On the canonical energy of weak gravitational fields with a cosmological constant $\Lambda \in \mathbb{R}$

We analyse the canonical energy of vacuum linearised gravitational fields on light cones on a de Sitter, Minkowski, and Anti de Sitter backgrounds in Bondi gauge. We derive the associated asymptotic symmetries. When $\Lambda>0$ the energy diverges, but a renormalised formula with well defined flux is obtained. We show that the renormalised energy in the asymptotically off-diagonal gauge coincides with the quadratisation of the generalisation of the Trautman-Bondi mass proposed in [13].


Introduction
A question of current interest is the amount of energy that can be radiated by a gravitating system in the presence of a positive cosmological constant. This problem has been addressed in [13] using an approach based on the characteristic a Preprint UWThPh-2020-27 b co-author of Sections 1, 3-5 c co-author of Sections 1 and 4 constraint equations and involving Bondi coordinates. The analysis there showed ambiguities in the resulting expression. The question then arises, whether some insight into the problem at hand could be gained by considering linearised gravity on the de Sitter background. The aim of this work is to carry out this project.
We start, in Section 2 with a general analysis of the canonical energy in linearised Lagrangian theories. The main point is to derive a formula for the canonical energy including all boundary terms, which are usually neglected and which play a key role in general relativity. The results presented in this section are essentially known [41,56], but a coherent and systematic presentation for linearised field theories, keeping track of all terms in the integrals, does not exist in the literature. One of the main results in this section is Proposition 1, which does not seem to have appeared in the literature in this generality.
In Section 3 we show how to put a linearised gravitational field into Bondi gauge, analyse the small-r behaviour of the fields and derive the freedom remaining. This gives a unified treatment for all Λ ∈ R of asymptotic symmetriesà la Bondi-Metzner-Sachs in the linearised regime, which leads to a clear view of the differences arising from the sign of the cosmological constant and from the boundary conditions imposed. We note that the current approach to asymptotic symmetries, based on characteristic initial data and their evolution, gives a perspective different from the one based e.g. on Fefferman-Graham expansions, as it introduces naturally a foliation of the conformal boundary I by spheres obtained by intersecting I with the light cones.
In Section 4 we analyse the large-r behaviour of vacuum metric perturbation in the Bondi gauge and show explicitly how the formalism works for a class of linearised solutions of the vacuum Einstein equations discovered by Blanchet and Damour [8]. Our asymptotic conditions on the linearised perturbations of the metric are modelled on the asymptotic behaviour of the full solutions of the Einstein vacuum field equations with positive cosmological constant and with smooth initial Cauchy data on S 3 , as derived by Friedrich in [29]. Here some comments might be in order. In [29] Friedrich shows that small perturbations of de Sitter Cauchy data on S 3 lead to vacuum spacetimes with smooth conformal boundaries at infinity. He isolates a set of data on the spacelike boundary at infinity which parameterise uniquely all vacuum spacetimes with a positive cosmological constant and with smooth conformal completions at infinity. His analysis carries over without difficulties to the linearised equations; a convenient way to proceed is to linearise the equations in [47]. The results of Friedrich provide a rigorous justification of the asymptotic expansions proposed by Starobinsky [50], revisited later from a more general perspective in [26]. The readers familiar with the Fefferman-Graham expansions can view the results in this Section as a translation of these expansions to characteristic Cauchy data in the linearised setting.
It should be emphasised that requiring asymptotic conditions more stringent than ours will lead to non-generic fields, in the sense that generic smooth changes of initial data on a Cauchy surface in de Sitter spacetime will lead to solutions which do not satisfy the more stringent conditions. And allowing less stringent conditions will lead to linearised metric fields which are not smoothly conformally extendable along, and in a neighborhood of, the initial light cone. In other words, our asymptotic conditions are optimal for linearised fields which are smoothly conformally extendable.

The energy of linearised fields by PTC and TS
Our aim in this section is to establish that the canonical energy of the linearised theory can be calculated, up-to-divergence, by means of the "presymplectic current" (using the Lee-Wald terminology) of the original theory; see Proposition 1 below. This is well known, but we revisit the proof as we will need the formula for the divergence term.
We further show gauge-independence up-to-boundary term of the canonical energy of the linearised theory. This is also well known (cf., e.g., [22,28,56]), and closely related to the above, but here again our focus is on the explicit form of the boundary term.

General formalism
We consider a first-order Lagrangian field theory for a collection of fields φ ≡ (φ A ), where A runs over a finite set. We write Given a Lagrangian density L (φ, ∂φ, ·), where · denotes background fields (which might or might not be present), the field equations are For λ ∈ I, where I is an open interval containing 0, let φ(λ) ≡ φ A (λ) be a one-parameter family of fields differentiable with respect to λ, set φ ≡ (φ A ) := dφ A dλ . (2.2) Thenφ satisfies the set of equations (2.3) To continue, it is convenient to introduce some notation. We set (2.5) In this notation, (2.3) can be rewritten somewhat more concisely as (2. 6) If dEA dλ has been prescribed (e.g., equal to zero, in which case the fieldsφ A satisfy the linearised field equations), or is known and isφ-independent, this set of equations can be derived from a Lagrangian density L given by (2.7) (To avoid ambiguities: we see that (2.7) with dEA dλ ≡ 0 provides the Lagrangian for the linearised field equations at φ(λ)| λ=0 , regardless of whether or not φ(λ)| λ=0 itself satisfies any field equations, though typically one would be interested in situations where E A (φ(0)) = 0. ) We remark that the field φ := φ(λ)| λ=0 plays a role of a background field in L , so even if we assumed that L depends only upon φ and ∂φ, we end up with a Lagrangian density where background dependence has to be taken into consideration.
The following identities are useful, assuming dEA dλ = 0: (2.9) From now on, we consider a theory which satisfies the following: H1. L is a scalar density. H2. There exists a notion of derivation with respect to a family of vector fields X, which we will denote by L X , which coincides with the usual Lie derivative on vector densities, and which we will call Lie derivative regardless of whether or not this is the usual Lie derivative on the remaining fields, such that the following holds: a) L X preserves the type of a field, thus L X of a scalar density is a scalar density, etc.; b) the field π A µ L X φ A is a vector density; c) in a coordinate system in which X = ∂ 0 we have L X = ∂ 0 ; d) L X satisfies the Leibnitz rule.
The above holds if the fields φ A are tensor fields and L is of the form | det g|L, where L is a scalar, with L the standard Lie derivative.
Let us denote by H the Hamiltonian density vector (called "Noether current" in [45]) associated with the Lagrangian density L and X: (2. 10) Let H µ [X] be the corresponding vector density associated with the original field φ: (2.11) We wish to calculate the variation of H µ . Typically one assumes that the background fields, if any, are λ-independent. This might, however, not be the case for some variations, e.g. if the variations correspond to coordinate transformation which do not leave the background invariant. In order to allow for such situations let us denote by ψ := (ψ I ) the collection of all background fields; if L depends both upon a background and some derivatives thereof, we include the derivatives of the background as part of components of ψ. For completeness we will carry out the usual calculation of . For this, recall the formula for the Lie derivative of a vector density Z: (2. 12) Keeping in mind that π A µ is a tensor density by H1), and our remaining hypotheses H2.a)-H2.d), we are led to the following identity: (2. 13) We are ready now to calculate the variation of H : (2.14) (One should keep in mind that, when the background is not invariant under the flow of X, there might be a contribution from the background when calculating L X π A µ .) When integrated over a compact hypersurface S with boundary, (2.14) leads to the usual field-theoretical version of the generating formula of Hamilton: Indeed, for solutions of the field equations and for λ-independent vector fields X and background fields the last line in (2.14) vanishes and, using the notation δφ A := dφ A dλ , δπ A µ := dπA µ dλ , (2. 16) dSµ = ∂µ⌋dx 0 ∧ · · · ∧ dx n , dSµν = −∂µ⌋dSν , (2. 17) after integration of (2.14) over S one obtains (2. 18) where the boundary term might or might not vanish depending upon the boundary conditions satisfied by the fields at hand. These terms do not vanish, and play a key role for the problems at hand in this work. Given two one-parameter families of fields φ A (λ) and φ A (τ ), the "presymplectic current" ω µ is defined as (2. 19) with a similar definition forω µ : but we will stick to the notationπ A µ .
Whileω µ and ω µ look identical, one should keep in mind that they are not defined on the same spaces: the arguments of ω µ are sections of the bundle tangent to the bundle of fields, while the arguments ofω µ are sections of the bundle of tangents to the tangents. The difference is, however, somewhat esoteric in any case.

