Gravitational radiation condition at infinity with a positive cosmological constant
aa r X i v : . [ g r- q c ] J u l Gravitational radiation condition at infinitywith a positive cosmological constant
Francisco Fernández-Álvarez i and José M. M. Senovilla ii Departamento de Física Teórica e Historia de la Ciencia,Universidad del País Vasco UPV/EHUApartado 644, 48080 Bilbao, Spain ( July 31, 2020)
Gravitational waves have been directly detected and astronomical observations indicatethat our Universe has a positive cosmological constant Λ. Nevertheless, a theoretical gauge-invariant notion of gravitational waves arriving at infinity (escaping from the space-time)in the presence of a positive Λ has been elusive. We find the answer to this long-standinggravitational puzzle, and present a geometric, gauge-invariant radiation condition at infinity.
INTRODUCTION
Physics entered the new era of gravitational-wave astronomy in 2016, following the announce-ment of the first gravitational-wave direct detection ever [1]. This was the successful culminationof a scientific adventure launched theoretically by Einstein a century before, having the cele-brated quadrupole formula [2] in the weak-field limit as a solid first step, then followed by un-certainties whether gravitational waves constitute a feature of the full non-linear theory, finallysettled with various theoretical developments in the 1950-60’s [3–8], see [9], and the discoveryand analysis of the binary pulsar PSR B1913+16 [10]. To analyze isolated systems one focuseson regions of the space-time far away from the radiating sources, which formally correspondto ‘infinity’. The conformal geometric representation of these asymptotic regions by Penrose[11] was particularly pioneering: carried out in full non-linear General Relativity in a covari-ant manner [12], it contributed to dispel remaining doubts about the theoretical description ofgravitational waves.While the conformal completion [11] can be built for any value of the cosmological constantΛ, its relationship with Bondi’s fundamental quantities (news function and energy-momentum)[6] has only been established in the asymptotically flat case with Λ = 0 [12]. Observational data[13, 14] reveal, however, that we inhabit an accelerated-expanding universe. This empirical factevince the presence of a positive (bare or effective) cosmological constant. Thus, annoyingly wedo not have a rigorous theoretical description of radiation escaping to infinity in the presence ofa positive Λ, no matter how tiny Λ may be . Signs of attention to this situation date back to [15],and were amplified in [16] where the predicament was clearly presented. Some advances havebeen made [17–24] (see [25, 26] for reviews), usually trying to adapt techniques from the Λ = 0case to the new scenario. Frustratingly, we still lack a fully satisfactory solution. One of thechallenging difficulties is to understand and describe unambiguously the directional dependencethat emerges when one approaches infinity in different lightlike directions [27]. Not to mentionthe absence of an asymptotic universal structure of infinity —which does exist for Λ = 0, allowingto isolate the two degrees of freedom associated to gravitational radiation [28].In summary, the next question remains open: How to tell when a space-time with positivecosmological constant contains gravitational radiation arriving at infinity?
In this letter, weanswer this fundamental question which underlies any other hypothetical deeper characterisa-tion, such as a formula for the energy carried away by the waves from an isolated source or thedefinition of a mass-energy. We do it by taking a fully new perspective of the problem, differentfrom the methods used so far in the literature. Namely, we ground our investigation in studyingtidal effects, motivated by the nature of the gravitational field and of actual gravitational-wavemeasurements. Our approach is supported by its successful application to the well-establishedasymptotically flat case, in which we recently put it at test [29]: we demonstrated that our tidalapproach is fully equivalent, in a precise sense, to the traditional scheme with Λ = 0. By thesetidal means, hitherto we arrive at a satisfactory radiation condition at infinity in the presenceof a positive cosmological constant. As far as we know, this is the first such criterion.
