A new approach to conserved charges of generic gravity in AdS Emel Altas∗ Department of Physics, Karamanoglu Mehmetbey University, 70100, Karaman, Turkey
Bayram Tekin†
arXiv:1811.11525v1 [hep-th] 28 Nov 2018
Department of Physics, Middle East Technical University, 06800, Ankara, Turkey (Dated: November 29, 2018) Starting from a divergence-free rank-four tensor whose trace is the cosmological Einstein tensor, we give a construction of conserved charges in Einstein’s gravity and its higher derivative extensions for asymptotically anti-de Sitter (AdS) spacetimes. The current yielding the charge is explicitly gauge-invariant and the charge expression involves the linearized Riemann tensor at the boundary. Hence to compute the mass and angular momenta in these spacetimes, one just needs to compute the linearized Riemann tensor. We give two examples.
I.
INTRODUCTION
Let us start with a seemingly innocent question which will have far-reaching consequences for the conserved charges of gravity theories. Given the Riemann tensor Rν µβσ , its single trace (over the first and third indices) yields the Ricci tensor Rµσ ; is there a rank-four tensor whose single trace is not the Ricci tensor but the (cosmological) Einstein tensor, G µσ = Rµσ − 12 Rgµσ + Λgµσ , with the condition that this four-index tensor has the symmetries of the Riemann tensor and it is divergence-free just like the Einstein tensor? Remarkably the answer is affirmative, the following tensor (whose construction will be given below) P νµβσ := Rνµβσ +gσν Rβµ −gβν Rσµ +gβµ Rσν −gσµ Rβν +
R Λ (n − 3)) βν σµ − g g − gσν gβµ (1) 2 n−1
is divergence-free for all smooth metrics, i.e. without the use of any field equations ∇ν P ν µβσ = 0,
(2)
and its trace is the cosmological Einstein tensor as desired P ν µνσ = (3 − n)G µσ .
(3)
Clearly, the interesting exception is that one cannot do this construction in three dimensions. What happens for n = 3 is that, the P-tensor vanishes identically since, due to the vanishing of the Weyl tensor, the Riemann and the Ricci tensors carry the same amount of information and the Riemann tensor can be expressed in terms of the Ricci tensor as Rνµβσ = Rνβ gµσ + Rµσ gνβ − Rνσ gµβ − Rµβ gνσ − ∗ †
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R νβ µσ g g − gµβ gνσ . 2
(4)
2 Therefore in some sense, the P-tensor (1) is an obstruction for a smooth generically curved metric to be 3-dimensional. This can also be seen from the following identity: the Gauss-Bonnet combination χGB := Rνµβσ Rνµβσ − 4Rµν Rµν + R2 vanishes identically in 3 dimensions and it is easy to show that the contraction of the P-tensor with the Riemann tensor yields1 Rνµβσ P νµβσ = χGB − 2Λ
(n − 3) R, n−1
(5)
which vanishes in 3 dimensions; but gives the Einstein-Gauss-Bonnet Lagrangian (with a fixed relative coefficient) in generic n dimensions. The natural question is how does one arrive at the P-tensor (1)? We have found the P-tensor from the following construction: starting from the Bianchi identity ∇ν Rσβµρ + ∇σ Rβνµρ + ∇β Rνσµρ = 0,
(6)
and carrying out the gνρ multiplication, one arrives at the P ν µβσ as given in (1) after making use of ∇µ G µν = 0, ∇µ gαβ = 0. Note that this still leaves an ambiguity in the P-tensor, since one can add an arbitrary constant times gµσ gβν but that part can be fixed by demanding that the P-tensor has the symmetries of the Riemann tensor and also vanishes for constant curvature backgrounds, which we assumed. This tensor turns out to be extremely useful in finding conserved charges of Einstein’s gravity for asymptotically AdS spacetimes for n > 3 dimensions. Recently in [3], we gave a brief account of this formulation in Einstein’s theory and in the current work we shall extend this formulation to quadratic and generic gravity theories. The main motivation of our construction is the following: outside the localized sources, the properties of gravity are fully encoded in the Riemann tensor. One would naturally expect that the charge expression, which is an integral in the boundary of a spacelike surface, would also involve the Riemann tensor at infinity. But a straightforward computation shows that this is not the case, as we shall revisit in the next section. The existing formulae involve the first derivatives of the metric perturbation. The crux of the matter is that, the existing expressions are based on conserved currents which are only gauge-invariant up to a boundary term that vanishes. Our formalism remedies this and constructs an explicitly gauge invariant current and simplifies the charge expressions significantly. The layout of the paper is as follows: In section II, which is the bulk of the paper, we discuss the conserved Killing charges in generic gravity and give a compact expression that utilizes the Ptensor. In section III, we discuss the gauge invariance issue of the conserved currents. In section IV we study the n-dimensional Schwarzschild-AdS spacetime and the AdS soliton. In [3], we studied the Kerr-AdS solution an hence we shall not repeat it here. II.
