{'fraction_non_alphanumeric': 0.03872053872053872, 'fraction_numerical': 0.012449051922736133, 'mean_word_length': 4.298826291079812, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 2, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
A Wald-like Formula for Energy

We present a simple"Wald-like"formula for gravitational energy about a constant curvature background spacetime. The formula is derived following the Abbott-Deser-Tekin approach for the definition of conserved asymptotic charges in higher-derivative gravity.

In this paper, we present a simple formula for the computation of global charges (in particular energy) in asymptotically constant-curvature spacetimes for general gravitational theories. This is of particular interest for solutions of higher-derivative theories that approach a constant curvature solution in the asymptotic region.
Our formula is similar to Wald's formula for entropy [1][2][3] in the sense that both formulas involve a derivative of the Lagrangian of the theory with respect to the Riemann tensor. The main difference is that the entropy is computed as an integral over the horizon, while the energy is computed in the asymptotic region (where we regard the solution as a perturbation of the background). Another difference is that Wald's formula involves only first derivatives with respect to curvature, whereas the formula for energy involves second derivatives. Nevertheless, the two formulas should be related by the first law of black hole thermodynamics and its integrated forms, i.e., Komar integrals and the Smarr Here we present the formula for energy and explain how to use it. Let us consider a general d-dimensional theory of gravity whose action depends on the metric g µν and the curvature (through the Riemann tensor) (2.1) We will construct the energy for such theories following the approach of Refs. [12,13,15]. We assume that the action is invariant under diffeomorphisms. In order to define a gauge-invariant conserved charge we need the presence of an asymptotic Killing symmetry. The charge is then defined relative to a background solution, denoted asḡ µν , which admits a Killing vectorξ µ . We assume that the background is a homogeneous solution, namely, described by an "effective" cosmological constant Λ, which can be negative, positive, or zero (which is the asymptotically flat case).
In addition, the solutions are required to fall off sufficiently fast at infinity relative to the background 1 . For a large class of solutions [which includes the asymptotically Schwarzschild-(A)dS (SdS) solutions defined below], we can write the energy of a generic higher-derivative theory in the same form as applies to Einstein-Hilbert gravity (with a cosmological constant), but with an overall multiplicative factor that depends on the higher-curvature terms. This will later serve as a basis for the interpretation of higher-derivative corrections as effective modifications of the gravitational coupling constant (Newton's constant). The Lagrangian of Einstein-Hilbert gravity with a cosmological constant is where κ is the d-dimensional gravitational coupling constant. The ADT energy for solutions of this theory is denoted by E 0 and is given explicitly in Eq. (3.6). Then, the energy for the general Lagrangian (2.1) is [The complete energy formula for more general boundary conditions is given in Eq. (7.7).] Here Λ is the effective cosmological constant associated with the background solutionḡ µν , which in general is distinct from the "bare" cosmological constant Λ 0 that may appear in the action. The derivative of the Lagrangian with respect to R µν ρσ is performed formally, as if R µν ρσ and g µν are independent, and we impose the same tensor symmetries as R µν ρσ . For example, in the case of Einstein gravity (2.2) we get The (. . .)ḡ notation indicates that the expression in parentheses is to be evaluated on the background spacetimeḡ µν . The expressions in Eqs. (2.4) and (2.5) are "projection" tensors that pick out certain coefficients to give the correct energy. [The subscript (1) will be explained later.] While the contractions with the projectors might appear complicated, their main use is to formally write the final formula (2.3). When we take a derivative with respect to the Riemann tensor and evaluate on the (homogeneous) background, we always get an expression of the form where N is some constant coefficient. The projector (2.4) is defined to give precisely the coefficient N when acting on Eq. (2.7). Thus, in practice one often simply reads off the coefficient after computing the derivative on the background, rather than actually performing the contraction with P ρσ µν . Similarly, for the second derivative with respect to the Riemann tensor evaluated on the background, there are in general three terms, 8) where N 1 , N 2 , N 3 are constants. When the projector P γδ,µν (1)αβ ρσ acts on Eq. (2.8), it picks out the coefficient N 1 , but again, we can also simply read off this coefficient by writing the expression for the second derivative in the above form. For example, if L = C R αβγδ R αβγδ , (2.9) then N 1 = 2C. In other words, the second derivative term in Eq. (2.3) roughly corresponds to coefficients of terms with the same type of contractions as in Eq. (2.9). A typical example of solutions that fall off sufficiently fast at infinity are asymptotically SdS solutions. The asymptotic behavior of such solutions is where r 0 is a constant. For these solutions, the energy in the case of Einstein gravity is given by and for a general theory we get Another important example is the energy density of black branes in AdS. When the asymptotic behavior of the black brane solution is the energy density of the black brane is (2.14) Let us now look at two examples.

