Entanglement equilibrium for higher order gravity
We show that the linearized higher derivative gravitational field equations are equivalent to an equilibrium condition on the entanglement entropy of small spherical regions in vacuum. This extends Jacobson’s recent derivation of the Einstein equation using entanglement to include general higher derivative corrections. The corrections are naturally associated with the subleading divergences in the entanglement entropy, which take the form of a Wald entropy evaluated on the entangling surface. Variations of this Wald entropy are related to the field equations through an identity for causal diamonds in maximally symmetric spacetimes, which we derive for arbitrary higher derivative theories. If the variations are taken holding fixed a geometric functional that we call the generalized volume, the identity becomes an equivalence between the linearized constraints and the entanglement equilibrium condition. We note that the fully nonlinear higher curvature equations cannot be derived from the linearized equations applied to small balls, in contrast to the situation encountered in Einstein gravity. The generalized volume is a novel result of this work, and we speculate on its thermodynamic role in the first law of causal diamond mechanics, as well as its possible application to holographic complexity.
Black hole entropy remains one of the best windows into the nature of quantum gravity available to dwellers of the infrared. Bekenstein’s original motivation for introducing it was to avoid gross violations of the second law of thermodynamics by sending matter into the black hole, decreasing the entropy of the exterior Bekenstein (1972, 1973). The subsequent discovery by Hawking that black holes radiate thermally at a temperature , with the surface gravity, fixed the value of the entropy in terms of the area to be and suggested a deep connection to quantum properties of gravity Hawking (1974).
The appearance of area in is somewhat mysterious from a classical perspective; however, an intriguing explanation emerges by considering the entanglement entropy of quantum fields outside the horizon Sorkin (1983); Bombelli et al. (1986); Srednicki (1993); Frolov and Novikov (1993). Entanglement entropy is UV divergent, and upon regulation it takes the form
with a regulator and a constant. Identifying the coefficient with would allow to be attributed to the leading divergence in the entanglement entropy. The subleading divergences could similarly be associated with higher curvature gravitational couplings, which change the expression for the black hole entropy to the Wald entropy Wald (1993).
To motivate these identifications, one must assume that the quantum gravity theory is UV finite (as occurs in string theory), yielding a finite entanglement entropy, cut off near the Planck length, . Implementing this cutoff would seem to depend on a detailed knowledge of the UV theory, inaccessible from the vantage of low energy effective field theory. Interestingly, this issue can be resolved within the effective theory by the renormalization of the gravitational couplings by matter loop divergences. There is mounting evidence that these precisely match the entanglement entropy divergences, making the generalized entropy
independent of Susskind and Uglum (1994); Cooperman and Luty (2014); Solodukhin (2011); Bousso et al. (2016). Here is the Wald entropy expressed in terms of the renormalized gravitational couplings and is a renormalized entanglement entropy of matter fields that is related to in (1), although the precise relation depends on the renormalization scheme.111A covariant regulator must be used to ensure that the subleading divergences appear as a Wald entropy. Also, since power law divergences are not universal, when they are present the same renormalization scheme must be used for the entanglement entropy and the gravitational couplings. Additional subtleties for nonminimally coupled fields, gauge fields, and gravitons are discussed in Section V.4.3. The identification of gravitational couplings with entanglement entropy divergences is therefore consistent with the renormalization group (RG) flow in the low energy effective theory, and amounts to assuming that the bare gravitational couplings vanish Jacobson (1994). In this picture, , with acting as a placeholder for the UV degrees of freedom that have been integrated out.
When viewed as entanglement entropy, it is clear that generalized entropy can be assigned to surfaces other than black hole horizon cross sections Jacobson (1999); Jacobson and Parentani (2003); Bianchi and Myers (2012); Bousso et al. (2016). For example, in holography the generalized entropy of a minimal surface in the bulk is dual via the quantum-corrected Ryu-Takayanagi formula Ryu and Takayanagi (2006); Faulkner et al. (2013a) to the entanglement entropy of a region of the boundary CFT.222The UV divergences in the CFT entanglement entropy have no relation to the Planck length in the bulk, but instead are related to the infinite area of the minimal surface in AdS, courtesy of the UV/IR correspondence. Even without assuming holographic duality, the generalized entropy provides a link between the geometry of surfaces and entanglement entropy. When supplemented with thermodynamic information, this link can give rise to dynamical equations for gravity. The first demonstration of this was Jacobson’s derivation of the Einstein equation as an equation of state for local causal horizons possessing an entropy proportional to their area Jacobson (1995). Subsequent work using entropic arguments Verlinde (2010, 2016) and holographic entanglement entropy Lashkari et al. (2013); Faulkner et al. (2013b); Swingle and Van Raamsdonk (2014) confirmed that entanglement thermodynamics is connected to gravitational dynamics.
Recently, Jacobson has advanced a new viewpoint on the relation between geometry and entanglement that has been dubbed “entanglement equilibrium” Jacobson (2016). This proposal considers spherical spatial subregions in geometries that are a perturbation of a maximally symmetric spacetime (MSS). Each such subregion in the maximally symmetric background defines a causal diamond, which admits a conformal Killing vector whose flow preserves the diamond (see Figure 1). The entanglement equilibrium hypothesis states that any perturbation of the matter fields and geometry inside the ball leads to a decrease in entanglement, i.e., the vacuum is a maximal entropy state. This hypothesis applies holding the volume of fixed; even so, the introduction of curvature from the geometry variation can lead to a decrease in the area of the boundary . This affects the divergent terms in the entanglement entropy by changing Wald entropy, which at leading order is simply . The variation of the quantum state contributes a piece , and maximality implies the total variation of the entanglement entropy vanishes at first order,333The separation of the entanglement entropy into a divergent Wald piece and a finite matter piece is scheme-dependent, and can change under the RG flow Jacobson and Satz (2013). Also the matter variation can sometimes produce state-dependent divergences Marolf and Wall (2016), which appear as a variation of the Wald entropy. Since we only ever deal with total variations of the generalized entropy, these subtleties do not affect any results. For simplicity, we will refer to as coming from the geometry variation, and from the matter state variation.
When applied to small spheres, this maximal entropy condition was shown to be equivalent to imposing the Einstein equation at the center of the ball.
