Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited

0.08 MB·English

by  Glenn Barnich

ULB-TH/09-24
Symmetries of asymptotically flat 4 dimensional
spacetimes at null infinity revisited
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0 Glenn Barnicha and Ce´dric Troessaertb
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PhysiqueThe´oriqueetMathe´matique
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Universite´ LibredeBruxelles

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and
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International SolvayInstitutes

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c CampusPlaineC.P.231,B-1050Bruxelles, Belgium
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1 ABSTRACT. It is shownthat the symmetryalgebra of asymptoticallyflat space-
6 times at null infinity in 4 dimensions should be taken as the semi-direct sum of
2
. supertranslations with infinitesimal local conformal transformations and not, as
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0 usuallydone,withtheLorentzalgebra. Asaconsequence,twodimensionalcon-
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0 formal field theory techniques will play as fundamental a role in this context of
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v direct physicalinterest as theydointhree dimensionalanti-deSittergravity.
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r
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aResearchDirectoroftheFundforScientificResearch-FNRS.E-mail:[email protected]
bResearchFellowoftheFundforScientificResearch-FNRS.E-mail:[email protected]
2 BARNICH, TROESSAERT
Inthestudyofgravitationalwavesintheearlysixties[1,2],alotofeffortshavebeen
devotedtospecifyingbothlocalcoordinateandglobalboundaryconditionsatnullinfinity
that characterize asymptotically flat 4 dimensional spacetimes. The group of non singu-
lar transformations leaving these conditions invariant is the well-known Bondi-Metzner-
Sachs group. It consists of the semi-direct product of the group of globally defined con-
formal transformations of the unit 2-sphere, which is isomorphic to the orthochronous
homogeneousLorentzgroup, timestheabelian normalsubgroupof so-called supertrans-
lations.
What seems to have been largely overlookedso far is the fact that, when one focuses
oninfinitesimaltransformationsanddoesnotrequiretheassociatedfinitetransformations
tobegloballywell-defined,thesymmetryalgebraisthesemi-directsumoftheinfinitesi-
mal local conformal transformationsof the2-sphere withthe abelian ideal of supertrans-
lations, and now both factors are infinite-dimensional. This is already obvious from the
detailsofthederivationoftheasymptoticsymmetryalgebraby Sachs in1962 [3].
Letx0 = u,x1 = r,x2 = θ,x3 = φandA,B,··· = 2,3. Following[3]uptonotation,
themetricg ofan asymptoticallyflat spacetimecan bewrittenintheform
µν
V
ds2 = e2β du2 −2e2βdudr+g (dxA −UAdu)(dxB −UBdu) (1)
AB
r
where β,V,UA,g (detg )−1/2 are 6 functions of the coordinates, with detg = r4b
AB AB AB
for a function b(u,θ,φ). Sachs fixes b = sin2θ, but the geometrical analysis by Penrose
[4]suggeststokeepitarbitrarythroughouttheanalysis. Inordertostreamlinethederiva-
tion below, it turns out convenient to use the parametrization |b| = 1e4ϕe, which implies
4
in particularthatgAB∂ g = ∂ ln(r4e4ϕe).
α AB α 4
Thefall-offconditionsforg are
AB
g dxAdxB = r2γ¯ dxAdxB +O(r), (2)
AB AB
where the 2-dimensional metric γ¯ is conformal to the metric of the unit 2-sphere,
AB
γ¯ = e2ϕ γ and γ dxAdxB = dθ2 +sin2θdφ2. In terms of the standard complex
AB 0 AB 0 AB
coordinatesζ = eiφcot θ, themetriconthesphereisconformallyflat, dθ2+sin2θdφ2 =
2
P−2dζdζ¯,P(ζ,ζ¯) = 1(1+ζζ¯). Wethushaveγ¯ dxAdxB = e2ϕedζdζ¯withϕ = ϕ−lnP.
2 AB
In the following we denote by D¯ the covariant derivativewith respect to γ¯ and by ∆¯
A eAB
theassociatedLaplacian.
In thegeneral case, theremainingfall-offconditionsare
β = O(r−2), UA = O(r−2), V/r = −2r∂ ϕ+∆¯ϕ+O(r−1). (3)
u
e e
The transformations that leave the form of the metric (1) invariant up to a conformal
rescaling of g , i.e., up to a shift of ϕ by ω(u,xA), are generated by spacetime vectors
AB
e e
BMS ALGEBRA REVISITED 3
satisfying
L g = 0, L g = 0, gABL g = 4ω,
ξ rr ξ rA ξ AB
L g = O(r−2), L g = O(1), L g = O(r), (4)
ξ ur ξ uA ξ AB e
L g = −2r∂ ω −2ω∆¯ϕ+∆¯ω +O(r−1).
ξ uu u
Thegeneral solutionto theseequationis
e e e e
ξu = f,
 ξA = YA +IA, IA = −f ∞dr′(e2βgAB), (5)
 ,B r
ξr = −1r(D¯ ξA −f UB +2Rf∂ ϕ−2ω),
2 A ,B u


