Radiative observables for linearized gravity on asymptotically flat spacetimes and their boundary induced states
aa r X i v : . [ g r- q c ] A p r Radiative observables for linearized gravity on asymptotically flatspacetimes and their boundary induced states
Marco Benini ,a , Claudio Dappiaggi ,b and Simone Murro , ,c Dipartimento di FisicaUniversit`a di Pavia & INFN, sezione di Pavia – Via Bassi 6, I-27100 Pavia, Italia. Fakult¨at f¨ur Mathematik,Universit¨at Regensburg, D-93040 Regensburg, Germany a [email protected] , b [email protected] , c [email protected]
August 23, 2018
Abstract
We discuss the quantization of linearized gravity on globally hyperbolic, asymptotically flat,vacuum spacetimes and the construction of distinguished states which are both of Hadamardform and invariant under the action of all bulk isometries. The procedure, we follow, consistsof looking for a realization of the observables of the theory as a sub-algebra of an auxiliary,non-dynamical algebra constructed on future null infinity ℑ + . The applicability of this schemeis tantamount to proving that a solution of the equations of motion for linearized gravity can beextended smoothly to ℑ + . This has been claimed to be possible provided that a suitable gaugefixing condition, first written by Geroch and Xanthopoulos, is imposed. We review its definitioncritically showing that there exists a previously unnoticed obstruction in its implementationleading us to introducing the concept of radiative observables. These constitute an algebrafor which a Hadamard state induced from null infinity and invariant under the action of allspacetime isometries exists and it is explicitly constructed. Keywords: quantum field theory on curved spacetimes, Hadamard states, linearized gravity
MSC 2010:
The quantization of general relativity is one of the most debated, hard and treacherous topicsin theoretical and mathematical physics. Much has been written on this subject, several modelshave been proposed and yet no unanimous solution has been found. A general consensus has beenapparently reached in the form that, whatever is the correct answer, certainly it has either toaccount for non-perturbative effects or to realize Einstein’s theory as the low energy regime of amore fundamental theory. The standard perturbation scheme, which has been successfully appliedto many other cases, for instance quantum electrodynamics, has been thoroughly investigated andit is known since the eighties to be doomed to failure [HV74, GS85a, GS85b] on account of thenon renormalizability which becomes manifest at two loops.From then on, the linearization of Einstein’s equations has been often seen only in combinationwith an analysis of classical phenomena such as gravitational waves [Wal84, Section 4.4] and it ishardly advocated to play any deep foundational role at a quantum level. In the same years, whennew and mostly non perturbative approaches to quantum gravity were developed, several leapsforwards have been obtained also in our understanding of how to formulate rigorously free and1erturbative quantum field theories on arbitrary backgrounds. The leading approach nowadaysin this endeavour is known as algebraic quantum field theory, a framework which emphasizes thatquantization should best be seen as a two-step procedure: In the first one assigns to a classicaldynamical system a suitable ∗ -algebra of observables A , which encodes the mutual relationalproperties, locality, dynamics and causality in particular. In the second one constructs a state,namely a positive linear functional ω : A → C , from which one recovers the standard probabilisticinterpretation via the GNS theorem – see [BDH13, HW14] for two recent reviews on the algebraicapproach.If one is interested in free field theories, the Klein-Gordon or the Dirac field for instance, thefirst step is fully understood, insofar the underlying background is assumed to be globally hyper-bolic: The dynamics of the system can be studied in terms of an initial value problem and thecollection of all smooth solutions, i.e. configurations, can be constructed. Basic observables arethen introduced via smooth and compactly supported sections of a suitable vector bundle whichis dually paired to the collection of all configurations. With this procedure one associates unam-biguously to the whole system a ∗ -algebra of fields. More problematic is instead the identificationof a state since, in the plethora of all possible choices, not all can be deemed to be physicallyacceptable. While on Minkowski spacetime, such quandary is bypassed simply by exploiting thecovariance of the theory under the action of the Poincar´e group so to single out a unique vacuumstate, on curved backgrounds the situation is more complicated. In this case it is widely acceptedthat a state can be called physical if and only if it fulfils the Hadamard property, a conditionon the singular structure of the (truncated) two-point function [KW88, Rad96a, Rad96b]. Suchcondition guarantees, on the one hand, that the ultraviolet behaviour of all correlation functionsmimics that of the Poincar´e vacuum and, on the other hand, that the quantum fluctuations of allobservables are bounded. The importance of the Hadamard condition has been recently vigorouslyreaffirmed in [FV13]. Furthermore Hadamard states are of capital relevance in treating on curvedbackgrounds interactions at a perturbative level since they allow for an extension of the algebraof fields to encompass also Wick polynomials [HW01].Despite being structurally so important, Hadamard states are known to be rather elusiveto find unless the spacetime is static. Their existence is nonetheless guaranteed in most of thecases thanks to a deformation argument [FNW81]. Although such result is certainly important,for practical applications a constructive scheme is needed. Unless one considers very specificbackgrounds, such as the cosmological ones, few options are known. The first aims at workingdirectly on the initial value surface for the underlying equation of motion and it relies heavily ontechniques proper of pseudo-differential calculus [GW14a, GW14b]. The second is the one we willpursue and it is based on a procedure also dubbed bulk-to-boundary correspondence . It is a schemewhich identifies for the underlying spacetime a distinguished codimension 1, null submanifold, suchas for example the conformal boundary. Thereon one defines an auxiliary ∗ -algebra as well as adistinguished quasi-free state. Subsequently, via a suitable homomorphism, it suffices to realizethe algebra of fields as a sub-algebra of the auxiliary counterpart so to induce on the former, viapull-back, another state which turns out to be both Hadamard and invariant under all isometries.Such scheme has been applied successfully to several cases, ranging from the rigorous definition ofthe Unruh state for the wave equation on Schwarzschild spacetime [DMP11], to the identificationof distinguished local states of Hadamard form [DPP11], to the construction of Hadamard stateson asymptotically flat spacetimes for the free scalar [DMP05], Dirac [DPH11] and Maxwell field[DS13, Sie11] – for the latter there exists also another approach recently discussed in [FS13].Conceptually all these approaches follow the same path proposed for the first time in [Ho00].It is noteworthy that, until recently, all the investigations mentioned above never involvedlinearized gravity. Despite being a linear theory, it was not considered in algebraic quantum fieldtheory. On the one hand one of the key problem is local gauge invariance. It is often thought to bedifficult to reconcile with the algebraic approach and even the free Maxwell field suffered almostthe same fate for this reason. On the other hand, the general philosophy according to whichalmost no insight on quantum gravity can be earned from perturbation theory has discouraged2orking on this topic. Yet, it appears that recently this trend has inverted. Particularly relevantis [BFR13] in which not only perturbative quantum gravity is constructed in a generally covariantway, but a new light has been shed on the concept of observables for quantum gravity. In thiscontext linearized gravity is seen as an important step to extract information about the localgeometry of the full non-linear phase space. Most interestingly many results in [BFR13] areactually derived using Hadamard states for linearized gravity, thus prompting the question oftheir explicit construction. The relevance of this question is increased by the realization that thestandard deformation arguments, used for all other free fields adapting the analysis of [FNW81],cannot be applied in this context since one is constrained to working with spacetimes which aresolutions of the Einstein vacuum equations.Goal of this paper is thus to investigate an alternative mean to construct Hadamard statesfor linearized gravity. More precisely we will consider asymptotically flat vacuum spacetimes andwe will investigate whether the bulk-to-boundary procedure, used successfully for all other freefields, can be implemented also in this case – see also [Fro79] for an earlier related paper. Inour analysis we will benefit mostly from a very recent and thorough analysis on the constructionof the algebra of fields for linearized gravity on an arbitrary globally hyperbolic background andon the definition thereon of Hadamard states [FH12, Hun12]. The outcome of our investigationis rather surprising and it turns out to have far reaching consequences also for classical generalrelativity. As a matter of fact, the key point in the whole procedure is the following: Everyasymptotically flat vacuum spacetime ( M, g ) can be realized via an embedding ψ as an opensubset of a second auxiliary spacetime ( f M , e g ) so that, in between other properties, the metrics g and e g are related by a conformal rescaling Ξ on ψ ( M ) and the boundary of ψ ( M ) contains anull hypersurface, known as future (or past) null infinity ℑ + ( ℑ − ). This can also be seen as thelocus Ξ = 0. On account of this geometric construction, we will show that realizing the algebraof observables in the bulk as a sub-algebra of a counterpart living on ℑ + , entails in particularproving that every gauge equivalence class of spacelike compact solutions of linearized Einstein’sequations contains a representative which, up to a suitable conformal rescaling via Ξ, obeys in f M to a hyperbolic partial differential equation. While in the scenarios considered previously in theliterature, this feature was a direct consequence of the conformal invariance of the dynamics inthe physical spacetime, linearized gravity behaves differently. More precisely, upon a conformaltransformation, the equation of motion acquires terms which are proportional to inverse powers ofΞ leading, thus, to a blow-up of the coefficients on ℑ + . This pathology can be in principle avoidedexploiting the invariance of the theory under the action of the linearization of the diffeomorphismgroup of ( M, g ) and finding a suitable gauge fixing which cancels the unwanted contributions. Apositive answer to this question was found in the late seventies in [GX78] and it played a key roleboth in establishing the stability of the notion of asymptotic flatness under linear perturbationand in studying the symplectic space of general relativity [AM82].We shall review in detail this construction and we will unveil that, in general, there existsan obstruction in implementing the so-called Geroch-Xanthopoulos gauge which depends bothon the geometry and on the topology of the underlying background. As a matter of fact, asan example, we will show that Minkowski spacetime does not suffer from this problem whileaxisymmetric vacuum, asymptotically flat spacetimes do. In combination with the obstructiondiscovered by Fewester and Hunt in [FH12, Hun12] to implement the transverse-traceless gauge,our result suggests that linearized gravity might be very much akin to electromagnetism. As amatter of fact, as shown in a series of papers [BDHS13, BDS13, DL12, SDH12], also the latter isaffected by topological obstructions although these manifest explicitly as a consequence of Gauss’law when one tried to realize Maxwell equations as a locally covariant field theory. Although wewill not investigate this specific issue, it might happen that also linearized gravity suffers of thesame problem.As far as the construction of Hadamard states is concerned, our result suggests the introductionof the concept of radiative observables to indicate those which admit a counterpart on null infinity.In this way we identify a, not necessarily proper, sub-algebra of the full algebra of observables3ince, depending on the underlying spacetime, it can also coincide with the whole algebra ofobservables. Yet, for radiative observables there exists a bulk-to-boundary correspondence andwe can thus identify via ℑ + a distinguished state which is of Hadamard form and invariant underall bulk isometries.The paper is organized as follows: In Section 2 we discuss linearized gravity on an arbitraryglobally hyperbolic spacetime ( M, g ) which solves the vacuum Einstein’s equations. In particularwe construct the space of gauge equivalence classes of solutions via the de Donder gauge fixing andwe remark on the obstructions related to the transverse-traceless gauge. In Section 2.3 we definethe classical observables as suitable equivalence classes of compactly supported, smooth symmetric(2 ,
0) tensors of vanishing divergence, endowing this space with a presymplectic structure. Section3 deals with the bulk-to-boundary correspondence. In Section 3.1, first we review the notion ofan asymptotically flat spacetime and we outline the main geometric and structural propertiesof ℑ + , the conformal boundary. Afterwards we construct on ℑ + a suitable symplectic space ofsmooth (0 ,
2) tensors. In Section 3.2 we discuss the Geroch-Xanthopoulos gauge and we provethe existence of an obstruction in its implementation. Examples are given and the concept of aradiative observable is introduced. In Section 3.2.1 we show that there exists a symplectomorphismbetween the bulk radiative observables and the symplectic space on ℑ + constructed in Section3.1. In Section 3.3 we extend this correspondence at a level of ∗ -algebras and we exploit it toconstruct a state for radiative observables which is both of Hadamard form and invariant underthe action of all isometries of the bulk metric. In Section 4 we draw our conclusions. Aim of this section is to introduce the linearized version of Einstein’s equations and to discusshow to assign a gauge invariant algebra of observables. Here we shall mimic the point of view usedin [BDS13] to tackle the same problem for Abelian principal connections. Furthermore, in ourdiscussion we shall make use of the analyses both of Fewster and Hunt [FH12, Hun12] concerninglinearized gravity and of Hack and Schenkel concerning linear gauge theories [HS12].