Taken as an effective field theory, gravity is expected to contain higher curvature corrections that arise from matching to its UV completion. An important test of the entanglement equilibrium hypothesis is whether it can consistently accommodate these corrections. It is the purpose of this paper to demonstrate that a generalization to higher curvature theories is possible and relates to the subleading divergences appearing in (1).
i.1 Summary of results and outline
It is not a priori clear what the precise statement of the entanglement equilibrium condition should be for a higher curvature theory, and in particular what replaces the fixed-volume constraint. The formulation we propose here is advised by the first law of causal diamond mechanics, a purely geometrical identity that holds independently of any entanglement considerations. It was derived for Einstein gravity in the supplemental materials of Jacobson (2016), and one of the main results of this paper is to extend it to arbitrary higher derivative theories. As we show in section II, the first law is related to the off-shell identity
where is the surface gravity of Jacobson and Kang (1993), is the Wald entropy of given in equation (24) Wald (1993); Iyer and Wald (1994), is the matter Hamiltonian for flows along , defined in equation (9), and are the linearized constraint equations of the higher derivative theory. The Wald entropy is varied holding fixed a local geometric functional
with , and defined in Figure 1. is the variation of the gravitational Lagrangian scalar with respect to and is a constant determined by the value of in a MSS via . We refer to as the “generalized volume” since it reduces to the volume for Einstein gravity.
The Wald formalism contains ambiguities identified by Jacobson, Kang and Myers (JKM) Jacobson et al. (1994) that modify the Wald entropy and the generalized volume by the terms and given in (43) and (44). Using a modified generalized volume defined by
the identity (4) continues to hold with replacing . As discussed in section III.1, the subleading divergences for the entanglement entropy involve a particular resolution of the JKM ambiguity, while section II.4 argues that the first law of causal diamond mechanics applies for any resolution, as long as the appropriate generalized volume is held fixed.
Using the resolution of the JKM ambiguity required for the entanglement entropy calculation, the first law leads to the following statement of entanglement equilibrium, applicable to higher curvature theories:
Hypothesis (Entanglement Equilibrium).
In a quantum gravitational theory, the entanglement entropy of a spherical region with fixed generalized volume is maximal in vacuum.
This modifies the original equilibrium condition (3) by replacing the area variation with
In Section III, this equilibrium condition is shown to be equivalent to the linearized higher derivative field equations in the case that the matter fields are conformally invariant.444There is a proposal for including nonconformal matter that involves varying a local cosmological constant Jacobson (2016); Casini et al. (2016); Speranza (2016). If valid, that proposal applies in the higher curvature case as well, since it deals only with the matter variations. Facts about entanglement entropy divergences and the reduced density matrix for a sphere in a CFT are used to relate the total variation of the entanglement entropy to the left hand side of (4). Once this is done, it becomes clear that imposing the linearized constraint equations is equivalent to the entanglement equilibrium condition.
In Jacobson (2016), this condition was applied in the small ball limit, in which any geometry looks like a perturbation of a MSS. Using Riemann normal coordinates (RNC), the linearized equations were shown to impose the fully nonlinear equations for the case of Einstein gravity. We will discuss this argument in Section IV for higher curvature theories, and show that the nonlinear equations can not be obtained from the small ball limit, making general relativity unique in that regard.
In section V, we discuss several implications of this work. First, we describe how it compares to other approaches connecting geometry and entanglement. Following that, we provide a possible thermodynamic interpretation of the first law of causal diamond mechanics derived in section II. We then comment on a conjectural relation between our generalized volume and higher curvature holographic complexity. Finally, we lay out several future directions for the entanglement equilibrium program.
Note on conventions: we set , use metric signature , and use to refer to the spacetime dimension. We write the spacetime volume form as , and occasionally we will denote it or , suppressing all but its first one or two abstract indices.
Ii First law of causal diamond mechanics
Jacobson’s entanglement equilibrium argument Jacobson (2016) compares the surface area of a small spatial ball in a curved spacetime to the one that would be obtained in a MSS. The comparison is made using balls of equal volume , a choice justified by an Iyer-Wald variational identity Iyer and Wald (1994) for the conformal Killing vector of the causal diamond in the maximally symmetric background. When the Einstein equation holds, this identity implies the first law of causal diamond mechanics Jacobson (2016); Jacobson and Visser
where is the trace of the extrinsic curvature of embedded in , and the matter conformal Killing energy is constructed from the stress tensor by
The purpose of this section is to generalize the variational identity to higher derivative theories, and to clarify its relation to the equations of motion. This is done by focusing on an off-shell version of the identity, which reduces to the first law when the linearized constraint equations for the theory are satisfied. We begin by reviewing the Iyer-Wald formalism in subsection II.1, which also serves to establish notation. After describing the geometric setup in subsection II.2, we show in subsection II.3 how the quantities appearing in the identity can be written as variations of local geometric functionals of the surface and its boundary . As one might expect, the area is upgraded to the Wald entropy , and we derive the generalization of the volume given in equation (5). Subsection II.4 describes how the variational identity can instead be viewed as a variation at fixed generalized volume , as quoted in equation (4), and describes the effect that JKM ambiguities have on the setup.
ii.1 Iyer-Wald formalism
We begin by recalling the Iyer-Wald formalism Wald (1993); Iyer and Wald (1994). A general diffeomorphism invariant theory may be defined by its Lagrangian , a spacetime -form locally constructed from the dynamical fields , which include the metric and matter fields. A variation of this Lagrangian takes the form
where collectively denotes the equations of motion for the dynamical fields, and is the symplectic potential -form. Taking an antisymmetric variation of yields the symplectic current -form
whose integral over a Cauchy surface gives the symplectic form for the phase space description of the theory. Given an arbitrary vector field , evaluating the symplectic form on the Lie derivative gives the variation of the Hamiltonian that generates the flow of
Now consider a ball-shaped region , and take to be any future-pointed, timelike vector that vanishes on the boundary . Wald’s variational identity then reads
where the Noether current is defined by
Here denotes contraction of the vector on the first index of the differential form . The identity (13) holds when the background geometry satisfies the field equations , and it assumes that vanishes on . Next we note that the Noether current can always be expressed as Iyer and Wald (1995)
where is the Noether charge -form and are the constraint field equations, which arise as a consequence of the diffeomorphism gauge symmetry. For non-scalar matter, these constraints are a combination of the metric and matter field equations Seifert and Wald (2007); Jacobson (2011), but, assuming the matter equations are imposed, we can take , where is the variation of the Lagrangian density with respect to the metric. By combining equations (12), (13) and (15), one finds that
When the linearized constraints hold, , the variation of the Hamiltonian is a boundary integral of . This on-shell identity forms the basis for deriving the first law of causal diamond mechanics. Unlike the situation encountered in black hole thermodynamics, is not zero because below we take to be a conformal Killing vector as opposed to a true Killing vector.
