Membrane Paradigm, Gravitational Θ -Term and Gauge/Gravity Duality
UUTTG-21-15TCC-010-15
Membrane Paradigm, Gravitational Θ -Term and Gauge/Gravity Duality Willy Fischler a,b , Sandipan Kundu ca Theory Group, Department of Physics, University of Texas, Austin, TX 78712, USA b Texas Cosmology Center, University of Texas, Austin, TX 78712, USA c Department of Physics, Cornell University, Ithaca, New York, 14853, USA
E-mail: [email protected] , [email protected] Abstract:
Following the membrane paradigm, we explore the effect of the gravitationalΘ-term on the behavior of the stretched horizon of a black hole in (3 + 1)-dimensions.We reformulate the membrane paradigm from a quantum path-integral point of viewwhere we interpret the macroscopic properties of the horizon as effects of integratingout the region inside the horizon. The gravitational Θ-term is a total derivative, how-ever, using our framework we show that this term affects the transport properties of thehorizon. In particular, the horizon acquires a third order parity violating, dimensionlesstransport coefficient which affects the way localized perturbations scramble on the hori-zon. Then we consider a large-N gauge theory in (2 + 1) − dimensions which is dual toan asymptotically AdS background in (3 + 1) − dimensional spacetime to show that theΘ-term induces a non-trivial contact term in the energy-momentum tensor of the dualtheory. As a consequence, the dual gauge theory in the presence of the Θ-term acquiresthe same third order parity violating transport coefficient. a r X i v : . [ h e p - t h ] M a y ontents θ -term and stretched horizon 7 Θ -term 15 ϑ : Kubo formula 267 Conclusions 28A Near horizon metric 29B Stretched horizon: transport coefficients 30 Black holes are not only fascinating but they also provide us with a natural laboratoryto perform thought experiments to understand quantum gravity. String theory, MatrixTheory [1], and the AdS/CFT correspondence [2], which are the only models of quantumgravity over which we have mathematical control, have provided us with some insightinto different aspects of quantum gravity, e.g. the Bekenstein-Hawking entropy formulafor a large class of black holes. They also strongly indicate that black hole evolution– 1 –s seen by an external observer is unitary. However, none of these models give us acomprehensive microscopic description of the physics of black holes.Historically the membrane paradigm [3, 4] has also been successful at providing uswith a powerful framework to study macroscopic properties of black hole horizons. Inastrophysics the membrane paradigm has been used extensively as an efficient compu-tational tool to study phenomena in the vicinity of black holes (see [4–9] and referencestherein). The membrane paradigm has also been able to provide crucial hints aboutdetails of the microscopic physics of horizons. In particular, the membrane paradigmpredicts that black hole horizons are the fastest scramblers in nature. Fast-scramblingstrongly indicates that the microscopic description of scrambling of information on statichorizons must involve non-local degrees of freedom [10, 11]. In this paper, we will tryto understand the membrane paradigm from a quantum path-integral point of view.We will interpret the macroscopic properties of the horizon as effects of integrating outthe region inside the horizon. The semi-classical approximation of this path-integralapproach is equivalent to the action formulation [12] of the conventional membraneparadigm.We are mainly interested in figuring out how total derivative terms can affect themacroscopic properties of black hole horizons. Total derivative terms do not affect theclassical equations of motion and hence do not contribute even in perturbative quantumfield theory. However, it is well known that total derivative terms can have physicaleffects, e.g.
Lorentz and gauge invariance of Quantum chromodynamics (QCD) allow fora CP-violating topological θ QCD term which contributes to the electric dipole moment ofneutrons [13]. Similarly, the electrodynamics θ -term is also a total derivative, therefore,does not contribute for perturbative quantum electrodynamics (QED). However, in thepresence of the electrodynamics θ -angle a black hole horizon behaves as a Hall conductor,for an observer hovering outside [14]. As a consequence, the electrodynamics θ -angleaffects the way localized perturbations, created on the stretched horizon by dropping acharged particle, fast scramble on the horizon [14]. Later it was also shown that in-fallingelectric charges produce a non-trivial Berry phase in the QED wave function which canhave physical effects in the early universe [15].Another example comes from gravity in (3 + 1) − dimensions, where the topologicalGauss-Bonnet term contributes a correction term to the entropy of a black hole whichis proportional to the Euler number of the horizon. One can show that this correctionterm violates the second law of black hole thermodynamics and hence should be zero[16–19]. In (3 + 1) − dimensions, there exists another total derivative term, a parityviolating gravitational Θ-term S Θ = Θ8 (cid:90) d x (cid:15) µναβ R τσµν R σταβ . In this paper, we explore the effect of this Θ-term on black hole horizons. The membraneparadigm tells us that for an outside observer a black hole horizon effectively behaves like– 2 – viscous Newtonian fluid. Using our framework, we will show that the gravitational Θ-term affects the transport properties of the horizon fluid, in particular, the horizon fluidacquires a third order parity violating, dimensionless transport coefficient, which we willcall ϑ . This indicates that the Θ-term will affect the way perturbations scramble on thehorizon. Specifically, we can perform a thought experiment, in which an outside observerdrops a massive particle onto the black hole and watches how the perturbation scrambleson the black hole horizon. We will argue that the gravitational Θ-term, similar to theelectrodynamics θ -term, will also introduce vortices without changing the scramblingtime. This strongly suggests that in a sensible theory of quantum gravity the Θ-termwill play an important role, a claim that we will show is also supported by the AdS/CFTcorrespondence.The membrane paradigm has become even more relevant with the emergence of holography [1, 2], a remarkable idea that connects two cornerstones of theoretical physics:quantum gravity and gauge theory. The AdS/CFT correspondence [2], which is a con-crete realization of this idea of holography, has successfully provided us with theoreticalcontrol over a large class of strongly interacting field theories [2, 20–22]. This dualityenables us to compute observables of certain large-N gauge theories in d -dimensions byperforming some classical gravity calculations in ( d + 1)-dimensions. Gravity duals ofthese field theories at finite temperature contain black holes with horizons. It has beenshown that there is some connection between the low frequency limit of linear responseof a strongly coupled quantum field theory and the membrane paradigm fluid on theblack hole horizon of the dual gravity theory [23–27]. In this paper, we will considera large-N gauge theory in (2 + 1) − dimensions which is dual to a gravity theory in(3 + 1) − dimensions with the gravitational Θ-term and figure out the effect of the parityviolating Θ-term on the dual field theory. A reasonable guess is that the boundary the-ory, similar to the membrane paradigm fluid, will acquire the same third order parityviolating transport coefficient ϑ . We will confirm this guess by performing an explicitcomputation.It was argued in [28] that the two-point function of the energy-momentum tensor ina (2 + 1) − dimensional conformal field theory can have a non-trivial contact term (cid:104) T ij ( x ) T mn (0) (cid:105) = − i κ g π (cid:2)(cid:0) ε iml ∂ l (cid:0) ∂ j ∂ n − ∂ δ jn (cid:1) + ( i ↔ j ) (cid:1) + ( m ↔ n ) (cid:3) δ ( x ) . It is possible to shift κ g by an integer by adding a gravitational Chern-Simons countert-erm to the UV-Lagrangian and hence the integer part of κ g is scheme-dependent. On theother hand, the fractional part κ g mod 1 does not depend on the short distance physicsand hence it is a meaningful physical observable in (2 + 1) − dimensional conformal fieldtheory [28]. We will argue that a gravity theory in AdS (3+1) with the gravitationalΘ-term is dual to a conformal field theory with non-vanishing κ g , in particular κ g π = Θ– 3 –hich also suggests that only a fractional part of the Θ-term is a well-defined observable. The contact term κ g is also related to the transport coefficient ϑ , which to ourknowledge has never been studied before. It is a parity violating third order transportcoefficient in (2+1) − dimensions and hence forbidden in a parity-invariant theory. Undera small metric perturbation γ AB around flat Minkwoski metric, it contributes to theenergy-momentum tensor in the following way: T = − T = − ϑ ∂ γ ∂t , T = T = ϑ (cid:18) ∂ γ ∂t − ∂ γ ∂t (cid:19) and hence ϑ contributes to the retarded Green’s function of the energy-momentum tensorin order ω : G R , − ( ω, (cid:126)k →
0) = − iϑω .ϑ is dimensionless and it does not affect the trace of the energy-momentum tensor. In (2+1) − dimensional hydrodynamics, the Hall viscosity is another parity violating effect thatappears in the first order in derivative expansion. The Hall viscosity has been studiedextensively for both relativistic [30] and non-relativistic systems [31–33]. We believethat ϑ is a third order cousin of Hall viscosity and hence it is also an example of Berry-like transport [34]. We will show that for a holographic theory dual to asymptoticallyAdS spacetime in (3 + 1) − dimensions: ϑ = Θ = κ g / π . We will also speculate on thepossible covariant structure of the ϑ contribution to the energy-momentum tensor.The rest of the paper is organized as follows. We start with a discussion of themembrane paradigm in section 2. In section 3, we review the membrane paradigm forthe Einstein gravity. Then in section 4, we introduce gravitational Θ-term and discussits effect on the stretched horizon. In section 5, we discuss the effect of the Θ-termin the context of the AdS/CFT correspondence and make some comments on the ϑ -transport in section 6. Finally, we conclude in section 7. Some technical details havebeen relegated to appendices A and B. For readers only interested in the effect of theΘ-term in the context of the AdS/CFT correspondence, it is sufficient to read sections5 and 6. This also suggests that Θ, in our normalization is not an angle. However, one can work in thenormalization in which the gravitational Θ-term takes the form S Θ = Θ1536 π (cid:90) d x (cid:15) µναβ R τσµν R σταβ . In that case, adding an integer to κ g changes the new Θ by an integer times 2 π and hence in the abovenormalization Θ is an angle. Little is known about third-order transport coefficients in any dimensions. Very recently, thirdorder hydrodynamics for neutral fluids has been studied in (3 + 1)-dimensions [29]. – 4 –
Integrating out inside: membrane paradigm
The membrane paradigm provides a simple formalism to study macroscopic propertiesof horizons by replacing the true mathematical horizon by a stretched horizon , an effec-tive time-like membrane located roughly one Planck length away from the true horizon.Finiteness of the black hole entropy suggests that between the actual black hole horizonand the stretched horizon, the effective number of degrees of freedom should be vanish-ingly small. So, it is more natural as well as convenient to replace the true mathematicalhorizon by a stretched horizon.Predictions of the membrane paradigm are generally considered to be robust sincethey depend on some very general assumptions: • The effective number of degrees of freedom between the actual black hole horizonand the stretched horizon are vanishingly small. • Physics outside the black hole, classically must not be affected by the dynamicsinside the black hole.In this section, we will try to reformulate the membrane paradigm from a quantumpath-integral point of view. Our goal is to interpret the macroscopic properties of thestretched horizon as effects of integrating out the region inside the stretched horizon. We will not attempt to derive an effective action of the membrane, rather we will workin the semi-classical approximation where a lot can be learnt even without knowing theexact membrane effective action. However, our approach is somewhat similar to theapproach of [36] and it should be possible to derive a quantum mechanical version of themembrane paradigm from our approach by following [36].