As a starting point we introduce the basic geometric ingredients, we shall use in this paper. We call spacetime the collection (
M, g, o , t ), where M is a four dimensional, smooth, connected manifold,endowed with a smooth Lorentzian metric g of signature ( − , + , + , +), as well as with the choiceof an orientation o and a time orientation t . We require additionally both that the metric solvesvacuum Einstein’s equations, i.e. Ric( g ) = 0, and that ( M, g ) is globally hyperbolic. In otherwords, there exists an achronal closed subset Σ ⊂ M whose domain of dependence coincides with M itself. This class of backgrounds is distinguished since it allows to discuss the dynamics of allstandard free fields in terms of an initial value problem [BGP07].We recall, moreover, that a subset Ω of a globally hyperbolic spacetime ( M, g ) is said to be future compact ( resp. past compact ) if there exists a Cauchy surface Σ ⊂ M such that Ω ⊂ J − (Σ)( resp. J + (Σ)), where J ± stands for the causal future/past. At the same time Ω is called timelikecompact if it is both future and past compact, whereas we call it spacelike compact if thesupport of the intersection of Ω with any Cauchy surface is either empty or compact. In thispaper we shall employ the subscripts tc and sc to indicate maps whose support is timelike andspacelike compact respectively.For later convenience, we will denote the bundle of symmetric tensors of order n built outof a vector bundle V with S n V . We shall also need to work with Ω k ( M ) (Ω kc ( M )) the space ofsmooth (and compactly supported) k -differential forms on which we define the exterior derivatived : Ω k ( M ) → Ω k +1 ( M ) and the codifferential δ : Ω k ( M ) → Ω k − ( M ) where δ . = ( − k ∗ − d ∗ , ∗ being the metric induced Hodge dual operator. Notice that, on Ricci flat backgrounds, the so-called Laplace-de Rham wave operator d δ + δ d coincides with the metric induced wave operator4 if k = 0 ,
1. We shall indicate with Ω k d ( M ) and Ω kδ ( M ) the spaces of smooth forms which arerespectively closed, i.e. d ω = 0 and coclosed, i.e. δω = 0 with ω ∈ Ω k ( M ). The same definitionsare taken for those spaces of forms in which a suitable restriction on the support is assumed, e.g. compact, spacelike compact or timelike compact. We will also consider H k ( M ) (H kc ( M )), the k -th(compactly supported) de Rham cohomology group on M .In order to discuss linearized gravity, we shall follow the scheme outlined in [SW74] andrecently analyzed from the point of view of algebraic quantum field theory in [FH12, Hun12]. Thedynamical variable is a smooth symmetric tensor field of type (0 , h ∈ Γ( S T ∗ M ) whichfulfils the so-called linearized Einstein’s equations : − g ab (cid:16) ∇ c ∇ d h cd − (cid:3) h (cid:17) − (cid:3) h ab − ∇ a ∇ b h + ∇ c ∇ ( a h b ) c = 0 , (2.1)where ∇ stands for the Levi-Civita connection for the metric g , (cid:3) . = g ab ∇ a ∇ b and h . = g ab h ab .All indices are raised with respect to g . We also employ the standard symmetrization notationaccording to which ∇ ( a h b ) c = ( ∇ a h bc + ∇ b h ac ).As one can imagine already by looking at (2.1), it is advisable to employ a notation whereindices are not spelled out explicitly, so to avoid unreadable formulas. We will try to adhere tothis point of view as much as possible, although from time to time we will be forced to restoreindices to make certain concepts more clear to a reader. Remark 2.1.
Eq. (2.1) is slightly simpler than in other papers. As a matter of fact it wouldsuffice to require that (
M, g ) were an Einstein manifold. This perspective is assumed for instancein [FH12, Hun12], namely the metric g fulfils the cosmological vacuum Einstein’s equation G ab +Λ g ab = 0. In this setting additional terms, which are proportional to Λ, appear in (2.1). Inour case we are interested in constructing Hadamard states for asymptotically flat spacetimes, allfulfilling the vacuum Einstein’s equations. For this reason we set from the very beginning Λ = 0.Let us rewrite (2.1) in a more compact form. To start, we introduce four relevant operators: • The trace tr : Γ( S T ∗ M ) → C ∞ ( M ) with respect to the background metric g defined bytr h = g ab h ab , • The trace reversal I : Γ( S T ∗ M ) → Γ( S T ∗ M ), which in components reads( Ih ) ab . = h ab − g ab h cc , • The symmetrized covariant derivative, also known as the
Killing operator , namely ∇ S :Γ( S n T ∗ M ) → Γ( S n +1 T ∗ M ) defined out of the Levi-Civita connection as( ∇ S H ) i ...i n . = ∇ ( i H i ...i n ) , where the brackets stand for the normalized symmetrization of the indices. • The divergence operator div : Γ( S n +1 T ∗ M ) → Γ( S n T ∗ M ) such that(div H ) i ...i n . = g ab ∇ a H bi ...i n . Similar definitions are taken for sections of
T M and of its symmetric tensor powers. Let usconsider the standard pairing between sections of S n T ∗ M and S n T M :( · , · ) : Γ( S n T ∗ M ) ˇ × Γ( S n T M ) → R , ( H, Ξ) = Z M h H, Ξ i µ g , (2.2)5here h· , ·i is the dual pairing between S n T ∗ M and S n T M and ˇ × denotes the subset of theCartesian product whose elements are pairs with compact overlapping support. Accordingly ∇ S and − div are dual to each other, namely for each pair ( H, A ) ∈ Γ( S n +1 T ∗ M ) ˇ × Γ( S n T M ):( H, ∇ S A ) = ( − div H, A ) . (2.3)Since the background ( M, g ) is Ricci flat per assumption, the Riemann tensor R abcd is the onlynon vanishing geometric quantity which plays a distinguished role. We shall employ it to definean operation on symmetric (0 , S T ∗ M ) → Γ( S T ∗ M ) , Riem( h ) ab = R cda b h cd . The counterpart for (2 , · ♯ : T ∗ M ⊗ n → T M ⊗ n , ( H ♯ ) i ...i n = g i j · · · g i n j n H j ...j n · ♭ : T M ⊗ n → T ∗ M ⊗ n , ( A ♭ ) i ...i n = g i j · · · g i n j n A j ...j n Up to an irrelevant factor , we can rewrite (2.1) in terms of the operators introduced above: K : Γ( S T ∗ M ) → Γ( S T ∗ M ) , K = ( − (cid:3) + 2Riem + 2 I ∇ S div) I. (2.4)By direct inspection one can infer that K ◦ I is not a normally hyperbolic operator due to thepresence of the term I ∇ S div – for a detailed account of the definition and of the properties ofthese operators, refer to the monograph [BGP07]. In other words, even on globally hyperbolicspacetimes, the solutions of (2.4) cannot be constructed via a Cauchy problem with arbitraryinitial data. On the contrary, suitable initial data must satisfy appropriate constraints, see e.g. [FH12, Section 3.1] and [Hun12, Section 4.6]. Especially from the point of view of quantizing thetheory, this feature is rather problematic. Yet, as we will discuss in detail in the next section,we can circumvent this issue exploiting the underlying gauge invariance of (2.1). Since it plays adistinguished role in our calculations, we make explicit how (2.4) intertwines with the geometricoperators introduced above. The proof of the following lemma is a matter of using the standardstructural properties of the covariant derivative and it will be therefore omitted. Lemma 2.2.
The following tensorial identities hold true:1. ( (cid:3) − ∇ S = ∇ S (cid:3) on Γ( T M ) ;2. I ∇ S = (cid:3) on Γ( T M ) ;3. ( (cid:3) − I = I ( (cid:3) − on Γ( S T M ) ;4. tr( (cid:3) − (cid:3) tr on Γ( S T M ) .Similar identities hold true for T ∗ M via musical isomorphisms. Moreover one can derive dualidentities exploiting the pairing (2.2) . The well-known diffeomorphism invariance of general relativity translates at the level of the lin-earized theory in the following gauge freedom: Any solution h of (2.1) is equivalent to h + ∇ S χ ,for arbitrary χ ∈ Γ( T ∗ M ): h ∼ h ′ ∈ Γ( S T ∗ M ) ⇐⇒ ∃ χ ∈ Γ( T ∗ M ) : h ′ = h + ∇ S χ. (2.5)6otice that, consistently with gauge invariance, for all χ ∈ Γ( T ∗ M ), K ∇ S χ = 0 as it can be readilyverified via Lemma 2.2. Following the same line of reasoning as for other linear gauge theories[HS12], we can turn eq. (2.1) to an hyperbolic one by a suitable gauge fixing. For linearizedgravity the first choice is usually the so-called de Donder gauge , which plays the same role ofthe Lorenz gauge in electromagnetism. Although this is an overkilled topic, we summarize themain results in the following proposition [FH12, Hun12]: Proposition 2.3.
Let S / G be the space of gauge equivalence classes [ h ] of solutions of (2.1) , thatis S . = { h ∈ Γ( S T ∗ M ) | Kh = 0 } , where K has the form (2.4) , while G . = { h ∈ Γ( S T ∗ M ) | h = ∇ S χ, χ ∈ Γ( T ∗ M ) } . Then, for every [ h ] ∈ S , there exists a representative e h ∈ [ h ] satisfying P e h = ( (cid:3) − I e h = 0 , (2.6a)div I e h = 0 . (2.6b) Proof.
Let [ h ] ∈ S and let us choose any representative h . Suppose h does not satisfy (2.6b);we look therefore for another representative e h ∈ [ h ] such that div I e h = 0. To this end, consider χ ∈ Γ( T ∗ M ), solution of (cid:3) χ + 2div Ih = 0. The existence of such χ descends from applying [BF09,Section 3.5.3, Corollary 5] since (cid:3) is normally hyperbolic and div Ih is a smooth source term. Let e h = h + ∇ S χ . Since it is per construction a solution (2.4), which satisfies furthermore (2.6b), also(2.6a) holds true, as one can verify directly.We can use the last proposition to characterize the gauge equivalence classes of solutions of(2.1) via (2.6). As a starting point, notice that the operator e P . = (cid:3) − S T ∗ M )is normally hyperbolic. As explained in details in [Bar13, San13], this is tantamount to theexistence of unique retarded and advanced Green operators e E ± : Γ tc ( S T ∗ M ) → Γ( S T ∗ M ) suchthat e P e E ± = id and e E ± e P = id, where id stands for the identity operator on Γ tc ( S T ∗ M ). Thefollowing support properties hold true: supp( e E ± ( ǫ )) ⊆ J ± (supp( ǫ )) for any ǫ ∈ Γ tc ( S T ∗ M ).Since P = e P ◦ I , where I is the trace-reversal operator, item 3. of Lemma 2.2 guarantees usthat, although P is not normally hyperbolic, it is Green hyperbolic, that is one can associate toit retarded and advanced Green operators E ± . = I ◦ e E ± = e E ± ◦ I , which share the same supportproperties of e E ± and are both right and left inverses of P . Actually, E ± are also unique, P being formally self-adjoint, i.e. ( P h, k ♯ ) = ( h, ( P k ) ♯ ) for each h, k ∈ Γ( S T ∗ M ) with compactoverlapping support. Let E = E + − E − be the so-called causal propagator . On account of theexactness of the sequence0 −→ Γ tc ( S T ∗ M ) P −→ Γ tc ( S T ∗ M ) E −→ Γ( S T ∗ M ) P −→ Γ( S T ∗ M ) −→ , the causal propagator E induces an isomorphism between Γ tc ( S T ∗ M ) / Im tc P and the space ofsolutions of (2.6a) – see [Kha14a].Yet, since we are looking for gauge equivalence classes of solutions of (2.1), we are interestedonly in those solutions of (2.6a) fulfilling also (2.6b). To select them, we follow the same strategyas first outlined in [Dim92] for the free Maxwell equations, namely we translate the gauge fixingcondition in the restriction to a suitable subspace of Γ tc ( S T ( ∗ ) M ). As a preliminary result weprove the following lemma: Lemma 2.4.
Let E ± be the retarded (+) and advanced (-) Green operators of P . Then on Γ tc ( S T ∗ M ) it holds div IE ± = E ± (cid:3) div , where E ± (cid:3) denotes the retarded/advanced Green operator for (cid:3) acting on sections of T ∗ M .Proof. Via the dual pairing between
T M and T ∗ M , we deduce that ( E ± (cid:3) ξ, χ ) = ( ξ, E ∓ (cid:3) χ ) for each ξ ∈ Γ tc ( T ∗ M ) and each χ ∈ Γ c ( T M ). Notice that, with a slight abuse of notation, we use thesame symbol E ± (cid:3) for the Green operators of (cid:3) when it acts on sections of both T M and T ∗ M .7e can now refer to item 1. in Lemma 2.2 to conclude div IE ± = E ± (cid:3) div on Γ tc ( S T ∗ M ). As amatter of fact, for each v ∈ Γ tc ( S T ∗ M ) and ξ ∈ Γ c ( T M ) it holds(div IE ± v, ξ ) = − ( E ± v, I ∇ S ξ ) = − ( E ± v, I ∇ S (cid:3) E ∓ (cid:3) ξ ) = − ( E ± v, I ( (cid:3) − ∇ S E ∓ (cid:3) ξ )= − (( (cid:3) − IE ± v, ∇ S E ∓ (cid:3) ξ ) = − ( v, ∇ S E ∓ (cid:3) ξ ) = ( E ± (cid:3) div v, ξ ) . The arbitrariness of ξ entails the sought result. Remark 2.5.
Notice that the de Donder gauge fixing is not complete, namely there is a residualgauge freedom. In other words, for each [ h ] ∈ S , there exists more than one representative fulfillingboth (2.6a) and (2.6b). These representatives differ by pure gauge solutions of the form ∇ S χ ,where χ ∈ Γ( T ∗ M ) is such that (cid:3) χ = 0.We characterize the solutions of the equations of motion for linearized gravity in the de Dondergauge: Proposition 2.6.