ii.2 Geometric setup
Thus far, the only restriction that has been placed on the vector field is that it vanishes on . As such, the quantities and appearing in the identities depend rather explicitly on the fixed vector , and therefore these quantities are not written in terms of only the geometric properties of the surfaces and . A purely geometric description is desirable if the Hamiltonian and Noether charge are to be interpreted as thermodynamic state functions, which ultimately may be used to define the ensemble of geometries in any proposed quantum description of the microstates. This situation may be remedied by choosing the vector and the surface to have special properties in the background geometry. In particular, by choosing to be a conformal Killing vector for a causal diamond in the MSS, and picking to be the surface on which the conformal factor vanishes, one finds that the perturbations and have expressions in terms of local geometric functionals on the surfaces and , respectively.
Given a causal diamond in a MSS, there exists a conformal Killing vector which generates a flow within the diamond and vanishes at the bifurcation surface (see figure 1). The metric satisfies the conformal Killing equation
and the conformal factor vanishes on the spatial ball . The gradient of is hence proportional to the unit normal to ,
Note the vector is future pointing since the conformal factor decreases to the future of . In a MSS, the normalization function has the curious property that it is constant over , and is given by Jacobson and Visser
where is the trace of the extrinsic curvature of embedded in , and is the surface gravity of the conformal Killing horizon, defined momentarily. This constancy ends up being crucial to finding a local geometric functional for . Throughout this work, and will respectively denote constants equal to the normalization function and extrinsic curvature trace, both evaluated in the background spacetime.
Since vanishes on , is instantaneously a Killing vector. On the other hand, the covariant derivative of is nonzero, so
The fact that the covariant derivative is nonzero on is responsible for making nonvanishing.
A conformal Killing vector with a horizon has a well-defined surface gravity Jacobson and Kang (1993), and since vanishes on , we can conclude that
where is the binormal for the surface , and is the outward pointing spacelike unit normal to . Since is a bifurcation surface of a conformal Killing horizon, is constant everywhere on it. We provide an example of these constructions in appendix A where we discuss the conformal Killing vector for a causal diamond in flat space.
ii.3 Local geometric expressions
In this subsection we evaluate the Iyer-Wald identity (16) for an arbitrary higher derivative theory of gravity and for the geometric setup described above. The final on-shell result is given in (37), which is the first law of causal diamond mechanics for higher derivative gravity.
Throughout the computation we assume that the matter fields are minimally coupled, so that the Lagrangian splits into a metric and matter piece , and we take to be an arbitrary, diffeomorphism-invariant function of the metric, Riemann tensor, and its covariant derivatives. The symplectic potential and variation of the Hamiltonian then exhibit a similar separation, and , and so we can write equation (16) as
Below, we explicitly compute the two terms and
for the present geometric context.
where the Wald entropy is defined as
is the variation of the Lagrangian scalar with respect to the Riemann tensor taken as an independent field, given in (71), and is the volume form on , so that there. The equality (23) continues to hold at first order in perturbations, which can be shown following the same arguments as given in Iyer and Wald (1994), hence,
The minus sign is opposite the convention in Iyer and Wald (1994) since the unit normal
is outward pointing for the causal diamond.
The gravitational part of is related to the symplectic current via (12). The symplectic form has been computed on an arbitrary background for any higher curvature gravitational theory whose Lagrangian is a function of the Riemann tensor, but not its covariant derivatives Bueno et al. (2016). Here, we take advantage of the maximal symmetry of the background to compute the symplectic form and Hamiltonian for the causal diamond in any higher order theory, including those with derivatives of the Riemann tensor.
Recall that the symplectic current is defined in terms of the symplectic potential through (11). For a Lagrangian that depends on the Riemann tensor and its covariant derivatives, the symplectic potential is given in Lemma 3.1 of Iyer and Wald (1994)
where and the tensors and are locally constructed from the metric, its curvature, and covariant derivatives of the curvature. Due to the antisymmetry of in and , the symplectic current takes the form
Next we specialize to the geometric setup described in section II.2. We may thus employ the fact that we are perturbing around a maximally symmetric background. This means the background curvature tensor takes the form
with a constant Ricci scalar , so that , and also . Since the tensors , , and are all constructed from the metric and curvature, they will also have vanishing Lie derivative along when evaluated on .
We would like to write this as a variation of some scalar quantity. To do so, we split off the background value of by writing
The second term in this expression is the background value, and, due to maximal symmetry, the scalar must be a constant determined by the parameters appearing in the Lagrangian. By definition, is zero in the background, so any term in (II.3) that depends on its variation may be immediately written as a total variation, since variations of other tensors appearing in the formula would multiply the background value of , which vanishes. Hence, the piece involving becomes
The remaining terms simply involve replacing in (II.3) with . These terms then take exactly the same form as the terms that appear for general relativity, which we know from the appendix of Jacobson (2016) combine to give an overall variation of the volume. The precise form of this variation when restricted to is
where is the induced volume form on . Adding this to (31) produces
where we used that on . This leads us to define a generalized volume functional
and the variation of this quantity is related to the variation of the gravitational Hamiltonian by
where we have expressed in terms of and using (19). We have thus succeeded in writing in terms of a local geometric functional defined on the surface .
It is worth emphasizing that being constant over the ball was crucial to this derivation, since otherwise it could not be pulled out of the integral over and would define a non-diffeomorphism invariant structure on the surface. Note that the overall normalization of is arbitrary, since a different normalization would simply change the coefficient in front of in (35). As one can readily check, the normalization in (34) was chosen so that reduces to the volume in the case of Einstein gravity. In appendix B we provide explicit expressions for the generalized volume in gravity and quadratic gravity.
