Entanglement entropy, the Einstein equation and the Sparling construction
aa r X i v : . [ h e p - t h ] J un Entanglement entropy, the Einstein equation andthe Sparling construction
Mahdi Godazgar
Institut f¨ur Theoretische Physik,Eidgen¨ossische Technische Hochschule Z¨urich,Wolfgang-Pauli-Strasse 27, 8093 Z¨urich, Switzerland. [email protected] 7, 2018
Abstract
We relate the recent derivation of the linearised Einstein equation on an AdS backgroundfrom holographic entanglement entropy arguments to the Sparling construction: wederive the differential form whose exterior derivative gives the Einstein equation fromthe Sparling formalism. We develop the study of perturbations within the context of theSparling formalism and find that the Sparling form vanishes for linearised perturbationson flat space.
Introduction
One of the main puzzles of AdS/CFT, or holography in general, is how the bulk geometryand, in particular, bulk locality arises from data in the CFT in question. This has leadto much interest recently. One interesting string of ideas is whether considering entangle-ment between different regions on the boundary theory may be used to discern importantproperties of the bulk geometry. In the context of general relativity, a seemingly unrelated problem is the suitable defini-tion of energy. The equivalence principle precludes a meaningful local definition of energy.However, one would hope to be able to define a vector that measures the energy-momentumin some region enclosed by, say, a ball of radius r > The idea behind this paper is that these two important problems may, in fact, be related.In recent work [3–5], the authors argue that the linearised Einstein equations on a( d + 1)-dimensional anti-de Sitter background can be derived by considering the change inthe entanglement entropy for a ball-shaped region A under a perturbation of the vacuumstate of the boundary CFT in a holographic set-up. The starting point in the derivation isthe first law of entanglement entropy [4], which states that the change in the entanglemententropy associated to region A is equal to the change in the expectation value of the modular,or entanglement, Hamiltonian. This first law is then translated to a gravitational first lawthat applies to an AdS-Rindler horizon in the bulk constructed using the Ryu-Takayanagi [6]prescription. This gravitational first law can be thought of as the AdS-Rindler analogue ofthe Iyer-Wald first law for asymptotically flat black hole horizons [7]. Then, the challengeof deriving the Einstein equation essentially translates to reversing the Iyer-Wald theoremthat gives the gravitational first law from the Einstein equation [3,5]. The result is that thelinearised Einstein equation is given as the exterior derivative of a ( d − d − d − For example, see Ref. [1] and references therein. For example, see Ref. [2] and references therein.
In this section, following Ref. [5], we briefly review the recent derivation of the linearisedEinstein equation using holographic entanglement entropy ideas [3, 5]. The philosophy inthis construction is to address the bulk locality puzzle by arguing that the linearised Einsteinequation around an anti-de Sitter background follows from small perturbations of the CFTvacuum state. Although, it turns out that the actual derivation of the linearised equationis independent of holography, holography justifies the gravitational “first law” that is used2o derive the Einstein equation.The starting point is the first law of entanglement entropy [4] δS A = δ h H A i , (2.1)where δS A is the first-order change in the entanglement entropy for a region A , whilethe right hand side is the first-order variation in the expectation value of the modular,or