A. Example: Energy with a Gauss-Bonnet term
We start with the famous case of Gauss-Bonnet gravity. This term is topological (a total derivative) in four dimensions and leads to a ghost-free theory in any number of dimensions. The Lagrangain with the Gauss-Bonnet term reads and we have 18) and the projection P µν ρσ gives us the coefficient of δ ρ [µ δ σ ν] : The second derivative with respect to the Riemann tensor is (2.20) Since we only require the coefficient of the first type of term, namely δ Substituting Eqs. (2.19) and (2.21) into the formula for the energy (2.12), we obtain This agrees with the result in Ref. [13], which was obtained by a longer calculation.

B. Example: Energy with a six-derivative term
Here we will consider an example with six derivatives of the metric (cubic curvature), The ADT energy of this theory was previously given in Ref. [24], but required a much lengthier calculation. In this example, the first derivative with respect to the Riemann tensor gives and substituting the background metric leads to (2.25) The second derivative with respect to the Riemann tensor (which in this example is not just a constant) is When we substitute the background metric, the second derivative becomes proportional only to the first term in Ref. (2.8), so the coefficient is just N 1 , and N 2 = N 3 = 0. Substituting this and Eq. (2.25) into the energy formula (2.12), we finally obtain which agrees with the result in Ref. [24].

III. THE ADT METHOD
In this section we give a brief review of the ADT method. The ADT method is similar in spirit to the Landau-Lifshitz pseudotensor method for calculating energy [29] in asymptotically flat curved spacetime. In particular, one proceeds by linearizing the equations of motion with respect to a background spacetime. This leads to an effective stress-energy tensor that consists of matter sources and terms higher-order in the perturbation. This tensor turns out to be covariantly conserved and can thus be used to construct a conserved charge associated with an isometry of the background.
Let us consider some arbitrary gravitational theory with equations of motion of the form where κ is the gravitational coupling and τ µν is the matter stress-energy tensor. The symmetric tensor Φ µν , which is the analogue of the Einstein tensor, may depend on the metric, the curvature, derivatives of the curvature, and various combinations thereof. Assuming that the action is invariant under diffeomorphisms, we obtain the geometric identity ∇ µ Φ µν = 0 (the generalized Bianchi identity) and the covariant conservation of the stress tensor ∇ µ τ µν = 0. Now, we further assume that there exists a background solutionḡ µν to the equations (3.1) with τ µν = 0. Then we decompose the metric as where we note that the deviation h µν is not necessarily infinitesimal, but it is required to fall off sufficiently fast at infinity. Asymptotically SdS spacetimes are a typical example meeting this requirement. By expanding the left-hand side of Eq. (3.1) in h µν , the equations of motion may be expressed as µν denotes all terms in the expansion of Φ µν involving i powers of h µν , and we have defined the effective stress-tensor T µν . It then follows from the Bianchi identity of the full theory that∇ µ φ Suppose that the background spacetime admits a timelike Killing vectorξ µ , and let Σ be a constant-time hypersurface with unit normal n µ . Then we can construct a conserved energy in the standard way: whereḡ Σ denotes the determinant of the induced metric on Σ. Because∇ µ (T µνξ ν ) = 0, it follows that T µνξ ν =∇ ν F νµ for some antisymmetric tensor F νµ . The bulk integral (3.4) can therefore be rewritten as a surface integral over the boundary ∂Σ: where r µ is the unit normal to the boundary. For example, for the Einstein-Hilbert theory (2.2), the explicit expression for the energy is In summary, to apply the ADT method, one linearizes the equations of motion to obtain the stress-energy tensor, and then expresses the conserved current T µνξ ν as a total derivative to find the "potential" F νµ . Note that by construction, the background spacetimeḡ µν has E = 0.