Let us first consider some fields in a black hole background. We will assume that theback-reactions of the fields to the background is negligible. The action is given by S [ φ I ] = (cid:90) d d +1 x √ g L ( φ I , ∇ µ φ I ) , (2.1)where φ I with I = 1 , , ... stands for any fields. It is necessary to impose some boundaryconditions on the fields φ I in order to obtain equations of motion by varying this action.We will impose Dirichlet boundary conditions δφ I = 0 at the boundary of space-time.The stretched horizon M divides the whole space-time in regions: A : outside the membrane M ,B : inside the membrane M . A discussion on integrating out geometry in the context of the AdS/CFT correspondence can befound in [35]. – 5 –he total quantum field theory partition function for fields φ I is given by, Z = (cid:90) D [ φ I ] e iS [ φ I ] . (2.2)Now imagine an observer O who is hovering outside the horizon of a black hole.Observer O has access only to the region outside the black hole. We can write down theabove partition function in the following way Z = (cid:90) D [ φ BI ] D [ φ M I ] D [ φ AI ] e iS B [ φ I ] e iS A [ φ I ] . (2.3)Where, we have written S [ φ I ] = S A [ φ I ] + S B [ φ I ]. In the path integral, we have decom-posed every field φ such that φ B is the the field inside the stretched horizon M , φ A isthe the field outside the stretched horizon M and φ M is the the field on the stretchedhorizon M . Observer O has access only to the region outside the black hole and wewant to find out some effective action S O for the observer O . To that goal, we first fixthe values of all the fields φ I with I = 1 , , ... on M . Then in principle we can performthe path integral (cid:82) D [ φ BI ] e iS B [ φ I ] ≡ z ( φ M I ). After performing the path integral over φ BI ,partition function (2.3) becomes, Z = (cid:90) D [ φ M I ] D [ φ AI ] z ( φ M I ) e iS A [ φ I ] . (2.4)This partition function now depends only on quantities defined on or outside of thestretched horizon M . The non-trivial function z ( φ M I ) contains information about theinside. In practice, it is not possible to find z ( φ M I ) because that requires detailed knowl-edge of the physics inside the black hole. Our goal is not to compute z ( φ M I ) exactly,but to extract some information about z ( φ M I ) by demanding that the classical physicsoutside the black hole horizon must not be affected by the dynamics inside the blackhole.Let us now, re-write the partition function (2.4), in the following way: Z O = (cid:90) A D [ φ I ] e i ( S A [ φ I ]+ S surf [ φ I ]) , (2.5)where, now fields φ I ’s are defined only on or outside of the stretched horizon M and z ( φ M I ) ≡ exp( iS surf [ φ I ]). Observer O has access only to the region outside the blackhole and physics he observes, classically must not be affected by the dynamics insidethe black hole. That means observer O should be able to obtain the correct classicalequations of motion by varying action S O which is restricted only to the space-timeoutside the black hole: S O = S A [ φ I ] + S surf [ φ I ] . (2.6)Note that S surf [ φ I ] (cid:54) = 0 because the boundary terms generated on M from S A arein general non-vanishing. Surface terms S surf [ φ I ] obtained by integrating over fields– 6 –nside the stretched horizon must exactly cancel all these boundary terms. The factthat the correct classical equations of motion can be obtained by varying only S O = S A [ φ I ] + S surf [ φ I ], gives us certain information about S surf [ φ I ]. For the observer O , theaction S O for fields φ I now have sources residing on the stretched horizon S O = (cid:90) A d d +1 x √− g L ( φ I , ∇ µ φ I ) + (cid:88) I (cid:90) M d d x √− h J I M φ I (2.7)where, h is the determinant of the induced metric on the stretched horizon M and wehave written S surf [ φ I ] = (cid:88) I (cid:90) M d d x √− h J I M φ I . (2.8)It is important to note that one should interpret J I M as external sources such that δ J I M δφ J = 0. Now demanding that we obtain correct classical equations of motion for fields φ I by varying S O , we obtain J I M = (cid:20) n µ ∂ L ∂ ( ∇ µ φ I ) (cid:21) M (2.9)where, n µ is the outward pointing normal vector to the time-like stretched horizon M with n µ n µ = 1. The observer O can actually perform real experiments on the stretchedhorizon M to measure the sources J I M . θ -term and stretched horizon Several examples of the action principle of the membrane paradigm can be found in [12].However, we will focus on a particular example studied in [14] which shows that totalderivative terms can have important physical effects. The action for electromagneticfields with a θ -term in curved space-time in (3 + 1) − dimensions is S = (cid:90) √ gd x (cid:20) − F µν F µν + j µ A µ (cid:21) + θ (cid:90) d x(cid:15) αβµν F αβ F µν (2.10)where, F µν = ∂ µ A ν − ∂ ν A µ . Current j µ is conserved, i.e., ∇ µ j µ = 0. The electrodynam-ics θ -term does not affect the classical equations of motion because it is a total derivativeand hence the equations of motion obtained from the action (2.10) is ∇ µ F µν = − j ν . (2.11)Field strength tensor F µν also obeys ∂ [ µ F νλ ] = 0. Let us now write θ (cid:15) αβµν F αβ F µν = θ √ gF µν ∗ F µν . (2.12) Our convention of the metric is that the Minkowski metric has signature ( − + ++). – 7 –ur convention of the Levi-Civita tensor density (cid:15) αβµν is the following: (cid:15) = 1, (cid:15) = − g . We will also assume that the conserved current j µ is contained inside the membrane M and hence the observer O who has access only to the region outside the stretchedhorizon does not see the current j µ . However, the observer will see a surface current J µ M on the membrane. Let us start with the action for the observer O S O = (cid:90) A √ gd x (cid:20) − F µν F µν + θ F µν ∗ F µν (cid:21) + (cid:90) M √− hd x J M ; µ A µ . (2.13)The action is invariant under the gauge transformation: A µ → A µ + ∂ µ λ only if J M ; µ n µ =0, where n µ is the outgoing unit normal vector on M . In order for the observer O torecover the vacuum Maxwell’s equations, outside the horizon, the boundary terms on M should cancel out and from equation (2.9) we obtain J µ M = ( n ν F µν − θ n ν ∗ F µν ) | M . (2.14)Note that J M ; µ n µ = 0 and hence the action (2.13) is invariant under gauge transforma-tions.The electrodynamics θ -term is a total derivative, therefore, it does not contribute toperturbative quantum electrodynamics (QED), which indicates that the effects of the θ term in QED, if any, are non-perturbative and hence difficult to detect. But by couplingthis theory to gravity one finds that the θ -term can affect the electrical properties ofblack hole horizons, in particular, black hole horizons behave as Hall conductors [14].This strongly suggests that in a sensible quantum theory of black holes, a total derivativeterm like electrodynamics θ -term can play important role.The AdS/CFT correspondence which is one of the few models of quantum gravitywhich is well understood also supports the above claim [14]. In particular, let us considerU(1) gauge field in AdS-Schwarzschild in (3 + 1) − dimensions with the action S = (cid:90) d x (cid:20) − √ g g F µν F µν + θ (cid:15) αβµν F αβ F µν (cid:21) . (2.15)The U(1) gauge field in the bulk is dual to a conserved current j i in the boundary theory.DC conductivities are given by σ AB = i lim ω → lim k → G ABR ( k ) /ω , where, G ABR ( k ) is theretarded Green function of boundary current j and indices A, B = 1 , σ AB of the stronglycoupled (2 + 1) − dimensional dual theory is given by [26] σ = σ = 1 g , σ = − σ = θ (2.16)and hence in the presence of the θ -term, the boundary theory has nonzero Hall conduc-tivity. – 8 – Membrane paradigm and gravity
