Finite BMS transformations

0.38 MB·

by  Glenn Barnich

Finite BMS transformations
Glenn Barnicha and Ce´dric Troessaertb
aPhysiqueThe´oriqueetMathe´matique
Universite´ LibredeBruxelles andInternational SolvayInstitutes
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1 CampusPlaineC.P.231,B-1050Bruxelles, Belgium
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bCentrodeEstudiosCient´ıficos(CECs)
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a ArturoPrat514,Valdivia,Chile
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c ABSTRACT. The action of finiteBMS and Weyl transformationson thegravita-
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tionaldataat nullinfinityisworked outinthreeand fourdimensionsinthecase
-
r
g ofan arbitrary conformalfactor fortheboundary metricinducedon Scri.
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Contents
1 Introduction 3
2 AdaptedCartanformulation 4
3 Newman-Penroseformalismin3d 5
4 3dasymptotically AdSspacetimesatspatialinfinity 7
4.1 Fefferman-Graham solution space . . . . . . . . . . . . . . . . . . . . . . . . . 7
4.2 Residualgaugesymmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.3 ActionofconformalandWeylgroup . . . . . . . . . . . . . . . . . . . . . . . . 12
5 3dasymptotically flatspacetimesatnullinfinity 13
5.1 Solutionspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5.2 Residualgaugesymmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5.3 CombinedBMS3andWeylgroup . . . . . . . . . . . . . . . . . . . . . . . . . 17
5.4 Actiononsolutionspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
6 4dasymptotically flatspacetimesatnullinfinity 20
6.1 Newman-Penroseformalismin4d . . . . . . . . . . . . . . . . . . . . . . . . . 20
6.2 Newman-Untisolution space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
6.3 Residualgaugesymmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
6.4 Combinedextended BMS4groupwithcomplexrescalings . . . . . . . . . . . . 32
6.5 Actiononsolutionspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
7 Discussion 38
Acknowledgments 39
A Newman-Penrosefieldequationsin3d 39
B Additionaltransformation lawsin4d 40
References 40
FINITE BMS TRANSFORMATIONS 3
1 Introduction
There are two main applications of two dimensional conformal invariance [1]. The first
consists in using Ward identitiesassociated to infinitesimalsymmetry transformations in
order to constrain correlation functions. In the second application, starting from known
quantities in a given domain, the finite transformations are used to generate the corre-
sponding quantities pertaining to the transformed domain (see e.g. [2]). In this case, the
Schwarzianderivativeoccuringinthetransformationlawoftheenergy-momentumtensor
plays acrucial role.
For four-dimensional asymptotically flat spacetimes at null infinity, an extension of
the globally well-defined symmetry group [3–5] in terms of locally defined infinitesimal
transformations has been proposed and studied in [6–10]. In particular, their relevance
for gravitationalscattering has been conjectured. Physical implicationsin terms of Ward
identitiesforsoftgravitonshavesubsequentlybeen developedin[11–14].
The aim of the present paper is to derive the finite transformations necessary for the
second application. In particular for instance, if one knows the theory in the form of
an asymptotic solution to classical general relativity for the standard topology S2 R
ˆ
of I , one can use the transformation laws to get the solution on a cylinder times a
`
line. Particular aspects of such mappings in general relativity have been discussed pre-
viously for instance in [15–17]. More concretely, in the present paper we will work out
the transformation laws of asymptotic solution space and the analog of the Schwarzian
derivative for finite extended BMS transformations and local time-dependent complex
4
Weyl rescalings. Whereas the former corresponds to the residual symmetry group, the
latter represents the natural ambiguity in the definition of asymptotically flat spacetimes
in termsofconformalcompactifications[18,19].
As a warm-up, we start by re-deriving the known finite transformations in three di-
mensionsintheasymptoticallyanti-deSitterandflatcases. Intheformercase,onerecov-
erstheSchwarzianderivativeasanapplicationoftheAdS /CFT correspondence[20,21].