CONFORMAL SPACE-TIME AND SUPERENERGY
The suitable structure for the study of asymptotic gravitational radiation is the boundary J of a conformal completion (cid:16) M, g αβ (cid:17) of any physical space-time (cid:16) ˆ M , ˆ g αβ (cid:17) [11], where g αβ =Ω ˆ g αβ on ˆ M , the conformal factor Ω is strictly positive on ˆ M and ‘infinity’ lies at J := { Ω = 0 } ;for further details see e.g. [12, 30]. For Λ > J is a three-dimensional manifold endowedwith a Riemannian metric h ab inherited from (cid:16) M, g αβ (cid:17) . Generically J is not connected, itscomponents can be divided into ‘future’ and ‘past’ denoted by J ± respectively; our discussionis valid for any of them, but we will sometimes concentrate on J + for the sake of concreteness(escaping radiation). The normal N α := ∇ α Ω to the Ω = constant hypersurfaces is futuretimelike on a neighbourhood of J , for − N := N α N α J = − Λ /
3. In such a neighbourhood onecan normalise it n α := N − N α . There is a gauge freedom consisting on conformal rescaling of g αβ , Ω → Ω ω with ω >
0. It is common to restrict ω such that ∇ α N β J = 0, a ‘divergence-free’gauge that we adopt. Still, a huge gauge freedom persists, as any ω satisfying £ ~N ω J = 0 is stillallowed. Our main results are fully gauge independent.In order to motivate our novel approach, let us throw a quick sight into the electromagnetic(EM) field, whose energy-momentum tensor we denote by T EM µν . For any unit, future-pointing,vector v α , the four-momentum vector relative to v α is defined as P EM α := − T EM αµ v µ . This isalways causal and future, its v α -component gives the EM energy density measured by v α , whilethe remaining, three-dimensional, spatial vector is the Poynting vector. The latter points in thespatial direction along which EM energy propagates according to the observer described by v α . Null EM fields are characterized by having a non-zero Poynting vector for all possible v α .In General Relativity, things are more complicated. A great obstacle is the absence of a localnotion of energy-momentum associated to the gravitational field, although there are quasilocaldefinitions [31]. Instead of working at the energy-density level one can work with tidal energies.At this level the Bel-Robinson tensor [32] is defined T αβγδ := C ναµγ C µδνβ + C ∗ ναµγ C ∗ µδνβ (1)where C ναµγ is the Weyl tensor and C ∗ ναµγ its Hodge dual. T αβγδ is a totally symmetric,traceless, conformally invariant tensor whose physical units are M L − T − [31, 33, 34] and carriesinformation related to the tidal nature of the gravitational field. The objects defined with thistensor are usually referred to as superenergy quantities. Given an observer v α as before, onedefines the associated supermomentum P α := − v β v γ v δ T αβγδ = W v α + S α (2) The metric signature is ( − , + , + , +), and the curvature tensor is defined by ( ∇ α ∇ β − ∇ β ∇ α ) v γ = R µαβγ v µ ,where ∇ α is the covariant derivative on (cid:0) M, g αβ (cid:1) . Latin indices a = 1 , , J . where W and S α ( S α v α = 0) are the superenergy density and super-Poynting vector relative to v α W := v α v β v γ v δ T αβγδ ≥ , (3) S α := − ( δ αµ + v α v µ ) v β v γ v δ T µβγδ . (4)The Bel-Robinson tensor has many relevant properties [5, 34, 35] analogous to the ones fulfilledby T EM µν . Among them, a dominant property [34] ensuring that P α is causal and futurepointing. In analogy with the null EM fields, Bel [5] proposed to define a state of intrinsicgravitational radiation at a point q when S α | q = 0 for all possible v α . This is a local, observer-independent statement. THE RADIATION CRITERION AT J We aim at an observer-independent and gauge-invariant description of gravitational radiationat the conformal boundary J . For such purpose, and recalling that we are interested in studyingtidal effects, first we have to find a good definition of asymptotic supermomentum . Somethingas (2) does not work, as we know that the Weyl tensor vanishes at J [12, 30]. Nevertheless, therescaled Weyl tensor d δαβγ := Ω − C δαβγ is regular at infinity. Consequently, to carry out anasymptotic study in conformal space-time, it is natural to define a rescaled Bel-Robinson tensor , D αβγδ := Ω − T αβγδ = d ναµγ d µδνβ + d ∗ ναµγ d ∗ µδνβ (5)which shares most of the