First let us consider the Einstein-Hilbert action for gravity S EH = M P2 (cid:90) d x √− gR . (3.1)A theory of gravity contains higher derivatives in the action and hence we should bemore careful. In order to obtain equations of motion by performing a variation of thisaction, it is not sufficient to impose Dirichlet boundary conditions δg µν = 0 on the outerboundary of space-time Σ. With the Dirichlet boundary conditions on Σ, variation ofthe Einstein-Hilbert action gives rise to non-vanishing boundary terms. One solution isto further impose ∇ ρ δg µν = 0 on Σ. However, it is more useful to modify the action(3.1) by adding the standard Gibbons-Hawking-York boundary term to make the actionconsistent with the Dirichlet boundary condition on the boundary Σ S = S EH + S GHY (Σ) = M P2 (cid:90) d x √− gR + M P2 (cid:90) Σ d x (cid:112) | h | K , (3.2)where, K is the trace of the extrinsic curvature and h is the determinant of the inducedmetric on Σ.Let us again imagine a stretched horizon M that divides the whole space-time intwo regions A : outside the membrane M ,B : inside the membrane M and an observer O who has access only to the region outside the stretched horizon. Thetotal quantum partition function can in principle be written in the following way Z = (cid:90) D [ g Bµν ] D [ g M µν ] D [ g Aµν ] e iS B e iS A . (3.3)Where, we have written the total action S = S A + S B . Observer O has access only tothe region outside the black hole and we want to find out some effective action S O forthe observer O . Since, δS A (cid:54) = (cid:82) M ( ... ) δg µν , one must divide the total action (3.2) in thefollowing way: S = ( S A − S GHY ( M )) + ( S B + S GHY ( M )) (3.4)because otherwise the action is non-differentiable with respect to the metric on M . Thepath integral (3.3), can now be written as Z = (cid:90) A + M D [ g M µν ] D [ g Aµν ] e i ( S A − S GHY ( M )) (cid:90) B D [ g Bµν ] e i ( S B + S GHY ( M )) . (3.5)Now we first fix the metric on M and then in principle we can perform the path integralon B . Performing this path integral exactly is an impossible task without the detailed– 9 –nowledge of the physics inside the black hole. But fortunately we do not need toperform the path integral, instead we write (cid:90) B D [ g Bµν ] e i ( S B + S GHY ( M )) = e iS surf ( M ) . (3.6)Hence, the partition function (3.5) becomes, Z O = (cid:90) A + M D [ g µν ] e i ( S A − S GHY ( M )+ S surf ( M ) ) , (3.7)where, now the path integral is defined only on or outside of the stretched horizon M .Observer O has access only to the region outside the black hole and physics he observes,classically must not be affected by the dynamics inside the black hole. That meansobserver O should be able to obtain the correct classical equations of motion by varyingaction S O which is restricted only to the space-time outside the black hole, where, S O = S EH + S GHY (Σ) − S GHY ( M ) + S surf ( M ) (3.8)= (cid:90) A d x [ M P2 √− gR ] + M P2 (cid:90) Σ d x (cid:112) | h | K − M P2 (cid:90) M d x (cid:112) | h | K + S surf ( M ) . Note that the sign of the S GHY ( M ) term is negative because we choose outward pointingnormal vector to be positive. Possibly, one can interpret the above action in the followingway. The observer O has two boundaries: outer space-time boundary Σ and anotherboundary because of the membrane M and hence he needs the Gibbons-Hawking-Yorkboundary term for both Σ and M . The boundary conditions are fixed only at the outerboundary Σ and the surface term S surf ( M ) is necessary to obtain the correct classicalequations of motion.It is important to note that the division of the action (3.4) is not unique. We canalways add some intrinsic terms, S = ( S A − S GHY ( M ) + S i ( M )) + ( S B + S GHY ( M ) − S i ( M )) , (3.9)where, S i ( M ) = (cid:90) M d x √ hF ( h ij ) . (3.10) F ( h ij ) can be any scalar intrinsic to M . Since, it’s an intrinsic term, now (cid:90) B D [ g Bµν ] e i ( S B + S GHY ( M ) − S i ( M )) = e i ( S surf ( M ) − S i ( M )) (3.11)and hence S O does not depend on S i ( M ).In the rest of the section, we will mainly review some known results of the membraneparadigm. Experts can skip the rest of this section and move on to section 4.– 10 – .1 Energy momentum tensor on the stretched horizon Let us now consider variations of S EH and S GHY (Σ) with Dirichlet boundary conditionson Σ δ ( S EH + S GHY (Σ)) = M P2 (cid:90) A d x (cid:2) δ (cid:0) √− gg µν (cid:1) R µν + √− gg µν δR µν (cid:3) = M P2 (cid:90) A d x √− g (cid:2) G µν δg µν + √− gg µν δR µν (cid:3) , (3.12)where G µν = R µν − g µν R is the Einstein tensor. The variation of the Ricci tensor isgiven by, g µν δR µν = −∇ µ (cid:2) ∇ ν δg µν − g αν g µβ ∇ β δg αν (cid:3) . (3.13)Therefore, (cid:90) A d x √− gg µν δR µν = (cid:90) M d x (cid:112) | h | n µ g αν [ ∇ α δg µν − ∇ µ δg αν ]= (cid:90) M d x (cid:112) | h | n µ h αν [ ∇ α δg µν − ∇ µ δg αν ] . (3.15) n µ is the normal vector to the time-like surface M with n µ n µ = 1. Note that there is anegative sign because n µ is the outward pointing normal vector. h is the determinant ofthe induced 3-metric on the membrane M . The induced 3-metric has been written as a4-metric h µν = g µν − n µ n ν (3.16)that projects from the 4-dimensional space-time to the 3-dimensional membrane M .The membrane extrinsic curvature is defined as K µν ≡ h αµ h βν ∇ α n β . (3.17)Let us first note some of the properties of the extrinsic curvature tensor (3.17). One caneasily check that K µν n µ = K µν n ν = 0 . (3.18)One can also check that K µν = K νµ . Let us recall that the Gauss’ theorem in curved space-time is given by (cid:90) V √− gd x ∇ µ A µ = (cid:73) ∂V (cid:112) | h | d xn µ A µ , (3.14)where h is the induced metric on the surface ∂V and n µ is the normal vector with n µ n µ = 1 (assumingthe surface is timelike). Note that the vector n µ is orthogonal to M and hence it obeys the hypersurface orthogonalitycondition: n [ µ ∇ ν n λ ] = 0. – 11 –et us now compute the variation of S GHY ( M ) δS GHY ( M ) = M P2 (cid:90) M d x δ ( (cid:112) | h | h µν ∇ µ n ν )= M P2 (cid:90) M (cid:112) | h | d x (cid:20) − h µν Kδh µν + 12 K µν δh µν + ∇ µ c µ (cid:21) + 12 M P2 (cid:90) M d x (cid:112) | h | n µ h αν [ ∇ α δg µν − ∇ µ δg αν ] , (3.19)where, c µ = δn µ − n ν δg µν . (3.20)One can easily check that n µ c µ = 0 and as a consequence ∇ µ c µ | M = D (3) i c i | M , (3.21)where D (3) i is the covariant derivative in terms of the 3-dimensional induced metric h ij . Therefore, the term ∇ µ c µ in δS GHY ( M ) is a total derivative and hence does notcontribute. So, finally we obtain, δ ( S EH + S GHY (Σ) − δS GHY ( M )) = M P2 (cid:90) A d x √− gG µν δg µν + M P2 (cid:90) M (cid:112) | h | d x [ h µν K − K µν ] δh µν . (3.22)Therefore all the terms in the second line should be cancelled by δS surf . The membraneenergy-momentum tensor T µν is defined in the standard way δS surf ( M ) = − δ ( S EH + S GHY (Σ) − δS GHY ( M )) = − (cid:90) M (cid:112) | h | d x T µν δh µν . (3.23)Therefore, finally we obtain T µν = M P2 [ h µν K − K µν ] | M . (3.24)It is important to note that T µν n µ = 0. This is the famous result that was originallyobtained in [3] from equations of motion and later in [12] from action formulation ofmembrane paradigm. Now we will use equation (3.24) to review the claim that the stretched horizon behaves asa viscous Newtonian fluid. For that we only need to know the near horizon geometry. Letus denote the 3-dimensional absolute event horizon by H . We define a well-behaved timecoordinate ¯ t on the horizon as well as spatial coordinates x A with A = 1 ,
2. Thereforethe event horizon H has coordinates x i ≡ (cid:0) ¯ t, x A (cid:1) with a metric h ij . There exists a– 12 –nique null generator l µ through each point on the horizon (in the absence of horizoncaustics). We can always choose coordinates x i ≡ (cid:0) ¯ t, x A (cid:1) such that l = ∂∂ ¯ t . (3.25)Where we have chosen the spatial coordinates x A such that they comove with the horizongenerator l and h A = 0. Note that l µ ∇ µ l ν = g H l ν , (3.26)where g H is the surface gravity of the horizon H . We only restrict to the simpler case: g H (¯ t, x ) =constant on the horizon H . For the case of constant and non-zero g H , the nearhorizon geometry has the form (see appendix A for details) ds = − r d ¯ t + 2 rg H d ¯ tdr + γ AB (cid:18) dx A − Ω A (¯ t, x ) g H r d ¯ t (cid:19) (cid:18) dx B − Ω B (¯ t, x ) g H r d ¯ t (cid:19) + O ( r )(3.27)where the horizon is at r = 0 and the induced spatial metric on a constant ¯ t hypersurfaceis γ AB ≡ γ AB (¯ t, x, r ). Let us also note that in comoving coordinates ∂γ AB ∂ ¯ t = 2 σ HAB + θ H γ AB , (3.28)where, horizon expansion θ H and shear σ HAB are defined as σ HAB = θ AB − γ AB θ H , (3.29) θ H = γ AB θ AB = ∂∂ ¯ t ln √ γ . (3.30) θ AB is the 2-dimensional covariant derivative (with metric γ AB ) of the horizon nullgenerator θ AB = D (2) A l B . (3.31)We now have to choose a set of FIDO’s and a universal time. ¯ t is not a good choicefor the universal time because surfaces of constant ¯ t are null everywhere. However, thereis a preferred choice for the universal time t [3] t = ¯ t − g H ln r . (3.32)FIDO’s are chosen such that their world lines are orthogonal to constant t surfaces andhence velocity 4-vector can be written as U = − rdt = − rd ¯ t + drg H . (3.33)– 13 –herefore, U r = 0 , U ¯ t = 1 r , U A = − Ω A rg H . (3.34)Note that in the limit r →