3 2
In the latter case, one obtains the finite transformation laws for the Bondi mass and an-
gular momentum aspects that have been previously obtained by directly integrating the
infinitesimaltransformations[22]. Inboththesethreedimensionalcases,theseresultsare
generalizedtoincludelocalWeyltransformations. Inotherwords,weareworkingoutthe
action offinitePenrose-Brown-Henneaux transformationsintheterminologyof[23,24].
Explicit computations are done in the framework of the Newman-Penrose formalism
[25,26], as applied to asymptotically flat four dimensional spacetimes at null infinity
in [27,28]. Standard reviewsare[29–33].
To summarize the results for the simplest case when computations are done with re-
specttotheRiemannsphere,i.e.,whenthemetriconI istakenasds2 0du2 2dζdζ,
`
“ ´
r
4 G. BARNICH, C. TROESSAERT
the extended BMS4 group consists of superrotations ζ ζ ζ1 ,ζ ζ ζ1 together with
“ p q “ p q
Bζ Bζ 1
supertranslations u1 “ pBζ1 Bζ1q´2ru`βpζ,ζqs. In particular, the asymptoticpart of the
shear, thenews,and theBondimassaspect transformas
r r
σR10 “ pBBζζ1q´12pBBζζ1q32 σR0 `B2β ` 12tζ1,ζupu˜`βq , (1.1)
” ı
σ9R10 “ pBBζζ1q2 σ9R0 ` 12tζ1,ζu , (1.2)
” ı
p´4πGqMR1 “ pBBζζ1 BBζζ1q23 p´4πGqMR `B2B2β ` 12tζ1,ζupσ0R `B2βq`
” 1 1
0 2
2 ζ1,ζ σR β 4 ζ1,ζ ζ1,ζ u β , (1.3)
` t up `B q` t ut up ` q
ı
where , denotestheSchwarzian derivative.
t¨ ¨u r
2 Adapted Cartan formulation
In the Cartan formulation of general relativity, the fundamental fields are on the one
hand,avielbein,e µ,togetherwithitsinverseea andassociatedmetricg ea η eb ,
a µ µν µ ab ν
“
whereη isconstantand,ontheotherhand,aLorentzconnectionsatisfyingthemetricity
ab
condition ∇ η 0, Γ η Γd Γ . Indices are lowered and raised with
a bc abc ad bc abc
“ “ “ r s
η and g and their inverses. The associated connection 1-form is Γa Γa ec with
ab µν b bc
“
ec ec dxµ. The torsion and curvature 2-forms are given by Ta dea Γa eb,
µ b
“ “ ` ^
Ra dΓa Γa Γc .
b b c b
“ ` ^
Local Lorentz transformations are described by matrices Λ b x with Λa Λ c δa.
a p q c b “ b
Under combined frame and coordinate transformations, referred to as gauge transforma-
tionsbelow,thebasicvariablestransformas
e µ x Λ be νBx1µ x ,
1a p 1q “ a b Bxν p q (2.1)
´ ¯
Γ x Λ d Λ eΓ e Λ Λ f x ,
1abc 1 a b def f bd c
p q “ ` p q p q
´ ¯
where the last expression is equivalen“t to the transforma‰tion law for the connection 1-
B
form, Γa Λa Γc Λ d Λa dΛ c ande e µ .
1 b “ c d b ` c b a “ a Bxµ
Equationsofmotionderivingfromthevariationalprinciple
1
S e,Γ ddxe R ηacηbd 2Λ , (2.2)
r s “ 16πG p abcd ´ q
ż
are equivalent to Ta dea Γa eb 0 and Einstein’s equations, G Λη 0.
b ab ab
“ ` “ ` “
Togetherwiththemetricitycondition,theformerimplies
1
Γ D D D , (2.3)
abc 2 bac cab abc
“ p ` ´ q
FINITE BMS TRANSFORMATIONS 5
B
where the structure functions are defined by Dc e e e µ e e µ . Con-
ab c “ p ap b q ´ bp a qqBxµ
versely,Ta 0is equivalenttoD 2Γ .