Let
Ker tc (div) . = { ǫ ∈ Γ tc ( S T ∗ M ) | div ǫ = 0 } . Then Eǫ solves both (2.6a) and (2.6b) , where E is the causal propagator of P . Conversely, for each solution h ∈ Γ( S T ∗ M ) of the system (2.6) , there exists ǫ ∈ Ker tc (div) such that Eǫ differs from h by a residual gaugetransformation according to Remark 2.5.Proof. Let ǫ be in Ker tc (div). Per definition of E , we know that Eǫ solves (2.6a). On account ofLemma 2.4, it holds also that div IEǫ = E (cid:3) div ǫ = 0; hence Eǫ solves also (2.6b).Consider now any solution h of (2.6). As a consequence of (2.6a), there exists e ǫ ∈ Γ tc ( S T ∗ M )such that E ( e ǫ ) = h . Still on account of Lemma 2.4, (2.6b) translates into E (cid:3) div e ǫ = 0. Hencethere exists η ∈ Γ tc ( T ∗ M ) such that div e ǫ = (cid:3) η . Let ǫ . = e ǫ − I ∇ S η ∈ Γ tc ( S T ∗ M ). On accountof item 2. in Lemma 2.2, it holds that div ǫ = 0. Furthermore Eǫ = E e ǫ − EI ∇ S η = h − ∇ S E (cid:3) η. Therefore Eǫ differs from h by ∇ S χ , where χ . = − E (cid:3) η solves (cid:3) χ = 0 and, thus, it is a residualgauge transformation.We are ready to characterize the space of gauge equivalence classes of solutions of linearizedgravity: Theorem 2.7.
There exists a one-to-one correspondence between S / G , the set of gauge equiv-alence classes of solutions of (2.1) , and the quotient between Ker tc (div) and Im tc ( K ) . = { ǫ ∈ Γ tc ( S T ∗ M ) | ǫ = Kγ, γ ∈ Γ tc ( S T ∗ M ) } . The isomorphism is explicitly realized by the map [ ǫ ] [ Eǫ ] for [ ǫ ] ∈ Ker tc (div) / Im tc ( K ) .Proof. As a starting point, notice that the quotient between Ker tc (div) and Im tc ( K ) is meaningfulsince the identity div ◦ K = 0 holds on Γ( S T ∗ M ). This follows from eq. (2.3), K ◦ ∇ S = 0on Γ( S T ∗ M ) and K being formally self-adjoint, namely ( Kh, v ♯ ) = ( h, ( Kv ) ♯ ) for each h, v ∈ Γ( S T ∗ M ) with compact overlapping support.On account of Proposition 2.6 we know that E associates to each element in Ker tc (div) asolution of both (2.6a) and (2.6b). In turn this identifies a unique gauge equivalence class in S / G .Since E ◦ P = 0 and taking into account the dual of Lemma 2.4, one reads E ◦ K = 2 EI ∇ S div I =2 ∇ S E ± (cid:3) div I , it holds also that the map [ ǫ ] [ Eǫ ] is well-defined. Furthermore, Proposition 2.3and Proposition 2.6 together entail that this application is surjective since, for any [ h ] ∈ S / G ,one has a representative h in the de Donder gauge and there exists ǫ ∈ Ker tc (div) such that Eǫ differs from h at most by a residual gauge transformation. In other words we have found[ ǫ ] ∈ Ker tc (div) / Im tc ( K ) such that [ Eǫ ] = [ h ].Let us now prove that the map induced by E is injective. This is tantamount to provethe following statement: [ ǫ ] ∈ Ker tc (div) / Im tc ( K ) such that [ Eǫ ] = [0] ∈ S / G entails [ ǫ ] = 0.Accordingly, let us assume that ǫ ∈ Ker tc (div) is such that Eǫ = ∇ S χ for χ ∈ Γ( T ∗ M ). By (2.6b)8nd Lemma 2.4, this entails that (cid:3) χ = 0 and hence χ = E (cid:3) α , for a suitable α ∈ Γ tc ( T ∗ M ). Onaccount of the dual of Lemma 2.4, it holds Eǫ = EI ∇ S α , that is there exists β ∈ Γ tc ( S T ∗ M )such that ǫ = I ∇ S α + P β . Acting with div on both sides, one obtains 0 = (cid:3) α + 2 (cid:3) div Iβ , or inother words α = − Iβ . Hence, ǫ = P β − I ∇ S div Iβ = − Kβ . This concludes the proof. Remark 2.8.
In several discussions concerning linearized gravity, radiative degrees of freedomor even gravitational waves, it is customary to exploit the additional gauge freedom (see Remark2.5) to switch from the de Donder to the so-called transverse-traceless (TT) gauge. This consistsof adding one more constraint on the field configuration besides eq. (2.6b). The extra conditionreads tr h = 0, where tr : Γ( S T ∗ M ) → C ∞ ( M ) denotes the trace computed using the metric g , motivating the word traceless . It was noticed for the first time in [FH12, Hun12] that thiskind of gauge fixing is not always possible on account of a topological constraint. We discuss acomplementary approach to this problem: Considering any solution h of the system (2.6), we tryto exploit the residual gauge freedom explained in Remark 2.5 to find e h ∼ h such that e h is alsotraceless. Thus we look for χ ∈ Γ( T ∗ M ) such that (cid:3) χ = 0 and tr h + div χ = 0. Since χ is nothingbut a 1-form, χ ∈ Ω ( M ), it is convenient to read these conditions using the exterior derivatived and the codifferential δ defined on 1-forms. In other words we are looking for solutions of thefollowing two equations: (cid:3) χ = 0 , δχ = tr h, (2.7)where the D’Alembert wave operator (cid:3) coincides here with the Laplace-de Rham wave operator δ d + d δ since the background is Ricci flat. Here we follow the nomenclature of [SDH12] usingthe subscript ( p ) attached to E to indicate the causal propagator for the Laplace-de Rham waveoperator acting on p -forms. Actually, it is convenient to exploit Proposition 2.6 to express thegiven h in terms of a suitable ǫ ∈ Ker tc (div), namely h = Eǫ . In particular, we are interested inthe identity tr h = tr Eǫ = − E (0) (tr ǫ ), which is a consequence of item 4. in Lemma 2.2.We show below that the system of equations (2.7) admits a solution if and only if there exists λ ∈ Ω tc ( M ) such that tr ǫ = δλ . Assume first that there exists a solution χ of the system (2.7).From the first equation, χ is of the form χ = E (1) α for a suitable α ∈ Ω tc ( M ). Therefore,the second equation entails E (0) δα = − E (0) (tr ǫ ). In turn this fact entails δα = − tr ǫ + (cid:3) f for f ∈ C ∞ tc ( M ), that is to say tr ǫ = δ (d f − α ). Conversely, suppose tr ǫ = δλ for λ ∈ Ω tc ( M ). Thenone can directly check that χ = − E (1) λ is a solution of the system above.In conclusion, given a de Donder solution h = Eǫ , ǫ ∈ Γ c ( S T ∗ M ), it is possible to achieve theTT gauge if and only if ∗ (tr ǫ ) ∈ dΩ tc ( M ), that is to say [ ∗ (tr ǫ )] = 0 ∈ H tc ( M ), the fourth de Rhamcohomology group with timelike compact support – for a recent analysis see [Ben14, Kha14b].Following [Ben14, Theorem 5.5], H tc ( M ) ≃ H (Σ), Σ being a Cauchy surface for the globallyhyperbolic spacetime M . Therefore, the TT gauge can be always achieved for solutions withoutany restriction on the support provided H (Σ) = 0, namely, via Poincar´e duality, the Cauchysurface is non-compact. Otherwise there might be obstructions to the TT gauge for certainsolutions.Notice that, in principle, restricting our attention only to those asymptotically flat and globallyhyperbolic spacetimes having non-compact Cauchy surfaces might not be such a severe restrictionsince, actually, we are unaware of an explicit example falling outside the class just mentioned –see for example [Mur13]. Yet, in the next sections, in order to characterize a well-defined bulk-to-boundary projection, we shall employ another gauge fixing due to Geroch and Xanthopoulos. Thisprocedure displays a very similar obstruction, related both to the geometry and to the topologyof the underlying background. Our goal is to construct an algebra of observables which encompasses both the dynamics (2.1) andthe gauge symmetry (2.5). In this respect we employ a procedure partly different from the one in[FH12, Hun12]. We mimic, instead, the approach of [BDS12], subsequently applied in [BDS13] to9he quantization of principal Abelian connections. In our opinion this approach has several netadvantages. For instance, in the case of Yang-Mills theory with an Abelian gauge group containingat least a U (1) factor, it helps unveiling the optimal class of observables to test the collection ofgauge equivalent configurations, cfr. [BDS13, BDHS13]. Also in the case, we consider, this dual(in a sense specified below) approach is still worth following since it avoids any a priori gaugefixing to construct the functionals which define observables. This feature is particularly relevantin clarifying the key aspects of the bulk-to-boundary correspondence for linearized gravity onasymptotically flat spacetimes, for which several choices of gauge fixings appear at different stagesof the procedure.Following [BDS12, BDS13], we start from the space of off-shell field configurations, namelyΓ( S T ∗ M ). Recalling (2.2), a convenient space of sections which is dually paired to this one isgiven by compactly supported sections of the dual bundle, Γ c ( S T M ). For any ǫ ∈ Γ c ( S T M ),we can introduce a linear functional O ǫ as follows: O ǫ : Γ( S T ∗ M ) → R , O ǫ ( h ) = ( h, ǫ ) , where µ g is the metric induced volume form on M . Notice that the evaluation of O ǫ on the off-shellconfiguration h is nothing but the usual pairing between h ∈ Γ( S T ∗ M ) and ǫ ∈ Γ c ( S T M ). Thecollection of all functionals O ǫ , ǫ ∈ Γ c ( S T M ), forms a vector space which we indicate as E kin and, due to non-degeneracy of the pairing ( · , · ) introduced in (2.2), it is isomorphic to Γ c ( S T M ).Hence we will often identify E kin with Γ c ( S T M ) by writing ǫ ∈ E kin for any ǫ ∈ Γ c ( S T M ).Up to this point, a functional O ǫ , ǫ ∈ Γ c ( S T M ), is neither invariant under gauge transforma-tions nor on-shell, requirements which are both needed in order to interpret O ǫ as an observable forthe classical field theory describing linearized gravity. As a first step, we identify those functionalswhich behave properly under gauge transformations. Recalling (2.5), we realize that O ǫ ∈ E kin is gauge invariant if and only if O ǫ ( ∇ S χ ) = 0 for all χ ∈ Γ( T ∗ M ). The following lemmacharacterizes gauge invariant functionals: Lemma 2.9.
A functional O ǫ is invariant under gauge transformations if and only if div ǫ = 0 .Proof. This follows from eq. (2.3), which states that ∇ S and − div are the dual of each other.This entails that, for all ǫ ∈ Γ c ( S T M ) and for all χ ∈ Γ( T ∗ M ), O ǫ ( ∇ S χ ) = ( ∇ S χ, ǫ ) = ( χ, − div ǫ ) . Since the pairing between Γ c ( T M ) and Γ( T ∗ M ) is non-degenerate, we deduce that O ǫ is gaugeinvariant, namely O ǫ ( ∇ S χ ) = 0 for each χ ∈ Γ( T ∗ M ) if and only if div ǫ = 0.Lemma 2.9 motivates the definition given below for the space of gauge invariant linear functionals: E inv = n O ǫ ∈ E kin : div ǫ = 0 o = Ker c (div) . As a last step, we have to account for the dynamics. More precisely we wish to constructequivalence classes, identifying two elements in E inv whenever they differ by a third one whichyields 0 when evaluated on any configuration h ∈ Γ( S T ∗ M ) solving (2.1). This is achieved takingthe quotient of E inv by the image of the dual of the differential operator K ruling the dynamics.In this way we obtain (classes of) gauge invariant functionals whose evaluation is well-defined onlyon (gauge equivalence classes of) solutions to the field equation Kh = 0. We proceed as follows:First, we compute the dual of K with respect to the pairing ( · , · ) between sections of S T ∗ M and S T M defined in (2.2). For each h ∈ Γ( S T ∗ M ) and each ǫ ∈ Γ( S T M ) with compactoverlapping support, we have( h, K ∗ ǫ ) = ( Kh, ǫ ) = (( − (cid:3) + 2Riem + 2 I ∇ S div) Ih, ǫ ) = ( I ( − (cid:3) + 2Riem + 2 ∇ S div I ) h, ǫ )= ( h, ( − (cid:3) + 2Riem + 2 I ∇ S div) Iǫ ) = ( h, ( Kǫ ♭ ) ♯ ) , K ∗ of K coincides with K itself, barring the musical isomorphisms topass from T M to T ∗ M and vice versa. For this reason, with a slight abuse of notation, in thefollowing we will use K also to denote its dual K ∗ . We introduce the space of classical observablesas follows: E = E inv Im c ( K ) . (2.8)Notice that the quotient is well-defined on account of the identity div ◦ K = 0 on Γ( S T M ).The evaluation of an observable [ ǫ ] ∈ E on a gauge class of solutions [ h ] ∈ S / G is consistentlyobtained by an arbitrary choice of representatives, namely O [ ǫ ] ([ h ]) = O ǫ ( h ) for each ǫ ∈ [ ǫ ] and h ∈ [ h ]. The space of classical observables E can be endowed with a presymplectic form. Thisis introduced via the causal propagator E of the Green hyperbolic differential operator P , whichrules the gauge-fixed dynamics, see Proposition 2.3. Proposition 2.10.
