, C ( r ) = 1 + 6¯ r /(cid:96) is large forlarge black holes. We leave a fuller discussion and more reliable calculation for the future. RG flow.
Our bulk analysis is made convenient by the Weyl invariance of the masslessfermions, which correspond to boundary operators of particular dimensions. If we thinkof the interaction term as an interaction in a single CFT, this term appears to be exactlymarginal. It would be interesting to understand whether higher order corrections change thescaling dimension of this interaction, or more generally to understand the RG flow of oursystem.
Supersymmetry.
Related to the RG flow, it would add a degree of theoretical control torealize the initial extremal black hole as a BPS state in a supersymmetric theory. Accom-plishing this requires embedding our simple U (1) theory in a theory with more conservedcharges [46–49]. CFT state.
In the related construction of Cottrell et al [36], the state of the dual CFTwas identified with the thermofield double state, while the bulk geometry was not undersemiclassical control. In this paper, we have constructed a controlled traversable wormhole,but have not calculated the quantum state of the CFT in boundary variables. One may expectthat it is a thermofield double type state, but note that the wormhole is a zero temperaturesolution, so it cannot be exactly the TFD. On the other hand, the bulk geometry clearly lookslike a slightly superextremal Reissner Nordstrom black hole away from the wormhole mouth,giving a clear hint regarding the CFT state.
Multi-mouth wormholes.
In the present work, we focused on asymptotically AdS two-mouth traversable wormhole geometries. It might be interesting to extend our results andexplore in the future the possibility of fourth dimensional multi-mouth wormholes similar tothose studied in [28, 29]. In particular, the results of this paper might be used to understandexplicitly the role played by multiparty entanglement in the wormhole’s traversability.– 24 – nformation transfer. Moreover, it would be interesting to investigate the amount ofinformation that can be sent through this type of wormhole, in a similar fashion as in [14,20, 25].
Replica wormholes.
Finally, it has been found that two dimensional eternal traversablewormhole geometries contribute to the fine-grained entropy in the context of islands in deSitter spacetime [50] (see also [51, 52]). It would be interesting to understand whether moregeneral set-ups in higher dimensions can be described with similar methods to those employedin this paper.
Acknowledgements
We would like to thank Souvik Banerjee, Jan de Boer, Jackson R. Fliss, Victor Godet, DanielHarlow, Jeremy van der Heijden, Diego Hofman, Daniel Jafferis, Bahman Najian, and Markvan Raamsdonk for discussions. BF, RE, and DN are supported by the ERC ConsolidatorGrant QUANTIVIOL. SB is supported by the European Research Council under the Euro-pean Unions Seventh Framework Programme (FP7/2007-2013), ERC Grant agreement ADG834878. This work is part of the ∆ITP consortium, a program of the NWO that is fundedby the Dutch Ministry of Education, Culture and Science (OCW).
Appendices