properties of T αβγδ and is regular and –in general– different from zeroat J . Notice that the normal N α | J defines a privileged ‘asymptotic observer’ that is selectedby the geometry itself. Hence, it is natural to define the asymptotic supermomentum p α := − N µ N ν N ρ D αµνρ (6)and its canonical version P α := − n µ n ν n ρ D αµνρ . (7)Obviously, these two vector fields are proportional in a neighbourhood of J , P α = N − p α , andall the properties listed below hold for both of them. However, (7) is unsuitable for comparisonwith the Λ = 0 case, for which we will use (6) later on. Some important properties of thesevector fields are:1. P α is causal and future pointing at and around J .2. Under gauge transformations, it changes at J as P α J → ω − P α . (8)3. If the energy-momentum tensor of the physical space-time ( ˆ M , ˆ g µν ) behaves near J asˆ T αβ | J ∼ O (Ω ) (which includes the vacuum case ˆ T αβ = 0), then ∇ µ P µ J = 0 . Let { n α | J , e αa } be an orthonormal basis of M at J . The canonical asymptotic supermo-mentum decomposes as P α J = W n α + P α J = W n α + P a e αa , (9)where, W J := − n µ P µ ≥ , (10) P α J := − ( δ αµ + n α n µ ) P µ , P α n α = 0 , (11)are the asymptotic superenergy and super-Poynting vector field, respectively. At this stage, wecan already introduce the gravitational-radiation condition. Criterion 1 (Asymptotic gravitational-radiation condition with Λ > . Consider a three-dimensional open connected subset ∆ ⊂ J . There is no radiation on ∆ if and only if theasymptotic super-Poynting vanishes there P α ∆ = 0 ⇐⇒ No gravitational radiation on ∆ . Observe that an equivalent statement is that the super-momentum points along the normal N α at J . This allows to give two alternative but equivalent formulations of the criterion whichwill be valuable for later comparison.• No gravitational radiation on ∆ ⊂ J ⇐⇒ p α is orthogonal to all surfaces within ∆.• No gravitational radiation on ∆ ⊂ J ⇐⇒ N α | J is a principal vector (in the sense ofPirani [4, 5], see [36]) of d δαβγ | ∆ . Remarks:
1. This characterisation is gauge invariant, as follows from (8).2. In terms of the electric, D ab , and magnetic, C ab , parts of the rescaled Weyl tensor, definedwith respect to n α | J , the asymptotic superenergy and super-Poynting take the form [5, 37] W J = D ab D ab + C ab C ab , (12) P a J = [ C, D ] rs ǫ rsa J = 2 C tr D ts ǫ rsa , (13)where ǫ abc is the volume 3-form of ( J , h ab ). This means that there are no gravitationalwaves at J if and only if C ab and D ab commute. A necessary condition for this tooccur is that d δαβγ | J be of Petrov type I or D [5, 38]. Observe, though, that this is notsufficient and these two types can contain asymptotic radiation – later on, we will presentone example (Example 4).3. The existence of radiation depends on the interplay between D ab and C ab , and cannot bedetermined with only one of them –letting aside the trivial case in which either vanishes.The Cotton-York tensor of ( J , h ab ) can be shown to be − p Λ / C ab , and thus C ab isdetermined by the intrinsic geometry of J . Hence, this intrinsic geometry is not enoughto encode the presence of asymptotic radiation, and one needs to bring D ab into thepicture. This agrees with the next remark.4. A fundamental result by Friedrich [39, 40] states that a solution of the Λ-vacuum Einsteinfield equations is fully determined by initial/final data consisting of the conformal classof a 3-dimensional Riemmanian manifold ( J , h ab ) plus a traceless and divergence-freetensor D ab . Hence, from this perspective and given any particular conformal geometry for J + one only needs to add a TT-tensor D ab such that it does (not) commute with C ab ifescaping radiation is not (is) to be present.5. The condition as stated is computation friendly, and can be easily implemented in algebraiccomputing programs.The radiation detected with the above criterion at J + is always ‘escaping’ from the physicalspacetime, and P α points in its spatial direction of propagation there. However, at this stage,one cannot say if the radiation is originated by an isolated source, as there can be other pieces ofradiation coming to J + from beyond the cosmological horizon of such isolated source [16, 41].In order to show that Criterion 1 is reliable, we first compare with the Λ = 0 case and thenpresent relevant examples supporting it —with and without radiation. COMPARISON WITH THE ASYMPTOTICALLY FLAT CASE (