The space of classical observables E can be endowed with the presymplecticstructure defined below: τ : E ⊗ E → R , τ ([ ǫ ] , [ ζ ]) = 2( Eǫ ♭ , ζ ) , (2.9) where E is the causal propagator for P and where the right-hand-side is written in terms of anarbitrary choice of representatives in both equivalence classes.Proof. First, let us notice that ( E · ♭ , · ) is bilinear and skew-symmetric on E kin . Bilinearity canbe directly read from the formula, hence we take arbitrary η, ζ ∈ E kin and show that ( Eǫ ♭ , ζ ) = − ( Eζ ♭ , ǫ ): · ♭ intertwines both P and its dual (still denoted by P with a slight abuse of notationmotivated by the fact that the operators looks exactly the same up to musical isomorphisms).Therefore a similar property holds true for the corresponding causal propagators, both denoted by E with the same abuse of notation. Since we are dealing with P and its dual, the following relationbetween the corresponding Green operators holds: ( E ± h, η ) = ( h, E ∓ η ) for each h ∈ Γ c ( S T ∗ M )and η ∈ Γ c ( S T M ). To conclude this part of the proof, we stress that ( A ♭ , H ♯ ) = ( H, A ) for each A ∈ Γ( S n T M ) and H ∈ Γ( S n T ∗ M ) with compact overlapping support. All these considerationsentail that ( Eǫ ♭ , ζ ) = (( Eǫ ) ♭ , ζ ) = ( ζ ♭ , ( Eǫ ) ♭ ♯ ) = ( ζ ♭ , Eǫ ) = − ( Eζ ♭ , ǫ ) . Up to this point, we have a presymplectic form ( E · ♭ , · ) on E kin , hence in particular on E inv .We still have to prove that such structure descends to the quotient space E , thus providing τ as specified in the statement: To this end, we take ǫ ∈ E inv and η ∈ Γ c ( S T M ) and show that( Eǫ ♭ , Kη ) = 0. To proceed, we take into account that ( E · ♭ , · ) is skew-symmetric, we recall thedefinition of K , eq. (2.4), and we consider its dual acting on sections of S T M , still denoted by K with the usual abuse of notation. Exploiting item 1. of Lemma 2.2 and (2.3), we end up with( Eǫ ♭ , Kη ) = − ( EKη ♭ , ǫ ) = − ( EI ∇ S div Iη ♭ , ǫ ) = − ( ∇ S E (cid:3) div Iη ♭ , ǫ ) = ( E (cid:3) div Iη ♭ , div ǫ ) = 0 , where the last equality follows from gauge-invariance of ǫ . Therefore ( E · ♭ , · ) descends to thequotient E . This shows that τ is a well-defined presymplectic form on E , thus completing theproof.The factor 2 appearing in the expression of the presymplectic form might look unusual, as muchas the fact that the causal propagator appears in the left slot of the pairing. Here we are usingthe causal propagator E for P = ( (cid:3) − I in order to define the presymplectic form, whichdoes not take into account a factor − / From a geometric point of view, it would be more customary and appropriate to talk about a constant Poissonstructure – see for example [Kha14a]. We will stick to the nomenclature more commonly used in quantum fieldtheory on curved backgrounds.
Remark 2.11.
It is possible to obtain the same formula for the presymplectic structure in Proposi-tion 2.10 generalizing a method originally due to Peierls [Pei52] to gauge theories [Kha12, Kha14a].This approach was considered already in [Mar93] and was recently put on mathematically solidgrounds in [SDH12] for the vector potential of electromagnetism. In [BDS13] it was successfullyapplied also to principal connections for Abelian Yang-Mills models. We follow here a similarargument in order to motivate the definition of τ .Once a gauge invariant functional ǫ ∈ E inv is fixed, we are interested in studying how thepresence of ǫ affects the dynamics of the field. More precisely, we want to compare the retardedand the advanced effect produced by ǫ on any other gauge invariant functional ζ ∈ E inv . For eachon-shell configuration h , this is achieved by finding solutions h ± ǫ to the field equation modified bythe presence of ǫ such that h ± ǫ is gauge equivalent to h in the past/future of a Cauchy surface.In the end, the effect produced by ǫ on ζ is evaluated comparing O ζ ( h + ǫ ) with O ζ ( h − ǫ ). Wedefine the modified dynamics introducing the equation K ǫ h = Kh + 2 ǫ ♭ = 0. This is exactly theinhomogeneous differential equation we would obtain starting from the Lagrangian density forlinearized gravity, adding an external source ǫ and then looking for the associated Euler-Lagrangeequations. In particular this motivates the factor 2, which is due to the fact that Kh = 0 coincideswith eq. (2.1) up to such factor. Solutions to the equation K ǫ h = 0 can be obtained applying theGreen operators E ± for P to 2 ǫ ♭ : KE ± (2 ǫ ♭ ) = − ǫ ♭ + 4 I ∇ S div IE ± ǫ ♭ = − ǫ ♭ + 4 I ∇ S E ± (cid:3) (div ǫ ) ♭ = − ǫ ♭ , where we employed both Lemma 2.4 and div ǫ = 0. Consider an on-shell configuration h , namely Kh = 0, and look for h ± ǫ as above. Setting h ± ǫ = h + E ± (2 ǫ ♭ ), we read K ǫ h ± ǫ = 0. Moreover h ± ǫ differs from h only on J ± M (supp( ǫ )). Since ǫ has compact support, the requirement on theasymptotic behaviour is fulfilled as well. We are now ready to define the retarded/advanced E ± ǫ effect induced by ǫ on any gauge-invariant functional ζ ∈ E inv as E ± ǫ ζ = O ζ ( h ± ǫ ) − O ζ ( h ). Westress that, given h , the right-hand-side does not depend on our construction of h ± ǫ due to thegauge invariance of ζ . We now compare retarded and advanced effects: E + ǫ ζ − E − ǫ ζ = ( E + (2 ǫ ♭ ) , ζ ) − ( E − (2 ǫ ♭ ) , ζ ) = 2( Eǫ ♭ , ζ ) = τ ([ ǫ ] , [ ζ ]) . Therefore Peierls’ method yields exactly the presymplectic form used in Proposition 2.10.To conclude the section we establish an isomorphism between the presymplectic space of clas-sical observables E and spacelike compact solutions of the linearized Einstein’s equation up tospacelike compact gauge. Besides making contact with other treatments, see e.g. [FH12], thiscorrespondence will be exploited in the next section to construct the bulk-to-boundary correspon-dence. Let us first introduce some notation: We use the symbol S sc . = Ker sc ( K ) to indicate thespace of solutions h of the equation Kh = 0 with support included in a spacelike compact region,while we denote the space of spacelike compact gauge transformations ∇ S χ , χ ∈ Γ sc ( T ∗ M ), with G sc . = Im sc ( ∇ S ). Proposition 2.12.
There exists a one-to-one correspondence between E and S sc / G sc inducedby the causal propagator E for P , which is defined by [ ǫ ] [ Eǫ ♭ ] . Such map induces an iso-morphism of presymplectic spaces when S sc / G sc is endowed with the presymplectic form σ : S sc / G sc × S sc / G sc → R defined by σ ([ h ] , [ Eζ ]) = 2( h, ζ ♯ ) for each h ∈ S sc and ζ ∈ Γ c ( S T ∗ M ) . roof. The first part of the proof is a slavish copy of that of Proposition 2.6 and of Theorem2.7 where timelike compact sections are replaced by compactly supported ones whereas smoothsolutions of the linearized Einstein’s equations are replaced by those whose support is spacelikecompact and the same is done for gauge transformations.For the second part of the proof let us consider [ ǫ ] , [ ǫ ′ ] ∈ E . Then we have σ ([ Eǫ ♭ ] , [ Eǫ ′ ♭ ]) = 2( Eǫ ♭ , ǫ ′ ) = τ ([ ǫ ] , [ ǫ ′ ]) . This identity completes the proof.
Remark 2.13.
We may wonder whether the content of Remark 2.8, concerning the implemen-tation of the TT gauge, can be applied also to S sc / G sc . The point is the following: It is possibleto achieve the TT gauge for a spacelike compact de Donder solution h = Eǫ , ǫ ∈ Γ c ( S T ∗ M ),exploiting the spacelike compact gauge freedom ∇ S χ , χ ∈ Γ sc ( T ∗ M ) if and only if tr ǫ = δλ for a λ ∈ Ω c ( M ), that is to say [ ∗ (tr ǫ )] is the trivial class in H c ( M ). Therefore, the obstruction to theTT gauge is now ruled by H c ( M ), which is isomorphic to H ( M ) ≃ R c via Poincar´e duality, c be-ing the number of connected components of M . In particular, this means that on every spacetimeone may encounter obstructions in imposing the TT gauge for some spacelike compact solutions. Our present goal is to spell out explicitly the construction of a bulk-to-boundary correspondence forlinearized gravity on asymptotically flat spacetimes at a classical level. The quantum counterpartwill be discussed in the next section.
We focus our attention on a particular class of manifolds which are distinguished since they possessan asymptotic behaviour along null directions which mimics that of Minkowski spacetime. Usedextensively and successfully in the definition of black hole regions [Wal84], the most general class ofasymptotically flat spacetimes includes several important physical examples, such as for instancethe Schwarzschild and the Kerr solutions to Einstein’s equations. In this paper we will employthe definition of asymptotic flatness, as introduced by Friedrich in [Fri86]. To wit, we consideran asymptotically flat spacetime with future time infinity i + , i.e. a globally hyperbolicspacetime ( M, g ), solution of Einstein’s vacuum equations, hereby called physical spacetime , suchthat there exists a second globally hyperbolic spacetime ( f M , e g ), called unphysical spacetime , with apreferred point i + ∈ f M , a diffeomorphism ψ : M → ψ ( M ) ⊂ f M and a function Ξ : ψ ( M ) → (0 , ∞ )so that ψ ∗ (Ξ − e g ) = g . Moreover, the following requirements ought to be satisfied:a) If we call J − f M ( i + ) the causal past of i + , this is a closed set such that ψ ( M ) = J − f M ( i + ) \ ∂J − f M ( i + )and we have ∂M = ∂J − f M ( i + ) = I + ∪ { i + } , where I + is called future null infinity .b) Ξ can be extended to a smooth function on the whole f M and it vanishes on I + ∪ { i + } .Furthermore, d Ξ = 0 on I + while d Ξ = 0 on i + and e ∇ µ e ∇ ν Ξ = − e g µν at i + .c) Introducing n µ . = e ∇ µ Ξ, there exists a smooth and positive function ξ supported at least ina neighbourhood of I + such that e ∇ µ ( ξ n µ ) = 0 on I + and the integral curves of ξ − n arecomplete on future null infinity.Here e ∇ is the Levi-Civita connection built out of e g . Notice that, in the above definition, futuretimelike infinity plays a distinguished role, contrary to what happens in the more traditionaldefinition of asymptotically flat spacetimes where i + is replaced by i , spatial infinity – see forexample [Wal84, Section 11]. The reason for our choice is motivated by physics: We are interested13n the algebra of observables for linearized gravity which is constructed out of E , the causal prop-agator associated to the operator P as in (2.6a). This entails, that, for any smooth and compactlysupported symmetric rank 2 tensor ǫ , its image under the action of the causal propagator is sup-ported in the causal future and past of supp( ǫ ). Therefore it will be important in our investigationthat future timelike infinity is actually part of the unphysical spacetime, so to be able to controlthe behaviour of E ( ǫ ) thereon. Such requirement can be relaxed particularly if one is interestedin studying field theories on spacetimes like Schwarzschild where i + cannot be made part of theunphysical spacetime. The price to pay in this case is the necessity to make sure that any solutionof the classical dynamics falls off sufficiently fast as it approaches future timelike infinity. Thisline of reasoning has been pursued in [DMP11], though we shall not follow it here since it reliesheavily on the fact that a very specific manifold has been chosen. On the contrary we plan toconsider all at the same time a large class of backgrounds.Before focusing our attention on the field theoretic side, it is worth devoting a few linesto outlining the geometric properties of the null boundary of an asymptotically flat spacetime.Notice that the choice to work with ℑ + and not with ℑ − , past null infinity, is purely conventional.Everything can be translated slavishly to the other case. Here we will summarize what has beenalready discussed in detail in [Fri86, Ger77, Wal84] and in [DMP05, DS13, Sie11] for an applicationto quantum field theory: • ℑ + is a three dimensional submanifold of f M generated by the null geodesics emanating from i + , i.e. the integral curves of n . It is thus diffeomorphic to R × S although the possiblemetric structures are affected by the existence of a gauge freedom which corresponds to therescaling of Ξ to ξ Ξ, where ξ is a smooth function which is strictly positive in ψ ( M ) as wellas in a neighbourhood of ℑ + . • Null infinity is said to be both intrinsic and universal. In other words, if we introduce forany fixed asymptotically flat spacetime (