cab cab
“ “ ´ r s
3 Newman-Penrose formalism in 3d
In threedimensionswith , , signature,weuse
p` ´ ´q
0 1 0
η 1 0 0 , (3.1)
ab ¨ ˛
“
0 0 1
2
˚ ´ ‹
˝ ‚
and the triad e l,n,m with associated directional covariant derivatives denoted by
a
“ p q
D,∆,δ . In particular,
p q
gµν lµnν lνnµ 2mµmν, ∇ n D l ∆ 2m δ. (3.2)
a a a a
“ ` ´ “ ` ´
In this case, the spin connection can be dualized, ω 1Γab ǫ , Γab ǫabcω with
cµ 4 µ abc µ cµ
“ “
ǫ 1 andǫabc ηadηbeηcfǫ . The9 real spincoefficients aredefined by
123 def
“ “
∇ ma∇l na∇l ma∇n
a a a
´
D κ Γ ω2 ǫ Γ ω3 π Γ ω1
311 1 211 1 321 1
“ “ “ “ ´ “ ´ “ (3.3)
∆ τ Γ ω2 γ Γ ω3 ν Γ ω1
312 2 212 2 322 2
“ “ “ “ ´ “ ´ “
δ σ Γ ω2 β Γ ω3 µ Γ ω1
313 3 213 3 323 3
“ “ “ “ ´ “ ´ “
(see alsoe.g. [34]forslightlydifferentconventions). It followsthat
Dl ǫl 2κm, ∆l γl 2τm, δl βl 2σm,
“ ´ “ ´ “ ´
Dn ǫn 2πm, ∆n γn 2νm, δn βn 2µm, (3.4)
“ ´ ` “ ´ ` “ ´ `
Dm πl κn, ∆m νl τn, δm µl σn.
“ ´ “ ´ “ ´
In order todescribe Lorentztransformations,oneassociates toa real vectorv vae
a
“
a 2 2 symmetricmatrixvˆ vaj , wherej are chosenas
a a
ˆ “
1 0 0 0 1 0 1
p p
j , j , j , (3.5)
1 “ 0 0 2 “ 0 1 3 “ 2 1 0
˜ ¸ ˜ ¸ ˜ ¸
p p p
so that
1 1 1
det v η vavb, j ǫj η ǫ ǫ jc , jaj j ǫj ǫ, (3.6)
2 ab a b 2 ab abc b a 2 b
“ “ p ´ q “
where
p p p 0 1 p p pp p
ǫ . (3.7)
“ 1 0
˜ ¸
´
6 G. BARNICH, C. TROESSAERT
Forg SL 2,R , oneconsiders thetransformation
P p q
gj gTva j Λa vb, gTǫ ǫg 1. (3.8)
a a b ´
“ “
Moreexplicitly,if p p
a b
g , (3.9)
“ c d
˜ ¸
withad bc 1 and a,b,c,d R, then
´ “ P
a2 b2 ab d2 c2 2cd
´
Λa c2 d2 cd , Λ b b2 a2 2ab , (3.10)
b ¨ ˛ a ¨ ˛
“ “ ´
2ac 2bd ad bc bd ac ad bc
˚ ` ‹ ˚´ ´ ` ‹
˝ ‚ ˝ ‚
where thefirst indexisthelignindex. SL 2,R groupelementswillbeparametrized as
p q
1 0 1 A e E 2 0 e E 2 AeE 2
´ { ´ { {
g ´ ´ . (3.11)
“ B 1 0 1 0 eE 2 “ Be E 2 1 AB eE 2
˜ ¸˜ ¸˜ { ¸ ˜ ´ { { ¸
´ ´ p ` q
Using ωa 1ǫabcΓ and the transformation law of the Lorentz connection given below
2 bc
“
(2.1), wehave
1
ωa Λa ωb ǫabcΛ ddΛ . (3.12)