A Propagators
In this appendix we present the results for the propagators. Below, we show the derivationof (2.37) (cid:104) ψ † + ( x − ) ψ + ( x (cid:48)− ) (cid:105) = (cid:42) (cid:88) k,j ∈ Z π α † k α j e iω j x (cid:48)− − iω k x − (cid:43) = (cid:42) (cid:88) k,j ∈ Z ≥ π α † k α j e iω j x (cid:48)− − iω k x − (cid:43) = (cid:42) (cid:88) k,j ∈ Z ≥ π ( δ k,j − α j α † k ) e iω j x (cid:48)− − iω k x − (cid:43) = (cid:88) j ∈ Z ≥ π e iω j ( x (cid:48)− − x − ) = (cid:88) j ∈ Z ≥ π e i ( j +12 ) ( x (cid:48)− − x − ) (cid:18) − j iλ ( h ) π ( x (cid:48)− − x − ) (cid:19) + O (cid:0) ( x (cid:48)− − x − ) (cid:1) = 1 π e i ( x (cid:48)− − x − ) − e i ( x (cid:48)− − x − ) + e i ( x (cid:48)− − x − ) e i ( x (cid:48)− − x − ) iλ ( h ) π ( x (cid:48)− − x − ) + O (cid:0) ( x (cid:48)− − x − ) (cid:1) . (A.1)– 25 –he rest of the propagators can be derived in a similar fashion. We present the results (cid:104) ψ †− ( x + ) ψ − ( x (cid:48) + ) (cid:105) = 1 π e i ( x (cid:48) + − x + ) − e i ( x (cid:48) + − x + ) + e i ( x (cid:48) + − x + ) e i ( x (cid:48) + − x + ) iλ ( h ) π ( x (cid:48) + − x + ) + O (cid:0) ( x (cid:48)− − x − ) (cid:1) , (A.2) (cid:104) ψ † + ( x − ) ψ − ( x (cid:48) + ) (cid:105) = − π e i ( x (cid:48) + − x − ) e i ( x (cid:48) + − x − ) − e i ( x (cid:48) + − x − ) − e i ( x (cid:48) + − x − ) iλ ( h ) π ( x (cid:48) + − x − ) + O (cid:0) ( x (cid:48)− − x − ) (cid:1) , (A.3)and (cid:104) ψ − ( x + ) ψ + ( x (cid:48)− ) (cid:105) = (cid:104) ψ − ( x + ) ψ − ( x (cid:48) + ) (cid:105) = (cid:104) ψ + ( x − ) ψ + ( x (cid:48)− ) (cid:105) = 0 . (A.4) B Stress tensor
In this appendix we show the calculation of the stress tensor. First of all note that we canaverage over the angular directions by taking the spherical components on-shell in equation(2.38). Since the spherical components are normalized such that (cid:82) d Ω ¯ η m η n = δ m,n , aver-aging over the angular directions results in a factor of π . Using this and point-splitting, thefirst line of (2.38) becomes (cid:104) T (cid:105) = lim t (cid:48) → tx (cid:48) → x π (cid:88) m,n (cid:90) d Ω i R (cid:0) ∂ (cid:48) t − ∂ t (cid:1) (cid:16) (cid:104) ψ m + † ( x − ) ψ n + ( x (cid:48)− ) (cid:105) + (cid:104) ψ m †− ( x + ) ψ n − ( x (cid:48) + ) (cid:105) (cid:17) η m † η n = lim t (cid:48) → tx (cid:48) → x qi πR (cid:0) ∂ (cid:48) t − ∂ t (cid:1) (cid:16) (cid:104) ψ + † ( x − ) ψ + ( x (cid:48)− ) (cid:105) + (cid:104) ψ †− ( x + ) ψ − ( x (cid:48) + ) (cid:105) (cid:17) . (B.1)The renormalized (cid:104) T (cid:105) is found by subtracting the h = 0 contribution from the h (cid:54) = 0expression as follows (cid:104) T (cid:105) = (cid:104) T h (cid:54) =011 (cid:105) − (cid:104) T h =011 (cid:105) = lim t (cid:48) → tx (cid:48) → x qi πR (cid:0) ∂ (cid:48) t − ∂ t (cid:1) (cid:32) e i ( x (cid:48)− − x − ) e i ( x (cid:48)− − x − ) iλ ( h ) π ( x (cid:48)− − x − ) + e i ( x (cid:48) + − x + ) e i ( x (cid:48) + − x + ) iλ ( h ) π ( x (cid:48) + − x + ) (cid:33) = lim t (cid:48) → tx (cid:48) → x qi πR (cid:18) iλ ( h ) π + 3 iλ ( h )2 π (cid:0) ( t (cid:48) − t ) + ( x (cid:48) − x ) (cid:1)(cid:19) = − qλ ( h )2 π R , (B.2)where we have omitted combined factors of ( t (cid:48) − t ) and ( x (cid:48) − x ) to higher orders. Similarly,we find that the rest of the renormalized components of the stress tensor are given by (cid:104) T (cid:105) = − qλ ( h )2 π R , and (cid:104) T (cid:105) = (cid:104) T (cid:105) = 0 . (B.3)It is easy to see that this contribution to the stress tensor is traceless (cid:104) T µµ (cid:105) = g µν (cid:104) T µν (cid:105) = e − σ (cid:18) qλ ( h )2 π R − qλ ( h )2 π R (cid:19) = 0 . (B.4)– 26 –e can also check that the stress tensor is conserved. We will need the following componentsof ∇ µ (cid:104) T ρν (cid:105) ∇ x (cid:104) T xx (cid:105) = ∂ x (cid:104) T xx (cid:105) − Γ ρxx (cid:104) T ρx (cid:105) − Γ ρxx (cid:104) T xρ (cid:105) = qλ ( h ) R (cid:48) π R + qλ ( h ) σ (cid:48) π R , ∇ t (cid:104) T tx (cid:105) = ∂ t (cid:104) T tx (cid:105) − Γ ρtt (cid:104) T ρx (cid:105) − Γ ρtx (cid:104) T xρ (cid:105) = qλ ( h ) σ (cid:48) π R , ∇ θ (cid:104) T θx (cid:105) = ∂ θ (cid:104) T θx (cid:105) − Γ ρθθ (cid:104) T ρx (cid:105) − Γ ρθx (cid:104) T xρ (cid:105) = − qλ ( h ) R (cid:48) e − σ π R , ∇ φ (cid:104) T φx (cid:105) = ∂ φ (cid:104) T φx (cid:105) − Γ ρφφ (cid:104) T ρx (cid:105) − Γ ρφx (cid:104) T xρ (cid:105) = − qλ ( h ) R (cid:48) e − σ sin ( θ )2 π R , so that ∇ µ (cid:104) T µν (cid:105) = g µρ ∇ µ (cid:104) T ρν (cid:105) = g xx ∇ x (cid:104) T xx (cid:105) + g tt ∇ t (cid:104) T tx (cid:105) + g θθ ∇ θ (cid:104) T θx (cid:105) + g φφ ∇ φ (cid:104) T φx (cid:105) = e − σ (cid:18) qλ ( h ) R (cid:48) π R + qλ ( h ) hσ (cid:48) π R (cid:19) (B.5) − e − σ qλ ( h ) σ (cid:48) π R − R qλ ( h ) R (cid:48) e − σ π R − R sin ( θ ) qλ ( h ) R (cid:48) e − σ sin ( θ )2 π R =0 . (B.6)As a final check one can show that the t, x component of the stress tensor is indeed equal tozero. First note that T is equal to T = i R (cid:16) ψ † + ∂ − ψ + − ψ †− ∂ + ψ − − ∂ − ψ † + ψ + + ∂ + ψ †− ψ − (cid:17) ⊗ η † η. (B.7)Using point-splitting this becomes (cid:104) T (cid:105) = lim t (cid:48) → tx (cid:48) → x π (cid:88) m,n (cid:90) d Ω i R (cid:16) ∂ (cid:48)− (cid:104) ψ m † + ( x − ) ψ n + ( x (cid:48)− ) (cid:105) − ∂ − (cid:104) ψ m † + ( x − ) ψ n + ( x (cid:48)− ) (cid:105)− ∂ (cid:48) + (cid:104) ψ m †− ( x + ) ψ n − ( x (cid:48) + ) (cid:105) − ∂ + (cid:104) ψ m †− ( x + ) ψ n − ( x (cid:48) + ) (cid:105) (cid:17) η m † η n = lim t (cid:48) → tx (cid:48) → x qi πR (cid:16) ∂ (cid:48)− (cid:104) ψ † + ( x − ) ψ + ( x (cid:48)− ) (cid:105) − ∂ − (cid:104) ψ † + ( x − ) ψ + ( x (cid:48)− ) (cid:105)− ∂ (cid:48) + (cid:104) ψ †− ( x + ) ψ − ( x (cid:48) + ) (cid:105) − ∂ + (cid:104) ψ †− ( x + ) ψ − ( x (cid:48) + ) (cid:105) (cid:17) , (B.8)and we see that (cid:104) T (cid:105) = (cid:104) T h (cid:54) =012 (cid:105) − (cid:104) T h =012 (cid:105) – 27 – lim t (cid:48) → tx (cid:48) → x qi πR (cid:16) ∂ (cid:48)− e i ( x (cid:48)− − x − ) e i ( x (cid:48)− − x − ) λ ( h ) iπ ( x (cid:48)− − x − ) − ∂ − e i ( x (cid:48)− − x − ) e i ( x (cid:48)− − x − ) λ ( h ) iπ ( x (cid:48)− − x − ) − ∂ (cid:48) + e i ( x (cid:48) + − x + ) e i ( x (cid:48) + − x + ) λ ( h ) iπ ( x (cid:48) + − x + ) − ∂ + e i ( x (cid:48) + − x + ) e i ( x (cid:48) + − x + ) λ ( h ) iπ ( x (cid:48) + − x + ) (cid:17) + O (cid:0) ( x (cid:48)− − x − ) (cid:1) + O (cid:0) ( x (cid:48) + − x + ) (cid:1) =0 . (B.9) C Matching