By imposing the linearized constraints , this becomes the first law of causal diamond mechanics for higher derivative gravity
This reproduces (8) for Einstein gravity with Lagrangian , for which .
ii.4 Variation at fixed
We now show that the first two terms in (36) can be written in terms of the variation of the Wald entropy at fixed , defined as
Here we must specify what is meant by . We will take this partial derivative to refer to the changes that occur in both quantities when the size of the ball is deformed, but the metric and dynamical fields are held fixed. Take a vector that is everywhere tangent to that defines an infinitesimal change in the shape of . The first order change this produces in and can be computed by holding fixed, but varying the Noether current and Noether charge as and . Since the background field equations are satisfied and vanishes on , we have there that , without reference to the matter part of the Noether current. Recall that is related to the variation of the gravitational Hamiltonian, which can be expressed in terms of through (12) and (13). Then using the relations (23) and (35) and the fact that the Lie derivative commutes with the exterior derivative, we may compute
Finally, we comment on how JKM ambiguities Jacobson et al. (1994) affect this identity. The particular ambiguity we are concerned with comes from the fact that the symplectic potential in equation (10) is defined only up to addition of an exact form that is linear in the field variations and their derivatives. This has the effect of changing the Noether current and Noether charge by
This modifies both the entropy and the generalized volume by surface terms on given by
However, it is clear that this combined change in and leaves the left hand side of (40) unchanged, since the -dependent terms cancel out. In particular,
showing that any resolution of the JKM ambiguity gives the same first law, provided that the Wald entropy and generalized volume are modified by the terms (43) and (44). This should be expected, because the right hand side of (40) depends only on the field equations, which are unaffected by JKM ambiguities.
Iii Entanglement Equilibrium
The original entanglement equilibrium argument for Einstein gravity stated that the total variation away from the vacuum of the entanglement of a region at fixed volume is zero. This statement is encapsulated in equation (3), which shows both an area variation due to the change in geometry, and a matter piece from varying the quantum state. The area variation at fixed volume can equivalently be written
and the arguments of section II.4 relate this combination to the terms appearing in the first law of causal diamond mechanics (8). Since in (8) is related to in (3) for conformally invariant matter, the first law may be interpreted entirely in terms of entanglement entropy variations.
This section discusses the extension of the argument to higher derivative theories of gravity. Subsection III.1 explains how subleading divergences in the entanglement entropy are related to a Wald entropy, modified by a particular resolution of the JKM ambiguity. Paralleling the Einstein gravity derivation, we seek to relate variations of the subleading divergences to the higher derivative first law of causal diamond mechanics (37). Subsection III.2 shows that this can be done as long as the generalized volume [related to by a boundary JKM term as in (6)] is held fixed. Then, using the relation of the first law to the off-shell identity (40), we discuss how the entanglement equilibrium condition is equivalent to imposing the linearized constraint equations.
iii.1 Subleading entanglement entropy divergences
The structure of divergences in entanglement entropy is reviewed in Solodukhin (2011) and the appendix of Bousso et al. (2016). It is well-known that the leading divergence depends on the area of the entangling surface. More surprising, however, is the fact that this divergence precisely matches the matter field divergences that renormalize Newton’s constant. This ostensible coincidence arises because the two divergences have a common origin in the gravitational effective action , which includes both gravitational and matter pieces. Its relation to entanglement entropy comes from the replica trick, which defines the entropy as Callan and Wilczek (1994); Calabrese and Cardy (2009)
where the effective action is evaluated on a manifold with a conical singularity at the entangling surface whose excess angle is .
As long as a covariant regulator is used to define the theory, the effective action will consist of terms that are local, diffeomorphism invariant integrals over the manifold, as well as nonlocal contributions. All UV matter divergences must appear in the local piece of the effective action, and each combines with terms in the classical gravitational part of the action, renormalizing the gravitational coupling constants. Furthermore, each such local term contributes to the entanglement entropy in (47) only at the conical singularity, giving a local integral over the entangling surface Fursaev and Solodukhin (1996); Larsen and Wilczek (1996); Cooperman and Luty (2014).
When the entangling surface is the bifurcation surface of a stationary horizon, this local integral is simply the Wald entropy Nelson (1994); Iyer and Wald (1995). On nonstationary entangling surfaces, the computation can be done using the squashed cone techniques of Fursaev et al. (2013), which yield terms involving extrinsic curvatures that modify the Wald entropy. In holography, the squashed cone method plays a key role in the proof of the Ryu-Takayanagi formula Ryu and Takayanagi (2006); Lewkowycz and Maldacena (2013), and its higher curvature generalization Dong (2013); Camps (2013). The entropy functionals obtained in these works seem to also apply outside of holography, giving the extrinsic curvature terms in the entanglement entropy for general theories Fursaev et al. (2013); Bousso et al. (2016).555For terms involving four or more powers of extrinsic curvature, there are additional subtleties associated with the so called “splitting problem” Miao and Guo (2014); Miao (2015); Camps (2016).
The extrinsic curvature modifications to the Wald entropy in fact take the form of a JKM Noether charge ambiguity Jacobson et al. (1994); Sarkar and Wall (2013); Wall (2015). To see this, note the vector used to define the Noether charge vanishes at the entangling surface and its covariant derivative is antisymmetric and proportional to the binormal as in equation (21). This means it acts like a boost on the normal bundle at the entangling surface. General covariance requires that any extrinsic curvature contributions can be written as a sum of boost-invariant products,
where the superscript denotes the boost weight of that tensor, so that at the surface: . It is necessary that the terms consist of two pieces, each of which has nonzero boost weight, so that they can be written as
This is of the form of a Noether charge ambiguity from equation (42), with666This formula defines at the entangling surface, and allows for some arbitrariness in defining it off the surface. It is not clear that can always be defined as a covariant functional of the form without reference to additional structures, such as the normal vectors to the entangling surface. It would be interesting to understand better if and when lifts to such a spacetime covariant form off the surface. 777We thank Aron Wall for this explanation of JKM ambiguities.