The divergence of the presymplectic current
We wish to calculate the divergence of the pre-symplectic current (2.19). Consider thus, as before, a two-parameter family of fields φ A (λ, τ ). Assuming that the variations and the coordinate derivatives commute we have: (2.21) Changing the order of τ and λ leads to (2. 22) We conclude that the divergence will vanish if the linearised field equations (2.23) are satisfied.

The canonical energy of the linearised theory and the presymplectic form
Calculating directly from the definition (2.11) we have (2.24) Comparing (2.24) with (2.14) we obtain This is true for all fields φ,φ and X, regardless of whether or not the fields satisfy any equations.
We will differentiate (2.25) with respect to λ. Before doing this, we note first that a replacement in (2.25) of X by dX/dλ gives the identity (2. 26) Similarly, replacingφ by dφ/dλ in (2.25) gives (2.27) Differentiating (2.25) with respect to λ, after taking into account (2.26) and (2.27) one is led to (2.28) Adding this to twice the right-hand side of (2.10) one obtains This leads us to the following (compare [36,Appendix]); note the we are not assuming that the fields φ at which we are linearising satisfy any equations, nor that the background structures (if any) are invariant under the flow of X: Proposition 1 Consider a solutionφ of the linearised field equations and assume that the vector field X is independent of the fields considered. The Hamiltonian current of the linearised theory can be rewritten as (2.30) Here L is the Lagrangian density for the linearised equations, withπ A µ = ∂ L /∂(∂µφ A ), and ω µ is the presymplectic current (2.19).
In view of the above, the Hamiltonian H (S , X) for the linearised theory associated with a hypersurface S reads (2.31) When X is a time-translation, one often identifies the numerical value of (2.31) with the energy of the field contained in S . We will use this terminology, momentarily ignoring all the delicate issues associated with the boundary conditions satisfied by the fields, to which we will return in due course.

Energy flux
We wish to derive a formula for the flux of energy across ∂S . For this, define where we use the symbol Φτ [Y ] to denote both the flow of a vector field Y and its action on our field. We consider a family of fields obtaining by flowing along Y , and the resulting variational identity. We will require that X commutes with Y , that the background is invariant under the flow of Y , and that all the φ(τ )'s are solutions of the field equations. Equation (2.14) with λ replaced by τ reads (2.33) (2.34) This is the field-theoretical analogue of the statement that a Hamiltonian in mechanics is conserved along its flow, except that here one needs to take into account the boundary term. Indeed, given a hypersurface S set (2. 36) so that the integrand of (2.36) represents the flux of energy through ∂S when S is dragged along the flow of X.

The divergence of the Noether current
An important consequence of the hypotheses H1.-H2., p. 5, is the identity Note that the right-hand side is zero if a) either the field equations E A = 0 hold, or b) the solution is stationary in the sense that L X φ = 0.
The identity is easiest to establish by going to coordinates in which L X = ∂ 0 , so that which is the same as (2.37). Formula (2.37) provides an alternative derivation of (2.36), as follows: Let X be a vector field everywhere transversal to a hypersurface S with boundary ∂S . Let, as before, Sτ be obtained by flowing S with the vector field X for a time τ . Let us denote by Tτ the hypersurface obtained by flowing the boundary of S from time zero to time τ : Supposing that the right-hand side of (2.37) vanishes, and applying Stokes' theorem on the set bounded by S , Sτ and Tτ we obtain (2.39) Differentiating with respect to τ one obtains This coincides with (2.36), as the term X µ L in the definition of H µ drops out from the integral after antisymmetrisation.

The energy flux revisited
In the case of theory of fields linearised around a solution, there exists yet another way of computing the flux of energy across ∂S , as follows: 1 Since the divergence of ω µ vanishes for solutions of field equations, the calculation leading to (2.40) gives (2.41) Calculating the τ -derivative of H [S , X] as given by (2.31) we obtain (2.42) 1 The calculation here is due to J.Hoque.

Scalar fields on de Sitter spacetime
We apply the formalism to a linear scalar field in Minkowski spacetime and in de Sitter spacetime. In our signature the Lagrangian reads for a constant m. The theory coincides with its linearisation and we will therefore not make a distinction between the fields ϕ andφ. The canonical energymomentum current H µ equals (2.45) We consider simultaneously the Minkowski space-time and the de Sitter spacetime in Bondi coordinates, in which the metric takes the form with ǫ equal to one if 1 − α 2 r 2 < 0, and minus one otherwise. Hence (2.47) (Starting from a more usual form of the de Sitter metric, cf., e.g., [27].) We denote by Cu the light cone of constant u, and by C u,R its truncation in which the Bondi coordinate r ranges from zero to R. We wish to calculate the canonical energy associated with the Killing vector field X = ∂u and contained in C u,R . Letting We see that (∂rφ) 2 gives a positive contribution to the energy integral in the region where guu = ǫN 2 is negative, and a negative contribution otherwise.
The presymplectic current is defined as with obviously vanishing divergence on solutions of field equations: The u-component of the presymplectic current on Cu reads (2.53) When X = ∂u, this results in the following volume integrand in (2.30) while the boundary integrand equals (2.55) Equation (2.30) leads to the following alternative form of (2.51): (2.60) Let us momentarily assume that Λ = 0 = m. As is well known, there exists a large class of solutions of the wave equation on Minkowski spacetime with full asymptotic expansions, for large r, (2.61) After passing to the limit R → ∞, for such solutions (2.60) becomes the usual energy-loss formula for the scalar field: (2.62) We now turn our attention to the de Sitter case. We first consider a massive scalar field, with the mass chosen so that the equation is conformally covariant, where d is the dimension of spacetime and R(g) is the scalar curvature of g. After a conformal transformation g → Ω 2 g the field Ω d/2−1 φ satisfies again (2.63), with g there replaced by Ω 2 g. This implies that solutions of (2.63) with smooth initial data on a Cauchy surface in de Sitter spacetime behave asymptotically for large r, in spacetime dimension four, again as in (2.61). We return to this in Section 2.2.1 below.
For such solutions (2.57) becomes where the volume integral converges when passing with the radius R to infinity, but the boundary integral diverges linearly in R in general. Indeed, given φ| Cu one can integrate (2.59) to where ∆γ is the Laplace operator associated with the metricγ, and where we have used which shows that (-1) φ will not be zero at later times in general even if it is initially. As for (2.60), we find (2.68) This diverges again when R tends to infinity. However, we see that the divergent term in Ec has a dynamics of its own, so that a renormalised energy can be obtained by subtracting the divergent term in Ec and passing with R to infinity, (2.69) The renormalised energy satisfies again an energy-loss formula identical to (2.62), formally coinciding with that in Minkowski spacetime. A similar behaviour is observed for the massless scalar field. It follows from [55] (cf. Section 2.2.1) that scalar fields evolving out of smooth initial data on a Cauchy surface have an asymptotic expansion of the form (2.70) compare (2.92).
It turns out that the volume part of Ec given by (2.57) does not converge anymore as R tends to infinity under (2.70). Indeed, using (2.92)-(2.93) one finds whereD is the covariant derivative associated with the metricγ and where O(1) denotes terms which have a finite limit as R tends to infinity. Note that the logarithmic term integrates-out to zero over S 2 . This leads us to define the finite part, say E V , of the volume integral as (2.72) One now finds which continues to diverge as R tends to infinity in general. The associated energy flux formula reads (2.74) Using ∂u (2.93)), one finds (unsurprisingly) that the divergent term has a dynamics of its own, so that the finite renormalised energy, defined as has a finite and well-defined flux: (2.76)