0, these world lines approach the null generator of the horizon.Let us now replace the actual horizon by a stretched horizon at r = (cid:15) . In the limit (cid:15) →
0, using the near horizon metric (3.27), components of the extrinsic curvature (3.17)are given by (up to relevant orders in (cid:15) ) K | M = − g H (cid:15) + O ( (cid:15) ) , (3.35) K A | M = O ( (cid:15) ) , (3.36) K AB | M = 1 (cid:15) θ AB + O ( (cid:15) ) . (3.37)Therefore, the trace is given by K ≡ h ij K ij = 1 (cid:15) ( g H + θ H ) + O ( (cid:15) ) . (3.38) The energy-momentum tensor of a 3-dimensional viscous Newtonian fluid with pressure p and energy density ρ is given by T ij = ( ρ + p ) u i u j + ph ij − τ ij , (3.39)where, u i is the fluid velocity and τ ij is the the dissipative part of the energy-momentumtensor τ ij = − P ik P jl (cid:2) ηf kl + ζh kl D (3) m u m (cid:3) , (3.40)where P ij = h ij + u i u j is the projection operator and f kl = D (3) k u l + D (3) l u k − h kl D (3) m u m . (3.41) η is the shear viscosity and ζ is the bulk viscosity. Therefore, we can write the fullenergy-momentum tensor in the following way T ij = ρu i u j + P ij (cid:0) p − ζD (3) m u m (cid:1) − ηP ik P jl f kl . (3.42)Comparing the energy momentum tensor on the stretched horizon (3.24) with (3.42)we obtain ρ = − M P2 (cid:15) θ H , p = M P2 (cid:15) g H , (3.43) ζ = − M P2 , η = M P2 . (3.44) Let us recall that indices i, j run over all the coordinates on the stretched horizon, whereas indices
A, B run over only the spatial coordinates on the stretched horizon. See appendix B for details. We will also provide an alternative derivation of these relations insection 4.2. – 14 –tretched horizon of a black hole indeed behaves like a viscous Newtonian fluid. Fewcomments are in order: both energy density and pressure diverge in the limit (cid:15) → ρ H = (cid:15)ρ and renormalized pressure p H = (cid:15)p which areenergy and pressure as measured at infinity [3]. One can check that for a Schwarzschildor Kerr black hole θ H = 0 and hence ρ H = 0. Also note that the stretched horizon hasnegative bulk viscosity. In ordinary fluid, negative bulk viscosity indicates instability,however, it is alright for a horizon to have negative bulk viscosity. Negative bulk viscosityof the horizon indicates that expansion of the horizon is acausal and one must imposeteleological boundary conditions [3, 4]. Θ -term In any sensible theory of quantum gravity, the Einstein-Hilbert action should be theleading low energy term. However, it is expected that the low energy limit will alsogenerate higher derivative correction terms. The only ghost-free combination, in order R , is the Gauss-Bonnet term. Effects of the Gauss-Bonnet term on the stretchedhorizon have already been studied in [39]. In (3 + 1) − dimensions, the Gauss-Bonnetterm is a total derivative and hence does not contribute to the equations of motion.However, in the presence of this topological term the entropy of a black hole receives acorrection term proportional to the Euler number of the horizon. One can show thatthis correction term violates the second law of black hole thermodynamics and henceshould be zero [16–19]. In (3 + 1) − dimensions, there can exist another total derivativeterm, a parity violating gravitational Θ-term. We are interested in figuring out theeffects of this term on the stretched horizon which are forbidden in a parity-invarianttheory.Let us now introduce the parity violating Θ-term to the (3 + 1) − dimensional gravityaction S = S EH + S GHY + S Θ = (cid:90) d x √− g (cid:18) M P2 R + Θ4 R ∗ R (cid:19) + M P2 (cid:90) Σ d x (cid:112) | h | K , (4.1)where, Θ is a constant and the quantity R ∗ R is the Chern-Pontryagin density which isdefined as R ∗ R = R τσµν ∗ R σ µντ (4.2) If we allow higher derivative corrections then the Lanczos-Lovelock gravity is the unique extensionof the Einstein gravity with equations of motion containing only up to two time derivatives [37]. Inthe Lanczos-Lovelock gravity, the Gauss-Bonnet term is the first (and in (3 + 1) − dimensions the only)correction to the Einstein gravity. The membrane paradigm for the Lanczos-Lovelock gravity has beenstudied in [38]. Possibility of gravitationally induced CP-violation because of the Θ-term was already explored in1980 [40]. – 15 –here ∗ R σ µντ ≡ e µναβ R σταβ , (4.3)and e µναβ is the Levi-Civita tensor. Unlike the Chern-Simons modified gravity dis-cussed in [41, 42], we are interested in the constant Θ case because for constant Θ, theChern-Pontryagin density term is a total derivative. Applying the membrane paradigmwe will show that this total derivative term has non-trivial effect on the stretched hori-zon.We will impose the Dirichlet boundary conditions on the boundary of space-time Σ,i.e. δg µν = 0 over the outer boundary of space-time. Variation of the Θ-term generatesboundary terms on Σ which do not vanish with the Dirichlet boundary conditions onΣ. The action consistent with the Dirichlet boundary condition on the boundary isobtained by adding a Gibbons-Hawking-York like boundary term for the Θ-term [43] S b Θ = Θ (cid:90) Σ d x (cid:112) | h | n α e αβγδ K ρβ ∇ γ K δρ . (4.5)The action (5.1) should now be replaced by the following action which is well behavedwith the Dirichlet boundary condition S = (cid:90) d x √− g (cid:18) M P2 g µν R µν + Θ8 e µναβ R τσµν R σταβ (cid:19) + S GHY (Σ) + S b Θ (Σ) . (4.6)The Θ-term does not affect the classical equations of motion because it is a total deriva-tive. However, similar to the electrodynamics θ -term [14], the gravitational Θ-term alsogenerates non-trivial boundary terms which can affect the dynamics of the stretchedhorizon. Let us again imagine a stretched horizon M that divides the whole space-time in tworegions A : outside the membrane M ,B : inside the membrane M and an observer O who has access only to the region outside the membrane. Followingthe discussion of the previous section we can write down the path integral in the followingway Z = (cid:90) A + M D [ g M µν ] D [ g Aµν ] e i ( S A − S GHY ( M ) − S b Θ ( M )) (cid:90) B D [ g Bµν ] e i ( S B + S GHY ( M )+ S b Θ ( M )) , (4.7) Levi-Civita tensor e µναβ is related to the Levi-Civita tensor density (cid:15) µναβ in the following way e µναβ = (cid:15) µναβ √ g . (4.4) – 16 –here, the total action (4.6) has been divided as S = S A + S B . Now we fix the metricon M and then write the path integral on B as (cid:90) B D [ g Bµν ] e i ( S B + S GHY ( M )+ S b Θ ( M )) = e i S ( M ) . (4.8)Hence, the partition function (4.7) becomes, Z O = (cid:90) A + M D [ g µν ] e i ( S A − S b Θ ( M ) − S GHY ( M )+ S ( M )) , (4.9)where, now the path integral is defined only on or outside of the stretched horizon M .The physics outside the black hole classically must not be affected by the dynamics insidethe black hole. That means the observer O should be able to obtain the correct classicalequations of motion by varying action S O which is restricted only to the space-timeoutside the black hole, where, S O = (cid:90) A d x √− g (cid:18) M P2 g µν R µν + Θ8 e µναβ R τσµν R σταβ (cid:19) + M P2 (cid:90) Σ M d x (cid:112) | h | K + Θ (cid:90) Σ M d x (cid:112) | h | n α e αβγδ K ρβ ∇ γ K δρ + S ( M )= S EH + S Θ + S GHY (Σ) − S GHY ( M ) + S b Θ (Σ) − S b Θ ( M ) + S ( M ) . (4.10)Before we proceed note that S Θ + S b Θ is equivalent to the three-dimensional gravitationalChern-Simons term[43] (cid:90) d x √− g Θ4 R ∗ R + Θ (cid:90) Σ d x (cid:112) | h | n α e αβγδ K ρβ ∇ γ K δρ = Θ (cid:90) Σ d x (cid:112) | h | e ijk (cid:20) γ lim ∂ j γ mkl + 13 γ lim γ mjp γ pkl (cid:21) (4.11)where e ijk ≡ n µ e µijk . (4.12)The three-dimensional action (4.11) is exactly the gravitational Chern-Simons action in(2 + 1)-dimensions. Now the action for the observer O is S O = M P2 (cid:90) A d x √− gg µν R µν + M P2 (cid:90) Σ M d x (cid:112) | h | K + Θ (cid:90) Σ M d x (cid:112) | h | e ijk (cid:20) γ lim ∂ j γ mkl + 13 γ lim γ mjp γ pkl (cid:21) + S ( M ) . (4.13)Variation of the gravitational Chern-Simons action is well known δ (cid:90) Σ M d x (cid:112) | h | e ijk (cid:20) γ lim ∂ j γ mkl + 13 γ lim γ mjp γ pkl (cid:21) = − (cid:90) Σ M d x (cid:112) | h | C mn δh mn (4.14) Note that we are using the notation (cid:82) Σ M d x (cid:112) | h | K = (cid:82) Σ d x (cid:112) | h | K − (cid:82) M d x (cid:112) | h | K . – 17 –here C mn is the Cotton-York tensor C mn = − e ijk (cid:104) h mk D (3) i R (3) nj + h nk D (3) i R (3) mj (cid:105) , (4.15)where, R (3) ij is the Ricci tensor of the three-dimensional surface M (or Σ) and D (3) i is the three-dimensional covariant derivative. The Cotton-York tensor C mn is sym-metric, traceless and covariantly conserved in three-dimensions. Therefore, the three-dimensional stretched horizon energy-momentum tensor receives a correction because ofthe gravitational Θ-term T ij = M P2 [ h ij K − K ij ] | M − C ij | M . (4.16) We already know that a black hole horizon behaves like a fluid for an observer hoveringoutside. In section (3.3), we have derived different first order transport coefficients ofthis fluid for Einstein gravity. Let us now figure out how the Θ-term in (4.16) affects thetransport property of the fluid living on a stretched horizon. Transport coefficients arethe measure of the response of a fluid after hydrodynamic perturbations. For example,one can disturb a hydrodynamic system by perturbing the background metric and thenobserve the change in the energy momentum tensor of the fluid as a result of thisperturbation. In order to do that we will study linearized perturbation of the metricnear the horizon of a stationary black hole solution. Again we consider an observer O hovering just outside a (3 + 1) − dimensional black hole. For such an observer, theunperturbed near horizon metric is Rindler-like ds = − r d ¯ t + 2 rg H d ¯ tdr + dx + dx . (4.17)For the observer O , there is a horizon at the edge of the Rindler wedge r = 0. Notethat we are restricting to the case of constant g H on the horizon. Let us now perturbthe metric: g µν = g µν + g µν , where g µν is the unperturbed metric (4.17). We perturb themetric such that very close to the horizon only g AB ≡ γ AB (¯ t, r, x , x ) components arenon-zero ds = − r d ¯ t + 2 rg H d ¯ tdr + dx + dx + γ AB (¯ t, r, x , x ) dx A dx B , (4.18)where indices A, B = 1 , r = (cid:15) ) in linear order in perturbations, is given by K AB | M = 12 (cid:15) (cid:18) ∂γ AB ∂ ¯ t (cid:19) , K | M = 1 (cid:15) (cid:18) g H + 12 ∂γ ∂ ¯ t + 12 ∂γ ∂ ¯ t (cid:19) , (4.19)– 18 –here, we have ignored terms with positive powers of (cid:15) . Similarly, we can calculate theCotton-York tensor (4.15) on the stretched horizon. In linear order in perturbations andin the leading order in (cid:15) , we obtain C = − C = 12 (cid:15) ∂ γ ∂ ¯ t , (4.20) C = C = − (cid:15) (cid:18) ∂ γ ∂ ¯ t − ∂ γ ∂ ¯ t (cid:19) (4.21)and all the other components vanish. Before we proceed, we want to note that theCotton-York tensor contains derivatives only with respect to the coordinates on thestretched horizon. Hence, we do not need to solve the Einstein equations in order towrite down the Θ-contributions to the energy-momentum tensor in terms of quantitiesintrinsic to the stretched horizon.Let us now study the effect of the metric perturbation (4.18), order by order, onthe energy-momentum tensor on the stretched horizon. In the 0 th order in derivativeexpansion, from (4.16), we obtain T = O ( (cid:15) ) , T AB = M P2 g H (cid:15) ( δ AB + γ AB ) . (4.22)Comparing the 0 th order energy-momentum tensor with that of an ideal fluid, we findthat energy density ρ = 0 and pressure p = M P2 g H /(cid:15) . The renormalized pressure whichis the pressure as measured at infinity, is given by p H = M P2 g H .In the 1 st order in derivative expansion, from (4.16) and (4.19), we obtain T = M P2 (cid:15) (cid:18) ∂γ ∂ ¯ t (cid:19) , T = M P2 (cid:15) (cid:18) ∂γ ∂ ¯ t (cid:19) , (4.23) T = T = − M P2 (cid:15) (cid:18) ∂γ ∂ ¯ t (cid:19) . (4.24)For a Newtonian viscous fluid in (2 + 1) − dimensions, in the 1 st order in derivativeexpansion, the energy momentum tensor has the form: T AB = − η ∂γ AB ∂t − ζδ AB (cid:18) ∂γ ∂t + ∂γ ∂t (cid:19) , (4.25)where we have assumed that the spatial part of the unperturbed metric is flat. Hence, thestretched horizon, for an observer hovering outside, has shear viscosity η = M P2 / ζ = − M P2 /
2, which agrees with (3.44). Note that the 1 st order energy-momentum tensor (4.23-4.24) diverges in the limit (cid:15) → t = (cid:15) ¯ t .– 19 –nergy-momentum tensor on the stretched horizon (4.16) does not contain any termin the 2 nd order in derivative expansion. However, from equations (4.20-4.21), it is clearthat the Θ-term contributes in the 3 rd order in derivative expansion, yielding T = −T = − Θ (cid:15) ∂ γ ∂ ¯ t , (4.26) T = T = Θ2 (cid:15) (cid:18) ∂ γ ∂ ¯ t − ∂ γ ∂ ¯ t (cid:19) . (4.27)Let us now define a new third order, parity violating transport coefficient in (2 +1) − dimensions ϑ which contributes to the energy-momentum tensor in the followingway T = − T = − ϑ ∂ γ ∂t , (4.28) T = T = ϑ (cid:18) ∂ γ ∂t − ∂ γ ∂t (cid:19) . (4.29)Hence, the stretched horizon, for an observer hovering outside, has ϑ = Θ . (4.30)Note that this is not Hall viscosity but a third order cousin of Hall viscosity. The linearresponse theory dictates that the retarded Green’s functions of the energy-momentumtensor on the stretched horizon are G R , ( ω, k ) = − G R , ( ω, k ) = − i Θ ω (cid:15) , (4.33)where, the presence of 1 /(cid:15) factor is again a consequence of large blueshift near thehorizon. Therefore, the gravitational Θ-term affects the transport property of thestretched horizon and in principle it can be found from the coefficient of the ω term of G R , − ( ω, k ). We will end this section with a discussion of the effect of the gravitational Θ-termon the way perturbations scramble on the horizon. The process by which a localized We will discuss about the transport coefficient ϑ in more details in section 6. Hall viscosity contributes in the first order in derivative expansion: T = − T = − η A ∂γ ∂t , (4.31) T = T = η A (cid:18) ∂γ ∂t − ∂γ ∂t (cid:19) . (4.32) – 20 –erturbation spreads out into the whole system is known as scrambling . In quantummechanics, information contained inside a small subsystem of a larger system is fullyscrambled when the small subsystem becomes entangled with the rest of the system andafter scrambling time t s the information can only be retrieved by examining practicallyall the degrees of freedom. In a local quantum field theory, it is expected that thescrambling time t s is at least as long as the diffusion time. Consequently, one can showthat the scrambling time for a strongly correlated quantum fluid in d -spatial dimensionsand at temperature T , satisfies t s T ≥ c (cid:126) S /d , (4.34)where c is some dimensionless constant and S is the total entropy. In [10, 11], it hasbeen argued that (4.34) is a universal bound on the scrambling time. Hence, it is indeedremarkable, as shown in [10, 11], that information scrambles on black hole horizonsexponentially fast t s T ≈ (cid:126) ln S (4.35)violating the bound (4.34). This unusual process is known as “fast-scrambling” andit strongly suggests that the microscopic description of fast-scrambling on black holehorizons must involve non-local degrees of freedom [10, 11]. It is now well known thatnon-locality is indeed essential for fast scrambling [44–48] which is also supported by theobservation that non-local interactions can enhance the level of entanglement amongdifferent degrees of freedom [51–53].Recently it has been shown that in the presence of the electrodynamics θ -angle ablack hole horizon behaves as a Hall conductor, for an observer hovering outside [14].As a consequence, the electrodynamics θ -angle affects the way localized perturbationson the stretched horizon, created by dropping a charged particle, Hall-scramble on thehorizon [14].In presence of the gravitational Θ-term, it is reasonable to expect a similar con-clusion. We can perform a thought experiment, in which an outside observer drops amassive particle onto the black hole and watches how the perturbation scrambles onthe black hole horizon. Equations (4.28-4.29) indicate that the gravitational Θ-termwill also affect the way perturbations scramble on the horizon, in particular, it will in-troduce vortices without changing the scrambling time. This perhaps suggests that amicroscopic description of fast-scrambling needs to be able to explain the origin of thiseffect. Following [14] it is possible to perform an explicit calculation for this effect butwe will not attempt it in this paper.It is important to note that the Rindler approximation of the near horizon metric(4.17) has a crucial limitation. In a Schwarzschild black hole, any freely falling objectwill hit the singularity in finite proper time, the effect of which on the stretched horizon It will be very interesting to explore if microscopic models of fast-scrambling, such