IV. THE GENERAL STRUCTURE OF THE STRESS TENSOR
In Ref. [12], it was shown that the most general quadratic curvature theory has a stress tensor that is schematically of the form and the α i are constants. Here R L µν is the linearized Ricci tensor and R L is the linearized Ricci scalar Note that the tensors G L µν , H µν are each divergence free: It was later found in Ref. [24] that the stress tensor for a certain cubic curvature theory took exactly the same form (the only modification was to the values of the coefficients α i ) and it was suggested that this observation might hold more generally 2 . In this section, we will argue that this is indeed the case for any theory of the form (2.1). The basic idea is as follows. We saw in the previous section that the ADT stress tensor is given by the linearized (in h) equations of motion. This means that T µν only depends on the action to O(h 2 ). Now, expanding the general action (2.1) to O(h 2 ) involves expanding the Riemann tensor, which contains terms of the form∇∇h. Hence, this can yield terms of at most four derivatives. This suggests quite generally that the basic form of the stress tensor does not change from Eq. (4.1) even if the action contains more than two powers of the Riemann tensor 3 . We will show that there exists a basis of three different components for the stress-energy tensor [to O(h 2 ) and up to four derivatives], and that this basis can be chosen to correspond to G L µν , H µν , and H µν . To demonstrate this claim in more detail, we first consider the most general O(h 2 ) action with two derivatives: (4.8) For simplicity we work in the case of a flat background,ḡ µν = η µν . The generalization to a curved background will be discussed later. Varying I 2 with respect to h µν yields the stress tensor If we impose conservation of the stress tensor, we obtain so equating the coefficients to zero gives Substituting these relations into Eq. (4.9) gives T µν = −4β 1 G L µν . It follows that the Lagrangian yields a conserved stress tensor that is precisely G L µν (in a flat background). Note that this is also just the famous Fierz-Pauli Lagrangian [30]. Now let us repeat the same procedure for the most general O(h 2 ) action with four derivatives: The corresponding stress tensor is and imposing ∂ µ T µν = 0 gives We see that there are two independently conserved tensors, and substituting these relations into Eq. (4.14) gives µν . It follows that the Lagrangian yields a conserved stress tensor that is precisely H (1) µν and the Lagrangian yields a conserved stress tensor that is precisely H µν . Thus, we see that, in a flat background, there are at most two possible conserved combinations of terms with four derivatives in the Lagrangian.
The above calculation can in principle be repeated for the case of a curved background spacetime, but it becomes complicated by the fact that the covariant derivatives no longer commute. The key point, however, is that commuting two derivatives in a given expression only produces extra terms of lower differential order. Thus, the expressions for L H (i) now must include two-derivative and zero-derivative terms, but the highest derivative order (four) terms are the same as in a flat background. Once these are fixed, the two-derivative and zero-derivative terms are chosen by requiring that each H (i) µν is separately conserved. A similar argument shows that the unique conserved quantity consisting of two-derivative and 0-derivative terms is G L µν . In other words, the terms with the highest derivatives (four derivatives in H (i) µν and two derivatives in G L µν ) are the same for curved and flat backgrounds. In summary, we have just seen that any O(h 2 ) action with no more than four derivatives produces a conserved stress tensor with the same structure as Eq. (4.1). Combining this with the argument that any theory L = L (R µν ρσ , g µν ) expanded to O(h 2 ) cannot have terms with more than four derivatives, we conclude that any such theory also has a stress tensor of the form (4.1).