The upshot of this discussion is that all terms in the entanglement entropy that are local on the entangling surface, including all divergences, are given by a Wald entropy modified by specific JKM terms. The couplings for the Wald entropy are determined by matching to the UV completion, or, in the absence of the UV description, these are simply parameters characterizing the low energy effective theory. In induced gravity scenarios, the divergences are determined by the matter content of the theory, and the matching to gravitational couplings has been borne out in explicit examples Frolov et al. (1997); Myers et al. (2013); Pourhasan (2014).
iii.2 Equilibrium condition as gravitational constraints
We can now relate the variational identity (40) to entanglement entropy. The reduced density matrix for the ball in vacuum takes the form
where is the modular Hamiltonian and is the partition function, ensuring that is normalized. Since the matter is conformally invariant, the modular Hamiltonian takes a simple form in terms of the matter Hamiltonian defined in (9) Hislop and Longo (1982); Casini et al. (2011)
Next we apply the first law of entanglement entropy Blanco et al. (2013); Bhattacharya et al. (2012), which states that the first order perturbation to the entanglement entropy is given by the change in modular Hamiltonian expectation value
Note that this equation holds for a fixed geometry and entangling surface, and hence coincides with what was referred to as in section I. When varying the geometry, the divergent part of the entanglement entropy changes due to a change in the Wald entropy and JKM terms of the entangling surface. The total variation of the entanglement entropy is therefore
At this point, we must give a prescription for defining the surface in the perturbed geometry. Motivated by the first law of causal diamond mechanics, we require that has the same generalized volume as in vacuum, where differs from the quantity by a JKM term, as in equation (6). This provides a diffeomorphism-invariant criterion for defining the size of the ball. It does not fully fix all properties of the surface, but it is enough to derive the equilibrium condition for the entropy. As argued in section II.4, the first term in equation (54) can be written instead as when the variation is taken holding fixed. Thus, from equations (40), (52) and (54), we arrive at our main result, the equilibrium condition
valid for minimally coupled, conformally invariant matter fields.
The linearized constraint equations may therefore be interpreted as an equilibrium condition on entanglement entropy for the vacuum. Since all first variations of the entropy vanish when the linearized gravitational constraints are satisfied, the vacuum is an extremum of entropy for regions with fixed generalized volume , which is necessary for it to be an equilibrium state. Alternatively, postulating that entanglement entropy is maximal in vacuum for all balls and in all frames would allow one to conclude that the linearized higher derivative equations hold everywhere.
Iv Field equations from the equilibrium condition
The entanglement equilibrium hypothesis provides a clear connection between the linearized gravitational constraints and the maximality of entanglement entropy at fixed in the vacuum for conformally invariant matter. In this section, we will consider whether information about the fully nonlinear field equations can be gleaned from the equilibrium condition. Following the approach taken in Jacobson (2016), we employ a limit where the ball is taken to be much smaller than all relevant scales in the problem, but much larger than the cutoff scale of the effective field theory, which is set by the gravitational coupling constants. By expressing the linearized equations in Riemann normal coordinates, one can infer that the full nonlinear field equations hold in the case of Einstein gravity. As we discuss here, such a conclusion can not be reached for higher curvature theories. The main issue is that higher order terms in the RNC expansion are needed to capture the effect of higher curvature terms in the field equations, but these contribute at the same order as nonlinear corrections to the linearized equations.
We begin by reviewing the argument for Einstein gravity. Near any given point, the metric looks locally flat, and has an expansion in terms of Riemann normal coordinates that takes the form
where means evaluation at the center of the ball. At distances small compared to the radius of curvature, the second term in this expression is a small perturbation to the flat space metric . Hence, we may apply the off-shell identity (55), using the first order variation
and conclude that the linearized constraint holds for this metric perturbation. When restricted to the surface , this constraint in Einstein gravity is Seifert and Wald (2007)
Since the background constraint is assumed to hold, the perturbed constraint is
but in Riemann normal coordinates, we have that the linearized perturbation to the curvature is just , up to terms suppressed by the ball radius. Assuming that the ball is small enough so that the stress tensor may be taken constant over the ball, one concludes that the vanishing constraint implies the nonlinear field equation at the center of the ball888In this equation, should be thought of as a quantum expectation value of the stress tensor. Presumably, for sub-Planckian energy densities and in the small ball limit, this first order variation approximates the true energy density. However, there exist states for which the change in stress-energy is zero at first order in perturbations away from the vacuum, most notable for coherent states Varadarajan (2016). Analyzing how these states can be incorporated into the entanglement equilibrium story deserves further attention.
The procedure outlined above applies at all points and all frames, allowing us to obtain the full tensorial Einstein equation.
Since we have only been dealing with the linearized constraint, one could question whether it gives a good approximation to the field equations at all points within the small ball. This requires estimating the size of the nonlinear corrections to this field equation. When integrated over the ball, the corrections to the curvature in RNC are of order , where is the radius of the ball and is the radius of curvature. Since we took the ball size to be much smaller than the radius of curvature, these terms are already suppressed relative to the linear order terms in the field equation.
The situation in higher derivative theories of gravity is much different. It is no longer the case that the linearized equations evaluated in RNC imply the full nonlinear field equations in a small ball. To see this, consider an higher curvature theory.999Note that an analogous argument should hold for general higher derivative theories, which also involve covariant derivatives of the Riemann tensor. The equations of motion read
In appendix C we show that linearizing these equations around a Minkowski background leads to
where we split into its Einstein piece, which gives rise to the Einstein tensor, and a piece coming from higher derivative terms. As noted before, the variation of the Einstein tensor evaluated in RNC gives the nonlinear Einstein tensor, up to corrections that are suppressed by the ratio of the ball size to the radius of curvature. However, in a higher curvature theory of gravity, the equations of motion (61) contain terms that are nonlinear in the curvature. Linearization around a MSS background of these terms would produce, schematically, , where denotes evaluation in the MSS background. In Minkowski space, all such terms would vanish. This is not true in a general MSS, but evaluating the curvature tensors in the background still leads to a significant loss of information about the tensor structure of the equation. We conclude that the linearized equations cannot reproduce the full nonlinear field equations for higher curvature gravity, and it is only the linearity of the Einstein equation in the curvature that allows the nonlinear equations to be obtained for general relativity.
When linearizing around flat space, the higher curvature corrections to the Einstein equation are entirely captured by the second term in (62), which features four derivatives acting on the metric, since is constructed from curvatures that already contain two derivatives of the metric. Therefore, one is insensitive to higher curvature corrections unless at least corrections Brewin (2009) are added to the Riemann normal coordinates expansion (57)
Being quadratic in the Riemann tensor, this term contributes at the same order as the nonlinear corrections to the linearized field equations. Hence, linearization based on the RNC expansion up to terms is not fully self-consistent. This affirms the claim that for higher curvature theories, the nonlinear equations at a point cannot be derived by only imposing the linearized equations.
Maximal entanglement of the vacuum state was proposed in Jacobson (2016) as a new principle in quantum gravity. It hinges on the assumption that divergences in the entanglement entropy are cut off at short distances, so it is ultimately a statement about the UV complete quantum gravity theory. However, the principle can be phrased in terms of the generalized entropy, which is intrinsically UV finite and well-defined within the low energy effective theory. Therefore, if true, maximal vacuum entanglement provides a low energy constraint on any putative UV completion of a gravitational effective theory.