1 b 2 b cd
“ `
In terms ofωˆ ˆj ωa, thisisequivalentto
a
“
ωˆ gωˆgT gǫdgT (3.13)
1
“ ´
Explicitly,forthespincoefficients encoded in
1 1 1
π ǫ ν γ µ β
ωˆ1 1 ´2 , ωˆ2 1 ´2 , ωˆ3 1 ´2 , (3.14)
“ ǫ κ “ γ τ “ β σ
˜ 2 ¸ ˜ 2 ¸ ˜ 2 ¸
´ ´ ´
onefinds
ωˆ Λ cgωˆ gT gǫe gT . (3.15)
a1 “ a c ´ 1ap q
In thiscase, Einstein’sequationsare equivalentto
Λ
deˆ 2ωˆǫeˆ 0, dωˆ ωˆǫωˆ eˆǫeˆ 0. (3.16)
2
´ “ ´ ´ “
Alternatively, one can use v vˆǫ in order to describe real vectors by traceless 2 2
“ ˆ
matrices. Theassociatedbasisis
q
0 1 0 0 1 1 0
j1 “ 0 0 , j2 “ 1 0 , j3 “ 2 ´0 1 , (3.17)
˜ ¸ ˜ ¸ ˜ ¸
´
q q q
FINITE BMS TRANSFORMATIONS 7
so that
1 1
tr v2 η vavb, j j η ǫ jc , jaj j j . (3.18)
ab a b 2 ab abc b a 2 b
“ “ ´ p ` q “
In thiscase, we have
q q q q q qq q
gj g 1va j Λa vb, ω gωg 1 dgg 1, ω Λ cgω g 1 e g g 1, (3.19)
a ´ a b 1 ´ ´ a1 a c ´ 1a ´
“ “ ´ “ ´ p q
where
q q q q q q
1 1 1
ǫ π γ ν β µ
ω 2 , ω 2 , ω 2 , (3.20)
1 1 2 1 3 1
“ κ ǫ “ τ γ “ σ β
˜ 2 ¸ ˜ 2 ¸ ˜ 2 ¸
´ ´ ´ ´ ´ ´
and q q q
Λ
de 2ωe 0, dω ωω ee 0. (3.21)
2
´ “ ´ ´ “
q qq q qq qq
4 3d asymptotically AdS spacetimes at spatial infinity
4.1 Fefferman-Graham solution space
In the AdS case, Λ L 2 0, we start by rederiving the general solution to the
3 ´
“ ´ ‰
equations of motion in the context of the Newman-Penrose formalism. We will recover
the on-shell bulk metric of [35], but with an arbitrary conformal factor for the boundary
metric[21](seealsoSection 2 of[7]inthecurrent context).
TheanalogoftheFefferman-Graham gaugefixing isto assumethat
µ β σ 0. (4.1)
“ “ “
which is equivalent to Γ 0 and can be achieved by a local Lorentz transformation.
ab3
“
This means that the triad is parallely transported along m and that m is the generator of
an affinely parametrized spatial geodesic. In this case, ∇ m n l π τ so that m
a b a b
r s “ r sp ` q
is hypersurfaceorthonormalifand onlyitisagradient, whichin turnisequivalentto
π τ. (4.2)
“ ´
Thisconditionwillalsobeimposedinthefollowing.