V. THE GENERAL FORMULA FOR THE STRESS TENSOR (FLAT BACKGROUND)
In this section we derive an efficient method to extract the coefficients α i in the stress tensor (4.1) given a Lagrangian L = L (R µν ρσ , g µν ) for a flat background. The generalization for a curved background will be done in the next section. We wish to expand the action √ −g L to second order in the perturbation h. The Lagrangian can be expanded as The variation of the Riemann tensor is By convention, indices of Υ αβγ are raised/lowered with the background metric or its inverse. Note that each factor of δR µν ρσ contributes at least one h µν and two derivatives. In the remainder of this section we restrict to the case of a flat background,ḡ µν = η µν . Now, the terms in the action with two derivatives and two h's can only arise from expanding the term with one δR µν ρσ , that is Since L is a function only of g µν and R µν ρσ , it follows that ∂L /∂R µν ρσ evaluated on a homogeneous background can only be a function ofḡ µν . Furthermore, this quantity has the same symmetries as the Riemann tensor, so it must take the general form for some constant N . Formally, this constant can be expressed as a "projection" for the projection tensor Inserting Eq. (5.7) into Eq. (5.6), we see that we simply need the expansion of N √ −g R. This is of course just the Einstein-Hilbert action (up to the overall factor N ), whose expansion is well-known to give the Fierz-Pauli action (see, e.g.,Ref. [31]), N L G .
The O(h 2 ) terms in the action with four derivatives can only arise from the term 10) and for this we just need the linear term in δR µν ρσ , Now, the second derivative evaluated on a homogeneous background can only be a function ofḡ µν , and this quantity must have the same index symmetries as the product of two Riemann tensors. Hence, there can be three independent contributions for some constants N i . This is similar to the statement that there are only three independent curvature invariants formed from contracting two Riemann tensors. Formally, these constants can be expressed by acting with projectors (5.14) The coefficients are .
The next step is to insert Eq. (5.12) into Eq. (5.10) and use Eq. (5.11). We treat the three contractions separately. The first is analogous to R 2 µνρσ and gives The second contraction is analogous to R 2 and gives The last contraction is analogous to R 2 µν and gives Thus the relevant part of the expanded action is and the corresponding stress tensor is

VI. THE GENERAL FORMULA FOR THE STRESS TENSOR (CURVED BACKGROUND)
The procedure described previously for a flat background should in principle generalize to a curved background. The calculation becomes cumbersome, however, since the covariant derivatives no longer commute. Instead, we shall adopt a different approach that turns out to be much more straightforward.
It was argued in Refs. [10,11,[25][26][27] that any higher-curvature theory which is polynomial in the Riemann tensor and its contractions can be reduced to an "effective quadratic curvature" action with the same propagator. Since the propagator also only depends on the action up to order h 2 , we can adapt this procedure to determine the ADT stress-tensor for a general theory.
Consider expanding the Lagrangian of a generic higher-curvature theory of the form L = L (R µν ρσ ), Here the dots represent terms which are necessarily of order h 3 and therefore are not relevant to the ADT energy. Next we substitute the general expressions for the derivatives of the Lagrangian with respect to the Riemann tensor evaluated on the background, which were previously given in Eqs. (5.7) and (5.12). Using Eq. (2.17) and collecting coefficients of the full Riemann tensor terms, we obtain the effective quadratic theory Here we have defined and the "bare" cosmological constant for the effective theory is Now, the most general quadratic curvature theory has already been treated in Ref. [13]. The result is that the stress tensor is µν , and H (2) µν were given in Eqs. (4.2)-(4.4). Furthermore, the effective cosmological constant Λ is fixed by evaluating the equation of motion on the background solution: Substituting the above expressions forκ, α, β, γ into Eq. (6.8) yields Note that for Λ = 0, this agrees with the result of the previous section.