Higher curvature terms arise generically in any such effective field theory. Thus, it is important to understand how the entanglement equilibrium argument is modified by them. As explained in section II, the precise characterization of the entanglement equilibrium hypothesis relies on a classical variational identity for causal diamonds in maximally symmetric spacetimes. This identity leads to equation (40), which relates variations of the Wald entropy and matter energy density of the ball to the linearized constraints. The variations are taken holding fixed a new geometric functional , defined in (34), which we call the “generalized volume.”
We connected this identity to entanglement equilibrium in section III, invoking the fact that subleading entanglement entropy divergences are given by a Wald entropy, modified by specific JKM terms, which also modify by the boundary term (44). With the additional assumption that matter is conformally invariant, we arrived at our main result (55), showing that the equilibrium condition applied to small balls is equivalent to imposing the linearized constraints .
In section IV, we reviewed the argument that in the special case of Einstein gravity, imposing the linearized equations within small enough balls is equivalent to requiring that the fully nonlinear equations hold within the ball Jacobson (2016). Thus by considering spheres centered at each point and in all Lorentz frames, one could conclude that the full Einstein equations hold everywhere.101010There is a subtlety associated with whether the solutions within each small ball can be consistently glued together to give a solution over all of spacetime. One must solve for the gauge transformation relating the Riemann normal coordinates at different nearby points, and errors in the linearized approximation could accumulate as one moves from point to point. The question of whether the ball size can be made small enough so that the total accumulated error goes to zero deserves further attention. Such an argument cannot be made for a theory that involves higher curvature terms. One finds that higher order terms in the RNC expansion are needed to detect the higher curvature pieces of the field equations, but these terms enter at the same order as the nonlinear corrections to the linearized equations. This signals a breakdown of the perturbative expansion unless the curvature is small.
The fact that we obtain only linearized equations for the higher curvature theory is consistent with the effective field theory standpoint. One could take the viewpoint that higher curvature corrections are suppressed by powers of a UV scale, and the effective field theory is valid only when the curvature is small compared to this scale. This suppression would suggest that the linearized equations largely capture the effects of the higher curvature corrections in the regime where effective field theory is reliable.
v.1 Comparison to other “geometry from entanglement” approaches
Several proposals have been put forward to understand gravitational dynamics in terms of thermodynamics and entanglement. Here we will compare the entanglement equilibrium program considered in this paper to two other approaches: the equation of state for local causal horizons, and gravitational dynamics from holographic entanglement entropy (see Carroll and Remmen (2016) for a related discussion).
v.1.1 Causal horizon equation of state
By assigning an entropy proportional to the area of local causal horizons, Jacobson showed that the Einstein equation arises as an equation of state Jacobson (1995). This approach employs a physical process first law for the local causal horizon, defining a heat as the flux of local boost energy across the horizon. By assigning an entropy to the horizon proportional to its area, one finds that the Clausius relation applied to all such horizons is equivalent to the Einstein equation.
The entanglement equilibrium approach differs in that it employs an equilibrium state first law [equation (37)], instead of a physical process one Wald (1994). It therefore represents a different perspective that focuses on the steady-state behavior, as opposed to dynamics involved with evolution along the causal horizon. It is consistent therefore that we obtain constraint equations in the entanglement equilibrium setup, since one would not expect evolution equations to arise as an equilibrium condition.111111We thank Ted Jacobson for clarifying this point. That we can infer dynamical equations from the constraints is related to the fact that the dynamics of diffeomorphism-invariant theories is entirely determined by the constraints evaluated in all possible Lorentz frames.
Another difference comes from the focus on spacelike balls as opposed to local causal horizons. Dealing with a compact spatial region has the advantage of providing an IR finite entanglement entropy, whereas the entanglement associated with local causal horizons can depend on fields far away from the point of interest. This allows us to give a clear physical interpretation for the surface entropy functional as entanglement entropy, whereas such an interpretation is less precise in the equation of state approaches.
Finally, we note that both approaches attempt to obtain fully nonlinear equations by considering ultralocal regions of spacetime. In both cases the derivation of the field equations for Einstein gravity is fairly robust, however higher curvature corrections present some problems. Attempts have been made in the local causal horizon approach that involve modifying the entropy density functional for the horizon Eling et al. (2006); Elizalde and Silva (2008); Chirco et al. (2011); Brustein and Hadad (2009); Parikh and Sarkar (2016); Dey et al. (2016); Padmanabhan (2009a, b); Guedens et al. (2011), but they meet certain challenges. These include a need for a physical interpretation of the chosen entropy density functional, and dependence of the entropy on arbitrary features of the local Killing vector in the vicinity of the horizon Guedens et al. (2011); Jacobson (2012). While the entanglement equilibrium argument avoids these problems, it fails to get beyond linearized higher curvature equations, even after considering the small ball limit. The nonlinear equations in this case appear to involve information beyond first order perturbations, and hence may not be accessible based purely on an equilibrium argument.
v.1.2 Holographic entanglement entropy
A different approach comes from holography and the Ryu-Takayanagi formula Ryu and Takayanagi (2006). By demanding that areas of minimal surfaces in the bulk match the entanglement entropies of spherical regions in the boundary CFT, one can show that the linearized gravitational equations must hold Lashkari et al. (2013); Faulkner et al. (2013b); Swingle and Van Raamsdonk (2014). The argument employs an equilibrium state first law for the bulk geometry, utilizing the Killing symmetry associated with Rindler wedges in the bulk.
The holographic approach is quite similar to the entanglement equilibrium argument since both use equilibrium state first laws. One difference is that the holographic argument must utilize minimal surfaces in the bulk, which extend all the way to the boundary of AdS. This precludes using a small ball limit as can be done with the entanglement equilibrium derivation, and is the underlying reason that entanglement equilibrium can derive fully nonlinear field equations in the case of Einstein gravity, whereas the holographic approach has thus far only obtained linearized equations. Some progress has been made to go beyond linear order in the holographic approach by considering higher order perturbations in the bulk Faulkner (2014); Lashkari and Van Raamsdonk (2015); Beach et al. (2016). Higher order perturbations will prove useful in the entanglement equilibrium program as well, and has the potential to extend the higher curvature derivation to fully nonlinear equations. Due to the similarity between the holographic and entanglement equlibrium approaches, progress in one will complement and inform the other.
v.2 Thermodynamic interpretation of the first law of causal diamond mechanics
Apart from the entanglement equilibrium interpretation, the first law of causal diamond mechanics could also directly be interpreted as a thermodynamic relation. Note that the identity (8) for Einstein gravity bears a striking resemblance to the fundamental relation in thermodynamics
where is the internal energy, which is a function of the entropy and volume . The first law (8) turns into the thermodynamic relation (64), if one makes the following identifications for the temperature and pressure
Here we have restored fundamental constants, so that the quantities on the RHS have the standard units of temperature and pressure. The expression for the temperature is the well-known Unruh temperature Unruh (1976). The formula for the pressure lacks a microscopic understanding at the moment, although we emphasize the expression follows from consistency of the first law.