M, g ) the set C composed by the equivalence classesof triples ( ℑ + , h, n ), where h . = e g ↾ ℑ + and ( ℑ + , h, n ) ∼ ( ℑ + , ξ h, ξ − n ) for any choice of ξ satisfying c), there is no physical mean to select a preferred element in C . This is calledthe intrinsicness of ℑ + . Concerning universality , if we select any pair of asymptotically flatspacetimes, ( M , g ) and ( M , g ), together with the corresponding triples, say ( ℑ +1 , h , n )and ( ℑ +2 , h , n ), there always exists a diffeomorphism γ : ℑ +1 → ℑ +2 such that h = γ ∗ h and n = γ ∗ n . • In each equivalence class, element of C , there exists a choice of conformal gauge ξ B yieldinga coordinate system ( u, Ξ , θ, ϕ ) in a neighbourhood of ℑ + , called Bondi frame , such that the(rescaled) unphysical metric tensor becomes e g ↾ ℑ + = − u dΞ + d θ + sin θ d ϕ . (3.1)In this novel coordinate system future null infinity is the locus Ξ = 0, while u is the affineparameter of the null geodesics generating ℑ + . Thus, at each point on ℑ + the vector field n coincides with ∂ u . • A distinguished role both from a geometric and from a quantum theoretical point of viewis played by the subgroup of diffeomorphisms of ℑ + which maps each equivalence classlying in C into itself. This is the so-called Bondi-Metzner-Sachs (BMS) group whichcoincides, moreover, with the group of asymptotic symmetries of the physical spacetime(
M, g ) [Ger77]. It can be explicitly characterized in a Bondi frame as follows: Considerthe complex coordinates ( z, ¯ z ) obtained from ( θ, ϕ ) via the stereographic projection, z = e iϕ cot( θ/ u, z, ¯ z ) as the following map u u ′ . = K Λ ( z, ¯ z ) ( u + α ( z, ¯ z )) ,z z ′ . = az + bcz + d and c.c. , (3.2)14here a, b, c, d ∈ C with ad − bc = 1, whereas α ( z, ¯ z ) ∈ C ∞ ( S ) and K Λ ( z, ¯ z ) = 1 + | z | | az + b | + | cz + d | . From (3.2) it descends that the BMS group is the semidirect product SO (3 , ⋊ C ∞ ( S ),where SO (3 ,
1) is the component connected to the identity of the Lorentz group, hereacting on ( z, ¯ z ) via a M¨obius transformation. Hence each γ ∈ BM S identifies actually apair (Λ , α ) ∈ SO (3 , × C ∞ ( S ).We can now focus on the main goal of this section, namely constructing a ∗ -algebra intrinsicallydefined on ℑ + on which to encode the information of the bulk counterpart in a sense specifiedbelow. We start by defining a suitable “space of observables” on null infinity and, to this end,we follow a strategy very similar to the one employed in [DS13, Sie11], which we combine withearlier analysis, see [Ash81, AM82] in particular. Notice that, on account of the peculiar structureof ℑ + , it is more convenient to write explicitly the indices of all tensors involved in our analysis.Furthermore we need the following key ingredients:1. Let ( M, g ) be an asymptotically flat spacetime whose unphysical counterpart is ( f M , e g ). Thenwe call ι : ℑ + → f M the embedding of null infinity into the unphysical spacetime.2. Let q = ι ∗ e g be the pull-back of (3.1) to ℑ + . On account of the null direction on ℑ + , theoutcome is degenerate and, thus, a canonical inverse metric does not exist. Hence, followingthe historical convention – see for example [Ash81, AM82], we shall call q ab any symmetrictensor field satisfying the condition q ab q ac q bd = q cd . There is a large freedom in this choice,but, as we shall comment later, it does not play a role in our analysis.With these data and on account of the analysis of Ashtekar on the radiative degrees of freedomin general relativity at null infinity, we introduce the following space of sections: S ( ℑ + ) . = { λ ab ∈ Γ( S T ∗ ℑ + ) | λ ab n a = 0 , λ ab q ab = 0 , and ( λ, λ ) ℑ < ∞ , ( ∂ u λ, ∂ u λ ) ℑ < ∞} . (3.3)Here ( · , · ) ℑ denotes a pairing between sections of S T ∗ ℑ + defined by( λ, λ ′ ) ℑ . = Z ℑ + h λ, λ ′ i µ ℑ , h λ, λ ′ i . = λ ab λ ′ cd q ac q bd , (3.4)for all λ, λ ′ ∈ Γ( S T ∗ ℑ + ) such that λ ab n a = λ ′ ab n a = 0 and h λ, λ ′ i is an integrable function,where, in a Bondi reference frame, µ ℑ = sin θ d u d θ d ϕ . Notice that the inner product on ℑ + (and therefore the integrability condition too) does not depend on the choice of q ab since thefreedom in the choice of an inverse to q ab lies in the null direction, but the constraint λ ab n a = 0on elements of S ( ℑ + ) ensures that such components never contribute. Furthermore we can regard S ( ℑ + ) as a symplectic vector space after endowing it with the antisymmetric bilinear form below: σ ℑ ( λ, λ ′ ) = Z ℑ + (cid:0) λ ab L n λ ′ cd − λ ′ ab L n λ cd (cid:1) q ac q bd µ ℑ , (3.5)where L n is the Lie derivative along the null vector n . Notice that, repeating verbatim thesame argument of [DS13, Proposition 4.1], (3.5) is weakly non-degenerate as a consequence of thefiniteness of both ( λ, λ ) ℑ and ( ∂ u λ, ∂ u λ ) ℑ .We remark that (3.3) differs slightly from the one used in [AM82] since an explicit conditionon the square integrability of the symmetric (0 , ℑ + and of their derivatives along thenull-direction is spelled out. This is motivated by our desire to mimic the same analysis for theconformally coupled scalar field in [Mor05, Mor06] and for the vector potential in [DS13, Sie11].15urthermore, also the authors in [AM82] stress that a suitable fall-off condition of both λ and λ ′ towards i + , timelike infinity, is necessary so to ensure the finiteness of the integral in (3.5).To conclude our excursus on the boundary data, we need to specify how the BMS group actson our fields on ℑ + . Following [DS13, Section 2], we consider any vector bundle E on ℑ + and weintroduce a family of representations of the BMS group on the smooth sections Γ( E ) via the mapΠ ρ : BMS × Γ( E ) → Γ( E ) defined according to(Π ρ ( γ, s ))( u ′ , z ′ , ¯ z ′ ) = K Λ ( z, ¯ z ) ρ s ( u + α ( z, ¯ z ) , z, ¯ z ) , (3.6)where ρ ∈ R is a parameter, γ is presented in the form (Λ , α ) ∈ SO (3 , × C ∞ ( S ) (see thecomment below (3.2)) and ( u ′ , z ′ , ¯ z ′ ) are defined as functions of ( u, z, ¯ z ) according to (3.2). Withthese data we can prove the following: Proposition 3.1.
The symplectic space ( S ( ℑ + ) , σ ℑ ) is invariant under the representation Π ρ ofthe BMS group with ρ = 1 .Proof. To start with, let us consider ℑ + in the Bondi frame and let us pick any element γ =(Λ , α ) ∈ BMS. Let λ ∈ S ( ℑ + ) and let λ γ . = Π ρ ( γ, λ ) be the outcome after the action of γ on λ fora given value of ρ ∈ R . Per construction it holds that λ γ is symmetric and it fulfils both λ γ ab n a = 0and λ γ ab q ab = 0, as these properties are inherited directly from λ . Furthermore the measure onnull infinity can be rewritten in terms of the complex coordinates as µ ℑ = − i (1 + ¯ zz ) − d u d z d¯ z and, thus, (3.2) entails that µ ℑ is transformed by γ into K Λ ( z, ¯ z ) µ ℑ . Since K Λ ( z, ¯ z ) is a smooth,bounded and strictly positive function and since µ ℑ is translation invariant along the u -direction,finiteness of both ( λ γ , λ γ ) ℑ and ( ∂ u λ γ , ∂ u λ γ ) ℑ can be traded from the same property of λ . In otherwords, for each γ ∈ BMS, Π ρ ( γ, · ) maps S ( ℑ + ) to itself, for all ρ ∈ R . It remains to be proventhat the symplectic form is preserved under action of Π ( γ, · ) for each γ ∈ BMS. We notice that(3.5) can be written as σ ℑ ( λ, λ ′ ) = Z ℑ + (cid:0) λ ab ∂ u λ ′ cd − λ ′ ab ∂ u λ cd (cid:1) q ac q bd µ ℑ . Let us take any γ = (Λ , α ) ∈ BMS. At the same time, in a Bondi frame the line element reads ds = 0 · du + dθ + sin θdϕ , from which one can infer via (3.2) that any BMS group elementtransforms q ab in K Λ ( z, ¯ z ) − q ab . Furthermore, for any λ ∈ S ( ℑ + ) and for any γ = (Λ , α ) ∈ BMS,it holds that ∂ u λ ( u, z, ¯ z ) is transformed to ∂ u λ ( u + α ( z, ¯ z ) , z, ¯ z ). Gathering all data together, weinfer that, under the action of the BMS group σ ℑ ( λ, λ ′ ) Z ℑ + K (cid:0) K Λ λ ab ∂ u λ ′ cd − K Λ λ ′ ab ∂ u λ cd (cid:1) K − q ac q bd µ ℑ = σ ℑ ( λ, λ ′ ) , where for notational simplicity we have omitted the explicit dependence on the coordinates of thevarious factors in the integrand. The next goal of our analysis is to show whether there exists a linear map from the space ofclassical observables in the bulk to S ( ℑ + ), which is injective and preserves the (pre-)symplecticform. Such result can be extended directly to the quantum counterpart and used to constructHadamard states induced from null-infinity. It is worth stressing that such procedure has beenshown to work for massless and conformally coupled scalar fields [DMP05], for the Dirac field[DPH11, Hac10] and for the vector potential [DS13, Sie11]. We will prove that linearized gravitybehaves in an inherently different way from all other free fields.The first deviation from the other cases considered manifests itself when, in a given spacetime( M, g ), one starts to study the behaviour of (2.1) under conformal transformations of the metric.16hile, in all other scenarios, conformal invariance was guaranteed, in our case a lengthy andtedious calculation not only shows the lack of it but also the arise of terms proportional to inversepowers of Ξ, the conformal factor. These are potentially pernicious since we are interested inconsidering solutions of the equations of motion propagating to null infinity, which is defined asthe locus for which Ξ = 0. To highlight the problem, it is better to work explicitly with indices.Let us consider any solution h of (2.1) and let us set τ ab = Ξ h ′ ab , τ a = Ξ − n b h ′ ab , τ = e g ab τ ab , f = Ξ − n a n a and n a = ∇ a Ξ = e ∇ a Ξ. On ( f M , e g ) it holds that τ ab obeys to the following partialdifferential equation – see [HI03, Section 3] for an expression valid in all dimensions: − e (cid:3) τ ab + e ∇ a e ∇ b τ + 4 ∇ ( a τ b ) + 2 e ∇ ( a y b ) − e g ab e ∇ c τ c − e R acdb τ cd + 12 e R ab τ + 16 e Rτ ab − e g ab e Rτ + 2Ξ n ( a y b ) + e g ab Ξ (cid:18) n c τ c + n c e ∇ c τ + 12 f τ (cid:19) = 0 , (3.7)where we used the auxiliary quantity y a . = e ∇ b τ ab − e ∇ a τ − τ a .In order to avoid the ensuing singularities in the above expression, one can only follow thesame approach used when dealing with the vector potential in asymptotically flat spacetimes:exploit the gauge invariance of (2.1) in order to tame the unwanted terms. Despite the approachis morally the same, notice the sharp contrast with [DS13, Sie11], in which, on account of theconformal invariance of the Maxwell equations for the Faraday tensor, it suffices to work in thestandard Lorenz gauge, albeit it is not conformally invariant. One hopes that similarly one couldconsider either the de Donder gauge or the transverse-traceless gauge for linearized gravity. Earlierinvestigations, see for example [GY80], show that pathologies with these choices cannot be avoided.A way to circumvent these problems has been devised by Geroch and Xanthopoulos in [GX78] byintroducing a highly non trivial gauge fixing which cancels all potentially divergent terms. Thegoal of their investigation was to prove the stability under metric perturbations of the propertyof asymptotic simplicity, which is slightly more general than that of asymptotic flatness. We willreview critically this procedure and we will show that, despite the common belief, it cannot bealways applied and obstructions are present. While also in electromagnetism similar features arepresent [BDHS13, BDS13, DL12, SDH12], here the situation is different since the source of suchobstructions cannot be only ascribed to the topology of the spacetime but there is a non trivialinterplay with the geometry of the background. The latter plays a key role since all observablesof linearized gravity have vanishing divergence, a condition which explicitly involves both thecovariant derivative and the metric.Let ( M, g ) be an asymptotically flat spacetime in the sense specified in Section 3 and let( f M , e g ) be an associated unphysical spacetime. By omitting for the sake of notational simplicitythe diffeomorphism ψ : M → ψ ( M ) ⊂ f M , we know that, on M , e g = Ξ g , where Ξ ∈ C ∞ ( f M ) isstrictly positive in M and vanishing on null infinity. Definition 3.2.
