Entanglement entropy, the Einstein equation and the Sparling construction

We relate the recent derivation of the linearised Einstein equation on an AdS background from holographic entanglement entropy arguments to the Sparling construction: we derive the differential form whose exterior derivative gives the Einstein equation from the Sparling formalism. We develop the study of perturbations within the context of the Sparling formalism and find that the Sparling form vanishes for linearised perturbations on flat space.


Introduction
One of the main puzzles of AdS/CFT, or holography in general, is how the bulk geometry and, in particular, bulk locality arises from data in the CFT in question. This has lead to much interest recently. One interesting string of ideas is whether considering entanglement between different regions on the boundary theory may be used to discern important properties of the bulk geometry. 1 In the context of general relativity, a seemingly unrelated problem is the suitable definition 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-momentum in some region enclosed by, say, a ball of radius r > 0-a quasi-local definition of energy.
Beyond the scope of stationary asymptotically flat/AdS spaces, it is not even clear how to formulate a suitable definition of energy for a whole space. 2 The idea behind this paper is that these two important problems may, in fact, be related.
In recent work [3][4][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 in the entanglement entropy for a ball-shaped region A under a perturbation of the vacuum state of the boundary CFT in a holographic set-up. The starting point in the derivation is the first law of entanglement entropy [4], which states that the change in the entanglement entropy 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 law that 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 of the Iyer-Wald first law for asymptotically flat black hole horizons [7]. Then, the challenge of deriving the Einstein equation essentially translates to reversing the Iyer-Wald theorem that gives the gravitational first law from the Einstein equation [3,5]. The result is that the linearised Einstein equation is given as the exterior derivative of a (d − 1)-form. The fact that the Einstein equation can be written in terms of the exterior derivative of a differential form is characteristic of the Sparling construction [8][9][10][11].
The Sparling construction can be best thought of, at least for our purposes, as a construction in the orthonormal frame bundle that gives the Einstein tensor as the sum of the exterior derivative of a (d−2)-form, the Witten-Nester form and a (d−1)-form, the Sparling form. Importantly, the two differential forms that enter this equation are not horizontal. 1 For example, see Ref. [1] and references therein. 2 For example, see Ref. [2] and references therein.
Hence, the construction depends non-covariantly on the choice of section. An important corollary of this construction is that given an orthonormal frame such that the associated metric satisfies the vacuum Einstein equation, the Sparling form becomes exact and equal to the exterior derivative of the Witten-Nester form. Both of these pseudo-tensors play an important rôle in the understanding of energy in general relativity, such as the Witten proof of the positive ADM mass theorem and similar positivity of mass proofs, Penrose's definition of quasi-local energy and the Hamiltonian of general relativity as expressed through the Ashtekar variables (see Ref. [10]).
In this paper, we relate the recent holographic derivation of the linearised Einstein equation to the Sparling construction by rederiving the differential form whose exterior derivative gives the linearised Einstein equation from the Sparling formalism. We demonstrate this correspondence explicitly for four dimensions only in the interests of clarity. A similar analysis for higher dimensions will be straightforward and along essentially the same lines as that used in four dimensions. An important ingredient in demonstrating this relation is formulating linearised perturbation theory within the Sparling formalism. Rather interestingly, though not surprisingly, we find that for linearised perturbations around flat space the Sparling form vanishes identically and the linearised Einstein equation is always given as the exterior derivative of the perturbed Witten-Nester form.
In section 2, following Ref. [5] we review the derivation of the linearised Einstein equation from holographic entropy arguments [3,5] and in section 3, we review the Sparling construction. In section 4, we develop the formalism needed to consider perturbations in the context of the Sparling construction and briefly consider perturbations on flat space in section 4.1, before moving on to the main interest of the paper in section 4.2: perturbations on AdS space, where we also derive the linearised Einstein equation as the exterior derivative of a two-form and show that this is the same two-form as that found in Ref. [5] up to an exact form. We end with some conclusions and outlook.