The thermodynamic interpretation motivates the name “first law” assigned to (8), and arguably it justifies the terminology “generalized volume” used for in this paper, since it enters into the first law for higher curvature gravity (37) in the place of the volume. The only difference with the fundamental relation in thermodynamics is the minus sign in front of the energy variation. This different sign also enters into the first law for de Sitter horizons Gibbons and Hawking (1977). In the latter case the sign appears because empty de Sitter spacetime has maximal entropy, and adding matter only decreases the horizon entropy. Causal diamonds are rather similar in that respect.
v.3 Generalized volume and holographic complexity
The emergence of a generalized notion of volume in this analysis is interesting in its own right. We showed that when perturbing around a maximally symmetric background, the variation of the generalized volume is proportional to the variation of the gravitational part of the Hamiltonian. The fact that the Hamiltonian could be written in terms of a local, geometric functional of the surface was a nontrivial consequence of the background geometry being maximally symmetric and being a conformal Killing vector whose conformal factor vanishes on . The local geometric nature of makes it a useful, diffeomorphism invariant quantity with which to characterize the region under consideration, and thus should be a good state function in the thermodynamic description of an ensemble of quantum geometry microstates. One might hope that such a microscopic description would also justify the fixed- constraint in the entanglement equilibrium derivation, which was only motivated macroscopically by the first law of causal diamond mechanics.
Volume has recently been identified as an important quantity in holography, where it is conjectured to be related to complexity Susskind (2014); Brown et al. (2015), or fidelity susceptibility Miyaji et al. (2015). The complexityvolume conjecture states that the complexity of some boundary state on a time slice is proportional to the volume of the extremal codimension-one bulk hypersurface which meets the asymptotic boundary on the corresponding time slice.121212A similar expression has also been proposed for the complexity of subregions of the boundary time slice. In that case, is the bulk hypersurface bounded by the corresponding subregion on the asymptotic boundary and the Ryu-Takayanagi surface Ryu and Takayanagi (2006) in the bulk Alishahiha (2015); Ben-Ami and Carmi (2016), or, more generally, the Hubeny-Rangamani-Takayanagi surface Hubeny et al. (2007) if the spacetime is time-dependent Carmi et al. (2016).
While volume is the natural functional to consider for Einstein gravity, Alishahiha (2015) noted that this should be generalized for higher curvature theories. The functional proposed in that work resembles our generalized volume , but suffers from an arbitrary dependence on the choice of foliation of the codimension-one hypersurface on which it is evaluated. We therefore suggest that , as defined in (34), may provide a suitable generalization of volume in the context of higher curvature holographic complexity.
Observe however that our derivation of using the Iyer-Wald formalism was carried out in the particular case of spherical regions whose causal diamond is preserved by a conformal Killing vector. On more general grounds, one could speculate that the holographic complexity functional in higher derivative gravities should involve contractions of with the geometric quantities characterizing , namely the induced metric and the normal vector . The most general functional involving at most one factor of can be written as
for some constants , and which should be such that for Einstein gravity. It would be interesting to explore the validity of this proposal in particular holographic setups, e.g., along the lines of Carmi et al. (2016).
v.4 Future work
We conclude by laying out future directions for the entanglement equilibrium program.
v.4.1 Higher order perturbations
In this work we restricted attention only to first order perturbations of the entanglement entropy and the geometry. Working to higher order in perturbation theory could yield several interesting results. One such possibility would be proving that the vacuum entanglement entropy is maximal, as opposed to merely extremal. The second order change in entanglement entropy is no longer just the change in modular Hamiltonian expectation value. The difference is given by the relative entropy, so a proof of maximality will likely invoke the positivity of relative entropy. On the geometrical side, a second order variational identity would need to be derived, along the lines of Hollands and Wald (2013). One would expect that graviton contributions would appear at this order, and it would be interesting to examine how they play into the entanglement equilibrium story. Also, by considering small balls and using the higher order terms in the Riemann normal coordinate expansion (63), in addition to higher order perturbations, it is possible that one could derive the fully nonlinear field equations of any higher curvature theory. Finally, coherent states pose a puzzle for the entanglement equilibrium hypothesis, since they change the energy within the ball without changing the entanglement Varadarajan (2016). However, their effect on the energy density only appears at second order in perturbations, so carrying the entanglement equilibrium argument to higher order could shed light on this puzzle.
v.4.2 Nonconformal matter
The arguments deriving the entanglement equilibrium condition in section III.2 were restricted to matter that is conformally invariant. For nonconformal matter, there are corrections to the modular Hamiltonian that spoil the relation between and the matter Hamiltonian . Nevertheless, in the small ball limit these corrections take on a simple form, and one possible solution for extending the entanglement equilibrium argument introduces a local cosmological constant to absorb the effects of the modular Hamiltonian corrections Jacobson (2016); Casini et al. (2016); Speranza (2016). Allowing variations of the local cosmological constant would result in a modified first law Jacobson and Visser , and may have connections to the black hole chemistry program Kastor et al. (2009); Kubiznak et al. (2016). It is also possible that some other resolution exists to this apparent conflict, perhaps involving the RG properties of the matter field theory when taking the small ball limit.
v.4.3 Nonminimal couplings and gauge fields
We restricted attention to minimally coupled matter throughout this work. Allowing for nonminimal coupling can lead to new, state-dependent divergences in the entanglement entropy Marolf and Wall (2016). As before, these divergences will be localized on the entangling surface, taking the form of a Wald entropy. It therefore seems plausible that an entanglement equilibrium argument will go through in this case, reproducing the field equations involving the nonminimally coupled field. Note the state-dependent divergences could lead to variations of the couplings in the higher curvature theory, which may connect to the entanglement chemistry program, which considers Iyer-Wald first laws involving variations of the couplings Caceres et al. (2016).