Asymptotics of scalar fields on de Sitter spacetime
In order to understand the behaviour for large r, in Bondi coordinates, of solutions of the massive or massless wave equation, on de Sitter spacetime it is most convenient to work using a foliation of (part of) de Sitter spacetime by flat submanifolds, so that the metric takes the form (2.78) where (x, y, z) are as in (2.78), thus (2.80) We start our discussion with the case m = 0. According to [55], smooth solutions of the massless scalar wave equation on de Sitter spacetime extend through the conformal boundary {T = 0} as where f andf are smooth functions of (T, x, y, z). (By matching coefficients as below one finds in fact thatf ≡ 0; in other words, φ extends smoothly across the conformal boundary.) The coordinate transformation (2.83) In terms of the coordinate u of (2.49), (2.86) We thus have the expansions as r → ∞, absolutely convergent for r > α −1 , (2.87) This shows that a Taylor-series in T , near T = 0, for a function f , translates into a full asymptotic expansion in 1/r, for large r: (2.89) Using ln(αT ) = −αu − ln(1 + αr) = ln(αr) 1 + 1 Equation (2.89) translates to the following asymptotic expansion for solutions φ of the massless wave equation: (2.90) Here some words of caution are in order. When considering the characteristic Cauchy problem for the wave equation with initial data on a light cone C (u 0 ), any function φ| C (u0) can be used as initial data. In particular we can prescribe φ of the form (2.90) at u = u 0 with arbitrary expansion functions (-k) φ . However, some relations will have to be satisfied by the coefficients so that φ extends as in (2.81). This can be seen by inserting the Taylor expansion (5.65) in the massless wave equation in the coordinates of (2.78) to find that solutions take the following form near T = 0: with arbitrary functions f andf , where ∆ δ is the Laplacian on Euclidean R 3 , and where no logarithmic terms occur, so that the functionf in (2.81) is zero. Using (2.91), the expansion (2.90) becomes (2.94) We finish by a short remark on the conformally covariant case. The corresponding wave equation in the metric (2.79) becomes the massless Minkowskian wave equation in the coordinates (T, x, y, z) for the function A Taylor expansion near T = 0 of a solutionφ giveŝ where ∆ δ is the Laplace operator of the flat metric δ on R 3 , and where f andf are arbitrary functions. Hence We see that the initial data for a solution which extends smoothly through the conformal boundary at infinity will have, in Bondi coordinates, arbitrary expansion coefficients (-1) φ and (-2) φ , with all the remaining expansion coefficients determined uniquely by these first two.

Linearised gravity
We apply the results of Section 2.1 to vacuum general relativity with cosmological constant Λ, using the background metric approach of [11]. Thus We note the usual ambiguity related to the question, how to differentiate with respect to a symmetric tensor field. When performing variations we resolve this by allowing gµν not to have any symmetries, with all geometric quantities such as the Christoffel symbols, the Ricci tensor, or the volume form defined using the symmetric part g (µν) of gµν . The tensor field gµν is assumed to be symmetric in all final formulae and in all unrelated calculations.
In [11] the Lagrangian density is obtained by removing from the Hilbert one a divergence which is made covariant by using a background metric b. After allowing for a cosmological constant, in space-time dimension d this leads to the Lagrangian (see [15,Section 5.1]) where rµν is the Ricci tensor of b, and where the B ϕ ψγ 's are the Christoffel symbols of the background metric b. Consider the field Viewing momentarily Γ ϕ µν and gµν as independent variables we have (2.100) The point is that L depends upon the derivatives of the metric only through δΓ ϕ χρ , so that the formula allows us to calculate π βγα : (2.101) Denoting by∇ the covariant derivative associated with the background metric b, one has A somewhat lengthy calculation allows one to rewrite (2.101) as Note that most expressions that follow in this paper involve contractions of P αβγδǫσ with tensors which are symmetric both in the pairs βγ and ǫσ, and in such expressions there is no need to symmetrise P αβγδǫσ as in (2.103), since such a symmetrisation is done automatically when the contraction is performed.
Incidentally, since L is quadratic in∇g, (2.104) implies which shows that it makes sense to require This last equation is not obvious by staring at (2.104), but can be checked by a direct calculation.
Recall that we are interested in the linearised theory. For this, it is clearly convenient to choose the background metric b to be the metric g at which we are linearising. Denoting by hµν the linearised metric field, the Lagrangian L for the linearised theory is thus where Q arises from the quadratic terms in the Taylor expansion, at the background metric, of (2.108) We have, ignoring the usual issues related to the symmetry of g αβ as this will be taken care of by itself when calculating the Taylor expansion below, )g αβ g ρσ −(g αρ g µσ g βν + g αµ g βρ g νσ )rµν − Λg αρ g βσ + 1 2 g µρ g νσ g αβ + g µν g αρ g βσ )rµν . (2.109) Replacing g by g + h in the right-hand side of (2.108) we thus obtain the Taylor expansion, after replacing rµν by Rµν in the result, (2.111) Assuming that the background satisfies the Einstein vacuum equations, In view of (2.103), the variation of π βγα at g = b equals (2.114) Given two solutions δ i g, i = 1, 2, of linearised Einstein equations, the presymplectic current of vacuum Einstein gravity with a cosmological constant therefore reads (2.115) (Because of the symmetrisations occurring in (2.115), one can use there instead an equivalent version of (2.104) given by Wald and Zoupas in [56]: (2.116) One can check that the part of the Lagrangian density which contains Christoffel symbols can be reduced to four terms. Indeed, we have where P αβγδǫσ := g ασ g βǫ g γδ − g αβ g γδ g ǫσ + 1 2 g αδ g βγ g ǫσ − 1 2 g αδ g βσ g γǫ . (2.118) Using (2.117), it follows from (2.105) that Note that one of the terms constituting P αβγδǫσ is not invariant under exchange of the first three indices with the three last ones: which prevents us to express the canonical momenta in a simple form using P αβγδǫσ . The following relations hold: (2.123)

Canonical energy of weak gravitational fields
In the gravitational case (2.31) reads (2.124) From (2.8) and (2.114) we find (2.125) If ∂S is a spacelike surface given by the equation and if X equals ∂u, then the boundary integral in (2.124) reads (2.126) If we denote by hµν the linearised metric field, the "generating equation" (2.18) reads, again with X = ∂u, (2.127)

Energy flux
In the setting just described, the linearised-fields version of the flux formula (2.36) takes the form (2.128)