CONSERVED CHARGES
Conserved charges of generic gravity theory in asymptotically AdS spacetimes were constructed in [4] as an extension of the Abbott-Deser charges [5] of the cosmological Einstein’s theory. The latter is an generalization of the ADM charges [6] which are valid for asymptotically flat spacetimes. 1
As a curious note, one can see that the square of this tensor yields a particular Einstein plus quadratic gravity in generic n ≥ 4-dimensions P 2νµβσ
= χGB + (n − 3)
2 4Rµν
(n − 3) R2 (n − 6) − 2Λ ((n − 2)R − Λn) , + 2 n−1
which is not the Lagrangian of critical gravity [1, 2]
3 A detailed account of these constructions were recently given in [7] and for related constructions see [8] and [9]. Here for the sake of completeness, we will briefly summarize the salient parts of this construction. Consider a generic gravity theory defined by the field equations depending on the Riemann tensor (R), its derivatives and contractions
Eµν g, R, ∇R, R2 , ... = κτµν ,
(7)
where ∇µ E µν = 0, and κ is the n-dimensional Newton’s constant while τµν represents a localized conserved source. A non-trivial, partially conserved current arises after one splits the metric as gµν = g¯µν + κhµν ,
(8)
which yields a splitting of the field equations as
κ(Eµν )(1) (h) = κτµν − κ2 (Eµν )(2) (h) + O κ3 ,
(9)
where we assumed that g¯ solves the field equations, Eµν (¯ g) = 0, exactly in the absence of any d (1) g + κh) |κ=0 . Hence defining (Eµν )(1) := Tµν , one has the source τµν and (Eµν ) (h) := dκ Eµν (¯ ¯ desired partially conserved current, if the background admits a Killing vector ξ: J µ :=
−¯ g ξ¯ν (Eµν )(1) .
p
(10)
¯ one has the conserved As usual, making use of the Stokes’ theorem, given a spacelike hypersurface Σ, charge for each background Killing vector ˆ p ¯ dn−1 y γ¯ n Q ξ := ¯ µ ξ¯ν (Eµν )(1) , (11) ¯ Σ
where we assumed the that J µ vanishes at spacelike infinity. To proceed further and reduce this ¯ to an integral over the boundary ∂ Σ, ¯ one must know the field equations and express integral over Σ (1) ξ¯ν (Eµν ) as a divergence of an antisymmetric two tensor. Recently [3], we have shown that using the P-tensor of the previous section, one can reformulate this problem in the cosmological Einstein’s theory in AdS spacetimes without using the explicit form of the linearized cosmological Einstein tensor. This is possible because in Einstein spaces (that are not Ricci-flat such as the AdS), one has the nice property that the Killing vector can be derived from an antisymmetric "potential" F¯µν as ¯ ν F¯ νµ , ξ¯µ = ∇
(12)
¯ ν ξ¯µ with R ¯ being the constant scalar curvature. Although this result is valid where F¯ νµ = − R2¯ ∇ for any Einstein space as a background, for concreteness, we shall work in the AdS background, for which we have 2Λ ¯ µν = 2Λ g¯µν , ¯ = 2nΛ . ¯ µανβ = (¯ gµν g¯αβ − g¯µβ g¯αν ) , R R (13) R (n − 2) (n − 1) n−2 n−2 To find the conserved charges of a gravity theory defined on an asymptotically AdS spacetime M , let us assume that there is an antisymmetric two form, Fµν , on the manifold. Then one has the exact equation for any smooth metric ∇ν (F βσ P νµβσ ) − P νµβσ ∇ν F βσ = 0.
(14)
Linearization of (14) about the AdS background yields
¯ βσ − (P νµβσ )(1) ∇ ¯ νF ¯ βσ = 0, ∇¯ν (P νµβσ )(1) F which is the main equation from which we will read the conserved current.