Introducing coordinates xµ x ,x ,ρ , µ 1,2,3 such that m is normal to the
` ´
“ p q “
surfaces ρ cte and the coordinate ρ is the suitably normalized affine parameter on the
“
geodesicgenerated by m, thetriad takes theform
m B , l la B , n na B . (4.3)
“ ρ “ xa “ xa
B B B
where a , . Theassociatedcotriad is
“ p` ´q
ǫ nbdxa ǫ ladxb
e1 ab , e2 ab , e3 dρ, e ǫ lanb, (4.4)
ab
“ e “ e “ “
8 G. BARNICH, C. TROESSAERT
where ǫ 1 ǫ and ǫ 0. In order to compare with the general solution
`´ “ “ ´ ´` ˘˘ “ ρ
givenin[7], oneintroducesan alternativeradial coordinater e?2L, interms ofwhich
“
r L
m B , e3 ?2 dr. (4.5)
“ ?2L r “ r
B
Under these assumptions, the Newman-Penrose field equations (A.1)-(A.12) can be
solvedexactly. Indeed, thethreeequations(A.1), (A.7)and(A.9)reduce tothesystem
1
δκ 2τκ, δν 2τν, δτ τ2 κν , (4.6)
“ “ “ ´ ´ 2L2
which is solved by introducingthe complex combinationsL τ i?νκ. The general
˘ “ ˘
solutionis givenby
1 ?2C r2 ?2C r2
4 2 2 3
τ “ ´π “ ?´2Lkpr ´C1 `C2C3q, κ “ ´ Lk , ν “ Lk , (4.7)
k r4 2C r2 C2 C C .
1 1 2 3
“ ´ ` ´
The last two radial equations involving the spin coefficients, equations (A.3) and (A.8),
simplifyto
δǫ τǫ κγ, δγ τγ νǫ, (4.8)
“ ` “ ´
and are solvedthrough
r3 C r C r r3 C r C r
1 2 1 3
ǫ C4 ´ C5 , γ C5 ´ C4 . (4.9)
“ k ` k “ k ` k
The last radial equations are (A.11) and (A.12). Their r-componentare triviallysatisfied
whiletheircomponentsalongx are ofthesameform than(4.8),
˘
δl τl κn , δn τn νl , (4.10)
˘ ˘ ˘ ˘ ˘ ˘
“ ` “ ´
which leadsto
r3 C r C r r3 C r C r
1 2 1 3
l K ´ K , n K ´ K . (4.11)
˘ 1˘ 2˘ ˘ 2˘ 1˘
“ k ` k “ k ` k
In theseequations,C ,K ,K are functionsofxa x alone.
i 1˘ 2˘ ˘
“
Note that asymptotic invertibility of the triad is controlled by the invertibility of the
matrixformed by thesefunctions,
K K
1` 2` , ǫ KaKb 0. (4.12)
ab 1 2
K K ‰
˜ 1´ 2´ ¸
Using the radial form of the various quantities, equations (A.2) and (A.6) are equivalent
to
Ka C Ka C 2C C 0,
1 a 1 2 a 2 2 5
B ´ B ` “ (4.13)
Ka C Ka C 2C C 0,
1 a 3 2 a 1 3 4
B ´ B ` “
FINITE BMS TRANSFORMATIONS 9
which thenimpliesthatequation(A.4)reduces to
4
Ka C Ka C 2C C C 0, (4.14)
1Ba 5 ´ 2Ba 4 ` 4 5 ` L2 1 “
whilethecomponentsalongx ofequation(A.10)become
˘
Ka K Ka K C K C K 0. (4.15)
1 a 2˘ 2 a 1˘ 5 1˘ 4 2˘
B ´ B ` ` “
Because of invertibility of the matrix (4.12), equations (4.15) and (4.14) can be used to
express C ,C and C in terms of Ka and Ka. The two equations in (4.13) then become
4 5 1 1 2
dynamical equations for C and C . Since we now have treated all Newman-Penrose
2 3
equations, the solution space is parametrized by Ka,Ka and by initial conditions for C
1 2 2
and C .
3
Inthelimitr goingtoinfinity,thetriadelementsl andngivenin(4.11)taketheform
l r 1K O r 3 , n r 1K O r 3 . With a change of coordinates on the
˘ ´ 1˘ ´ ˘ ´ 2˘ ´
“ ` p q “ ` p q
cylinder, we can make the associated asymptoticmetric explicitly conformally flat. This
amountstothechoice
K 0, K ?2e ϕ, K ?2e ϕ, K 0. (4.16)
1` 1´ ´ 2` ´ 2´
“ “ “ “
Introducingthisintoequations(4.14)and (4.15), weget
C ?2e ϕ ϕ, C ?2e ϕ ϕ, C L2e 2ϕ ϕ, (4.17)
4 ´ 5 ´ 1 ´
“ B´ “ ´ B` “ B´B`
whilethedynamicalequations(4.13)reduceto
C 2 ϕC L2 e 2ϕ ϕ ,
2 2 ´
B` ` B` “ B´ B´B` (4.18)
C 2 ϕC L2 e 2ϕ ϕ .