Gauge invariance
It is shown in [15,Equations (5.19)-(5.20)] that for solutions of the field equations and for all vector fields Y the current H µ [Y ] takes the form H µ = ∂αU µα + G µ , where where a semicolon denotes the covariant derivative of the metric b, with and where G µ does not depend upon the derivatives of g. Under our conditions, inspection of the analysis in [15, Section 5.1] leads to the formula A direct proof of (2.132) will be provided shortly. Thus, for all vector fields Y which vanish together with their first derivatives at ∂S and for all variations δg of the metric satisfying the linearised field equations it holds that S ω µ (L Y g, δg) dSµ = 0 , (2.134) as already established by different arguments in [22,28,45]. Since L Y g is the variation of the metric g corresponding to infinitesimal coordinate-transformations, this is interpreted as the statement that the form obtained by integrating the presymplectic current is gauge-invariant.
For our purposes the key significance of (2.134) is: Theorem 2 The total Noether charge H [S , X] of the linearised gravitational field associated with a compact hypersurface S with smooth boundary is invariant under the "gauge transformation" . (2.135) The result follows now from (2.134).
Remark 3 There is an obvious version of Theorem 2 for non-compact S 's, when suitable asymptotic conditions, ensuring the vanishing of the boundary integrals at the right-hand side of (2.132), are imposed on all objects involved. ⊓ ⊔ 2.3.4 Proof of (2.132) The remainder of this section will be devoted to the proof of (2.132). For this it is convenient to write (2.136) The presymplectic form on a light cone Cu, which we denote by Ω Cu , is obtained by integrating the presymplectic current: where the light cone is given by the equation {u = 0}, and coordinatised by coordinates (r, x A ). Thus, to determine Ω Cu (δ 1 g, δ 2 g), which in turn determines the volume part of the Noether charge (2.124), we need to calculate w u . We define b µ (δ 1 g, δ 2 g) := P µ(βγ)δ(ǫσ) δ 1 g βγ ∇ δ δ 2 gǫσ , (2.138) so that the vector field w of (2.136) equals We consider the following gauge transformations A gauge transformation of the vector field w of (2.136) leads to (2.142) In order to avoid a notational confusion between fields such as δg µν , understood as a variation of g µν , and g µα g νβ δg αβ , as before we will write hµν for δgµν . It is convenient introduce (2.143) Indices on hµν and hµν are of course raised and lowered with the metric g. Each term in square brackets in (2.142) can be rewritten using the identity (2.145) Note that the last line in (2.144) vanishes on a background which satisfies the vacuum Einstein equations (2.112) and for metric perturbation satisfying the linearised vacuum Einstein equations: (2.148) In order to find b α (L ξ g, h), we calculate ∇ β ∇ δ U αβγδ and use (2.146) to obtain (2.149) (2.150) (2.152) The linearisation of (2.130) gives which shows that δH boundary equals the second line in (2.144).

Adding matter fields
We consider now Einstein equations interacting with matter fields. The fields φ A under consideration take the form where φ a are matter fields. We write the Lagrangian in the form where Lg is the Lagrangian (2.98) and Lm is the Lagrangian describing matter fields, which is assumed to depend upon the metric but not its derivatives. The examples of main interest in the current context would be the Einstein-Maxwell equations, as well as the equations for gravitating elastic bodies. Assuming that φ a and ∂µφ a are independent fields, the variation of L reads where δLg is given by (2.100).
The momenta π A split into gravitational and matter parts, with the gravitational momenta given by (2.103), and the matter ones defined as before: (2.159) The Hamiltonian density of linearised fields equals now (2.160) According to (2.31), the contribution to the total energy arising from the linearised matter fields equals where ω µ (φ, L Xφ ) = L Xφ aπ a µ − L Xπa µφa . From (2.36), the energy flux formula for matter fields is equal to While we are mainly interested in families of light cones in this work, the calculations that follow apply to any null hypersurfaces. Hence we consider a set of coordinates which is adapted to a space-time foliation by null hypersurfaces. The null generator of each surface is proportional to ∂r. We will write the space-time metric in adapted coordinates as (2.162) In terms of coordinate components, the field b u defined in (2.138) takes the form (2.163) While we use the notation (2.162) for the components of the metric at which the variations are taking place, most of the time we simply use δgµν for the variations, which are assumed to satisfy Note that ∇ A δ 2 g rB will not be zero in general even though δg rA vanishes. Writingout the Christoffel symbols, we find (2.164) Antisymmetrising over δ 1 g and δ 2 g to obtain the field w u of (2.136), the first term at the right-hand side drops out, which is the only obvious simplification. If we assume moreover the Bondi condition we obtain (2.167) Explicit formulae for the field b r can be found in Appendix B, see also (5.21).

The nonlinear theory
Let us denote by K a family of future directed generators of N . We choose the orientation of Bondi coordinates so that K = f ∂r where f > 0. For each point p ∈ N the tangent space TpN may be quotiented by the subspace spanned by K. This quotient space TpN /K carries a non-degenerate Riemannian metric h and, therefore, is equipped with a volume form ω. Consider a two-form L which is equal to the pull-back of ω from the quotient space TpN /K to TpN We choose a one-form α on N , such that < K, α >≡ 1, and define a three- (2.168) We define the following vector density which is equivalent to the equation where Q is given by The reader is warned that the field Q a b is not invariant under rescalings of the null generator K. Linearising (2.172)-(2.173) provides another derivation of (2.167). Compare [44]. In this section we show how to put a linearised metric perturbation in Bondi gauge, and analyse the gauge freedom remaining.

Coordinate transformations, gauge freedom
Linearised gravitational fields are defined up to a gauge transformation determined by a vector field ζ. The aim of this section is to analyse the gauge transformations which bring a smooth linearised solution h of the vacuum Einstein equations to the Bondi gauge. We will assume that, near the conformal boundary at future infinity, the linearised solution behaves as if it arose from a one-parameter of smoothly conformally compactifiable solutions near the de Sitter metric. Thus we take a background of the form where N depends only upon r, with ǫ ∈ {±1}, and where ∂uγ AB = 0 = ∂rγ AB . It turns out that the transformation to Bondi gauge introduces singularities at the vertex. For this reason in this section in formulae where ambiguities might arise, and only these formulae, we will write h reg for the metric perturbation in the original manifestly smooth gauge (where "reg" stands for "regular") in the coneadapted coordinates (u, r, x A ), and we will write h Bo for the metric in the Bondi gauge. For instance, in order not to overburden the notation we will continue to write h tt , h ti and h ij instead of h reg tt , h reg ti and h reg ij in the original manifestly smooth coordinates (t, x i ), since the metric in the Bondi coordinates will only be considered in the (u, r, x A )-coordinate system.
The "infinitesimal coordinate transformations" (3.1) should transform the metric perturbation to the Bondi gauge: The last condition deserves a justification. For this, consider a one-parameter family of metrics, say λ → g(λ) in Bondi coordinates. (In the current case of interest λ is the flow parameter along the vector field ζ, but the argument applies to any such family.) We then have (3.6) Differentiating with respect to λ one finds After performing a gauge transformation (3.1), in the new gauge we must likewise have which explains (3.5). The conditions (3.3)-(3.4) are equivalent to: which is solved by , and where r 0 can be chosen conveniently according to the context. In order to address (3.5) we will use the symbol We have where in the last line we used (3.5). Hence (3.12) We will denote by the part of ζ which depends explicitly upon h, and write We note thatζ still contains a part which depends upon h, as needed to satisfy the asymptotic boundary conditions. This is discussed in more detail in Section 3.1.2. The remaining part ofζ describes asymptotic symmetries, we return to this in Section 3.2.1.

Small r
An analysis of the behavior of the metric at the tip of the light cone is in order. For definiteness consider a smooth metric perturbation of Minkowski, Anti de Sitter or de Sitter spacetime. After transforming to Bondi coordinates of the background metric we have for small r h reg (3.14) Equation (3.9) gives, for small r, Now, there could be a 1/r term for small r in the integral defining ζ A , which could lead to logarithmic terms. To see that there is a cancellation, we note that (3.18) and (3.10) gives, again for small r, Equations (3.9)-(3.10) together with (3.12) lead to, again for small r, Inserting (3.10) in (3.12) we find where ∆γ is a Laplace operator associated with the metricγ AB .
The behaviour of various derivatives should be clear from the above. We emphasise that the original behaviour (3.14) of h near the vertex will not be true in general for the metric coefficients in Bondi coordinates. For instance: Summarising, for small r, For further reference we emphasise that (3.36)

From smooth to Bondi
In view of the formulae so far, one proceeds as follows. Given a smooth linearised metric perturbation h we perform the "infinitesimal coordinate transformation" ζ[h] as defined above to obtain a metric in Bondi gauge, still denoted by h but occasionally by h Bo . It still remains to take care of the boundary conditions. In particular, after the above, the asymptotic field h AB (u, ·) was zero before the ζ[h]-transformation. In order to remedy this we take the u-parameterised family of covector fields ξ B (u, ·) to be any family of solutions, smooth in u, of the equations (cf., e.g., [10, Théorème 3.4]) It follows from the first line of (3.29) that the gauge-transformed fields (0) h AB (u, ·) will vanish. Equation (3.37) determines ξ A [h](u, x B ) up to a u-dependent family of conformal Killing vectors of the round two-dimensional sphere; we will return to this freedom shortly.
We let ξ u [h] be a solution of (compare (3.27) and (3.36)) with smooth initial data ξ u (u 0 , ·) for some u 0 . In vacuum (cf. (4.12) below) this leads to a gauge-transformed field for which This procedure leads to a metric perturbation satisfying all Bondi gauge conditions together with where the second equality requires Tur ≡ 0.