Let us turn to the time component of the metrics. At large ρ we can expand γ ( ρ ) in thefollowing way γ ( ρ ) = ζ C (¯ r ) (cid:18) r (cid:96) (cid:19) (cid:18) − π ρ − π ρ + 2 log( ρ ) (cid:19) + · · · . (C.1)In order to match the cubic term in ρ to the (cid:0) r − ¯ r ¯ r (cid:1) term in (3.21) we will consider thefollowing limit. We consider ρ to be large, but ζρ small and fixed. In this limit ψ is stillgiven by (3.25). Furthermore, from (3.27) we see that we should consider (cid:15) to be of order ζ .This limit corresponds to expanding f at small r − ¯ r ¯ r , but even smaller ζ ∝ (cid:15) . More precisely,compared to (3.21) we still expand in r − ¯ r ¯ r up to third order. However, we only consider (cid:15) upto zeroth order f ( r ) = C (¯ r ) (cid:18) r − ¯ r ¯ r (cid:19) − (cid:18) r (cid:96) (cid:19) (cid:18) r − ¯ r ¯ r (cid:19) + O (cid:32)(cid:18) r − ¯ r ¯ r (cid:19) (cid:33) . (C.2)We should now match the expansion (C.2) to the following expression, where we will assumeto be in the limit discussed above¯ r C (¯ r ) (1 + ρ + γ ( ρ )) dt dτ = C (¯ r ) (cid:18) ρ − π ρ ζ C (¯ r ) (cid:18) r (cid:96) (cid:19)(cid:19) ρ (cid:18) r − ¯ r ¯ r (cid:19) = C (¯ r ) (cid:18) r − ¯ r ¯ r (cid:19) − π ρ (cid:18) r − ¯ r ¯ r (cid:19) ζ (cid:18) r (cid:96) (cid:19) = C (¯ r ) (cid:18) r − ¯ r ¯ r (cid:19) − (cid:18) r (cid:96) (cid:19) (cid:18) r − ¯ r ¯ r (cid:19) , (C.3)where in the third line we used (3.25). We see that we recover (C.2) up to third order. D Correlators in (cid:104) H int (cid:105) In this appendix, we show explicitly the equal-time correlators involved in the computationof the semi-classical interacting Hamiltonian in terms of the 2 D propagators presented inAppendix A. Note that in Appendix A, the propagators are expanded in x − x (cid:48) , while herewe need the location of the fields to approach opposing boundaries. However, the expressions– 28 –btained are still valid if we take h small, and expand in h instead. The equal-time correletorspresent in the Hamiltonian are given by (cid:104) H int (cid:105) ⊃ − ih C (¯ r ) (cid:15) ¯ r (cid:90) d Ω (cid:104) ¯Ψ R − Ψ L + (cid:105) = − ih(cid:15) r C (¯ r ) (cid:90) d Ω (cid:16) (cid:104)− ψ R † + ψ L + (cid:105) + i (cid:104) ψ R † + ψ L − (cid:105) + i (cid:104) ψ R †− ψ L + (cid:105) + (cid:104) ψ R †− ψ L − (cid:105) (cid:17) η †− η − = qh(cid:15)π ¯ r C (¯ r ) (cid:18) − hπ (cid:19) + O (cid:0) h (cid:1) , (D.1)and (cid:104) H int (cid:105) ⊃ − ih C (¯ r ) (cid:15) ¯ r (cid:90) d Ω (cid:104) ¯Ψ L + Ψ R − (cid:105) = − ih(cid:15) r C (¯ r ) (cid:90) d Ω (cid:16) (cid:104) ψ L † + ψ R + (cid:105) + i (cid:104) ψ L † + ψ R − (cid:105) + i (cid:104) ψ L †− ψ R + (cid:105) − (cid:104) ψ L †− ψ R − (cid:105) (cid:17) η †− η − = qh(cid:15)π ¯ r C (¯ r ) (cid:18) − hπ (cid:19) + O (cid:0) h (cid:1) . (D.2) References [1] M.S. Morris and K.S. Thorne,
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