A. Examples
Let us look at some examples for the formula (6.10) for the stress-energy tensor. Let us start with the simple theory The coefficients N, N i are computed as described previously by taking derivatives with respect to the Riemann tensor and evaluating on the background AdS solution. We find that (6.12) and 13) so N 1 = 2 , N 2 = N 3 = 0. Using Eq. (6.10), we obtain µν , (6.14) which matches the result of Ref. [13]. For a more complicated example, consider the six-derivative theory 15) whose stress tensor was computed explicitly in Ref. [24]. The results are summarized in the following table: . (6.16) Substituting the above results into Eq. (6.10) gives where .
This reproduces precisely the stress tensor obtained in Ref. [24].

VII. THE DERIVATION OF THE ENERGY FORMULA
Given the result (6.10), the final step in the derivation of the energy formula is to writeξ ν T µν as a total derivative. For this purpose, we follow the steps in Ref. [12]. For the first term,ξ ν G µν L , the result has already been given in Eq. (3.6). It is straightforward to show that the second term can be written as The third term,ξ ν H (2) µν , is more complicated and turns out to give an additional contribution of the formξ ν G µν L . To see this, we can rewritē Sinceξ ν is a Killing vector, it satisfies Then the last terms of Eq. (7.2) simplify tō Using these results, we find that the final form of the conserved energy is In asymptotically SdS spacetimes [see Eq. (2.10)], the last two terms in Eq. (7.7) fall off too fast at large r to contribute and the total energy is given by 8) or in the full explicit form as in Eq. (2.3).

VIII. DISCUSSION
In this paper, we have derived a simple formula (2.3) for the ADT energy of any gravitational theory of the form L = L (R µν ρσ , g µν ). We gave a detailed argument that the energy of such a theory takes the same basic form as in quadratic curvature gravity, but with coefficients modified by the higher-curvature terms. The coefficients are given by taking derivatives of the Lagrangian with respect to the Riemann tensor, and in this sense our energy formula is reminiscent of Wald's entropy formula. We have demonstrated in a number of examples that our formula correctly reproduces previous results, but with significantly less computations. For more complicated theories in which following the full ADT procedure would be unmanageable in practice, it seems that our formula could still be applied relatively easily.
We note from the final formula for energy (2.3) that only N and N 1 appear, and it would be interesting to understand why this is the case. We also see that in d = 3, the contribution of the second derivative of the Lagrangian completely drops out. In the case of three-dimensional topologically massive gravity (TMG) [32][33][34], the action contains a gravitational Chern-Simons term so it is not of the form L = L (R µν ρσ , g µν ). Indeed, the ADT energy for TMG has a different structure than (2.3) [35]. It is also known that Wald's entropy formula has to be modified in TMG, since the Chern-Simons term does not satisfy the diffeomorphism-covariance requirement in the original construction (see, e.g., Refs. [36,37]).
Given the final expression for the energy (7.8), it seems natural to define the effective gravitational coupling as Then the energy can be written succinctly in terms of the Einstein gravity result as This is analogous to the way the entropy was written in Ref. [8] as where A is the black hole area. However, the effective coupling constant also has an interpretation in the tree-level scattering amplitude via the exchange of a graviton. When one looks at a similar process on the background [10], the effective coupling turns out to be the coefficient of G L µν in the stress-energy tensor: The two definitions for the effective coupling coincide for Lanczos-Lovelock gravity, since any Lagrangian of the Lanczos-Lovelock type can be reduced to a Gauss-Bonnet quadratic theory [11]. This coincidence might be related to the fact that higher-derivative theories which are not of the Lanczos-Lovelock type exhibit ghosts and other inconsistencies [38,39]. In future work, it would be interesting to further understand this ambiguity in the definition of the effective gravitational coupling.
Note Added: One day after this paper appeared on the arXiv, the paper [40] appeared with some overlapping material.

IX. ACKNOWLEDGMENTS
We thank Ramy Brustein for a discussion and Stanley Deser for comments.