(15)
4 A.
Einstein’s Theory
Let us recap the main points of [3]. Using the following equivalent form of the P-tensor, written in terms of the cosmological Einstein’s tensor, P ν µβσ := Rν µβσ + δσν G βµ − δβν G σµ + G νσ gβµ − G νβ gσµ + (
R Λ (n + 1) ν − ) δσ gβµ − δβν gσµ , 2 n−1
(16)
one arrives at its linearized form (P νµβσ )(1) =
(Rνµβσ )1 + 2(G µ[β )(1) gσ]ν + 2(G ν[σ )(1) g β]µ + (R)(1) g µ[β g σ]ν 4Λ + (hµ[σ g β]ν + g µ[σ hβ]ν ), (n − 1)(n − 2)
(17)
where the square brackets denote anti-symmetrization with a factor of 1/2. For the particular antisymmetric background tensor ¯ αβ := ∇ ¯ α ξ¯β , F
(18)
where ξ¯β is an AdS Killing vector, one finds from (15) the following conserved current ¯ξ (G λµ )(1) = (n − 1)(n − 2) ∇¯ν (P νµβσ )(1) F ¯ βσ . λ 4Λ (n − 3)
(19)
Comparing this with the integrand of (11), and using the Stokes’ theorem one more time, we find the desired result (n − 1)(n − 2) Q ξ¯ = 8(n − 3)ΛGΩn−2
ˆ
¯ ∂Σ
dn−2 x
¯ βσ , γ¯ ¯ǫµν (Rνµ βσ )(1) F
p
(20)
where (Rνµ βσ )(1) is the linearized part of the Riemann tensor about the AdS background. Observe that on the boundary (P νµ βσ )(1) = (Rνµ βσ )(1) , since the linearized Einstein tensor and the linearized scalar curvature vanish. The barred quantities refer to the background spacetime M¯ with the boundary ∂ M¯. The Killing vector is ξ¯σ from which one defines the antisymmetric tensor ¯ is not equal to ∂ M¯, hence Σ ¯ can have a boundary of ¯ β ξ¯σ . The spatial hypersurface Σ as F¯ βσ = ∇ ¯ nµ σ ¯ν − n ¯ν σ ¯µ ), its own, that is ∂ Σ. Here the antisymmetric two-form ǫ has components ǫ¯µν := 12 (¯ ¯ ¯ where n ¯ µ is a normal one form on ∂ M and σ ¯ν is the unit normal one form on ∂ Σ and γ¯ is the induced metric on the boundary. This is sufficient for the conserved charges of the cosmological Einstein’s theory in AdS. But for a generic theory, one must carry out an analogous computation which is what we do next. But before that, let us note that for the energy of the spacetime, we have ξ¯ = ∂t and (20) becomes ˆ (1) p (n − 1)(n − 2) ¯ r ξ¯t , dn−2 x γ¯ Rrt rt ∇ (21) E := Q (∂t ) = 2(n − 3)ΛGΩn−2 ∂ Σ¯ where r is the radial coordinate and one takes r → ∞ at the end of the computation. Similarly, for the angular momentum, one can take the Killing vector ξ¯µ = (0, ..., 1, 0, ..., 0) and carry out the computation.
5 B.
Generic Theory
Consider a generic gravity theory which starts with the Einsteinian part as Eµν =
1 1 Rµν − Rgµν + Λ0 gµν + σEµν = τµν , κ 2
(22)
where at this stage all we know about the Eµν tensor is that, it is a symmetric divergence-free tensor (which can come from an action) and σ is a dimensionful parameter. To proceed further, it is better to recast the equation as Eµν =
Λ0 − Λ 1 G µν + gµν + σEµν = τµν , κ κ
(23)
whose (A)dS vacua are determined by Λ0 − Λ ¯µν = 0, E¯µν = g¯µν + σ E κ
(24)
¯µν tensor. We shall assume that which in general has many vacua depending on the details of the E Λ represents any one of the viable vacua. To find the conserved charges in this theory, we use the same procedure as the one in the previous section and define (Eµν )(1) = Tµν ,
(25)
where the right-hand side has all the higher order terms Tµν = τµν − κ(Eµν )(2) − κ2 (Eµν )(3) − ... .