Let h ∈ Γ sc ( S T ∗ M ) be any spacelike compact solution of (2.1). We say that h isa solution of the linearized Einstein’s equations in the Geroch-Xanthopoulos gauge (GX-gauge) if, setting τ ab = Ξ h ′ ab , τ a = Ξ − n b h ′ ab , τ = e g ab τ ab and f = Ξ − n a n a and n a = ∇ a Ξ = e ∇ a Ξ, itholds that y a = e ∇ b τ ab − e ∇ a τ − τ a = 0 , (3.8a) (cid:18) n a e ∇ a + 16 Ξ e R + 32 f (cid:19) e (cid:3) τ = 112 e Rf τ − τ e (cid:3) f − e Rn a τ a + 4Ξ e C abcd τ bd n a n c , (3.8b)where e · refers to quantities computed with respect to e g , e.g. e C abcd is the Weyl tensor for e g .17s proven in [GX78], working with the GX-gauge reduces (3.7) to the following set of equations: e (cid:3) τ ab = e ∇ a e ∇ b τ + 4 e ∇ ( a τ b ) − e C acbd τ cd − e R τ ab + e R τ e g ab − τ e R ab + 2 e R c ( a τ cb ) − u e g ab , (3.9) e (cid:3) τ a e ∇ a σ + e R ac e ∇ c τ + R e ∇ a τ − e R cd e ∇ c τ ad − τ ac e ∇ c e R + τ cd e ∇ [ c e R a ] d + τ c e R ac + e R τ a + τ e ∇ a e R, e (cid:3) σ = − e R cd e ∇ c e ∇ d τ − e R cd e ∇ c τ d − e ∇ c e R e ∇ c τ + e Rσ + e R τ − τ cd e R cm e R dm − τ c e ∇ c e R, where σ . = Ξ − ( n a τ a + n a e ∇ a τ + f τ ). Remark 3.3. espite being quite complicated, this set of equations is rather advantageous notonly because no singularity occurs in the coefficients as Ξ →
0, but also because the system,together with (3.8b), admits a well-posed initial value problem. This feature was first remarkedand exploited in [GX78], but it can be also inferred from the analysis of [Bar13, Section 5]. Moreprecisely, in the language of this paper and with the notation introduced already by Geroch andXanthopoulos, we can rewrite the whole system as a PDE of the form QF = 0 where F is avector whose entries are the following fields F = τ ab , F = τ a , F = σ , F = τ , F = ∇ S τ and F = (cid:3) τ . The explicit form of Q , which can be constructed out of (3.9) and (3.8b), is not ofparticular interest. Noteworthy is instead the following: We are dealing with a linear system ofmutually coupled equations, each of which has a principal symbol of hyperbolic type, except forone equation of first order whose principal symbol is of the type described in [Bar13, Definition5.1]. This entails two important properties. On the one hand, for given smooth and compactlysupported initial data, the associated solution is unique, smooth and spacelike compact. On theother hand, one can associate to such a system retarded e G + and advanced e G − Green operatorswith the same properties enjoyed by e E as in Section 2.2.Notice that for our aims we want to introduce a procedure to map observables from the bulkspacetime M to null infinity ℑ + . Recalling Proposition 2.12, any observable [ ǫ ] ∈ E can beequivalently regarded as a spacelike compact solution up to gauge, [ h ] ∈ S sc / G sc . Our strategyconsists of extending beyond the boundary a representative h ∈ [ h ] which fulfils the GX-gauge,namely such that (3.8a) holds. More precisely, we want to exploit the well-posedness of the Cauchyproblem for (3.9) together with (3.8b) to extend in a unique way the above mentioned h to f M .Nonetheless this last statement has a direct consequence on the GX-gauge, namely, we need tomake sure that, given any smooth and spacelike compact solution of (2.1), there exists a gaugetransformation in G sc such that the gauge-transformed solution fulfils also (3.8a). A key remarkin this direction, mentioned partly in [GX78] and more precisely in [AM82] is that, indeed forany solution h of (2.1) with smooth and compactly supported initial data, one can always find χ ∈ Γ sc ( T ∗ M ) such that h ′ = h + ∇ S χ lies in the GX-gauge. We show why there is a loopholein this statement. More precisely the problem lies in (3.8a). Let us then rewrite it in terms ofgeometric quantities and operations defined with respect to the physical metric g . It holds: y a ( h ′ ) = Ξ − v a ( h ′ ) , v a ( h ′ ) = ∇ b h ′ ab − ∇ a h ′ , where we have made explicit the dependence from h ′ . In other words, if we consider h ∈ Γ sc ( S T ∗ M ) which solves (2.1) but fails to fulfil (3.8a), we need to look for χ ∈ Γ sc ( T ∗ M ) suchthat y a ( h ′ ) = 0, where h ′ = h + ∇ S χ . This is tantamount to solving ∇ b ∇ [ b χ a ] = − v a ( h ) . (3.10)This equation is supplemented by the identity ∇ a v a ( h ) = 0, which follows from (2.1) by takingits trace. It is not guaranteed that (3.10) has a solution. In order to realize it, let us rewrite theequation using the differential d and the codifferential δ , namely δ d χ = 2 v ( h ) , δv ( h ) = 0 . χ . On the otherhand it also imposes a constraint on v , namely there cannot exist a solution if v is not coexact.Actually, a solution χ exists if and only if v is coexact. Unfortunately the second equation doesonly guarantee that v is coclosed. Furthermore it also imperative that χ ∈ Γ sc ( T ∗ M ) is not onlya solution of (3.10), but also that its restriction to any Cauchy surface is compactly supported. We stress once more that such requirement is paramount to guarantee the applicability of theprocedure to uniquely extend Ξ h ′ to τ ∈ Γ sc ( S T ∗ f M ) such that τ | M = Ξ h ′ by solving the system(3.9) supplemented with (3.8b). In the next theorem we show which constraint has to be imposedon the solution of (2.1) for this to happen: Theorem 3.4.
Let ǫ ∈ E inv . Denote with E the causal propagator for P = ( (cid:3) − I . Then v ( Eǫ ♭ ) ∈ δ Ω sc ( M ) if and only if tr ǫ ∈ δ Ω c ( M ) .Proof. Let us start from the sufficient condition. Suppose ǫ ∈ E inv is such that there exists β ∈ Ω c ( M ) for which δβ = tr ǫ . From Lemma 2.2 it follows that tr Eǫ = − E (cid:3) tr ǫ = − δE (1) β .Furthermore, since div ǫ = 0, div Iǫ = − (1 / ∇ S tr ǫ = − (1 / δβ and hence div Eǫ = div IEIǫ = E (cid:3) div Iǫ = − (1 / E (1) d δβ . Therefore we deduce the following: v ( Eǫ ♭ ) = div Eǫ − ∇ S tr Eǫ = − E (1) d δβ + d δE (1) β = − δ d E (1) β, which shows that v ( Eǫ ♭ ) lies in δ Ω sc ( M ).It remains to check that the condition is also necessary. To this avail, assume ǫ ∈ E inv is suchthat there exists α ∈ δ Ω sc ( M ) for which δα = v ( Eǫ ♭ ). Along the same lines of the first part weget div Eǫ = − E (cid:3) ∇ S tr ǫ = −
12 d E (0) (tr ǫ ) , ∇ S tr Eǫ = −∇ S E (cid:3) tr ǫ = − d E (0) (tr ǫ ) . Therefore our hypothesis entails δα = v ( Eǫ ♭ ) = (1 / E (0) (tr ǫ ). From this identity it follows thatd δα = 0. According to [Ben14, Section 7.2], there exist ξ ∈ Ω c d ( M ) and η ∈ Ω sc ( M ) such that Eξ + δη = α , hence we deduce that δξ = (1 / ǫ ) + (cid:3) ζ for a suitable ζ ∈ Ω c ( M ). Applying δ to both sides of the last identity we get (cid:3) (tr ǫ + 2 δζ ) = 0, therefore tr ǫ = − δζ ∈ δ Ω c ( M ), thusconcluding the proof.Theorem 3.4, together with the previous discussion, prompts us to introduce the following defini-tion: Definition 3.5.
We say that [ ǫ ] ∈ E is a radiative classical observable for linearized gravity if there exists a representative ǫ ∈ [ ǫ ] such that tr ǫ = δβ for a suitable β ∈ Ω c ( M ). The collection E rad of these equivalence classes forms a vector subspace of E .Notice that the definition does not depend on the choice of the representative since, for any ǫ ofthe form Kα , α ∈ Γ c ( S T ∗ M ), where K is as in (2.4), it holds that v ( EKα ) ∈ δ Ω sc ( M ). Asusual, E denotes the causal propagator for P = ( (cid:3) − I .At this stage we need to answer an important question: Is there a spacetime where E rad issmaller than E ? We show that, contrary to what implicitly assumed in [GX78] and [AM82] this isindeed possible. Actually we show two explicit cases: Minkowski spacetime where E rad = E andan axisymmetric spacetime where instead E rad ( E . Dropping such requirement would milden the obstruction we have pointed out. As a matter of fact, every co-closed 1-form is coexact without further constraints on the support if H ( M ) is trivial, which amounts to consideringonly globally hyperbolic spacetimes with non-compact Cauchy surfaces. inkowski spacetime Let us thus consider the simplest example of an asymptotically flatspacetime and let us work with the standard global Cartesian coordinates x i , i = 0 , ...,
3, so that M ≡ R endowed with metric η = diag( − , , , ǫ ∈ E inv on Minkowski spacetime. Ourgoal is to prove that tr ǫ = δβ with β ∈ Ω c ( R ). An alternative way to rewrite this condition is torequire that ∗ (tr ǫ ) is an exact compactly supported 4-form. On account of the non-degeneracy ofthe pairing between H c ( R ) and H ( R ), this is equivalent to state that R R ∗ (tr ǫ ) = 0. To showthat this is indeed the case, it is better to work explicitly with indices. Since ǫ ∈ E inv , ∂ a ǫ ab = 0for all b = 0 , . . . , b = 1, it holds ∂ ǫ = − ∂ ǫ − ∂ ǫ − ∂ ǫ . Therefore it follows that ǫ ( x , x , x , x ) = − x Z −∞ (cid:0) ∂ ǫ ( x , y , x , x ) + ∂ ǫ ( x , y , x , x ) + ∂ ǫ ( x , y , x , x ) (cid:1) d y . Adapting the values of the indices, a similar formula can be written for ǫ aa , a = 0 , ,
3. We cannow compute Z R ∗ (tr ǫ ) = Z R η aa ǫ aa d x, as the sum of four integrals, which actually do vanish separately. To show this, let us just focuson the contribution from ǫ : Z R ǫ d x = − Z R d x x Z −∞ (cid:0) ∂ ǫ ( x , y , x , x ) + ∂ ǫ ( x , y , x , x ) + ∂ ǫ ( x , y , x , x ) (cid:1) d y . Each of the three contributions to R R ǫ d x vanishes. For example, the first contribution can berewritten as Z R d x x Z −∞ ∂ ǫ ( x , y , x , x ) d y = Z R d x Z R d x Z R d x x Z −∞ d y Z R ∂ ǫ ( x , y , x , x ) d x = 0 . The integral along the variable of derivation entails evaluation of the components of ǫ ab at ±∞ .On account of the compact support this always vanishes. Therefore R R ǫ d x vanishes as welland the same holds true for ǫ aa , a = 0 , ,
3, by the same argument. Hence R R ∗ (tr ǫ ) = 0 or, inother words, all observables for linearized gravity on Minkowski spacetime are of radiative type.Notice that in our analysis a key role is played by the geometry of the background. Even mildchanges in the metric coefficients would invalidate our line of reasoning and hence no positiveresult could be obtained. Axisymmetric spacetime
Let M be a globally hyperbolic spacetime, which topologically lookslike R × S , and let us consider thereon the standard coordinates ( t, x, y, ϕ ). Let us suppose thatthe line element is of the form d s = g ij d x i d x j + g ϕϕ d ϕ , i = t, x, y , where all coefficients aresmooth and independent of ϕ . Hence ( M, g ) admits a Killing field along S . Let us further noticethat the tangent bundle is trivial and thus it is legitimate to consider the components of any ǫ ∈ Γ c ( S T M ) as global sections. Since we want to consider an element of ǫ ∈ E inv , we set ǫ ϕϕ = 12 π p | g | g ϕϕ f ( t ) f ( x ) f ( y ) , ǫ ab = 0 for( a, b ) = ( ϕ, ϕ ) , where | g | here stands for the absolute value of the determinant of the metric. Notice that non-degeneracy of the metric entails that g ϕϕ is nowhere vanishing. The function f ∈ C ∞ c ( R ) is20hosen such that its integral along R is equal to 1. Notice that per construction ǫ is compactlysupported and gauge invariant, namely div ǫ = 0. In fact the only component which might havea non vanishing contribution is (div ǫ ) ϕ , for which we have (div ǫ ) ϕ = ∇ ϕ ǫ ϕϕ = 0 since ǫ ϕϕ isindependent of ϕ and because of the form of the line element d s . Let us now consider tr ǫ = g ϕϕ ǫ ϕϕ = (2 π p | g | ) − f ( t ) f ( x ) f ( y ). In order to show that tr ǫ is not coexact, we can use the sameargument as in the previous example, namely we compute Z M ∗ (tr ǫ ) = Z R Z R Z R Z S π f ( x ) f ( y ) f ( t ) d ϕ d y d x d t = 1 . In other words we have constructed explicitly an equivalence class [ ǫ ] ∈ E which is not of ra-diative type. We stress that, as far as (3.10) is concerned, asymptotic flatness or simplicity ofthe background is not a necessary prerequisite and one could look for solutions of such equationindependently. Yet, for the sake of completeness, we mention that axisymmetric asymptoticallysimple spacetimes are known to exist and they have been extensively studied in the literature, see[BS84] and references therein.To conclude the section, we stress that (3.8b), that is the residual gauge fixing, is nothingbut a rather involved partial differential equation which can be solved with the argument givenin the appendix of [GX78] and it does not yield any problem in terms of implementation andsupport properties. Hence we can slightly adapt the result of [GX78] in order to account for theobstructions written above: Theorem 3.6.
Let ( M, g ) be a an asymptotically flat spacetime whose associated unphysical space-time is ( f M , e g ) with associated conformal factor Ξ . Let [ Eǫ ] ∈ S sc / G sc be a gauge equivalence classof spacelike compact solutions of the linearized Einstein’s equations, where ǫ is any representativeof [ ǫ ] ∈ E rad and E is the causal propagator for P = ( (cid:3) − I . Then there exists h ′ ∈ [ Eǫ ♭ ] which is asymptotically regular , that is τ = Ξ h ′ admits an extension to f M whose restrictionto ℑ + is smooth. Furthermore both τ ab n a and τ ab n a n b do admit a vanishing limit to null infinity. We remark that our concept of radiative observables is related to that of radiative degrees offreedom as used for example in [Ash81] although, in this paper, the focus is on the structure ofthe kinematical arena on null infinity, whereas we are interested more on those observables whichcan be mapped to the boundary compatibly with the dynamics and with gauge invariance.
We have all ingredients to define a well-behaved projection of the classical radiative observablesfrom the bulk to the boundary. Our approach extends the one already discussed in [AM82]although we take into account the obstruction outlined in Theorem 3.4:
Theorem 3.7.