entanglement, Hamiltonian H A . Both the entanglement entropy and Hamiltonian aredefined via the reduced density matrix associated with region Aρ A = tr ¯ A ( ρ ); (2.2) S A = − tr ( ρ A log ρ A ) , ρ A = e − H A tr (e − H A ) . (2.3)It is not too difficult to derive the first law (2.1) from the definitions above as well as theconstraint that the reduced density matrix is unit trace [5].Now that we have such a law, which resembles/generalises the first law of thermodynam-ics, an obvious question in the context of holography is how this translates to a gravitationalstatement in the bulk. More precisely, we assume that the gravitational state correspondingto the CFT vacuum state is ( d + 1)-dimensional anti-de Sitter space in Poincar´e coordinatesd s = g µν d x µ d x ν = ℓ z (cid:16) d z + η αβ d X α d X β (cid:17) (2.4)with index µ = ( z, α ) and X α = ( t, x ˆ α ). A perturbation of the CFT vacuum state is goingto correspond, holographically, to a perturbed geometry about the AdS background. Thequestion, then, is what does the first law above, which constrains the admissible perturba-tions on the boundary, imply for the gravitational perturbations in the bulk?In general, this turns out to be a difficult problem. However, the case where the region A = B ( R, x ), corresponding to a ball of radius R , centre x , is well-understood [12, 13].Here, one identifies S A with S grav , the entropy associated with an AdS-Rindler wedge ˜ B at“temperature” T = 1 / (2 πR ), such that ∂ ˜ B = ∂B. Note that the boundary surfaces B and˜ B enclose a constant t hypersurface Σ. Moreover, the gravitational analogue of h H A i is thecanonical energy associated with the Killing vector ξ that generates the Rindler horizon,which we denote E [ ξ ]. On the hypersurface Σ, ξ ∝ ∂ t . Thus, in conclusion, we have agravitational statement to the effect that δS grav = δE [ ξ ] . (2.5)Were we considering an asymptotically flat stationary black hole solution with a bifurcateKilling horizon, i.e. non-zero surface gravity κ , normalised to κ = 2 π , and a static solution of3he linearised Einstein equation around the black hole background, then the above identityis the content of the Iyer-Wald theorem [7]. Therefore, essentially, what we are hoping toachieve is the reverse of the Iyer-Wald theorem applied to an AdS-Rindler background.A clue as to how to proceed is that the entropy, whether in the context of Einsteingravity where it corresponds to the area of the horizon or more generally, where it is givenby the Wald prescription, is given by an integral over the horizon, in this case ˜ B . Similarly,the canonical energy, as is to be expected of energy definitions in gravity, is given by anintegral over the boundary of the space; in this case anti-de Sitter space. As long as itis independent of the surface of integration, then we may define it as an integral over thesurface B. If the integrands in the two integrals are the same then we can use Stokes’theorem to relate their difference to an integral on Σ . The Iyer-Wald formalism provides a ( d − χ , the integrand of the presymplecticform, such that δS grav = 116 πG N Z ˜ B χ, δE [ ξ ] = 116 πG N Z B χ (2.6)and d χ = − ξ µ δG µν ǫ ν , (2.7)where δG µν is the linearised Einstein equation and ǫ ν is the volume form on a surface withnormal vector ∂/∂X ν . Moreover, the conservation and tracelessness of the CFT stress tensor gives thatd χ = 0 (2.8)on the AdS boundary, corresponding to the surface z = 0 , so that δE [ ξ ] is independent ofthe surface of integration. The above ingredients imply that0 = δS grav − δE [ ξ ]= 116 πG N Z ˜ B χ − πG N Z B χ = 116 πG N Z Σ d χ = − πG N Z Σ ξ t δG tt ǫ t , (2.9)where in the last line we have only the t -components of ξ µ and ǫ ν contributing, becausethese