as [49, 50] canbe used to implement these parity violating scrambling processes. – 21 –an not be analyzed in the near horizon approximation. However, one can argue thatwhen a massive particle hits the singularity, the spherical symmetry will be restored[54]. This should not be surprising because an order one perturbation will decay to size ∼ M P /m in one scrambling time t s , where m is the mass of the black hole [10]. Hence,this should be the time scale for any classical fields on a spherically symmetric horizonto become spherically symmetric. The AdS/CFT correspondence [2, 20–22] has successfully provided us with theoreticalcontrol over a large class of strongly interacting field theories. It is indeed remarkablethat observables of certain large-N gauge theories in d -dimensions can be calculated byperforming some classical gravity computations in ( d + 1)-dimensions. Gravity dualsof these field theories at finite temperature contain black holes, where the field theorytemperature is given by the Hawking temperature of black holes. It is well known that atvery long length scales the most dominant contributions to different non-local observablescome from the near horizon region of the dual black hole geometry [55, 56]. So, it isnot very surprising that there is some connection between the low energy hydrodynamicdescription of a strongly coupled quantum field theory and the membrane paradigm fluidon the black hole horizon of the dual gravity theory [23–27]. Long before the emergence ofgauge/gravity duality, it was shown that the membrane paradigm fluid on the stretchedhorizon of a black hole has a shear viscosity to entropy density ratio of 1 / π [3, 4]. Laterit was found that the shear viscosity to entropy density ratio of a gauge theory with agravity dual is indeed 1 / π [24, 27]. Interestingly, the strong coupling physics of quark-gluon plasma has been experimentally explored in the Relativistic Heavy Ion Collider(RHIC), where the shear viscosity to entropy density has been measured to be close to1 / π . On the other hand, it is also known that there are membrane paradigm resultswhich differ from the AdS/CFT values. For example, the bulk viscosity of a conformalfluid is exactly zero, whereas the membrane paradigm bulk viscosity is not only nonzerobut negative. Let us now consider a large-N gauge theory in (2 + 1) − dimensions which is dual to agravity theory in (3 + 1) − dimensions with the gravitational Θ-term S = (cid:90) d x √− g (cid:20) M P2 (cid:18) R − L (cid:19) + Θ4 R ∗ R (cid:21) , (5.1) See [57] for another example when membrane paradigm is incomplete in the context of the AdS/CFTcorrespondence. – 22 –here, cosmological constant is − /L . Note that this is different from the case con-sidered in [30], where Θ was dynamical. In the presence of a dynamical Θ( r ), theboundary theory exhibits Hall viscosity, where η A is proportional to Θ (cid:48) ( r ) at the hori-zon r = r H [30]. However, we are interested in the constant Θ case for which theΘ-term is a total derivative and hence η A = 0.Now we figure out the effect of the parity violating Θ-term on the dual field theory.A reasonable guess is that the boundary theory, similar to the membrane paradigm fluid,will acquire the third order parity violating “Hall viscosity-like” transport coefficient ϑ asdefined in equations (4.28-4.29), with ϑ = Θ. We will confirm this guess by performingan explicit computation. We will also show that in the presence of the gravitationalΘ-term, the two-point function of the energy-momentum tensor of the boundary theoryacquires a non-trivial contact term. The fractional part of this contact term does notdepend on the short distance physics and hence it is a meaningful physical observablein (2 + 1) − dimensional conformal field theory.The action (5.1) is not consistent with Dirichlet boundary condition, so we arerequired to add boundary terms for the variational principle to be well defined S = (cid:90) d x √− g (cid:20) M P2 (cid:18) R − L (cid:19) + Θ4 R ∗ R (cid:21) + S GHY + S b Θ + S ct . (5.3) S b Θ is a Gibbons-Hawking-York like boundary term for the Θ-term, defined in equation(4.5). Counterterm S ct that we need to add in order to get a finite boundary energymomentum tensor [65, 66] is the constructed with quantities intrinsic to the boundarygeometry. For (3 + 1) − dimensions, S ct is given by [66] S ct = 2 M P2 L (cid:90) d x √− h (cid:18) − L R (3) (cid:19) . (5.4)We are mainly interested in the contribution of the Θ-term to the boundary energymomentum tensor. From the membrane paradigm calculation (4.14), we already knowthat δ (cid:18)(cid:90) d x √− g Θ4 R ∗ R + S b Θ (cid:19) = − (cid:90) Σ d x (cid:112) | h | C mn δh mn (5.5) In [30], it was also assumed that Θ vanishes asymptotically. In (2+1) dimensions, the Hall viscosity contributes to the energy-momentum tensor in the firstorder in derivative expansion T ijH = − η A (cid:16) e ikl u k f jl + e jkl u k f il (cid:17) , (5.2)where, u i is the fluid velocity, e ikl is the Levi-Civita tensor and f kl = D (3) k u l + D (3) l u k − h kl D (3) m u m . Hall viscosity has been studied extensively in the context of Holography, e.g. see [58–64, 69] – 23 –here C mn is the Cotton-York tensor (4.15). Therefore, following [66], the Θ-termcontribution to the energy-momentum tensor of the dual field theory is given by T Θ ij = lim r b →∞ (cid:18) − r b Θ C ij L (cid:19) boundary , (5.6)where, the boundary of the asymptotically AdS (3+1) is at r = r b . Let us now consider the asymptotically AdS (3+1) spacetime, ds = 2 H ( r ) dvdr − r L f ( r ) dv + r L (cid:0) dx + dx (cid:1) . (5.7)where, both f ( r ) and H ( r ) go to 1 near the boundary. One can easily check that forthis metric, R ∗ R vanishes. Now we will perturb the metric in the following way, ds = 2 H ( r ) dvdr − r L f ( r ) dv + r L (cid:0) dx + dx (cid:1) + r L h AB dx A dx B , (5.8)where A, B = 1 , h ( r, v ) = h ( r, v ) , h ( r, v ) = − h ( r, v ) . (5.9)One can easily check that this set of perturbations decouple from the rest of the com-ponents. Now using equation (5.6), we obtain the contribution of the Θ-term on theenergy-momentum tensor of the (2 + 1) − dimensional boundary theory T Θ11 = − Θ f ( r b ) / (cid:18) ∂ h ∂v (cid:19) boundary , (5.10) T Θ12 = T Θ21 = Θ2 f ( r b ) / (cid:18) ∂ h ∂v − ∂ h ∂v (cid:19) boundary , (5.11) T Θ22 = Θ f ( r b ) / (cid:18) ∂ h ∂v (cid:19) boundary . (5.12)Note that we have obtained the energy-momentum tensor off-shell because we do notneed to solve the Einstein equations in order to write down the Θ-contributions to theenergy-momentum tensor. Similar observation was also made by [70]. Note that in [67], authors constructed (3 + 1) − dimensional bulk geometries for which the boundaryCotton-York tensor has the form of the energy momentum tensor of a perfect fluid. These solutions havenon-trivial boundary geometries which lead to interesting effects even in equilibrium. However, we areinterested in asymptotically AdS (3+1) spacetime where the Θ-term only affects the out of equilibriumdynamics. – 24 –luctuations of the boundary metric is given by: γ AB = h AB ( r = r b ). In theboundary theory, we are considering zero-momentum modes: γ AB ∼ e − iωt , where at theboundary v = t and hence T Θ11 = − T Θ22 = − Θ ∂ γ ∂t , T Θ12 = T Θ21 = Θ2 (cid:18) ∂ γ ∂t − ∂ γ ∂t (cid:19) . (5.13)In the last equation, we have used the fact that both f ( r ) and H ( r ) go to 1 nearthe boundary. Therefore, comparing equation (5.13) with (4.28-4.29), we find that the(2 + 1)-dimensional boundary theory has a nonzero ϑ , in particular ϑ = Θ , (5.14)which agrees with the membrane paradigm result. Interestingly, for a holographic the-ory ϑ is independent of the temperature. This is a consequence of the fact that theΘ-contribution to the boundary theory energy-momentum tensor is always a local func-tional of the boundary metric and does not depend on the interior geometry. This alsostrongly indicates that the transport coefficient ϑ is completely independent of the quan-tum state of the field theory. It is important to note that the most transport coefficientsmake sense only at finite temperature and in the low frequency limit ω/T →