Let (cid:0) E rad , τ (cid:1) be the space of radiative classical observables as in (2.8) endowed withthe presymplectic form defined in Proposition 2.10. Then there exists a map Υ : E rad → S ( ℑ + ) defined by Υ([ ǫ ]) ab = γ ab − γ cd q cd q ab , (3.11) where ι : ℑ + → f M is the embedding of ℑ + into f M , γ = ι ∗ τ , τ is the extension to f M of Ξ h ′ obtained solving (3.9) together with (3.8b) and h ′ ∈ [ Eǫ ♭ ] is a solution of (2.1) in the GX-gaugebuilt out of [ ǫ ] . Furthermore, for all [ ǫ ] , [ ǫ ′ ] ∈ E rad , it holds that σ ℑ (Υ[ ǫ ] , Υ[ ǫ ′ ]) = τ ([ ǫ ] , [ ǫ ′ ]) .Proof. The proof is a recollection of already known results. To start with, on account of theconstruction of Geroch and Xanthopoulos we know that for all [ ǫ ] ∈ E rad , Υ([ ǫ ]) ∈ Γ( S T ∗ ℑ + )and Υ([ ǫ ]) ab n a = 0 and Υ([ ǫ ]) ab q ab = 0. Furthermore, since every representative of [ ǫ ] is compactlysupported, the support properties of the causal propagator E as well as the extensibility of the21olution to f M entail that there exists u ∈ R such that, in the Bondi frame, Υ([ ǫ ]) = 0 on( −∞ , u ) × S . Such property is useful to prove that both ( τ, τ ) ℑ and ( ∂ u τ, ∂ u τ ) ℑ are finite.As a matter of fact we can now following slavishly the same proof used in [DS13, Theorem 4.4]for the vector potential from which the sought statement descends. Hence Υ maps E in S ( ℑ + ).Furthermore, on account of [AM82, Theorem 2], τ ([ ǫ ] , [ ǫ ′ ]) = σ ℑ (Υ[ ǫ ] , Υ[ ǫ ′ ]).Notice that Υ([ ǫ ]) has only two independent components compatibly with our expectations onthe degrees of freedom for linearized gravity. Goal of this section is to extend the classical bulk-to-boundary correspondence defined in Theorem3.7 to the quantum level and to exploit the outcome to construct explicitly Hadamard states forlinearized gravity. The first part of this programme is rather straightforward with all the buildingblocks we have. As a starting point we construct the algebra of observables both for the bulktheory and for the one living intrinsically on null infinity.
Definition 3.8.
Let T ( E ) be the tensor algebra built out of (2.8) as T ( E ) = L n E ⊗ n C , wherethe zeroth-tensor power is nothing but C and the subscript C denotes complexification. We call algebra of observables for linearized gravity the quotient F ( E ) between T ( E ) and I , the idealgenerated by elements of the form − iτ ([ ǫ ] , [ ǫ ′ ]) ⊕ [ ǫ ] ⊗ ([ ǫ ′ ] − [ ǫ ′ ] ⊗ [ ǫ ]), where τ is the presymplecticform (2.9). This is a ∗ -algebra if endowed with the ∗ -operation induced by complex conjugation.At the same time we call algebra of radiative observables for linearized gravity the algebra F ( E rad )built replacing E with E rad .Notice that, since τ is possibly degenerate, F ( E ) is not guaranteed to be a simple algebra. Inother words it may possess a non trivial center. This feature gives rise to potential problems ininterpreting the theory in the framework of the principle of general local covariance, as it has beenalready thoroughly discussed for Abelian gauge theories [BDHS13, BDS12, SDH12]. Yet, for thesake of constructing states, central elements do not play a distinguished role. We can define acounterpart of F ( E rad ) on the boundary out of the symplectic space S ( ℑ + ) defined in (3.3). Definition 3.9.
Let T ( ℑ + ) be the tensor algebra built out of (3.3) as T ( ℑ + ) = L n S ( ℑ + ) ⊗ n C , where the zeroth-tensor power is nothing but C and the subscript C denotes complexification. Wecall algebra of observables on null infinity the quotient F ( ℑ + ) between T ( ℑ + ) and I ℑ , the idealgenerated by elements of the form − iσ ℑ ( λ, λ ′ ) ⊕ ( λ ⊗ λ ′ − λ ′ ⊗ λ ), where σ ℑ is the symplectic form(3.5). This is a ∗ -algebra if endowed with the ∗ -operation induced by complex conjugation.We can relate the two algebra of observables we built as follows: Proposition 3.10.
There exists a ∗ -homomorphism F ( E rad ) → F ( ℑ + ) specified on the generatorsby [ ǫ ] Υ[ ǫ ] , where Υ is defined in (3.11) . With a slight abuse of notation, we use Υ to denotealso the ∗ -homomorphism defined here.Proof. In order to prove that ι is an homomorphism it suffices to show that it preserves thepresymplectic form when evaluated on all generators. Both these conditions have been alreadyproven in Theorem 3.7. To conclude we notice that all operations involved do not affect thecomplex conjugations and thus we have constructed a ∗ -homomorphism.Having set up the bulk-to-boundary correspondence for the algebra of observables, we are readyto discuss the construction of Hadamard states. Let us recall that, for any unital ∗ -algebra A , analgebraic state is a map ω : A → C such that ω ( a ∗ a ) ≥ a ∈ A and ω ( e ) = 1, where e ∈ A is the identity element. The role of a state is to allow us to recover the standard probabilisticinterpretation of a quantum system via the GNS theorem which associates to each pair ( A , ω ) atriple ( D ω , π ω , Ω ω ) consisting of a dense subspace D ω of a Hilbert space H ω , a representation π ω of22 in terms of linear operators on D ω and a cyclic unit-norm vector Ω ω such that D ω = π ω ( A )Ω ω .This triple is unique up to unitary equivalence and, for any a ∈ A , ω ( a ) = (Ω ω , π ω ( a )Ω ω ) H ω ,where ( · , · ) H ω is the inner product in H ω . If the role of A is played by the algebra of fields, such asin our case either F ( E ) or F ( E rad ), we can focus our attention on a special subclass which is oftenused in theoretical and mathematical physics. We are referring to the quasi-free/Gaussian stateswhich are completely defined in terms of their n-point correlation functions ω n . In particular ω n = 0 if n is odd whereas, if n is even, than ω n ( λ ⊗ ... ⊗ λ n ) = X π n ∈ S n n/ Y i =1 ω ( λ π n (2 i − ⊗ λ π n (2 i ) ) , (3.12)where λ i ∈ E , i = 1 , , ..., n , whereas S n denotes the ordered permutations of n elements.The explicit identification of a state for a quantum field theory is usually a rather dauntingquest unless the symmetries of the background are sufficient to help us in singling out a pre-ferred candidate, e.g. the vacuum in Minkowski spacetime, whose existence and uniqueness isa by-product of Poincar´e invariance. Yet, since a generic curved background might even have atrivial isometry group, one has to look for a different procedure to construct explicitly a quantumstate. On the class of globally hyperbolic and asymptotically flat spacetimes, we will show thatProposition 3.10 provides a tool to induce states for the bulk theory starting from the boundarycounterpart. The advantage of working with a theory defined on null-infinity is two-fold: On theone hand ℑ + is nothing but R × S and thus along the null R -direction, one can perform a Fouriertransform, thus working in terms of modes. On the other hand, as discussed in Section 3.1, atheory on ℑ + is invariant under a suitable action of the BMS group. The latter plays the samerole of the Poincar´e group in Minkowski spacetime in helping us to single out a distinguished stateat null infinity. More precisely, along the same lines of [DMP05, DPH11, DS13], the followingproposition holds true: Proposition 3.11.
The map ω ℑ : S ( ℑ + ) C ⊗ S ( ℑ + ) C → R such that ω ℑ ( λ ⊗ λ ′ ) = − π lim ǫ → Z R × S λ ab ( u, θ, ϕ ) λ ′ cd ( u ′ , θ, ϕ ) q ac q bd ( u − u ′ − iǫ ) d u d u ′ d S ( θ, ϕ ) , (3.13) where d S ( θ, ϕ ) is the standard line element on the unit -sphere, unambiguously defines a quasi-free state ω ℑ : F ( ℑ + ) → C . Furthermore:1. ω ℑ induces via pull-back a quasi-free bulk state ω M : F ( E rad ) → C such that ω M . = ω ℑ ◦ Υ ,2. ω ℑ is invariant under the action Π of the BMS group induced on F ( ℑ + ) by (3.2) .Proof. As a starting point, we notice that from ω ℑ we can define unambiguously a Gaussian state ω ℑ on F ( ℑ + ) via (3.12). Yet one needs to ensure positivity of ω ℑ which is equivalent to showingthat this property holds true for ω ℑ . To this avail, we notice that every element λ ∈ S ( ℑ + )tends to 0 as u tends to ±∞ as one can prove readapting the argument of [Mor05, Footnote 7,p. 52] and exploiting that both ( λ, λ ) ℑ < ∞ and ( ∂ u λ, ∂ u λ ) ℑ < ∞ . Hence we can perform aFourier-Plancherel transform along the u-direction – see [Mor06, Appendix C] – and eventuallyuse the convolution theorem to obtain ω ℑ ( λ ⊗ λ ′ ) = 1 π lim ǫ → Z R × S k Θ( k ) b λ ab ( k, θ, ϕ ) b λ ′ cd ( − k, θ, ϕ ) q ac q bd d k d S ( θ, ϕ ) , Here we use the symbol Π with a slight abuse of notation since we have already introduced it to indicate inProposition 3.1 the representation of the BMS group on S ( ℑ + ). Since F ( ℑ + ) is built out of S ( ℑ + ) we feel that noconfusion can arise. k ) is the Heaviside step function. From this last formula it follows that ω ℑ ( λ ⊗ ¯ λ ) ≥ ω ℑ ( λ ⊗ λ ′ ) − ω ℑ ( λ ′ ⊗ λ ) = iσ ℑ ( λ, λ ′ ) where σ ℑ is the symplectic form definedin (3.5). Hence ω ℑ is indeed a state on F ( ℑ + ). Proposition 3.10 ensures that ω M is in turn awell-defined and, per construction, quasi-free state on F ( E rad ). Only BMS invariance remains tobe proven. Since ω ℑ is quasi-free, it suffices to show the statement for the two-point function. Letus first collect all ingredients we need. On account of Proposition 3.1, we know already that (3.2)induces the action Π on each λ ∈ S ( ℑ + ) – see (3.6). Furthermore, for each γ = (Λ , α ) ∈ BMS,suppressing the dependence on the coordinates on the 2-sphere for the BMS group elements andactions, q ab K − q ab , whereas d u K Λ d u and d S ( θ, ϕ ) K d S ( θ, ϕ ). Putting everythingtogether, it holds that, for every λ, λ ′ ∈ S ( ℑ + )( ω ℑ ◦ Π)( λ ⊗ λ ′ ) = − π lim ǫ → Z R × S K Λ λ ab ( u, θ, ϕ ) K Λ λ ′ cd ( u ′ , θ, ϕ ) K − q ac K − q bd ( K Λ u − K Λ u ′ − iǫ ) K d u d u ′ d S ( θ, ϕ ) , which coincides with ω ℑ since the factor K Λ is bounded and thus we are free to redefine the ǫ -termaccordingly.In the previous proposition we have displayed a state for the bulk algebra of fields whichis constructed out of the boundary counterpart. A genuine question at this stage would be whyshould one consider such candidate. To answer, notice that, in between the plethora of all possiblestates for the algebra of fields of a quantum theory, not all of them can be considered as physicallysensible. While in Minkowski spacetime, it is always possible to resort for free fields to the Poincar´evacuum, which is, moreover, unique, this luxury is not at our disposal on curved backgrounds.The main reason is due to the absence in general of a sufficiently large isometry group. In thiscase it is paramount to find a criterion to select in between all possible states those which areacceptable. After long debates, there is now a wide consensus that such statement translatesin the request that the ultraviolet behaviour mimics that of the Poincar´e vacuum on Minkowskispacetime and that the quantum fluctuations of all observables, such as, for example, the smearedcomponents of the stress energy tensor, are bounded. From a mathematical point of view, thistranslates in choosing Hadamard states ω , namely states which satisfy a condition on the singularstructure of the bi-distribution Ω associated to their two-point function ω . Before spelling itout explicitly, we remark that, as discussed in [HW01] for scalar field theories, Hadamard statescan be used to define a locally covariant notion of Wick products of fields and, thus, interactionscan be discussed at least at a perturbative level. Since the main feature which is exploited in theanalysis of Hollands and Wald is essentially that in every normal neighborhood of the underlyingmanifold the singular structure of Ω depends only on the geometry of the spacetime, we expectthat their results can be extended also to the case we are considering. Yet we shall not dwell intothis topic since it would lead us far from our goal.On the contrary we review now the tools necessary to define rigorously Hadamard states,adapting them to the case at hand. All basic notions and definitions related to microlocal analysisare taken as in [Hor90], to which we refer. Let us thus consider a bi-distribution Ω : Γ c ( S T M ) × Γ c ( S T M ) → R which is furthermore a bi-solution of e P = (cid:3) − isgeneric and not necessarily stemming from a two-point function of a quasi-free state, we employthe same symbol for the sake of notational simplicity. The latter could be in principle any normallyhyperbolic operator and, not necessarily e P = (cid:3) − E , the wavefront set W F ( u ) of a distribution u ∈ D ′ ( E ) is defined locally as the union of W F ( u i ), the wavefront setof each component of u in a local trivialization of E . Hence we can extend also to this scenariothe definition given in [Rad96a, Rad96b, SV00]: Definition 3.12.