Gauge fields introduce additional subtleties related to the existence of edge modes Donnelly and Wall (2012, 2015a, 2015b), and since these affect the renormalization of the gravitational couplings, they require special attention. Gravitons are even more problematic due to difficulties in defining the entangling surface in a diffeomorphism-invariant manner and in finding a covariant regulator Fursaev and Miele (1997); Cooperman and Luty (2014); Solodukhin (2015); Bousso et al. (2016). It would be interesting to analyze how to handle these issues in the entanglement equilibrium argument.
v.4.4 Nonspherical subregions
The entanglement equilibrium condition was shown to hold for spherical subregions and conformally invariant matter. One question that arises is whether an analogous equilibrium statement holds for linear perturbations to the vacuum in an arbitrarily shaped region. Nonspherical regions present a challenge because there is no longer a simple relation between the modular Hamiltonian and the matter stress tensor. Furthermore, nonspherical regions do not admit a conformal Killing vector which preserves its causal development. Since many properties of the conformal Killing vector were used when deriving the generalized volume , it may need to be modified to apply to nonspherical regions and their perturbations.
Adapting the entanglement equilibrium arguments to nonspherical regions may involve shifting the focus to evolution under the modular flow, as opposed to a geometrical evolution generated by a vector field. Modular flows are complicated in general, but one may be able to use general properties of the flow to determine whether the Einstein equations still imply maximality of the vacuum entanglement for the region. Understanding the modular flow may also shed light on the behavior of the entanglement entropy for nonconformal matter, and whether some version of the entanglement equilibrium hypothesis continues to hold.
v.4.5 Physical process
As emphasized above, the first law of causal diamond mechanics is an equilibrium state construction since it compares the entropy of on two infinitesimally related geometries Wald (1994). One could ask whether there exists a physical process version of this story, which deals with entropy changes and energy fluxes as you evolve along the null boundary of the causal diamond. For this, the notion of quantum expansion for the null surface introduced in Bousso et al. (2016) would be a useful concept, which is defined by the derivative of the generalized entropy along the generators of the surface. One possible subtlety in formulating a physical process first law for the causal diamond is that the (classical) expansion of the null boundary is nonvanishing, so it would appear that this setup does not correspond to a dynamical equilibrium configuration. Nevertheless, it may be possible to gain useful information about the dynamics of semiclassical gravity by considering these nonequilibrium physical processes. An alternative that avoids this issue is to focus on quantum extremal surfaces Engelhardt and Wall (2015) whose quantum expansion vanishes, and therefore may lend themselves to an equilibrium physical process first law.
Acknowledgements.We would like to thank Joan Camps, Ted Jacobson, Arif Mohd, Rob Myers, Erik Verlinde and Aron Wall for helpful discussions, Fernando Rejon-Barrera for an early collaboration on this project, and Ted Jacobson for comments on a draft of this work. AJS is grateful to the Maryland Center for Fundamental Physics and Aron Wall for organizing the “Minicourse on Spacetime Thermodynamics.” VSM, AJS and MRV thank the organizers of the “Amsterdam String Workshop,” hosted by the Delta Institute for Theoretical Physics, and PB and AJS are grateful to the organizers of the “It from Qubit Summer School” held at the Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. The work of PB is supported by a postdoctoral fellowship from the Fund for Scientific Research - Flanders (FWO). PB also acknowledges support from the Delta ITP Visitors Programme. VSM is supported by a PhD fellowship from the FWO and by the ERC grant 616732-HoloQosmos. AJS is supported by the National Science Foundation under grant No. PHY-1407744. MRV acknowledges support from the ERC Advanced Grant 268088-EMERGRAV, the Spinoza Grant of the Dutch Science Organisation (NWO), and the NWO Gravitation Program for the Delta Institute for Theoretical Physics.
Appendix A Conformal Killing vector in flat space
Here we make explicit the geometric quantities introduced in section II.2 in the case of a Minkowski background, whose metric we write in spherical coordinates, i.e., . Let be a spatial ball of radius in the time slice and with center at . The conformal Killing vector which preserves the causal diamond of is given by Jacobson (2016)
where we have chosen the normalization in a way such that at the center of the ball, which then gives the usual notion of energy for (i.e. the correct units). It is straightforward to check that i.e., the tips of the causal diamond and the maximal sphere at its waist are fixed points of , as expected. Similarly, is null on the boundary of the diamond. In particular, The vectors and (respectively normal to and to both and ) read , , so that the binormal to is given by It is also easy to check that holds, where Hence, we immediately see that on , which implies that the gradient of is proportional to the unit normal . Indeed, one finds , so in this case . It is also easy to show that holds, where the surface gravity reads .
As shown in Jacobson and Kang (1993), given some metric with a conformal Killing field , it is possible to construct other metrics conformally related to it, for which is a true Killing field. More explicitly, if , then as long as and are related through , where satisfies
For the vector (67), this equation has the general solution
Here, can be any function. Hence, in (67) is a true Killing vector for all metrics conformally related to Minkowski’s with a conformal factor given by (69). For example, setting , for some constant , one obtains the metric of AdS with equal radii, namely: . Another simple case corresponds to . Through the change of variables Casini et al. (2011): , , this choice leads to the metric (where is the hyperbolic plane): .
Appendix B Generalized volume in higher order gravity
The generalized volume is defined in (34). We restate the expression here
where is a theory-dependent constant defined by the tensor in a maximally symmetric solution to the field equations through . Moreover, is the variation of the Lagrangian scalar with respect to the Riemann tensor if we were to treat it as an independent field Iyer and Wald (1994),
where is then defined through . In this section we provide explicit expressions for in gravity, quadratic gravity and Gauss-Bonnet gravity. Observe that throughout this section we use the bar on to denote evaluation on a MSS. Imposing a MSS to solve the field equations of a given higher derivative theory gives rise to a constraint between the theory couplings and the background curvature . This reads Bueno et al. (2016)
where denotes the Lagrangian scalar evaluated on the background.
A simple higher curvature gravity is obtained by replacing in the Einstein-Hilbert action by a function of
To obtain the generalized volume we need
The generalized volume then reads
A general quadratic theory of gravity is given by the Lagrangian
Taking the derivative of the Lagrangian with respect to the Riemann tensor leaves us with
and using (28) one finds
The generalized volume for quadratic gravity thus reads