3 3 ` ´ ˘
B´ ` B´ “ B` B´B`
` ˘
Withtheextraconditions(4.16), thespaceofsolutionsisparametrisedbythreefunc-
tionsϕ,C andC definedonthecylinderwithcoordinatesx suchthatequations(4.18)
2 3 ˘
are valid. These two equations can be integrated directly but we will derive the explicit
form of C and C in a different way using the action of the asymptoticsymmetry group
2 3
below.
4.2 Residual gauge symmetries
The residual gaugetransformationsare the finitegauge transformationsthat preservethe
set ofasymptoticsolutions. Since thesetransformationsmap solutionsto solutions,once
the conditions that determine the asymptotic solution space are preserved, no further re-
strictions can arise. A gaugetransformation is a combinationof a local Lorentz transfor-
mationand achangeofcoordinatesoftheform
r r r ,x , x x r ,x . (4.19)
1 1˘ ˘ ˘ 1 1˘
“ p q “ p q
10 G. BARNICH, C. TROESSAERT
Theunknownsare A,B,E,r and x as functionsofr ,x .
˘ 1 1˘
Usingthea 3componentofthetransformationlawofthetriad,
“
xν
eµ B Λ beν, (4.20)
1a xµ “ a b
1
B
therequirementmµ r1 δµ isequivalentto
1 “ ?2L r1
r
1 xµ bdlµ acnµ ad bc mµ. (4.21)
?2LBr1 “ ´ ´ `p ` q
Expandingforeach coordinate, weget
r x ?2r
?21LB r` “ k e´ϕ Ap1`ABqeEC2 `Be´Epr2 ´C1q ,
1
B
r x ?2r ` ˘
?21LB r´ “ k e´ϕ Ap1`ABqeEpr2 ´C1q`Be´EC3 , (4.22)
1
B
r r ` r ˘
1 B 1 2AB .
?2L r “ p ` q?2L
1
B
In ordertoimplementthegaugefixingconditiononthenewspincoefficientsω 0,we
31
“
first rewritethelastequationof(3.19)as
q
g 1e g Λ bω g 1ω g. (4.23)
´ 1ap q “ a b ´ ´ a1
When a 3thisbecomes
q q
“
g 1δ g Λ bω , (4.24)
´ 1 3 b
“
and is equivalenttothreeconditionson theLorentzparameters,
q
dδ a bδ c Λ b ω , dδ b bδ d Λ b ω , aδ c cδ a Λ b ω . (4.25)
1 1 3 b 11 1 1 3 b 12 1 1 3 b 21
´ “ p q ´ “ p q ´ “ p q
When suitablycombiningtheseequations,onefinds
q q q
r B
1 B A 1 AB e2Eκ Bτ,
?2L r “ p ` q `
1
B
r A
1 B A 1 AB π Be 2Eν A3 1 AB e2Eκ A2Bτ, (4.26)
?2L r “ ´ p ` q ´ ´ ` p ` q `
1
B
r E
1 B A 1 AB eEǫ Be Eγ 2A2 1 AB e2Eκ 2ABτ.
?2L r “ p ` q ´ ´ ´ p ` q ´
1
B
Thesetofequations(4.22)and(4.26)formsasystemofdifferentialequationsforthe
radial dependence of the unknown functions. In order to solveit asymptotically,we will
assumethatthefunctionshavethefollowingasymptoticbehavior,
r O r , x ,E O 1 , A,B O r 1 . (4.27)
1 ˘ 1´
“ p q “ p q “ p q