with∂ f = 0 = ∂ Y. In addition, e e
r r
1 1 u
∂ f = f∂ ϕ+ ψ −ω ⇐⇒ f = eϕe T + du′e−ϕe(ψ −2ω) , (6)
u u 2 2 Z
(cid:2) 0 (cid:3)
e e e
where we use the notation ψ = D¯ YA and where ∂ T = 0 = ∂ YA. Finally YA is
A u u
required to beaconformalKillingvectorofγ¯ .
AB
The Lie algebra bms can be defined as the semi-direct sum of the Lie algebra of
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∂
conformalKillingvectorsYA oftheRiemannspherewiththeabelianidealconsisting
∂xA
of functions T(xA) on the Riemann sphere. The bracket is defined through (Y,T) =
[(Y ,T ),(Y ,T )]
1 1 2 2 b b
YA = YB∂ YA −YB∂ YA,
1 B 2 1 B 2
1 (7)
bT = YA∂ T −YA∂ T + (T ∂ YA −T ∂ YA).
1 A 2 2 A 1 2 1 A 2 2 A 1
b
Considerthen themodifiedLiebracket
[ξ ,ξ ] = [ξ ,ξ ]−δg ξ +δg ξ , (8)
1 2 M 1 2 ξ1 2 ξ2 1
where δg ξ denotes the variation in ξ under the variation of the metric induced by ξ ,
ξ1 2 2 1
δg g = L g .
ξ1 µν ξ1 µν
Let I be the real line times the Riemann sphere with coordinates u,xA = (ζ,ζ¯). On
∂ ∂
I, consider the scalar field ϕ,ω and the vectors fields ξ¯(ϕ,ω,T,Y) = f + YA ,
∂u ∂xA
with f given in (6) and YA an u-independent conformal Killing vector of the Riemann
e e e e
sphere.
When equipped with the modified bracket, both the vector fields ξ¯and the spacetime
vectors (5) provide a faithful representation of the direct sum of bms with the abelian
4
algebra of conformal rescalings, i.e., the space of elements of the form (Y,T,ω) where
[(Y ,T ,ω ),(Y ,T ,ω )] = (Y,T,ω), with Y,T as beforeandω = 0.
1 1 1 2 2 2
Depeendingontheespaceofbfunbctbeionsundebrcbonsideration,theberearethenbasicallytwo
optionswhichdefine whatisactually meantby bms .
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4 BARNICH, TROESSAERT
Thefirstchoiceconsistsinrestrictingoneselftogloballywell-definedtransformations
on the unit or, equivalently, the Riemann sphere. This singles out the global conformal
transformations, also called projective transformations, and the associated group is iso-
morphictoSL(2,C)/Z ,whichisitselfisomorphictotheproper,orthochronousLorentz
2
group. Associatedwiththischoice,thefunctionsT(θ,φ),whicharethegeneratorsofthe
so-called supertranslations, have been expanded into spherical harmonics. This choice
has been adopted in the original work by Bondi, van der Burg, Metzner and Sachs and
followed ever since, most notably in the work of Penrose and Newman-Penrose [4, 5].
A lot of attention has been devoted to the conformal rescalings and the “edth” operator
togetherwithspin-weightedsphericalharmonicshavebeenintroduced. Afterattemptsto
cutthisversionoftheBMSgroupdowntothestandardPoincare´ group,ithasbeentaken
seriouslyasaninvariancegroupofasymptoticallyflatspacetimes. Itsconsequenceshave
been investigated, but we believe that it is fair to say that this version of the BMS group
has had onlyalimitedamountofsuccess.
The second choice that we would like to advocate here is motivated by exactly the
sameconsiderationsthatareattheoriginofthebreakthroughintwodimensionalconfor-
malquantumfieldtheories[6]. Itconsistsinfocusingonlocalpropertiesandallowingthe
setofall,notnecessarilyinvertibleholomorphicmappings. Inthiscase,Laurentserieson
theRiemannsphereareallowed. Thegeneral solutiontotheconformalKillingequations
is Yζ = Yζ(ζ),Yζ¯ = Yζ¯(ζ¯) and thestandard basisvectorsare choosenas
∂ ∂
l = −ζn+1 , ¯l = −ζ¯n+1 , n ∈ Z (9)
n ∂ζ n ∂ζ¯