Residual gauge
The freedom of choosing the vector fieldζ of (3.13) describes the freedom to perform coordinate transformations preserving the Bondi form of the metric. These will be referred to as residual gauge-transformations. An example is provided by the vector field ξ[h] just defined. Under theseζ-transformations, the linearised metric components acquire the following terms: (3.40) The residual gauge transformations are thus defined by a u-parameterised family of vector fields ξ A (u, ·) on S 2 together with (3. 44) and (3.25). Explicitly: with an arbitrary functionξ u (x A ).

Asymptotic symmetries
Unless explicitly indicated otherwise, in the remainder of this work we suppose that the metric has been transformed to Bondi form with h AB ≡ 0. The residual gauge transformation which preserve this condition will take the form (3.45) with, at each u, ξ B (u, x A )∂ B being a conformal Killing vector field ofγ.
The conformal Killing vectors of S 2 are related to the Lorentz group, and the remaining freedom in ξ u corresponds to translations and supertranslations.
All these gauge transformations are interpreted as governing asymptotic symmetries, defined here as transformation which preserve the Bondi gauge as well as the asymptotic fall off condition for linearised fields.
In the asymptotically flat case (thus Λ = 0) the fields ξ A become u-independent by (consistently) requiring in addition that However, when Λ > 0, the requirement h uA ≡ 0 is not consistent with (3.39) for general metric perturbations considered so far. Indeed, it follows from (3.42) that under the ξ-gauge transformations, the r 2 -terms in h uA transform as h AB ≡ 0 in general when Λ = 0. Indeed, we have the transformation law (3.47) One could therefore choose ξ B so that but there does not seem to be any reason why h AB (u, ·) should then be zero in general for u = u 0 .
In the gauge (3.48) the canonical energy on C u,R diverges as R 3 when R tends to infinity for u = u 0 , while R 2 is replaced by R when (3.39) holds. This property makes the gauge (0) h AB ≡ 0 more attractive from our perspective.
3.2.2 Rigid transport of sections of I + When Λ = 0, a prescription to reduce the set of asymptotic symmetries has been presented in [12]. For the sake of completeness we reproduce the construction here.
The Hodge-Kodaira decomposition of one-forms on S 2 shows that there exist functionsχ(u) andχ(u) on S 2 such that where ε B C is the two-dimensional Levi-Civita tensor. We can similarly write ξ B as where the functions ι(u) and υ(u) are linear combinations of ℓ = 1 spherical harmonics (see Appendix A). Equation (3.46) can be rewritten aŝ (3.52) Let P 1 denote the L 2 (S 2 )-orthogonal projection on the space of ℓ = 1 spherical harmonics. We can arrange that P 1 χ vanishes by solving the linear ODE which leaves the freedom of choosing υ(u 0 ). Next, using (3.51) and (3.44) we obtain ∂uχ(u) → ∂uχ(u) + ∂ 2 u ι(u) + α 2 ∂uξ u (u, ·) = ∂uχ(u) + ∂ 2 u ι(u) − α 2 ι(u) . (3.54) We can arrange that ∂u P 1 (χ) vanishes by solving the equation (3.55) Equation (3.51) shows that P 1χ will vanish if ∂uι(u 0 ) + P 1 χ(u 0 ) + α 2 ξ u (u 0 , ·) = 0 . (3.56) There remains the freedom of choosing ι(u 0 ), with the solutions of the homogeneous equation (3.55) taking the form where ι ± are linear combinations for ℓ = 1 spherical harmonics. Summarising, we can achieve a rigid transport of the Bondi coordinates from one sphere to the other by requiring that the potentialsχ andχ of (3.50) satisfy (3.58) We will refer to (3.58) as the rigid transport condition. Under the rigid transport conditions, we have the freedom of choosing ι(u 0 ), ∂uι(u 0 ), υ(u 0 ) , which is related to the freedom of rotating and boosting the initial light cone Cu 0 , and of choosing ξ u (u 0 , ·), which is the equivalent of the supertranslations that arise in the case Λ = 0, subject to the constraint ∂uι(u 0 ) + α 2 P 1 ξ u (u 0 , ·) = 0 . (3.59) After imposing (3.58), the residual gauge transformations which also preserve the rigid transport condition (3.58) take the form (3.45) with an arbitrary function ξ u (x A ), and whereξ B (u, x A )∂ B is the angular part of a Killing vector field of de Sitter spacetime as in (3.50), thus υ is a u-independent linear combination of ℓ = 1 spherical harmonics, the potential ι takes the form (3.57), with ∂uι(u 0 ) satisfying (3.59).

Linearised metric perturbations in Bondi coordinates
Let N be a null hypersurface given by u = const. We will use Bondi-type coordinates and a Bondi parameterisation of the metric on N : Here it is also assumed that det γ AB takes a canonical, r-and u-independent value, namely det γ = sin 2 θ . In spacetime dimension four, the Euler-Lagrange equations for the Lagrangian (2.107) with∇ replaced by ∇ are Assuming δg κ κ = 0, which is the case in the Bondi gauge, we obtain Taking a trace we find which simplifies (4.4) further to These equations are still unpleasant enough so that it appears simpler to instead linearise the equations as written down in [46], and we will do so. Nevertheless, to be on the safe side, we have checked, using Maple, that the set of equations E u r = E u u = E u A = 0 is equivalent to the linearised equations (4.12), (4.14) and (4.32) below, obtained from the equations in [46].
Our asymptotic conditions on the linearised perturbations of the metric will be modelled on the asymptotic behaviour of the full solutions of the Einstein vacuum field equations with positive cosmological constant and with smooth initial Cauchy data on S 3 , as constructed by Friedrich in [29]. The resulting spacetimes have a smooth conformal completion with a (necessarily spacelike) boundary at (timelike) infinity I + (denoted by I + by some authors). It is shown in [13, Section 2.1] that, given a foliation of I + by a function y 0 (in our case, this foliation will be provided by the intersections of a family of null light cones emanating from a world-line in spacetime), there exists a neighborhood of I + on which the metric takes the Bondi form (4.1), where u| I + = y 0 . Now, Bondi et al. consider the case Λ = 0 and assume whereγ AB is the standard metric on S 2 . It follows e.g. from [13, Section 2.1] that the last equation in (4.7) is justified under the hypothesis of existence of a smooth conformal completion at infinity regardless of the value of Λ. However, it is not clear at all whether the first two equations (4.7) can be assumed to hold for all retarded times in general: When Λ < 0 this is part of asymptotic conditions which one is free to impose, and which are usually imposed in this context, but which one might not want to impose in some situations. When Λ = 0 these conditions can be realised by choosing the function y 0 suitably. However, when Λ > 0 there is little doubt that all three conditions in (4.7) can be simultaneously satisfied for all retarded times by a restricted class of metrics only. In the linearised theory this will be clear from the calculations that follow.
We have mentioned above the results of Friedrich on the spacelike relativistic Cauchy problem, as they guarantee existence of a large class of vacuum spacetimes with a positive cosmological constant, near the de Sitter or Anti de Sitter spacetime, with a smooth conformal completion at Scri. As such, in our context it is more natural to think of the characteristic rather than the spacelike Cauchy problem (cf., e.g., [18]). In this context, in the linearised theory we are free to prescribe arbitrarily the angular part h AB dx A dx B of the linearised metric perturbation on the light cone, with the remaining fields, and their asymptotics, determined by these free data and the residual gauge conditions to which we return in Section 3.1. This follows quite generally from the analysis in e.g. [18], but can also be deduced directly from the considerations that we are about to present. Friedrich's results just mentioned guarantee, e.g. by taking data induced on light cones from his solutions, that there exists a large class of free data h AB dx A dx B on the initial light cone with an evolution which is smoothly conformally compactifiable at I + , and we restrict our attention to such data.
Writing interchangeably hµν for δgµν , andȟµν for r −2 hµν , we thus assume the following large-r expansion where the expansion tensors (1) h AB ≡ (-1) h AB , etc., are independent of r.
A comment on our hypothesis that (0) h AB ≡ 0 is in order. As discussed in detail in Section 3.1, after transforming the metric perturbation to the Bondi form we can always use the remaining coordinate freedom to achieve h AB = 0 in general) leads to energy integrals on balls of radius R which diverge as R 3 , and therefore we have opted for (0) h AB = 0 which leads to a slower divergence.
Smooth compactifiability of the solution guarantees that there will be no logarithmic terms in the asymptotic expansion, which in turn requires (-2) h AB ≡ 0 , (4.9) as follows from our calculations below (compare [13]). However, we will not assume (4.9) at this stage, to be able to track down the role of this term in the equations that follow.