(26)
So we have the background conserved current
¯ ν ξ¯µ (E µν )(1) = 0, ∇ and the partially conserved current is J ν =
√
(27)
−¯ g ξ¯µ (E µν )(1) . Hence we must compute2
1 Λ0 − Λ ¯ µν ξ¯µ (E µν )(1) = ξ¯µ (G µν )(1) − ξµ h + σ ξ¯µ (E µν )(1) . (28) κ κ We have already computed the first part in the previous subsection, hence, the new parts are the second and the third terms. But, when the theory is not given, one cannot proceed further from this point. For this reason, let us consider the quadratic theory as an example which also covers all the f (Riemann) type theories. The action of the quadratic theory is ˆ √ 1 (R − 2Λ0 ) + αR2 + βRµν Rµν + γ Rµνρσ Rµνρσ − 4Rµν Rµν + R2 , (29) I = dn x −g κ
and the field equations are [4]
1 1 1 Rµν − gµν R + gµν Λ0 + 2αR Rµν − gµν R + (2α + β) (gµν ✷ − ∇µ ∇ν ) R κ 2 4 1 + 2γ RRµν − 2Rµσνρ Rσρ + Rµσρτ Rν σρτ − 2Rµσ Rσν − gµν Rαβρσ Rαβρσ − 4Rαβ Rαβ + R2 4 1 1 (30) + β✷ Rµν − gµν R + 2β Rµσνρ − gµν Rσρ Rσρ = τµν . 2 4
2
At this stage, we can search for a modified version of the P tensor which is conserved and whose trace is the E tensor. One can find such a tensor but it does not have the symmetries of the Riemann tensor anymore and hence it does not make the ensuing computation any simpler.
6 Inserting (13) in the last equation, one finds the equation satisfied by Λ: (n − 4) (n − 3)(n − 4) 2 Λ − Λ0 + (nα + β) +γ Λ 2 2κ (n − 2) (n − 1)(n − 2)
!
= 0.
(31)
Defining the constant c :=
4Λn 4Λ 4Λ (n − 3) (n − 4) 1 + α+ β+ γ, κ n−2 n−1 (n − 1) (n − 2)
(32)
one can show that the linearized expressions read 4Λ Λ0 − Λ ¯ µν 1 β ξ¯ν (G µν )(1) σ ξ¯ν (E µν )(1) − ξν h = c − + κ κ (n − 1) (n − 2) ¯ [α (G µ]ν )(1) + (G ν[α )(1) ∇ ¯ µ] ξ¯ν ¯ α ξ¯ν ∇ + 2β ∇
¯ α] (R)(1) + (R)(1) ∇ ¯ µ ξ¯α , ¯ α 2ξ¯[µ ∇ + (2α + β)∇
which then yields the desired result
4Λ β ξ¯ν (G µν )(1) (n − 1) (n − 2) ¯ [α (G µ]ν )(1) + (G ν[α )(1) ∇ ¯ µ] ξ¯ν ¯ α ξ¯ν ∇ + 2β ∇
ξ¯ν (E µν )(1) = c +
(34)
¯ α] (R)(1) + (R)(1) ∇ ¯ α 2ξ¯[µ ∇ ¯ µ ξ¯α . + (2α + β)∇
Therefore the conserved charges in quadratic gravity in (A)dS read as 4Λβ (n − 1) (n − 2) c+ Q ξ¯ = 4Λ (n − 3) (n − 1) (n − 2)
ˆ
¯ ∂Σ
dn−2 x
p
(33)
¯ γ¯ ǫ¯µν (Rνµ βσ )(1) F
(35)
βσ
.
(36)
Observe that for asymptotically AdS spacetimes, the second and third lines in (35) do not contribute. But, if one tries to generalize the above procedure to asymptotically non-constant curvature spacetimes, those parts will also contribute generically. Therefore, for asymptotically AdS spacetimes, the only difference between the conserved charges in Einstein’s theory (20) and the quadratic theory is the numerical factor in (36). Using the ideas presented in [10], the above construction can be extended to any f (Riemann) theory, where f is a smooth function. This is because, as far as the energy, vacua and particle contents are considered, any f (Riemann) theory ¯ µν ), has an equivalent quadratic action formulation in which one computes only 3 quantities f (R αβ 2 ∂f , ∂R∂µν fRρσ ∂Rµν αβ αβ ηδ
and their contractions to find the κ, α, β and γ of the theory to insert in (36). As
this issue is dealt in [10] and [11], we refer the reader to these works. So the crucial part is the Einsteinian part which we have studied in the previous section. III.