0. However,this is not true for this new transport coefficient ϑ , which as we will show next arisesfrom a state-independent contact term in the energy-momentum tensor and hence insome sense is only probing the physics of the vacuum. We now compute the two-point function of the energy-momentum tensor of a (2 +1) − dimensional quantum field theory dual to a gravity theory in (3 + 1) − dimensionswith the gravitational Θ-term (5.1). We again consider asymptotically AdS (3+1) space-time (5.7) and calculate the contribution of the Θ-term to the two-point function of theenergy-momentum tensor. We perturb the metric: g µν + δg µν , such that δg rr = δg ri = 0at the boundary. The on-shell action ( S Θ + S b Θ ) in the second order in metric pertur-bations is obtained to be S Θ + S b Θ = Θ16 (cid:90) d p T ij ; mn γ ij ( p ) γ mn ( − p ) (5.15)where, γ ij = L r δg ij is the boundary metric perturbation and T ij ; mn = (cid:0) ε iml p l (cid:0) p j p n − p δ jn (cid:1) + ( i ↔ j ) (cid:1) + ( m ↔ n ) . (5.16) We are grateful to T. Hartman for a discussion that led to this subsection. In this subsection, we will be working in the Euclidean signature and we will use the convention γ ij ( x ) = π ) / (cid:82) d k e ip.x γ ij ( p ) for the Fourier transform. – 25 –ollowing the AdS/CFT dictionary, we can obtain the two-point function of the bound-ary energy-momentum tensor by varying the above quadratic action with respect to γ ij : (cid:104) T ij ( p ) T mn ( − p ) (cid:105) Θ = Θ2 T ij ; mn . (5.17)One can easily check that this gives rise to a conformally invariant contact term [28] (cid:104) T ij ( x ) T mn (0) (cid:105) Θ = − i Θ2 (cid:2)(cid:0) ε iml ∂ l (cid:0) ∂ j ∂ n − ∂ δ jn (cid:1) + ( i ↔ j ) (cid:1) + ( m ↔ n ) (cid:3) δ ( x ) . (5.18)It was shown in [28] that the two-point function of the energy-momentum tensor ina (2 + 1) − dimensional conformal field theory must have the following form (cid:104) T ij ( p ) T mn ( − p ) (cid:105) = − τ g | p | ( p ij p mn + p im p jn + p jm p in ) + κ g π T ij ; mn , (5.19)where, p ij = ( p i p j − p δ ij ) and T ij ; mn is given in (5.16). The term proportional to τ g controls the behavior of the correlation function at separated points, whereas, the termproportional to κ g leads to a pure contact term of the form (5.18). The coefficient κ g can take up any value, however, it is possible to shift κ g → κ g + δκ g by addinga gravitational Chern-Simons counterterm with coefficient δκ g to the UV Lagrangian.The gravitational Chern-Simons term, as argued in [28], is a valid counterterm only if δκ g is an integer. Therefore, the integer part of κ g is scheme-dependent, however, thefractional part κ g mod 1 does not depend on the short distance physics and hence itis a meaningful physical observable in (2 + 1) − dimensional conformal field theory [28].By comparing, (5.19) with (5.17), we conclude that a gravity theory in AdS (3+1) withthe gravitational Θ-term is dual to a conformal field theory with non-vanishing κ g , inparticular κ g π = Θ = ϑ . (5.20)This also suggests that Θ can take up any value, however, only a fractional part of theΘ-parameter is a well-defined observable. It is important to note that our result is anonlinear generalization of the contact term (5.18) and it is straight forward to obtainthe contribution of the Θ-term to the higher-point functions of the energy-momentumtensor of the boundary field theory. ϑ : Kubo formula Before we conclude, let us make some comments on the transport coefficient ϑ . It isa new third order, parity violating transport coefficient in (2 + 1) − dimensions whichunder a small metric perturbation γ AB around flat Minkwoski metric, contributes to the– 26 –nergy-momentum tensor (in the low momentum limit) in the following way: T = − T = − ϑ ∂ γ ∂t , (6.1) T = T = ϑ (cid:18) ∂ γ ∂t − ∂ γ ∂t (cid:19) . (6.2)Note that ϑ is dimensionless and it does not contribute to the trace of the energy-momentum tensor. It is also related to the contact term κ g , in particular ϑ = κ g / π .Very little is known about third-order transport coefficients in any dimensions and thetransport coefficient ϑ to our knowledge has never been studied before. This is a niceexample where gravity teaches us about a new hydrodynamic effect.In (2 + 1) − dimensional hydrodynamics, parity violating effect can appear in thefirst order in derivative expansion [68]. Hall viscosity is an example of such effect and ithas been studied for both relativistic [30, 69] and non-relativistic systems [31–33]. Webelieve that ϑ is a third order cousin of Hall viscosity and hence it should also contributeto Berry-like transport [34].Let us now derive the Kubo formula for ϑ . First, note that [69] (cid:104) T ij ( x ) (cid:105) γ = (cid:104) T ij ( x ) (cid:105) γ =0 − (cid:90) d x (cid:48) G Rij,kl ( x, x (cid:48) ) γ kl ( x (cid:48) ) + O ( γ ) , (6.3)where, i, j, k, l = 0 , , G Rij,kl ( x, x (cid:48) ) is the retardedGreen’s function of energy-momentum tensor G Rij,kl ( x, x (cid:48) ) = − iθ ( t − t (cid:48) ) (cid:104) [ T ij ( x ) , T kl ( x (cid:48) )] (cid:105) . (6.4)Similarly, one can define the retarded Green’s function in the momentum space simplyby performing a Fourier transformation G Rij,kl ( ω, (cid:126)k ) = (cid:90) d xe iωt − i(cid:126)k.(cid:126)x G Rij,kl ( x, . (6.5)Therefore, from (6.1-6.2), we find that ϑ contributes to the retarded Green’s function inorder ω : G R , − ( ω, (cid:126)k →
0) = − iϑω (6.6)which gives the Kubo formula for ϑ . Note that the Kubo formula (6.6) can also bederived directly from equation (5.19).We end this section by commenting on the possible covariant structure of the ϑ -contribution to the energy-momentum tensor. It is reasonable to guess that the ϑ -contribution to the energy-momentum tensor in the Landau gauge has the followingform: T ijϑ = − ϑ (cid:18) P ik P jl C kl − P ij P kl C kl (cid:19) (6.7)– 27 –ith P ij = g ij + u i u j , where g ij is the (2 + 1) − dimensional metric. One can easily checkthat the energy-momentum tensor (6.7) under a small metric perturbation γ AB aroundflat Minkwoski metric leads to (6.1-6.2) and hence will reproduce the retarded Green’sfunction formula (6.6). It will be nice to derive (6.7) directly by using the formalism offluid/gravity correspondence, however we will not attempt it here. We have shown that the gravitational Θ-term can have physical effect on the horizonof a black hole in (3 + 1)-dimensions. In particular, in the presence of the Θ-term, thehorizon acquires a third order parity violating, dimensionless transport coefficient ϑ ,which affects the way localized perturbations scramble on the horizon. This stronglysuggests that a sensible theory of quantum gravity should be able to provide a micro-scopic description of this effect. It will be very interesting to explore if the Θ-term hasany physical effect in the early universe.In the context of the AdS/CFT correspondence, the gravitational Θ-term is dual tofield theories with non-vanishing contact terms of the energy-momentum tensor. As aconsequence, in the presence of the Θ-term the (2 + 1) − dimensional dual gauge theoryacquires the same third order parity violating transport coefficient ϑ . We have studiedvarious properties of this new transport coefficient ϑ . Historically, gauge/gravity dualityhas played a significant role in hydrodynamics by discovering new universal effects [71–73]. This is another nice example where gauge/gravity duality teaches us about a newhydrodynamic effect. However, we would like to note that one could have found this hy-drodynamic effect even without knowing anything about the AdS/CFT correspondence,simply by studying the effect of the Θ-term on the stretched horizon.It is important to note that our conclusion about the effect of the Θ-term on thestretched horizon depends only on the near horizon geometry but not on the details ofthe metric and hence it can easily be generalized for arbitrary cosmological horizons.The AdS/CFT correspondence has taught us that the membrane paradigm fluid on theblack hole horizon and linear response of a strongly coupled quantum field theory in thelow frequency limit are related. However, it is not at all clear if this connection betweenthe membrane paradigm and holography goes beyond the AdS/CFT correspondence. Inparticular, it will be extremely interesting to figure out if the same conclusion is true forholographic models of cosmological spacetime. Acknowledgments
We would like to thank T. Hartman, T. Jacobson, S. Jain and M. Rangamani for theuseful discussions. The work of WF was supported by the National Science Foundationunder Grant Numbers PHY-1316033 and by the Texas Cosmology Center, which is– 28 –upported by the College of Natural Sciences and the Department of Astronomy at theUniversity of Texas at Austin and the McDonald Observatory. The work of SK wassupported by NSF grant PHY-1316222.