Holographic entanglement and the Einstein equation
In this section, following Ref. [5], we briefly review the recent derivation of the linearised Einstein equation using holographic entanglement entropy ideas [3,5]. The philosophy in this construction is to address the bulk locality puzzle by arguing that the linearised Einstein equation around an anti-de Sitter background follows from small perturbations of the CFT vacuum state. Although, it turns out that the actual derivation of the linearised equation is independent of holography, holography justifies the gravitational "first law" that is used to derive the Einstein equation.
The starting point is the first law of entanglement entropy [4] where δS A is the first-order change in the entanglement entropy for a region A, while the 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 are defined via the reduced density matrix associated with region A It is not too difficult to derive the first law (2.1) from the definitions above as well as the constraint that the reduced density matrix is unit trace [5].
Now that we have such a law, which resembles/generalises the first law of thermodynamics, an obvious question in the context of holography is how this translates to a gravitational statement in the bulk. More precisely, we assume that the gravitational state corresponding to the CFT vacuum state is (d + 1)-dimensional anti-de Sitter space in Poincaré coordinates with index µ = (z, α) and X α = (t, xα). A perturbation of the CFT vacuum state is going to correspond, holographically, to a perturbed geometry about the AdS background. The question, then, is what does the first law above, which constrains the admissible perturbations 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 0 ), corresponding to a ball of radius R, centre x 0 , is well-understood [12,13].
Here, one identifies S A with S grav , the entropy associated with an AdS-Rindler wedgeB at Were we considering an asymptotically flat stationary black hole solution with a bifurcate Killing horizon, i.e. non-zero surface gravity κ, normalised to κ = 2π, and a static solution of the linearised Einstein equation around the black hole background, then the above identity is the content of the Iyer-Wald theorem [7]. Therefore, essentially, what we are hoping to achieve 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 Einstein gravity where it corresponds to the area of the horizon or more generally, where it is given by the Wald prescription, is given by an integral over the horizon, in this caseB. Similarly, the canonical energy, as is to be expected of energy definitions in gravity, is given by an integral over the boundary of the space; in this case anti-de Sitter space. As long as it is independent of the surface of integration, then we may define it as an integral over the surface 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 − 1)-form χ, the integrand of the presymplectic form, such that where δG µν is the linearised Einstein equation and ǫ ν is the volume form on a surface with normal vector ∂/∂X ν .
Moreover, the conservation and tracelessness of the CFT stress tensor gives that dχ = 0 (2.8) on the AdS boundary, corresponding to the surface z = 0, so that δE[ξ] is independent of the surface of integration. The above ingredients imply that where in the last line we have only the t-components of ξ µ and ǫ ν contributing, because these are the only non-zero components on Σ. Since, Σ is arbitrary, we conclude that δG tt = 0. (2.10) The above result was derived by considering a ball B in a constant t slice. However, we can equivalently consider another frame of reference and the above argument will go through all the same. Thus, The remaining components of the linearised Einstein equation are constraint equations in a radial slicing of the space formulated as an initial value problem. Thus, as long as they are satisfied on the z = 0 surface, which they can be shown to be [5], For Einstein gravity [5] where h µν = δg µν , the traceless and transverse perturbed metric, is defined via with background metric 3 • g µν and (2.14) Choosing to work in a radial gauge, where we have used the fact that in Poincaré coordinates Note that on Σ, only the t-component of ξ is non-zero.
Upon taking the exterior derivative of χ, the second term on the right hand side of eqn. (2.16) cancels the derivative of ξ t in the first term, so that

The Sparling form
In general relativity, the equivalence principle means that a local definition of energy is impossible. Given that the equations are second-order, one would expect the energymomentum density at a point to be first order in the gravitational field. However, a local coordinate transformation can then be used to set this to zero. Thus, a reasonable expectation is that a quasi-local definition of energy-momentum ought to be pseudo-tensorial. As overwhelming as this may seem, one could view the pseudo-tensors in the different frames as being pull-backs in different local sections of some bundle on which a canonical expression for the energy-momentum is defined. Indeed, this was the motivation for Sparling's construction [8][9][10], which we review in this section. 4 Although the original construction is for a four-dimensional space, one can simply construct similar objects in higher dimensions [11]. However, here, we keep to four dimensions, since this is sufficient to get the main ideas across without introducing more notation, albeit simple.
Consider an orthonormal frame θ a . The Cartan equations, for vanishing torsion read where ω ab is the spin connection, which we choose to be anti-symmetric. This corresponds to a choice of a metric compatible connection with Γ ρ µν , the Christoffel symbols { ρ µν }. The two-form Ω a b parametrises the Riemann tensor The main observation in the Sparling construction is that the vanishing of the Einstein tensor, i.e. the vacuum Einstein equation, is equivalent to the vanishing of Expanding out the expression above gives * Ω ab ∧ θ Now, substituting the fact that 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 a c , where G ab is the Einstein tensor contracted into the frame components. In conclusion, On the other hand, making use of the Cartan equations (3.1) and (3.2), one can show that where the two-form (or more generally (d − 2)-form) is known as the Witten-Nester form. It corresponds to the two-form integrated on the asymptotic 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 − 1)form) denoted S a is the Sparling form