GAUGE INVARIANCE ISSUE
The problem of the gauge transformations of the charge and the current that yields the charge is important. Clearly one expects the charge to be gauge-invariant in any valid formulation, but the current need not be. In fact earlier constructions of conserved charges [4, 5] used gauge-variant currents which yield gauge-invariant charges. Of course, for the charges to be gauge-invariant, the
7 non-invariance of the current is only up to a boundary term that vanishes in the boundary. Let us show this in the expression of [4] for the cosmological Einstein’s theory: ¯ α J αµ , 2ξ¯ν (G µν )(1) = ∇
(37)
where the antisymmetric current is ¯ β hµβ − ξ¯µ ∇ ¯ β hαβ + ξ¯ν ∇ ¯ µ hαν − ξ¯ν ∇ ¯ α hµν + ξ¯µ ∇ ¯ α h − ξ¯α ∇ ¯ µh J αµ :=ξ¯α ∇ ¯ α ξ¯ν − hαν ∇ ¯ µ ξ¯ν − h∇ ¯ α ξ¯µ . + hµν ∇
(38)
Consider an infinitesimal coordinate transformation generated by a vector field ζ (not to be confused with the Killing field ξ), one has ¯ µ ζν + ∇ ¯ ν ζµ = Lζ g¯µν , δζ hµν = ∇
(39)
where L ζ denotes the Lie derivative and hence δζ hµν = −L ζ g¯µν . It is easy to see that δζ (G µν )(1) = Lζ G¯µν = 0. But this only implies from (37) that one has the divergence of the gauge-transformed current to vanish ¯ α δζ J αµ = 0, ∇
(40)
and hence J αµ is not necessarily gauge invariant. In fact one can show that J αµ varies, under the gauge transformations (39), as
¯ ν ζ µ + ξ¯µ ∇ ¯ α ζ ν + ξ¯ν ∇ ¯ µ ζ α + 2ζ α ∇ ¯ ν ξ¯µ + ζ ν ∇ ¯ µ ξ¯α − (µ ↔ α) . ¯ ν ξ¯α ∇ δζ J αµ = ∇
(41)
Clearly since the variation is a boundary term and since J αµ is the integrand on the boundary of ¯ = 0) and hence the the spatial slice, the boundary term does not contribute to the charges (as ∂∂ Σ charge is gauge invariant. But this exercise shows us that the current (38) is only gauge invariant up to a boundary term. On the other hand, since δζ (Rνµ βσ )(1) is gauge invariant, our charge expression (20) is explicitly gauge invariant without an additional boundary term. Let us show this: ¯ ν αβσ δζ hαµ . δζ (Rνµ βσ )(1) = g¯αµ δζ (Rν αβσ )(1) − R
(42)
Given the linearized Riemann tensor as ¯ β (Γνσα )(1) − ∇ ¯ σ (Γνβα )(1) , (Rν αβσ )(1) = ∇
(43)
¯ σ∇ ¯ αζ ν + R ¯ ν αρσ ζ ρ . δζ (Γνσα )(1) = ∇
(44)
one needs
Collecting all the pieces together, one arrives at ¯ νµ βσ . δζ (Rνµ βσ )(1) = L ζ R
(45)
¯ νµ βσ = 0 and hence δζ (Rνµ βσ )(1) = 0 and so δζ Q = 0 For the AdS background one clearly has L ζ R as expected. So in our formalism, not only the charge is explicitly gauge-invariant, but also the current is explicitly gauge invariant. IV.
SOME ASYMPTOTICALLY ADS SPACETIMES
We have given the computation of the energy and the angular momentum of the four dimensional Kerr-AdS solution in [3], here, let us give 2 more examples.