Λ = 0 ) The first distinguishable property of the asymptotically flat scenario is that J is a lightlikehypersurface, as the normal N α is lightlike and thus N α is tangent to J —and cannot benormalized. Also, the topology of J is determined to be R × S [11, 12]. In this Λ = 0case the presence of escaping gravitational radiation –i.e. transversal to J + – is successfullydetermined by the news tensor [12, 28] N ab —equivalent to the Bondi news function [6] in aparticular gauge—: a symmetric, traceless, gauge-invariant tensor field on J + orthogonal toits null generators. Specifically, the standard criterion for the existence of radiation on an openportion ∆ ⊂ J + with same topology as J + reads for Λ = 0 [12, 28] N ab ∆ = 0 ⇐⇒ No gravitational radiation on ∆ . Actually, using the results in [29] we can show that this criterion agrees with ours. To prove itwe must use the analog of p α when Λ = 0, which was introduced in [29] as Q α := − N µ N ν N ρ D αµνρ (Λ = 0) . (14)Observe that this is formally the same definition as (6); in fact, for spacetimes in which the limitΛ → one has lim Λ → p α | J = Q α | J . (15) Q α | J is observer independent, geometrically distinguished and, due to known properties [34, 44],is future ligthlike . This is why we call it the asymptotic radiant supermomentum . In [29] weproved N ab ∆ = 0 ⇐⇒ Q α ∆ = 0 . In simpler words, the standard radiation condition for Λ = 0 can be re-stated as Q α ∆ = 0 ⇐⇒ No gravitational radiation on ∆ . An example where such a limit does not exist is the a → ∞ limit of Kerr-de Sitter [42, 43]. It might seem at first sight that there is a slight difference between this and Criterion 1, but thisis only apparent: Q α | J admits decompositions into a vector field Q α tangent to J –‘radiantsuper-Poynting’– and a transverse component. The vanishing of the former on ∆ is equivalentto the vanishing of the whole Q α on that open portion [29].To reinforce the agreement between Criterion 1 for Λ > ⊂ J ⇐⇒ Q α is orthogonal to all surfaces within ∆.• No gravitational radiation on ∆ ⊂ J ⇐⇒ N α | J is a principal vector of d δαβγ | ∆ .These are the same characterizations as those given previously for the case Λ > > escaping from the physical space-time isdetermined by the asymptotic supermomentum –(6) or (14)– defined with the normal to J + ,no matter if Λ = 0 or Λ > J + when Λ = 0 excludesany possible incoming waves, which would propagate tangentially to J + , and thus the radiationdetected by N ab , or equivalently by Q α | J , is entirely due to an isolated system, in contrast withthe Λ > > J + . A worth-exploring possibility arising naturally in our scheme is partly related to thealgebraic structure of the rescaled Weyl tensor in relation with N α [45]. In any case, when Λ > J + , h ab , D ab ) with further structure, essentially a distinguishedor chosen congruence of curves which depends on the physical properties of the spacetime underconsideration (see also [46]). This matter, however, excesses the scope of this letter and will bepresented elsewhere. EXAMPLES
Example 1.
As an obvious example, every space-time whose metric h ab on J is conformallyflat has C ab J = 0 and, consequently, it does not contain gravitational radiation at infinity.Similarly, every spacetime with D ab = 0 does not have radiation either. Likewise, conformallyflat physical spacetimes (that is, with a vanishing Weyl tensor) have no radiation as d δαβγ J = 0.Friedman-Lemaître-Robertson-Walker models admitting J are relevant examples including, ofcourse, de Sitter (dS) spacetime. Example 2.
If the physical spacetime is spherically symmetric and the conformal completionrespects the symmetry then C ab and D ab inherit the symmetry too, implying that in a preferredbasis they are both diagonal, ergo they conmute, and therefore there is no gravitational radiation.This holds true also for all spacetimes with a 3-dimensional group of symmetries —not onlySO(3)— with spacelike 2-dimensional orbits, as long as Ω is invariant by the group. Relevantexamples are the generalized Kottler spacetimes [42, 47] or the Vaidya metric. Example 3.