are the only non-zero components on Σ . Since, Σ is arbitrary, we conclude that δG tt = 0 . (2.10)4he above result was derived by considering a ball B in a constant t slice. However, we canequivalently consider another frame of reference and the above argument will go throughall the same. Thus, δG αβ = 0 . (2.11)The remaining components of the linearised Einstein equation are constraint equations ina radial slicing of the space formulated as an initial value problem. Thus, as long as theyare satisfied on the z = 0 surface, which they can be shown to be [5], then they hold for allvalues of z . This completes the derivation of the linearised Einstein equation from the firstlaw of entanglement entropy, but most importantly, as far as we are concerned, it relatesthe linearised Einstein equation to the exterior derivative of some ( d − χ = δ ( ∇ µ ξ ν ) ǫ µν , (2.12)where h µν = δg µν , the traceless and transverse perturbed metric, is defined via g µν = ◦ g µν + h µν (2.13)with background metric ◦ g µν and ǫ µν = 1( d − ǫ µνρ ...ρ ( d − dX ρ ∧ . . . ∧ dX ρ ( d − . (2.14)Choosing to work in a radial gauge, h µz = 0 (2.15)we find that on Σ = { t = t } [5] χ | Σ = − ξ t n ∂ z h ˆ α ˆ α ǫ tz − (cid:16) ∂ ˆ α h ˆ β ˆ β − ∂ ˆ β h ˆ α ˆ β (cid:17) ǫ t ˆ α o − πR n zh ˆ α ˆ α ǫ tz + h ( x ˆ α − x ˆ α ) h ˆ β ˆ β − ( x ˆ β − x ˆ β ) h ˆ α ˆ β i ǫ t ˆ α o , (2.16)where we have used the fact that in Poincar´e coordinates ξ = πR n(cid:2) R − z − ( t − t ) − ( x − x ) (cid:3) ∂ t − t − t ) h z∂ z + ( x ˆ α − x ˆ α ) ∂ ˆ α io . (2.17)Note that on Σ, only the t -component of ξ is non-zero. In general, we denote all background quantities with a circle on top, except when it is clear from thecontext. For example, in the expression ∇ µ h νρ , it is clear that the covariant derivative is with respect tothe background metric in order for the expression to remain first order. χ , the second term on the right hand side ofeqn. (2.16) cancels the derivative of ξ t in the first term, so thatd χ | Σ = − ξ t d n ∂ z h ˆ α ˆ α ǫ tz − (cid:16) ∂ ˆ α h ˆ β ˆ β − ∂ ˆ β h ˆ α ˆ β (cid:17) ǫ t ˆ α o . (2.18)In summary, on Σ δG tt ξ t ǫ t = ξ t d n ∂ z h ˆ α ˆ α ǫ tz − (cid:16) ∂ ˆ α h ˆ β ˆ β − ∂ ˆ β h ˆ α ˆ β (cid:17) ǫ t ˆ α o . (2.19) In general relativity, the equivalence principle means that a local definition of energy isimpossible. Given that the equations are second-order, one would expect the energy-momentum density at a point to be first order in the gravitational field. However, a localcoordinate transformation can then be used to set this to zero. Thus, a reasonable expecta-tion is that a quasi-local definition of energy-momentum ought to be pseudo-tensorial. Asoverwhelming as this may seem, one could view the pseudo-tensors in the different framesas being pull-backs in different local sections of some bundle on which a canonical expres-sion for the energy-momentum is defined. Indeed, this was the motivation for Sparling’sconstruction [8–10], which we review in this section. Although the original constructionis for a four-dimensional space, one can simply construct similar objects in higher dimen-sions [11]. However, here, we keep to four dimensions, since this is sufficient to get the mainideas across without introducing more notation, albeit simple.Consider an orthonormal frame θ a . The Cartan equations, for vanishing torsion readd θ a + ω ab ∧ θ b = 0 , (3.1)d ω ab + ω ac ∧ ω cb = Ω ab , (3.2)where ω ab is the spin connection, which we choose to be anti-symmetric. This correspondsto a choice of a metric compatible connection ∂ µ θ νa − Γ ρµν θ ρa + ( ω µ ) ab θ νb = 0 (3.3)with Γ ρµν , the Christoffel symbols { ρµν } . The two-form Ω ab parametrises the Riemann tensorΩ ab = R abcd θ c ∧ θ d . (3.4) The original construction of Sparling’s is defined on the spin bundle. However, for our purposes it willbe more useful to work with the orthonormal frame bundle [14]. E a = ∗ Ω ab ∧ θ b . (3.5)Expanding out the expression above gives ∗ Ω ab ∧ θ b = 14 η abcd R cdef θ b ∧ θ e ∧ θ f . (3.6)Now, substituting the fact that θ b ∧ θ e ∧ θ f = η befg ζ g (3.7)for some one form ζ a gives ∗ Ω ab ∧ θ b = ∗ R ∗ ab bc ζ c . (3.8)But, of course, ∗ R ∗ ab bc = − G ac , where G ab is the Einstein tensor contracted into the framecomponents. In conclusion, E a = − G ab ζ b . (3.9)On the other hand, making use of the Cartan equations (3.1) and (3.2), one can show that E a = d W a + S a , (3.10)where the two-form (or more generally ( d − W a = η abcd ω bc ∧ θ d (3.11)is known as the Witten-Nester form. It corresponds to the two-form integrated on theasymptotic boundary of a general spacelike hypersurface in Witten [15] and Nester’s [16]proofs of the positive ADM mass theorem [17]. The three-form (or more generally ( d − S a is the Sparling form S a = η abcd (cid:0) ω ce ∧ ω ed ∧ θ b − ω cd ∧ ω be ∧ θ e (cid:1) . (3.12)Note that while E a is clearly horizontal, W a and S a are not. They depend on the particularchoice of the orthonormal frame θ a . From eqn. (3.10), we conclude that the Sparling formis exact if, and only if, the vacuum Einstein equation is satisfied.7
The linearised Einstein equation in the Sparling construc-tion
In sections 2 and 3, we found that the Einstein equation (or its linearisation) can be relatedto the exterior derivative of a two-form in four dimensions and ( d − θ µa = e µa + f µa (4.1)so that the perturbed part of the metric h µν , defined via g µν = ◦ g µν + h µν , (4.2)is equal to h µν = 2 e ( µa f ν ) a . (4.3)Henceforth, all equations will be written to first order in the perturbation parameter. Theinverse vielbein θ µa = e µa − e µb e ν a f νb . (4.4)Similarly, the spin connection decomposes as ω ab = ◦ ω ab + a ab , (4.5)where the perturbed piece( a µ ) ab = e ν c e τ b f τ c ∂ µ e νa − e ν b ∂ µ f νa + ◦ Γ ρµν f τ c ( e ρa e ν c e τ b + e τ a e νb e ρ c )+ e σ a e ν b ( ∂ µ h σν + ∂ ν h σµ − ∂ σ h µν ) (4.6)has been calculated using the metric compatibility condition (3.3).The objects in the Sparling equation (3.10) are constructed from the vielbein θ a and thespin connection ω ab . Hence, also we can decompose these into background and perturbedpieces E a = ◦ E a + δE a , W a = ◦ W a + w a , S a = ◦ S a + s a (4.7)8ith ◦ E a = − ◦ G ab ◦ ζ b , ◦ W a = η abcd ◦ ω bc ∧ e d , ◦ S a = η abcd (cid:16) ◦ ω ce ∧ ◦ ω ed ∧ e b − ◦ ω cd ∧ ◦ ω be ∧ e e (cid:17) (4.8)and δE a = − ◦ G ab δζ b − δG ab ◦ ζ b , (4.9) w a = η abcd (cid:16) ◦ ω bc ∧ f d + a bc ∧ e d (cid:17) , (4.10) s a = η abcd (cid:16) a ce ∧ ◦ ω ed ∧ e b + ◦ ω ce ∧ a ed ∧ e b + ◦ ω ce ∧ ◦ ω ed ∧ f b − a cd ∧ ◦ ω be ∧ e e − ◦ ω cd ∧ a be ∧ e e − ◦ ω cd ∧ ◦ ω be ∧ f e (cid:17) . (4.11)This splits the Sparling equation into a background piece, which the background quanti-ties will satisfy, and most importantly a perturbed piece that gives the linearised Einsteinequation δG ab , which appears in the expression for δE a , in terms of the exterior derivativeof w a and the perturbed Sparling form s a − δG ab ◦ ζ b = ◦ G ab δζ b + d w a + s a . (4.12)For a traceless, transverse perturbation the linearised Einstein tensor is simply the Lich-nerowitz operator on h µν , which coincides with the background wave equation for the com-ponents of h µν . Before we consider the anti-de Sitter case, which will allow us to relate the Sparling con-struction to the linearised Einstein equation derived in Ref. [5], we consider first the simplestcase of a perturbation on a flat background. Recall that the Sparling construction