h uA
The linearisation of (4.11) at the de Sitter metric gives now, in vacuum, ∂r r 4 ∂r(r −2 δg uA ) = r 2D E γ EF ∂r r −2 δg AF . (4.14) Let ψ A denote the right-hand side of the last equation, (4.15) Equation (4.8) gives, for large r, while for small r we have, from (3.29), Here ξ is the gauge field of (3.9), cf. (3.1). As ψ A tends to a non-zero covector field as r goes to infinity in general, it turns out to be convenient to write the general solution of (4.14) as with fields µ A and λ A depending upon the arguments indicated. Here ψ A (s) stands for ψ A (u, s, x A ). It follows from (3.42) that the requirement, that δg is obtained by an infinitesimal coordinate transformation from a metric perturbation which is smooth near the vertex of the cone, enforces boundedness of r −2 δg uA . This, together with (4.17) shows that λ A ≡ 0 . The field µ A has a gauge character and can be determined by imposing convenient conditions at infinity, as follows from the results in Section 3.2. We have for large ρ whereλ h AB ln r . This leads to the following expansion, for large r, resulting in 2ȟ (4.23)

∂uh AB
We continue with an analysis of the asymptotics of ∂uh AB , which can be determined from [46, Equation (32)]. Denoting the traceless symmetric part of a tensor on the sphere by we have Integrating in vacuum, we find for large r ∂uȟ AB (r, ·) = − 1 where 29) and note that the limit in (4.28) exists and is finite under the current conditions. Also note that (4.26) has no r −2 terms, which shows that the expansion coefficients (-2) h AB are constants of motion.
The choice of asymptotic gauge (4.30)

huu
The V function occurring in the Bondi form of the metric solves the equation [46] 2e where R is the Ricci scalar of the conformal two-metric γ AB . Assuming δβ ≡ 0, the linearised version of (4.31) reads (4.32) LetR AB denote the Ricci tensor of the metricγ AB . As h AB isγ-traceless we have  It follows from (4.23) that the r −1 terms in the large-r expansion of the right-hand side of (4.34) cancel out, and that for large r we have as r goes to infinity, where we used that δV | r=0 = 0 vanishes when transforming to Bondi coordinates a field which was originally smooth near the tip of the light cone; compare (3.40). (4.37) We note that the individual terms in the integrands of (4.36) and (4.37) have, for small r, potentially dangerous 1/r terms which could lead to a logarithmic divergence of the integral near r = 0. However, these terms have to cancel out for fields obtained by a coordinate transformation from metric perturbations which are smooth near the vertex. This can be seen by a direct calculation from (3.42) and (3.43). A simpler argument is to notice that a pure gauge field satisfies the linearised field equations, and that there are no logarithmic terms in the pure gauge fields (3.40)-(3.43).

The remaining Einstein equations
The remaining Einstein equations are irrelevant for our analysis in this paper, in the following sense (see [46]): 1. The uu part of the Einstein equations is an equation involving ∂uV , which did not occur in the equations above and therefore cannot put further constraints on the expansion coefficients so far. 2. Similarly for the uA part of the Einstein equations, which involves ∂uU A . 3. The trace part of the angular part of the Einstein equations is automatically satisfied in Bondi coordinates once the remaining equations are satisfied.

An example: the Blanchet-Damour solutions
An interesting class of linearised solutions of the linearised Einstein equations with Λ = 0 has been introduced in [8]. We use these solutions to provide an explicit example of the behaviour both at the origin and at infinity of a vacuum metric perturbation in Bondi coordinates. In particular we calculate the news tensor for the Blanchet-Damour metrics, see (4.82).
Given a collection of smooth functions I ij : R → R such that I ij = I ji , the tensor fieldh where each dot represents a derivative with respect to the argument of I ij , is a smooth tensor field on Minkowski spacetime solving (4.60).
Transforming to the cone-adapted coordinates (u = t − r, r, x A ) one has It follows that Finally (with all expansions in the last equation and below for large r), To continue, for simplicity we assume that all the I ij 's vanish for sufficiently large arguments, and that r is so large that all the I ij (u + 2r)'s vanish. The vector field ξ which brings the metric perturbation to the Bondi form is (cf. Section 3.1) (4.71) To obtain the mass aspect function one needs the gauge-transformed huu After applying a u-derivative to (4.71), we will obtain a solution for which huu tends to zero as r tends to infinity in Bondi gauge if and only if In the calculations that follow the identities (3.32) are useful; we repeat them here for the convenience of the reader: (4.75) One then finds that in the linearised Bondi gauge the original field huu becomes Let δµ denote the linearised mass aspect function, thus huu = 2δµ/r + O(r −2 ). We see that (4.77) The function χ of (D.4) can be calculated by inspecting the asymptotic behaviour, for large r, of the functions appearing there. One finds It then follows from (D.3) that This equation looks surprising at first sight, since the left-hand side is zero for a smooth tensor field in the (u, r, x A ) coordinates. However, (4.79) makes it clear that the linearised Bondi gauge introduces a singular behaviour of the gaugetransformed metric perturbation at r = 0. This singularity can be removed on some chosen light cone, say Cu 0 , but cannot be removed for all u by a residual gauge transformation (in other words, asymptotic symmetry) in general.
Recall the formula for the gauge-transformed h AB : whereL ζ denotes the two-dimensional Lie derivative with respect to the field ζ A ∂ A . This leads to from which the news tensor ∂u (1) h AB is readily obtained: (4.82) 5 The energy of weak gravitational fields by PTC, JH, TS