A Near horizon metric
We will denote the 3-dimensional absolute event horizon by H . We have defined a well-behaved time coordinate ¯ t on the horizon as well as spatial coordinates x A with A = 1 , H has coordinates x i ≡ (cid:0) ¯ t, x A (cid:1) with a metric h ij . We canalways choose coordinates x i ≡ (cid:0) ¯ t, x A (cid:1) such that null generator l = ∂∂ ¯ t . (A.1)Where we have chosen the spatial coordinates x A such that they comove with the horizongenerator l and h A = 0. The basis e i ≡ ( l, e A ) spans the horizon and l.e A = 0. Theinduced spatial metric on a constant ¯ t hypersurface is γ AB . Following [3], let us nowintroduce a future directed ingoing null vector k µ at each point on H which obeys l.k = − , k.e A = 0 . (A.2)There exists a unique congruence of null ingoing geodesics that are tangent to k µ onthe horizon. Horizon coordinates x i ≡ (cid:0) ¯ t, x A (cid:1) can be carried on these geodesics intothe near horizon region. We can use the affine parameter λ on the null geodesics as thefourth coordinate, where on the horizon H λ = 0 and k = − ∂∂λ . (A.3)We will use these coordinates ( λ, ¯ t, x A ) to explore the near horizon region of a blackhole. Since k µ is tangent to affinely parametrized null geodesic, it obeys k µ ∇ µ k ν = 0 (A.4)and this equation leads to ∂ λ g µλ = 0 . (A.5)Therefore, g µλ are λ − independent and from equation (A.2) in this coordinate system weobtain g λλ = 0 , g λA = 0 , g ¯ tλ = 1 . (A.6)On the horizon H : g ¯ t ¯ t = g ¯ tA = 0. From the equation (3.26) one can easily show that12 ∂ λ g ¯ t ¯ t = − g H (A.7)– 29 –nd hence near the horizon g ¯ t ¯ t = − g H λ + O ( λ ) , (A.8)where, g H is the surface gravity defined by equation (3.26). Similarly, one can write g ¯ tA = − A (¯ t, x ) λ + O ( λ ) (A.9)where Ω A (¯ t, x ) is the Hajicek field defined in the following wayΩ A = (cid:104) d ¯ t, ∇ A l (cid:105) . (A.10)This quantity is related to the angular momentum of a black hole. Therefore, near thehorizon space-time metric has the following form ds = − g H (¯ t, x ) λ d ¯ t + 2 d ¯ tdλ + γ AB ( λ, ¯ t, x ) (cid:0) dx A − A (¯ t, x ) λd ¯ t (cid:1) (cid:0) dx B − B (¯ t, x ) λd ¯ t (cid:1) + O ( λ ) . (A.11)We will now restrict to the simpler case: g H (¯ t, x ) =constant on the horizon H . For thecase of constant g H (and non-zero), we can define a new radial coordinate 2 g H λ = r and the near horizon geometry has the form ds = − r d ¯ t + 2 rg H d ¯ tdr + γ AB ( r, ¯ t, x ) (cid:18) dx A − Ω A (¯ t, x ) g H r d ¯ t (cid:19) (cid:18) dx B − Ω B (¯ t, x ) g H r d ¯ t (cid:19) + O ( r )(A.12)where the horizon is at r = 0. B Stretched horizon: transport coefficients
The fluid living on the stretched horizon is almost at rest in the comoving coordinates,i.e., u ¯ t = 1 (cid:15) , u A = O ( (cid:15) ) . (B.1)Therefore, u i = U i + O ( (cid:15) ), where U µ is the velocity of the FIDOs. We can now easilyshow that in the limit (cid:15) → D (3) i u i = 1 (cid:15) θ H (B.2)and f = f A = O ( (cid:15) ) , f AB = 2 (cid:15) σ HAB + O ( (cid:15) ) . (B.3)Let us now investigate the energy momentum tensor on the stretched horizon (3.24).We can rewrite equation (3.24) in the following form T ij = M P2 (cid:20)(cid:0) − K − K kl u k u l (cid:1) u i u j + 12 (cid:0) K − K kl u k u l (cid:1) P ij (cid:21) M + M P2 (cid:20) KP ij − K ij + (cid:0) K kl u k u l (cid:1) (cid:18) u i u j + 12 P ij (cid:19)(cid:21) M . (B.4)– 30 –herefore, comparing equation (3.42) with the last equation, we obtain, ρ = M P2 (cid:2) − K − K kl u k u l (cid:3) M , (B.5) p − ζD (3) m u m = M P2 (cid:20) (cid:0) K − K kl u k u l (cid:1)(cid:21) M , (B.6) − ηP ik P jl f kl = M P2 (cid:20) KP ij − K ij + (cid:0) K kl u k u l (cid:1) (cid:18) u i u j + 12 P ij (cid:19)(cid:21) M . (B.7)From equations (B.5) and (B.6), we can find out ρ , p and ζ easily. For η one needs tolook at the spatial components (i.e. ( ... ) AB components) of equations (B.7). So finallywe obtain [3] ρ = − M P2 (cid:15) θ H , (B.8) p = M P2 (cid:15) g H , (B.9) ζ = − M P2 , (B.10) η = M P2 . (B.11)We provide an alternative derivation of these relations in section 4.2. References [1] T. Banks, W. Fischler, S. H. Shenker and L. Susskind, “M theory as a matrix model: AConjecture,” Phys. Rev. D , 5112 (1997) [hep-th/9610043].[2] J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity,”Adv. Theor. Math. Phys. , 231 (1998) [hep-th/9711200].[3] R. H. Price and K. S. Thorne, “Membrane Viewpoint on Black Holes: Properties andEvolution of the Stretched Horizon,” Phys. Rev. D , 915 (1986).http://resolver.caltech.edu/CaltechAUTHORS:PRIprd86[4] K. S. Thorne, R. H. Price and D. A. Macdonald, “Black Holes: The MembraneParadigm,” NEW HAVEN, USA: YALE UNIV. PR. (1986) 367p[5] P. Anninos, D. Hobill, E. Seidel, L. Smarr and W. M. Suen, “The Headon collision oftwo equal mass black holes,” Phys. Rev. D , 2044 (1995)doi:10.1103/PhysRevD.52.2044 [gr-qc/9408041].[6] J. Masso, E. Seidel, W. M. Suen and P. Walker, “Event horizons in numerical relativity2.: Analyzing the horizon,” Phys. Rev. D , 064015 (1999)doi:10.1103/PhysRevD.59.064015 [gr-qc/9804059].[7] S. S. Komissarov, “Electrodynamics of black hole magnetospheres,” Mon. Not. Roy.Astron. Soc. , 407 (2004) doi:10.1111/j.1365-2966.2004.07446.x [astro-ph/0402403]. – 31 –
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