8 A.
n-dimensional AdS-Schwarzschild spacetime
Consider a spherically symmetric metric in n-dimensions: ds2 = −f (r)dt2 +
1 dr 2 + r 2 dΩn−2 . f (r)
(46)
For the choice f (r) = 1 −
r0 r
n−3
+
r2 , ℓ2
ℓ2 ≡ −
(n − 1)(n − 2) , 2Λ
(47)
the metric (46) is an Einstein spacetime with the Ricci tensor n−1 gµν . (48) ℓ2 As such, it solves the n-dimensional vacuum Einstein’s equations with a cosmological constant. For the Killing vector ξ µ = (−1, 0, ..., 0), one needs to compute the integrand in (21) which boils ¯ r ξ¯t . It is easy to see that the relevant component of down to computing the expression (Rrt rt )(1) ∇ the full Riemann tensor reads Rµν = −
1 Rrt rt = − f ′′ (r), 2
(Rrt rt )(1) = −(n − 3)(n − 2)
r0n−3 , r n−2
(49)
¯ r ξ¯t = − f ′ |r →0 = − r2 . Combining all these in (21) for where f ′ (r) = dfdr(r) . Similarly one has ∇ 2 0 ℓ G = 1, one finds the energy of (46) as E=
n − 2 n−3 r , 4 0
(50)
which is exactly the one computed in [4]. In four dimensions, one has r0 = 2m and E = m. The AdS Soliton
B.
The metric of the "AdS Soliton" was found by Horowitz-Myers [12] and reads as 2
ds =
r2 ℓ2
1−
r0p+1 r p+1
!
2
dτ +
p−1 X i=1
i 2
(dx ) − dt
2
r p+1 + 1 − 0p+1 r
!−1
ℓ2 2 dr . r2
(51)
We shall not go into the physical meaning of this solution which is obtained from a p-brane metric: the Cartesian coordinates xi (i = 1, ..., p − 1) and the t denote the coordinates on the “brane" and r ≥ r0 . Solution does not have a singularity if the coordinate τ is periodic with a period β = 4πℓ2 /(r0 (p + 1)). Consider the timelike Killing vector ξ¯µ = (−1, 0, ..., 0), ¯ r ξ¯t = then ∇
−r . ℓ2
(52)
The relevant linearized Riemann tensor component can be computed to be (Rrt rt )(1) = −
(n − 3) r0n−1 , 2ℓ2 r n−1
(53)
which also shows that there is no n = 3 AdS-soliton. Making use of (21), one obtains E=−
Vn−3 π r0n−2 , (n − 1) Ωn−2 ℓn−2
(54)
where Vn−3 is the volume of the compact dimensions. This result matches the one obtained in [13].
9 V.
CONCLUSIONS
In a gauge or gravity theory, the conserved charges make sense if they are gauge or coodinate invariant (at least for small transformations). The celebrated ADM [6], AD [5] charges for asymptotically flat and AdS spacetimes, respectively, and their generalizations to higher order gravity [4], are all gauge invariant. Namely, they are invariant under small diffeomorphisms. [Large diffeomorphisms are a different story, even the flat Minkowski space, while remaining flat, can be assigned any mass value in a coordinate system that does not have proper asymptotics. See [7] for a brief review of this issue.] However the explicit expressions of these charges do not involve the relevant gauge-invariant quantity; that is the linearized Riemann tensor with two up and two down indices, (Rµν σρ )(1) , but instead they involve the first covariant derivative of the metric perturbation as ¯ α hµν contracted with the Killing vector in such a way that the final result is gauge invariant ∇ only up to a divergence term which vanishes in the boundary. The obvious question is to try to understand if the gauge-invariant charges can be written in an explicitly gauge-invariant way with the help of the Riemann tensor. There is a stronger motivation for such a search: outside the sources, the Riemann tensor carries all the information about gravity. Naturally, it must carry the information about the conserved charges. It turns out, as we have shown recently [3] and here, this is indeed the case and the conserved charge is basically a flux of the Riemann tensor at spatial infinity contracted with an antisymmetric two tensor. The construction is somewhat non-trivial and is valid only for asymptotically AdS spacetimes (which can be generalized to Einstein spacetimes with a non-zero scalar curvature). The reason that one can write the conserved charges as a flux of the linearized Riemann tensor at all, is that for AdS spacetimes, a given Killing vector ξ¯µ has an antisymmetric two-form potential ¯ ν F¯ µν , which helps bring another covariant derivative in the conserved charges whenever as ξ¯µ = ∇ the Killing vector appears, converting the expression to the linearized Riemann tensor that has two covariant derivatives of the metric perturbation. To find the charge expression, we used a divergence-free rank-four tensor whose trace is the Einstein tensor. Interestingly this construction is valid only for n ≥ 4 dimensions and is not valid in 3 dimensions, since the Riemann tensor can be expressed directly in terms of the Einstein tensor in 3 dimensions, whose linearized version vanishes at spatial infinity.
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