Trautman-Bondi mass
When Λ = 0, the Trautman-Bondi mass is determined from the function V appearing in the metric. Based on the variational arguments so far, one expects that its linearised-theory equivalent will be determined by the second variation of V . Our aim in this section is to derive a formula for this second variation.
Recall that the one-parameter families of vacuum metrics gµν (λ, ·) which define the variations considered here satisfy Using (5.1) and our remaining asymptotic conditions, the second variation equation is obtained by differentiating (4.31) twice with respect to λ and reads (cf. (5.10) below) We need the second variation of (4.10), which reads ACγBD ∂rȟ AB ∂rȟ CD + 2πrδ 2 Trr , (5.3) and which can be explicitly integrated after requiring that δ 2 β goes to zero as r tends to infinity: ACγBD ∂rȟ AB ∂rȟ CD + 2πrδ 2 Trr dr . (5.4) (We note that this is negative if we impose the dominant energy condition.) In vacuum and for large r we obtain the expansion while for small r we have, using (3.29), We see that δ 2 β diverges badly at the origin unless ξ u is a linear combination of ℓ = 0 and ℓ = 1 spherical harmonics. Now, as explained in Section 3, the field ξ u is determined by the linearised metric up to the choice of a u-independent initial datum. Hence, given a light cone we can always find a gauge so that the leading-order singularity above vanishes. Equation (3.29) shows that in this gauge the integrand in (5.4) is bounded and δ 2 β is finite everywhere.
In the remainder of this section we assume a vacuum metric perturbation, and a gauge so that δ 2 β is bounded on the light cone under consideration.
We rewrite (5.2) as Recall that for Λ = 0 the Trautman-Bondi mass m TB is defined as [9,53,54] V is the r-independent coefficient in an asymptotic expansion of V . It was proposed in [13] to use this definition for Λ = 0, which was motivated by Cauchyproblem considerations.
Before proceeding further, recall that a one-parameter family of metrics λ → gµν (λ) in Bondi gauge satisfies the condition det g AB (λ) = det g AB (0) . (5.9) This implies in particular (5.10) Equivalently, δ det g AB = 0 = δ 2 det g AB . (5.11) It then follows from the Gauss-Bonnet theorem that Integrating (5.7) over a truncated cone When Λ ≤ 0 we have δ 2 β ≤ 0, which proves positivity of δ 2 m TB (C u,R ), and of its limit δ 2 m TB (Cu) with R → ∞. The notation in (5.13) is somewhat misleading, as the limit as R goes to infinity of (5.13) will only reproduce the Trautman-Bondi mass if this limit converges. This is the case when Λ ≤ 0 (compare (5.105), Section 5.8). However, when Λ > 0 the integrand for large r behaves as so that

The energy and its flux
We return now to a linearised vacuum gravitation field in Bondi gauge, Let, as before, C u,R denote a light cone Cu truncated at radius R, and let Ec[h, C u,R ] denote the canonical energy contained in C u,R , defined using the vector field ∂u: Recall that the Bondi gauge introduces singular behaviour at the tip of the light cone in general. When integrating formulae such as (2.30) over S one obtains a "boundary integral at r = 0". One can keep track of this but the resulting formulae do not seem to be very enlightening, so in what follows we can, and will, choose a Bondi gauge so that the (freely specifiable) gauge vector field ξ in (3.9), which is part of the transformation which takes the metric from a smooth gauge to a Bondi gauge, satisfiesD We note that this choice is tied to the chosen light cone Cu, and will not be satisfied by nearby light cones in general. This turns out to be irrelevant for the calculations in this section.
We further note that the Bondi gauge is mainly relevant for us for the analysis of the field at large distances. One can sweep under the carpet the problem of the singularity at the origin by using Bondi coordinates for large r, and any other coordinates for small r. It follows from Section 2.3.3 that the total energy does not depend upon the choice of the coordinates for small r. The volume integral will then not take the simple form presented here for "non-Bondi" values of r, but this would again be irrelevant from the point of view of the large-r analysis of the fields.
Putting together our calculations so far, namely (2.124)-(2.126), (2.136), (2.138), (2.167) and (5.27) we find: The associated energy flux formula follows from (2.36) together with the equations just listed: Let us first analyse the convergence, as R tends to infinity, of the volume integral in (5.32).
Consider the part of the boundary integral in (5.32) which diverges linearly with R; up to a multiplicative coefficient R/(64π) this term equals Using (4.23), the first term in (5.34) can be rewritten as h BC dµγ , (5.35) which cancels-out the last term in (5.34).
Incidentally, from (4.30) and the symmetries of h BD dµγ , (5.36) so all three terms in (5.34) are equal up to signs. Thus (5.32) can be rewritten as We finish this section by a remark concerning the vanishing of (0) h AB . If this were not the case, the insertion of the expansion (4.26) into the volume integral in (5.32) would lead to a divergent leading-order behaviour: Further, the boundary term in the energy would diverge as R 2 . As already pointed out, this provides the justification while it is natural to choose a gauge in which h AB vanishes.

Energy-loss revisited
In this section we rederive the energy-loss formula by a somewhat more direct calculation. For this we integrate over C u,R the identity One way to guarantee the vanishing of the last integral in (5.40) is to choose a gauge so that For simplicity this gauge choice will be made in the rest of this section.
From (2.167) and (5.28) this is equivalent to Letting δ 1 gµν = hµν , δ 2 gµν = ∂uhµν , using h AB (similarly for the u-derivatives), and keeping in mind thath := r −1 h andȟ = r −2 h, this becomes In order to obtain dEc[h, C u,R ]/du from this formula, we analyse the surface terms in the definition of canonical energy (5.32): Explicitly computing dB Ec /du, adding both sides to (5.43), after multiplying by 1/(32π) we recover the mass loss formula (5.33), keeping in mind that (-2) h AB = 0 when no logarithms occur.
In the case Λ = 0 things simplify as then α = (0) h uB = 0. One can integrate by parts in the first line above so that the left-hand side of (5.43) reads instead  Passing with R to infinity, after multiplying by 1/(64π) the right-hand side becomes (up to a multiplicative factor) the right-hand side of the familiar Trautman-Bondi mass loss formula. Hence the expression which is differentiated at the left-hand side must be the Trautman-Bondi energy, up to the addition of a time-independent functional of the fields. Now, it is known that there are no such functionals in the nonlinear theory which are gauge-independent and which vanish when hµν vanishes by [17], and it is clear that the argument there carries over to the linearised theory. However, gauge-independence of the functional being differentiated in (5.48), namely ABγCDh AC ∂uh BD dµγ , (5.49) is not clear at this stage. In Section 5.6 below we settle the issue by providing a direct proof of the equality ofÊc with half of the quadratisation of the Trautman-Bondi mass. When Λ vanishes this could have been anticipated by the results of [15], where it is shown that the canonical energy for the nonlinear field is the Trautman-Bondi mass. However, such statements involve a careful choice of boundary terms so that the correspondence is not automatic. And no such statement for Λ > 0 has been established so far in any case.

Renormalised energy
Equation (5.33) shows that the divergent term in Ec has a dynamics of its own, evolving separately from the remaining part of the canonical energy. It is therefore natural to introduce a renormalised canonical energy, sayÊc[h, C u,R ], by removing the divergent term in (5.37). After having done this, we can pass to the limit R → ∞ to obtain: h uB sin θ dθ dϕ . (5.50) This is our first main result here, and is our proposal how to calculate the total energy contained in a light cone of a weak gravitational wave on a de Sitter background. The flux equation for the renormalised energyÊc coincides with the one obtained by dropping the term linear in R in (5.33) and passing again to the limit R → ∞: h uB sin θ dθ dϕ .

(5.51)
This is our key new formula. When Λ = 0 we recover the weak-field version of the usual Trautman-Bondi mass loss formula since, as already pointed out, h uA ≡ 0 in the asymptotically Minkowskian case. Hence the last term in (5.51) shows how the cosmological constant affects the flux of energy emitted by a gravitating astrophysical system.