dependson the choice of basis. We choose to work with the simplest basis for which the vielbein,viewed as a matrix, is the identity θ µa = δ µa + f µa . (4.13)In this basis the background spin connection vanishes ◦ ω ab = 0 (4.14)and, of course ◦ G ab = 0 . (4.15)9lugging these expressions into the definitions above gives that the Sparling form vanishes S a = 0 (4.16)and δG ab ◦ ζ b = d (cid:0) − η abcd a bc ∧ δ d (cid:1) . (4.17)Hence, for perturbations on flat space we find that the linearised Einstein equation is givenby the exterior derivative of 2-form w a as given above.This result is related to the fact that in the weak field approximation that we areconsidering here, the energy-momentum tensor of the field h µν is second-order. Thus, atthe linearised level h µν does not contribute to the total energy [18]. Moving on to the AdS case, as before, we proceed by choosing a background vierbein. Byinspecting the background metric, AdS space in Poincar´e coordinates (2.4), we choose e µa = ℓz δ µa . (4.18)Moreover, we choose to work in radial gauge in which h µz = 0. Hence, we have the freedomto set f za = f µ ˆ z = 0 . (4.19)Moreover, we are free to set f t ˆ x = f t ˆ y = f x ˆ y = 0 . (4.20)In this basis, ◦ ω ab = − ℓ e i , a = i, b = ˆ z ℓ e i , a = ˆ z, b = i , otherwise , (4.21)where a = (ˆ z, i ). Similarly, the only non-vanishing components of ◦ Γ cab are ◦ Γ aaz = ◦ Γ aza = − z , η ii ◦ Γ zii = 1 z , (4.22)where we do not sum over repeated indices in the expressions above. The backgroundEinstein tensor ◦ G ab = 3 ℓ η ab . (4.23)Now, we can go ahead and substitute all these quantities into the eqns. (4.8)–(4.11) derivedbefore. However, the expressions we would obtain would not be as simple as those derived10or the flat case in the previous section. Therefore, we focus on the set-up considered insection 2. We consider a hypersurface Σ = { t = t } and investigate the ˆ t -component of thepseudo-tensors that appear in the Sparling equation (3.10). This will allow us to derive theanalogue of eqn. (2.19) in the Sparling formalism.Before we go on to look at the perturbed quantities, which includes the linearised Ein-stein equation, let us briefly verify that the background Sparling equation is indeed satis-fied, as expected. Working in conventions in which η = 1, where, henceforth we identify { ˆ t, ˆ x, ˆ y, ˆ z } with { , , , } , we find that on Σ ◦ E = − ℓ e ∧ e ∧ e , ◦ W = 2 ℓ e ∧ e , ◦ S = 1 ℓ e ∧ e ∧ e . (4.24)Using the fact that d ◦ W = − ℓ e ∧ e ∧ e (4.25)it is clear that ◦ E = d ◦ W + ◦ S . (4.26)Note, in particular, that ◦ S can be written as an exact form ◦ S = d (cid:2) − ℓ − e ∧ e (cid:3) . (4.27)Next, let us consider the perturbed quantities. From eqn. (4.9), δE = − δG ◦ ζ − z ( f x + f y ) d x ∧ d y ∧ d z. (4.28)Similarly, from eqn. (4.10) w = 3 a [12 ∧ e + 1 z ( f x + f y ) d x ∧ d y. (4.29)A straightforward calculation using the fact that( a x ) = zℓ ∂ z f x , ( a y ) = zℓ ∂ z f y , (4.30)gives that s as defined in eqn. (4.11) reduces to s = − z ∂ z ( f x + f y ) d x ∧ d y ∧ d z. (4.31)Substituting the above expressions into eqn. (4.12) with a = 0 and simplifying gives − δG ◦ ζ = d (cid:16) a [12 ∧ e (cid:17) + 2 z ( f x + f y ) d x ∧ d y ∧ d z. (4.32)From the definition of one-form a ab (4.6), we find that( a z ) a = 0 , ( a x ) = zℓ ( ∂ y f x − ∂ x f y ) , ( a y ) = − zℓ ∂ x f y . (4.33)11ogether with equations (4.30), they give that eqn. (4.32) reduces to − δG ◦ ζ = d h ( ∂ y f x − ∂ x f y ) d x ∧ d z − ∂ x f y d y ∧ d z − ∂ z ( f x + f y ) d x ∧ d y i + 2 z ( f x + f y ) d x ∧ d y ∧ d z. (4.34)In fact, the above equation simplifies to δG tt ◦ ζ t = d (cid:26) − ℓz (cid:20) ( ∂ y f x − ∂ x f y ) d x ∧ d z − ∂ x f y d y ∧ d z − ∂ z [ z ( f x + f y )] z d x ∧ d y (cid:21)(cid:27) . (4.35)Comparing the expression above for the linearised Einstein equation with that which appearsin section (2), eqn. (2.19), we identify ◦ ζ t with ǫ t , the volume form on the hypersurface Σ: ◦ ζ t = − ǫ t (4.36)Moreover, we expect the two two-forms that appear on the right hand side of these respectiveequations to be related, possibly up to an exact one-form and this is what we show in thefollowing.From the definition of h µν (4.3), it follows that h ˆ α ˆ α = 2 zℓ ( f x + f y ) . (4.37)Moreover, ∂ x h ˆ α ˆ α − ◦ g xx ∂ ˆ α h x ˆ α = z ℓ (cid:0) ∂ x f y − ∂ y f y (cid:1) ,∂ y h ˆ α ˆ α − ◦ g yy ∂ ˆ α h y ˆ α = z ℓ (cid:0) ∂ y f x − ∂ x f y (cid:1) . (4.38)and from eqn. (2.14)d x ∧ d z = − z ℓ ǫ ty , d y ∧ d z = z ℓ ǫ tx , d x ∧ d y = z ℓ ǫ tz . (4.39)Using the above equations, eqn. (4.35) can be written as δG tt ǫ t = d (cid:26) ∂ z h ˆ α ˆ α ǫ tz − (cid:16) ∂ ˆ α h ˆ β ˆ β − ∂ ˆ β h ˆ α ˆ β (cid:17) ǫ t ˆ α − d (cid:20) ℓz f y d z (cid:21)(cid:27) . (4.40)Now comparing the equation above, derived from the Sparling construction and eqn. (2.19),derived from the first law, we find that the two two-forms whose exterior derivatives givesthe linearised Einstein equation match up to an exact termd (cid:20) ℓz f y d z (cid:21) = d( h xy d z ) . (4.41)12 Conclusions
We have found the potential in Ref. [5] whose exterior derivative gives the linearised Einsteinequation (or more precisely its tt -component) from the Sparling formalism. Whereas in thecase of perturbations on flat space, we found that the Sparling form vanishes and theexterior derivative of the Witten-Nester form gives the linearised Einstein equation, evenfor a background as simple as anti-de Sitter we were not able to make as general a statementand had to instead consider an ADM slicing of the spacetime. In this case, the Sparlingform on the ADM hypersurface becomes (off-shell) exact. An obvious question is underwhat conditions the Sparling form vanishes or becomes off-shell exact? Furthermore, canone gain a geometric understanding of why this happens? An equivalent question is underwhat conditions (symmetry or otherwise) can the Einstein tensor, and hence the vacuumEinstein equation, be written as the exterior derivative of some ( d − On the other hand, from the Iyer-Wald formalism (see, in particular, equa-tion (2.6)) we know [7] that the integral of this two-form over asymptotic spacelike infinitygives the canonical energy, which coincides with the ADM mass [7]. Thus, we identify theIyer-Wald two-form with the Witten-Nester two-form. Beyond the scope of asymptoticallyflat spaces, we have demonstrated in this paper that the same correspondence holds forasymptotically AdS spaces. A possible application of these ideas could be in the context offlat space holography.Within the context of AdS holography, can the relation with the Sparling formalism,which gives the full non-linear Einstein tensor, allow one to do better than to derive sim-ply the linearised Einstein equation from holographic arguments? In many respects, thefull, non-linear, Sparling construction is much simpler and intuitive than the linearised ver-sion, which we derived here. This fact gives rise to reasonable optimism that the Sparling In fact, one may recognise that the second term in eqn. (4.40) multiplying the two-form ǫ t ˆ α is preciselythe same in structure as that which one would integrate to find the ADM mass. The coincidence of thesetwo expressions here is more than notational. Acknowledgements
I am indebted to David Skinner for discussions and remarks that initiated this study. Thiswork is partially supported by grant no. 615203 from the European Research Council underthe FP7.
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