A two-point distribution Ω : Γ c ( S T M ) × Γ c ( S T M ) → R , bi-solution of e P = (cid:3) − Hadamard form if it satisfies the following conditions:24.
W F (Ω ) = { ( x, k, x ′ , k ′ ) ∈ T ∗ ( M × M ) \ { } | ( x, k ) ∼ ( x ′ , − k ′ ) , k ⊲ } , where 0 is the zerosection of T ∗ ( M × M ), whereas ( x, k ) ∼ ( x ′ , − k ′ ) means that the point x is connected to x ′ by a lightlike geodesic γ so that k is cotangent to γ in x and − k ′ is the parallel transport of k from x to x ′ via γ . Furthermore k ⊲ k is future-directed;2. Ω ( ǫ, ζ ) − Ω ( ζ, ǫ ) = 2 i ( e Eǫ ♭ , ζ ) for each ǫ, ζ ∈ E kin , where the equality holds true up tosmooth terms which vanish when smeared on ǫ, ζ ∈ E inv .Contrary to what happens for electromagnetism in [DS13, Sie11], for linearized gravity we needto introduce an additional concept developed in [Hun12, Chapter 6] to cope with the operation oftrace-reversal which is present in (2.6a). Definition 3.13.
Let Ω : Γ c ( S T M ) × Γ c ( S T M ) → R be any bi-distribution. We call trace ofΩ the scalar bidistribution (tr Ω ) : C ∞ c ( M ) × C ∞ c ( M ) → R such that, for all f, f ′ ∈ C ∞ c ( M ),(tr Ω )( f, f ′ ) . = Ω ( f g − , f ′ g − ) , where g − is the inverse of the metric tensor. We call trace reversal of Ω the bidistribution I Ω : Γ c ( S T M ) × Γ c ( S T M ) → R , such that, for all ǫ, ζ ∈ Γ c ( S T M )( I Ω )( ǫ, ζ ) . = Ω ( ǫ, ζ ) −
18 (tr Ω )(tr ǫ, tr ζ ) , where tr ǫ = g µν ǫ µν .Notice that the apparently strange coefficient 1 / I Ω ) = − tr Ω andthat I ( I Ω ) = Ω . The trace reversal of a bidistribution plays a key role in understandingwhat is a Hadamard state for linearized gravity. As a matter of fact, the presymplectic form withwhich E ( E rad ) is endowed and, accordingly, the canonical commutation relations with which F ( E )( F ( E rad )) is constructed are built out of E the causal propagator of P = e P I , where e P = (cid:3) − I is the trace reversal on Γ( S T M ). As noted in [Hun12], every bisolution Ω of e P is alsoone for P and the same holds true for I Ω . Furthermore acting with I on Ω changes the secondrequirement of Definition 3.12 into a similar one where e E , the causal propagator for e P , is replacedby E , the causal propagator for P . In fact( I Ω )( ǫ, ζ ) − ( I Ω )( ζ, ǫ ) = Ω ( ǫ, ζ ) − Ω ( ζ, ǫ ) −
18 (tr Ω )( ǫ, ζ ) + 18 (tr Ω )( ζ, ǫ )= 2 i ( e Eǫ ♭ , ζ ) − i ( e E ( g − tr ǫ ) ♭ , g − tr ζ ) = 2 i ( e Eǫ ♭ , ζ ) − i ( g tr e Eǫ ♭ , ζ ) = 2 i ( Eǫ ♭ , ζ ) . We can now define the notion of a Hadamard state for linearized gravity, although we havestill to take care of the additional constraint that E is generated by compactly supported sectionswith vanishing divergence. Taking into account also Definition 3.12, we recall the definition ofHadamard states for linearized gravity given in [Hun12]. Definition 3.14.
A quasi-free state ω : F ( E ) → C is said to be a Hadamard state if there existsa bi-distribution Ω : Γ c ( S T M ) × Γ c ( S T M ) → C , which is a bi-solution of e P of Hadamard form(see Definition 3.12) such that, for every ǫ, ζ ∈ E inv , ω ([ ǫ ] ⊗ [ ζ ]) = ( I Ω )( ǫ, ζ ) , where I is as in Definition 3.13. A similar definition holds with E replaced by E rad simply restrictingto those gauge invariant functional ǫ ∈ E inv which fulfil the requirement of Definition 3.5, namelysuch that tr ǫ = δβ for a suitable β ∈ Ω c ( M ).As a last step, we have to show that the state for F ( E ) constructed via the bulk-to-boundaryprocedure fits indeed in the class characterized in the previous definition. In view of the previ-ous definitions we cannot work directly with ω M as in Proposition 3.11 and prove that it is of25adamard form. To this end we would need to show that there exists a bi-distribution on M ,hence defined on all pairs of elements in Γ c ( S T M ), which coincides with the two-point func-tion associated to ω M on E rad × E rad . The very same problem has been encountered alreadyin [DS13, Sie11] for the vector potential and, as in that case, we shall tackle it as follows: Weconstruct an auxiliary two-point function on f M , showing that this enjoys the correct wavefrontset condition. Although strictly speaking one should not be entitled to call ω M Hadamard, weare still convinced that it deserves this name since the ensuing wavefront set on f M is built onlyout of null geodesics which are invariant under conformal transformation. Hence we expect thatthe singular behaviour of ω M is genuinely the same as that of a full-fledged Hadamard state. Theorem 3.15.
The state ω M = ω ℑ ◦ Υ : F ( E rad ) → C , defined by the pull-back along Υ : F ( E rad ) → F ( ℑ + ) (see Proposition 3.10) of the state ω ℑ introduced in Proposition 3.11, enjoysthe following properties:1. Its two-point function is the restriction to E rad × E rad of a bi-distribution on f M whosewavefront set on ψ ( M ) is of Hadamard from, where ψ ( M ) is the image of M in f M ;2. It is invariant under the action of all isometries of the bulk metric g , that is ω M ◦ α φ = ω M .Here φ : M → M is any isometry and α φ represents the action of φ induced on F ( E rad ) bysetting α φ ([ ǫ ]) = [ φ ∗ ǫ ] on the algebra generators [ ǫ ] ∈ E rad ;3. It coincides with the Poincar´e vacuum on Minkowski spacetime.Proof. To prove 1 . , we follow the same strategy as in [DS13]. Denoting with ι the embedding of ℑ + into f M , we start introducing an auxiliary two-point function: ω f M ( f ⊗ f ′ ) . = ω ℑ ( ι ∗ e G − ( f ) ⊗ ι ∗ e G − ( f ′ )) , where e G ± are the retarded/advanced fundamental solutions of (3.9) together with (3.8b) andwhere f and f ′ are here any pair of smooth and compactly supported test sections generatingsolutions for (3.9) and (3.8b), seen as a Green hyperbolic PDE as per Remark 3.3. Notice that, perconstruction ω f M coincides with the two-point function of ω M when we consider initial data for (3.9)together with (3.8b) descending from E rad . By applying on both sides the operator Q and using e G − Q = id, for the arbitrariness of f and f ′ we obtain ω f M ( Qf ⊗ Qf ′ ) = ω ℑ ( ι ∗ f ⊗ ι ∗ f ′ ). Hence thepull-back of ω ℑ along ι : ℑ + → f M has the same wavefrontset as that of ( Q ∗ ⊗ Q ∗ ) ω f M . If we provethat the latter is the same as that of Hadamard states, we have reached the sought conclusion.This statements descends from the invariance of null geodesics under conformal transformationand from the fact that no null geodesic joining any x ∈ M to i + exists [Mor06, Lemma 4.3]. Letus start from the wavefront set of ω ℑ which have been already computed in [DMP09]: W F ( ω ℑ ) = { ( x, x, k, − k ) ∈ T ∗ ( ℑ + × ℑ + ) \ { } | k u > } , where k u is the component of the covector k along the null direction. If we apply the theorem ofpropagation of singularities to ( Q ∗ ⊗ Q ∗ ) ω f M we obtain: W F ( ω f M ) = { ( x, y, k x , − k y ) ∈ T ∗ ( f M × f M ) \ { } | ∃ p ∈ ℑ + , q ∈ T ∗ ι ( p ) f M | q u > x, x ′ ∈ J − f M ( i + ) \ { i + } , ( x, k ) ∼ ( x ′ , k ′ ) ∼ ( ι ( p ) , q ) } , where ∼ means that the points are connected by a lightlike geodesic, while the covectors areparallely transported along it. If we add to this result the fact that there does not exist any nullgeodesic joining a point x in ψ ( M ) ⊂ f M to i + , c.f. [Mor06, Lemma 4.2], then it holds that ω f M has a wavefront set of Hadamard form is ψ ( M ).We prove now 2 . As a consequence of [Mor06, Theorem 3.1] it suffices to prove the state-ment for any one-parameter group of isometries φ Xt with t ∈ R and X a Killing field. Per26efinition ω M ◦ α φ Xt = ω ℑ ◦ (Υ ◦ α φ Xt ). Let [ ǫ ] ∈ E rad be any generator of F ( E rad ), thenΥ ◦ α φ Xt ([ ǫ ]) = Υ([ φ Xt ∗ ǫ ]). Mimicking the same analysis as that of [Mor06, Proposition 3.4], onegets that Υ([ φ Xt ∗ ǫ ]) = Π e Φ e Xt Υ([ ǫ ]), where Π is the representation of the BMS group (3.6), while e Φ e Xt is the action on F ( ℑ + ) of a one-parameter group of BMS elements constructed via exponentialmap from e X , the unique extension of X to ℑ + [Ger77]. Yet Proposition 3.11 entails invariance of ω ℑ under the action of Π, from which it descends that ω M ◦ α φ Xt = ω ℑ ◦ Π e Φ e Xt Υ = ω ℑ ◦ Υ = ω M .Notice that point 3 . is a direct consequence of point 2 . since, on Minkowski spacetime, ω M is aquasi-free and Poincar´e invariant Hadamard state. Uniqueness of the vacuum yields the soughtresult. In this paper we discussed the quantization of linearized gravity on asymptotically flat, globallyhyperbolic, vacuum spacetimes within the framework of algebraic quantum field theory. Thegoal was to construct a distinguished Hadamard state which is invariant under the action of allspacetime isometries. To this end we exploited the existence of a conformal boundary which in-cludes ℑ + , future null infinity, a codimension 1 submanifold on which we defined an auxiliarynon-dynamical field theory and an associated ∗ -algebra F ( ℑ + ). The procedure we followed con-sists first of all of finding a map which associates to each element of the algebra of fields forlinearized gravity a counterpart in F ( ℑ + ). This step can be translated into proving that, up to agauge transformation, each solution of the linearized Einstein’s equations admits after a conformalrescaling a smooth extension to ℑ + . This operation was thought to be always possible thanks to asuitable gauge fixing, first written by Geroch and Xanthopoulos. We have proven that there existsan obstruction which depends both on the geometry and on the topology of the manifold. Hence,while on certain backgrounds such as for example Minkowski spacetime, all observables admit acounterpart at null infinity, in other scenarios, such as for example axisymmetric backgrounds, thisis not the case. We have therefore introduced the notion of radiative observables to indicate thosewhich admit an associated element in F ( ℑ + ). These form a not necessarily proper sub-algebra ofthe algebra of all observables and we have constructed for it a Hadamard state which is invariantunder the action of all spacetime isometries.We reckon that different follow-up to our analysis are conceivable. From the side of generalrelativity, the realization of the existence of an obstruction to implement the Geroch-Xanthopoulosgauge, suggests that a critical review of the claimed results about the stability of asymptoticflatness is due. A possible way out would be to find an alternative to the GX-gauge which stillallows to extend all spacelike compact solutions of linearized gravity to null infinity. We triedhard to find such alternative but to no avail. If, on the contrary, the obstruction is always present,then it would be interesting to understand whether radiative observables play a distinguished rolefrom a physical point of view.From the side of algebraic quantum field theory, our investigation, combined with that ofFewster and Hunt [FH12, Hun12] suggests strongly that linearized gravity might behave similarlyto the vector potential in electrodynamics with respect to its interplay with both general localcovariance and dynamical locality. It might also be interesting to explore other avenues to con-struct states of interest in physics, particularly following the approach advocated in recent works[GW14a, GW14b]. More generally, it might be worth trying to extend the notion itself of what isa Hadamard state in the following sense: For gauge theories, states and their two-point functionsin particular are defined on suitable gauge equivalence classes of observables, E or E rad in the caseat hand. Yet, in order to claim that a given two-point function is of Hadamard form, one has toshow that it can be seen as the restriction of a bi-distribution defined on a whole space of smoothand compactly supported sections so to be able to apply the tools proper of microlocal analysisto check the relevant wavefront sets. 27 cknowledgements We would like to thank Klaus Fredenhagen, Thomas-Paul Hack, Igor Khavkine, Katarzyna RejznerAlexander Schenkel and Daniel Siemssen for useful discussions and comments. S.M. is grateful toFrancesco Bonsante and Ludovico Pernazza for useful comments during the early stages of thiswork. We are greatly indebted to Chris Fewster for useful comments and for pointing us outreference [Hun12] and to Stefan Hollands for pointing us out references [GX78] and [HI03] as wellas for enlightening discussions on these papers. The work of C.D. has been supported partly by theUniversity of Pavia and partly by the Indam-GNFM project “Influenza della materia quantisticasulle fluttuazioni gravitazionali”. The work of M.B. is supported by a Ph.D. fellowship of theUniversity of Pavia.
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