At the same time, let us choose to expand the generators of the supertranslations with
respect to
T = ζmζ¯n, m,n ∈ Z. (10)
m,n
In terms of the basis vectors l ≡ (l ,0) and T ≡ (0,T ), the commutation relations
l l mn mn
forthecomplexifiedbms algebraread
4
¯ ¯ ¯ ¯
[l ,l ] = (m−n)l , [l ,l ] = (m−n)l , [l ,l ] = 0,
m n m+n m n m+n m n
(11)
l +1 l+1
¯
[l ,T ] = ( −m)T , [l ,T ] = ( −n)T .
l m,n m+l,n l m,n m,n+l
2 2
ThecomplexifiedPoincare´ algebra isthesubalgebraspannedby thegenerators
l−1, l0, l1, ¯l−1, ¯l0, ¯l1, T0,0, T1,0,T0,1, T1,1. (12)
The considerations above apply for all choices of ϕ which is freely at our disposal.
In the original work of Bondi, van der Burg, Metzner and Sachs, and in much of the
e
subsequent work, the choice ϕ = −lnP was preferred. From the conformal point of
e
BMS ALGEBRA REVISITED 5
view, the choice ϕ = 0 is interesting as it turns γ¯ into the standard flat metric on the
AB
Riemann sphere.
e
The consequences of local conformal invariance need to be taken into account when
studyingrepresentationsandourresultmeansthattwodimensionalconformalfieldtheory
techniques should play a major role both in the classical and quantum theory of gravita-
tionalradiation. Forinstance,therepresentationtheorybasedonthestandardBMSgroup
hasbeendiscussedin[7]andreferencestherein,whilerelatedholographicconsiderations
haveappeared in[8, 9]. Furthermore, implicationsofthe supertranslationsin thecontext
of asymptoticquantization [10, 11] have already been investigated. It should provemost
interestingtoextendtheseconsiderationstoincludethelocal conformaltransformations.
A new perspective also arises for the problem of angular momentum in general rel-
ativity [12] since the factor algebra of bms modulo the abelian ideal of infinitesimal
4
supertranslations is now the infinitedimensionalVirasoro algebra rather than the Lorentz
algebra.
Earlier work where the relevance of conformal field theories for asymptotically flat
spacetimes at null infinity has been discussed by starting out from the correspondence
in the (anti-) de Sitter case includes [13, 14, 15, 16, 17, 18]. In particular, a symmetry
algebra ofthekindthatwe havederivedhasbeen conjectured in [19].
A motivation for our investigation comes from Strominger’s derivation [20] of the
Bekenstein-Hawking entropy for black holes that have a near horizon geometry that is
locallyAdS . Morerecently,asimilaranalysishasbeenappliedinthecaseofanextreme
3
4-dimensionalKerrblackhole[21]. Ourhopeistomakeprogressalongtheselinesinthe
nonextremecase. Asafirststep,wehavecomputedthebehaviorofBondi’snewstensor
aswellasthemassandangularmomentumaspectsunderlocalconformaltransformations
in [22], where detailed proofs of all statements of this letter can also be found. The next
step consistsintheconstructionofthesurfacecharges, generators and central extensions
associated tobms .
4
Acknowledgements
The authors thank M. Ban˜ados, G. Compe`re, G. Giribet, A. Gomberoff, M. Henneaux,
C. Mart´ınez, R. Troncoso and A. Virmani for useful discussions. This work is supported
in parts by the Fund for Scientific Research-FNRS (Belgium), by the Belgian Federal
SciencePolicyOfficethroughtheInteruniversityAttractionPoleP6/11,byIISN-Belgium
and by Fondecytproject No. 1085322.
6 BARNICH, TROESSAERT
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BMS ALGEBRA REVISITED 7
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