Energy and gauge transformations
We have seen by general considerations that the energy integral is invariant under gauge transformations, up to boundary terms. It is instructive to rederive this result for the residual gauge transformations by a direct calculation. As a byproduct we obtain the explicit form of the boundary terms arising.
We focus attention on the volume part of the energy integral: It is convenient to define a new field hµν := r −1 hµν ≡ rȟµν . (5.53) In terms ofh the integral E V takes the form where the term linear in r vanishes since ξ A (u, ·) is a conformal Killing vector of S 2 . Whence Now, for each u the function ∂uξ u is a linear combination of ℓ = 0 and ℓ = 1 spherical harmonics (see Appendix A), which implies that the terms involving ∂uξ u in ∂uh AB vanish. We conclude that both ∂uh AB ≡ r∂uh AB and ∂r(r −1 h AB ) ≡ ∂rh AB are invariant under asymptotic symmetries.
Since the measure dµγ = detγ dx 2 ∧ dx 3 is r-independent, integration by parts in ( (5.58) and note that the second line above does not vanish in a general Bondi gauge, since then bothh AB and ∂uh AB are of order one for small r by (5.55). So where we assumed that the metric hµν has the usual asymptotic expansion for large r as considered elsewhere in this paper.
From what has been said the volume integral in (5.59) is invariant under residual gauge transformations, so that we have (5.60) The second line in (5.60) vanishes when ξ u is a linear combination of ℓ = 0 and ℓ = 1 spherical harmonics. It easily follows that the canononical energy is invariant under such residual gauge transformations.
In this section we show by a direct calculation that our renormalised energyÊc coincides with the Trautman-Bondi mass of the linearised theory. As before we fix a light cone Cu and use the gauge (5.31) on Cu.
We start with (5.13), which we repeat here for the convenience of the reader: We integrate by parts on the δ 2 β term and use the vacuum version of (5.3), ACγBD ∂rȟ AB ∂rȟ CD dr dµγ Denoting by E V [h, C u,R ] the volume term in (5.32) and after another integration by parts we find Inserting the vacuum version of (4.25), The last integral in the before-last line can be integrated by parts over the angles to become, using (5.67) to pass from ( We emphasise that no asymptotic conditions have been used in the analysis in this section so far. To continue we invoke the asymptotic conditions used in the preceding sections. From the expansion (5.5) of δ 2 β one obtains The last term in the second line can be integrated by parts as in (5.35)-(5.36) to obtain We see that the renormalised energyÊc coincides with one half of the quadratisation of the Trautman-Bondi mass.

Energy in the asymptotically block-diagonal gauge
So far we have assumed a gauge where the leading order corrections to the metric are encoded in the off-diagonal components h uA of the metric perturbation. As such, the residual gauge freedom of Section 3.2 allows us to pass between two natural choices of asymptotic gauge: h uA ≡ 0 . h AB ≡ 0 for all u for general metric perturbations considered in this work. We wish to revisit our analysis in the asymptotically block-diagonal gauge (I). Thus in the remainder of this section we assume

The energy and its flux
Repeating the analysis of Section 4.1, one finds: g AB = 0 does not change the expansion (4.23) of g uA . In particular δ (1) g uA remains related with the log terms in the asymptotic expansion of g AB .

When δ
(2) g AB = 0, Equation (4.36) becomes This differs from (4.37), but does not affect the mass because of the divergence structure of the right-hand side of (5.82).
The asymptotics of the linearised metric perturbations reads Let us denote byÊ c,II the renormalised canonical energy in the block-diagonal gauge. To determineÊ c,II we use (5.74) with the asymptotically block-diagonal boundary conditions. One checks that the asymptotic behaviour (5.5) of δ 2 β remains unchanged at the order needed; the analysis of the remaining terms in (5.74) is likewise straightforward, leading tô  V is gauge-independent (cf. (3.40)), and therefore so is δ 2 m TB (Cu). And we have seen that this last quantity coincides with 2Êc. We conclude that An alternative, direct way of calculatingÊ c,II invokes the boundary integrand b r in (5.21), which now reads: g AB ∂uδ 2 (2) g CD + α 2 δ 1 (2) g AB δ 2 (1) g CD + 2α 2 δ 1 (1) g AB δ 2 (2) g CD + 1 2 r δ 1 (2) g AB ∂uδ 2 (1) g CD + δ 1 (1) g AB ∂uδ 2 (2) g CD g AB δ 2 (2) g CD Proceedings as in the derivation of (5.32), after discarding the divergent terms both in the volume (compare (5.38)) and boundary integrals, the renormalised canonical energy in the block-diagonal gauge (II) is found to bê

Comparing with [20]
Compère et al. [20,21] proposed a version of mass in the non-linear theory, using the asymptotically block-diagonal gauge. In [20, Equation (2.39)] they define where M is an integration constant which appears when analysing the characteristic constraint equations, l is a gauge field which can be set to zero and C AB is, essentially, our field (-1) h AB . They propose to define a mass, which we will denote by E (Λ) , as (5.95) (Strictly speaking, a different multiplicative constant factor is probably used in [20].) Using our notation, the quadratised version of (5.94) reads Integrating over S 2 we find: where we used For completeness we note the following flux formula h CD dµγ .
(5.103) 5.8 Λ < 0 All our results so far are independent of the sign of Λ: it suffices to replace α 2 by −α 2 wherever relevant. Now, it should be pointed out that there is a key conceptual difference arising from the causal character of the boundary at infinity. While in the Λ ≥ 0 case the data on the initial light cone determine the remainder of the evolution uniquely, this is not the case anymore for Λ < 0, where we have the freedom to add and control boundary data on I (compare [30,34]).
In the nonlinear theory the "pre-holographic" approach is to require that the conformal boundary at infinity is the same as that of anti de Sitter spacetime. At a linearised level, this translates to the requirement that the evolution preserves the condition Under (5.107) the mass-loss formula (5.51) reduces, in view of (5.109), to the one known from the Λ = 0 case: ABγCD ∂u (-1) h AC ∂u (-1) h BD sin θ dθ dϕ . (5.111) For completeness we list some further simplifications which arise in the asymptotics of the field. Equation (4.52) gives ∂uD A (1) h AB = 0 . (5.112) Using (4.46) and (5.112) one obtains the following formula for the evolution of the linearised mass aspect function (not to be confused with the evolution of the quadritised mass aspect, which is relevant for (5.111)): (5.114) Comparing (5.113) with (5.114) we find The asymptotic symmetries of interest are now those preserving (5.107). Equations (3.43), (5.108)-(5.109) and (5.112) show that these satisfy shows that ∂uξ u is a linear combination of ℓ = 0 and ℓ = 1 spherical harmonics, and thus its gradient is a conformal Killing vector on S 2 . Equation (5.120) gives and (5.116) follows automatically by taking the divergence of (5.122). By (5.121) the vector field ∂uξ A is also a conformal vector field on S 2 . By (5.56) the right-hand side of (5.111) is gauge-invariant under the current asymptotic symmetries. We conclude that is gauge-invariant up do the addition of a functional which is u-independent.
Note that the flux of energy seems to be of more interest than the energy itself, since energy is always defined up to an additive constant anyway, so gaugeinvariance of the flux is the key for a physically meaningful quantity.
Let us return to (5.98)-(5.99): The first equation is quite general, while the second holds in a Bondi gauge which is "as regular at the origin as can be". But since dÊ c,I /du is gauge-independent, the flux ofÊ c,I coincides with that of δ 2 m TB (Cu)/2.
Next, consider the explicit integral term in (5.123). Under a residual gauge transformation we have, by (3.43) and by the above, Since δ 2 m TB (Cu) is invariant under all residual gauge-transformations, we conclude that both δ 2 m TB (Cu) and δ 2 E (Λ) are invariant under asymptotic symmetries preserving (5.106). But their fluxes differ. Which of the two fluxes is more relevant for specific physical applications requires further justification.

(D.17)
For further reference we note the following. In [37] Jezierski has introduced two gauge-invariant scalars describing linearised gravitational fields on spherically symmetric backgrounds. These scalars are closely related to the fieldsD B h uB and ε ABD A h uB . The time evolution of these last fields is governed by the divergence and co-divergence of (4.50): (D. 19)