The family of Kerr-de Sitter-like space-times comprises all vacuum space-timeswith positive cosmological constant admitting a conformal completion and whose MaSrs-Simontensor (see e.g. [42, 48, 49] and references therein) with respect to a given Killing vector vanishes.These spacetimes have been studied in [49], and they include in particular Kerr-dS, Kerr-NUT-dS, or Taub-NUT-dS. The Killing vector induces a conformal Killing vector Y a on J withoutfixed points and the explicit expressions for C ab and D ab were found to be both proportional to[49] Y a Y b − Y c Y c h ab . Hence, P a = 0 and these space-times contain no gravitational radiation at infinity according toour criterion, as expected. Example 4.
We deal now with the C-metric with Λ ≥
0, a solution which under certain rangeof parameters represents two accelerating black holes in a dS (Λ >
0) or flat (Λ = 0) background.Gravitational waves at infinity are thus expected; actually, the existence of a non-vanishing newstensor when Λ = 0 was demonstrated in [50]. For further insights on the properties of thesemetrics, see [51].We use a recent version of this metric [52], in which the conformal space-time metric in adivergence-free gauge readsd s = 1 S (cid:18) T d τ − T d q + 1 S d p + S d σ (cid:19) , (16)where T ( q ) := ( α − q )(1 + 2 mq ) + Λ / , (17) S ( p ) := (1 − p )(1 − αmp ) (18)and the conformal factor is Ω := ( q + αp ) /S ergo J is defined by q + αp = 0. There are twoKilling vectors: ∂ τ with R -orbits, and ∂ σ with cyclic orbits but conical singularities at p = ± p ∈ ( − ,
1) and thus
S > αm <
1. To describe a region containing J we must choose q ∈ ( − α, α ), so that T ( q ) > T ( q ) = 0 are horizons through which the metric can beregularly extended. Notice that these horizons do not intersect J . There are four constantparameters: α (acceleration), m (mass), Λ and C , the latter sometimes is used to remove oneof the conical singularities [51].The metric on J reads h = 1 S (cid:18) ( α S + Λ / τ + Λ3 S ( α S + Λ /
3) d p + S d σ (cid:19) . (19)This is clearly positive definite having a smooth limit when Λ → h ab . The electric and magnetic parts of the rescaled Weyl tensor on J , in the coordinate basis,have as non-zero components for Λ = 0 C τσ = 3Λ αmS / (3 Sα + Λ) , D σσ = m Λ S / (cid:16) Λ + 9 Sα (cid:17) , (20) D ττ = − m Λ √ S (cid:18) S α + 5Λ Sα + 23 Λ (cid:19) , D pp = m Λ √ S (Λ + 3 Sα ) (21)hence, the asymptotic canonical super-Poynting vector and superenergy have the following ex-pressions: P a J = r
3Λ 18 αm S / (cid:18) Sα (cid:19) δ ap , (22) W J = 6 m S (cid:18) S α + 18Λ Sα (cid:19) . (23)The super-Poynting vector field is different from zero everywhere on J , thus meaning thatgravitational radiation arrives there, as expected.To measure the effects of Λ with respect to the asymptotically-flat case, it is convenient tocompute the supermomentum (6) for Λ > ~p J = 2 m S / (cid:20)(cid:18) α S + Λ3 (cid:19) (cid:16) Λ + 9 Sα (cid:17) ∂ q + αS (cid:16)
2Λ + 9 Sα (cid:17) ∂ p (cid:21) , (24) ~ Q J = 18 m S / α ( α∂ q + ∂ p ) . (25)Observe that there are linear and quadratic terms in Λ in (24). Finally, it is straightforward toverify that (25) is lightlike for Λ = 0. DISCUSSION
We have presented a new geometric, covariant and gauge-invariant criterion to character-ize gravitational waves at infinity in the presence of a non-negative cosmological constant. Inrelevant examples, the criterion gives the expected results. More importantly, we have arguedthat the criterion is equivalent to the standard one –based on the news tensor– for the Λ = 0case. Adding appropriate structure on ( J , h ab , D ab ) which allows to deal with ‘incoming andoutgoing’ waves, we are developing a framework [45] in order to recover further properties of theasymptotically flat case. In particular, ways to define asymptotic symmetries at infinity, as wellas enlightening evidences for the existence of news-like tensor fields on J in the presence of apositive cosmological constant, emerge. Acknowledgments
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