The linearised Einstein equation in the Sparling construction
In sections 2 and 3, we found that the Einstein equation (or its linearisation) can be related to the exterior derivative of a two-form in four dimensions and (d − 2)-form in general. In this section, we relate these two constructions by considering perturbations in the Sparling construction. As in section 3, we work in four dimensions. However, our results will almost trivially generalise to higher dimensions.
We consider a linearised perturbation on a background solution. Accordingly, we split the vierbein into a background and perturbed piece so that the perturbed part of the metric h µν , defined via is equal to Henceforth, all equations will be written to first order in the perturbation parameter. The inverse vielbein Similarly, the spin connection decomposes as where the perturbed piece 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 the spin connection ω a b . Hence, also we can decompose these into background and perturbed pieces and 10)

Perturbations on a flat background
Before we consider the anti-de Sitter case, which will allow us to relate the Sparling construction to the linearised Einstein equation derived in Ref. [5], we consider first the simplest case of a perturbation on a flat background. Recall that the Sparling construction depends on the choice of basis. We choose to work with the simplest basis for which the vielbein, viewed as a matrix, is the identity In this basis the background spin connection vanishes • ω ab = 0 (4.14) and, of course Plugging these expressions into the definitions above gives that the Sparling form vanishes S a = 0 (4. 16) and Hence, for perturbations on flat space we find that the linearised Einstein equation is given by 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 are considering here, the energy-momentum tensor of the field h µν is second-order. Thus, at the linearised level h µν does not contribute to the total energy [18].

Perturbations on an AdS background
Moving on to the AdS case, as before, we proceed by choosing a background vierbein. By In this basis, where a = (ẑ, i). Similarly, the only non-vanishing components of (4.24) Using the fact that it is clear that Note, in particular, that • S 0 can be written as an exact form (4.27) Next, let us consider the perturbed quantities. From eqn. (4.9), Similarly, from eqn. (4.10) Together with equations (4.30), they give that eqn. (4.32) reduces to −δG 00 We have found the potential in Ref. [5]  Going back to the main motivation of the paper emphasised in the introduction, i.e.
the relationship between the holographic emergence of gravity on the one hand and suitable definitions of energy in general relativity on the other, there are hints of how one may proceed. Focusing on asymptotically flat spaces, we have shown that from the Sparling construction perspective, the two-form that yields the linearised Einstein equation is the Witten-Nester two-form (see section 4.1). However, we know [15][16][17] that for an asymptotically flat space the integral of this two-form over asymptotic spacelike infinity gives the ADM mass. 5 On the other hand, from the Iyer-Wald formalism (see, in particular, equation (2.6)) we know [7] that the integral of this two-form over asymptotic spacelike infinity gives the canonical energy, which coincides with the ADM mass [7]. Thus, we identify the Iyer-Wald two-form with the Witten-Nester two-form. Beyond the scope of asymptotically flat spaces, we have demonstrated in this paper that the same correspondence holds for asymptotically AdS spaces. A possible application of these ideas could be in the context of flat 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 simply the linearised Einstein equation from holographic arguments? In many respects, the full, non-linear, Sparling construction is much simpler and intuitive than the linearised version, which we derived here. This fact gives rise to reasonable optimism that the Sparling 5 In fact, one may recognise that the second term in eqn. (4.40) multiplying the two-form ǫ tα is precisely the same in structure as that which one would integrate to find the ADM mass. The coincidence of these two expressions here is more than notational.

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formalism has much more to say about holography and the emergence of bulk locality.