Table Of ContentAsymptotic properties of the development of
conformally flat data near spatial infinity
∗
Juan Antonio Valiente Kroon
School of Mathematical Sciences,
7 Queen Mary, University of London,
0 Mile End Road,
0
London E1 4NS,
2
United Kingdom.
n
a
J February 7, 2008
0
3
1 Abstract
v
6 Certainaspectsofthebehaviourofthegravitationalfieldnearnullandspatialinfinityfor
6 thedevelopmentsofasymptoticallyEuclidean,conformallyflatinitialdatasetsareanalysed.
1 Ideasandresultsfromtwodifferentapproachesarecombined: ontheonehandthenullinfin-
1 ityformalism related totheasymptoticcharacteristic initialvalueproblemandontheother
0
the regular Cauchy initial value problem at spatial infinity which uses Friedrich’s represen-
7
tationofspatialinfinityasacylinder. ThedecayoftheWeyltensorforthedevelopmentsof
0
theclass of initial dataunderconsideration is analysed undersome existenceand regularity
/
c assumptions for the asymptotic expansions obtained using the cylinder at spatial infinity.
q
Conditions on the initial data to obtain developments satisfying the Peeling Behaviour are
-
r identified. Further,thedecayoftheasymptoticshearonnullinfinityisalsoexaminedasone
g approaches spatial infinity. This decay is related to the possibility of selecting the Poincar´e
v: group out of the BMS group in a canonical fashion. It is found that for the class of initial
i data under consideration, if the development peels, then the asymptotic shear goes to zero
X
at spatial infinity. Expansions of the Bondi mass are also examined. Finally, the Newman-
r Penrose constants of the spacetime are written in terms of initial data quantities and it is
a
shownthattheconstantsdefinedatfuturenullinfinityareequaltothoseatpastnullinfinity.
PACS: 04.20.Ha, 04.20.Ex,04.20.Gz
1 Introduction
Penrose’sdefinitionofasymptoticsimplicity isanattempttoprovideacharacterisationofisolated
systems in General Relativity [37, 38, 39]. It offers a framework on which diverse notions of
physicalinterestcanbedefinedandhandledinapreciseandrigorousmanner. Thereis,however,
anotoriouslackofnontrivialexampleswheretheassociatedformalism—seee.g. [26,36,39,55]—
hasbeenactuallybeenusedtoextractsomephysics. Theobviousreasonforthisisthescarcityof
radiative exact solutions to the Einstein field equations other than the boost-rotationsymmetric
ones [2]. A more conspicuous consideration in this regard is the fact that radiative spacetimes
have to be constructed starting from an initial value problem where, say, some Cauchy initial
data are provided. Up to fairly recently, there has not been a way of relating the properties of
the initial data to the radiative properties of the development. For example, how does an initial
data set have to be so that the resulting spacetime peels? —that is, the components of the Weyl
tensor in an adapted gauge having a distinctive decay in terms of powers of an affine parameter
of the generators of outgoing light cones [39, 42]. Another natural question on these lines would
be how does the ADM mass and the Bondi mass relate? Similarly, is there any relation between
∗E-mailaddress: [email protected]
1
theangularmomentumdefinedatspatialinfinityandthatdefinedatnullinfinity? Whichclasses
ofinitialdataallowtoselectthePoincar´egroupoutofthegroupofasymptoticsymmetries—the
BMS group— in a canonical fashion?
Arguably, there is a shortage of general results about the evolution of initial data sets for
the Einstein field equations near spatial infinity. This particular has made the idea of relating
the structure of initial data with radiative properties a daunting endeavour. However, work by
Friedrich[20,23]onthe regular initial value problem at spatial infinity fortime symmetric initial
datasetshasprovidedatooltoaddressquestionssimilartotheonesraisedinthefirstparagraph
through a systematic approach. Friedrich has introduced a representationof spatial infinity as a
cylinder —the cylinder at spatial infinity, — in stark contrast to the usual representation as a
I
point. The cylinder at spatial infinity can be regarded as a limit set of incoming and outgoing
characteristicsofthe Einsteinfield equations, andas such it happens to be a total characteristic.
Assuch,itallowstotransportinformationfromtheCauchysurfacetonullinfinitywithoutbeing
contaminated by any sort of boundary conditions.
The construction which led to the cylinder at spatial infinity allows an unfolding of the evo-
lution process, which in turn can be analysed in all detail and to the desired order. The latter
can be thought of as enabling the construction of a certain kind of asymptotic expansions which
are completely determined by the Cauchy data near infinity. Friedrich’s seminal work has been
extended in several directions, automatising, in some sense, the calculation of the asymptotic
expansions of the relevant field quantities [50, 49] and to include more general classes of initial
data sets [52, 53]. The analysis described in these references has exhibited the existence of a
hierarchy of obstructions to the smoothness of null infinity which goes beyond that arising from
a mere considerationof the linear aspects of the field equations as described in e.g. [20, 46]. The
existence of the hierarchy of obstructions has nurtured a rigidity conjecture on the smoothness
of null infinity which, broadly speaking, states that the only Cauchy initial data sets which have
a development with a smooth null infinity1 are those which are stationary close to infinity. Sim-
ilarly, a given degree of differentiability at null infinity, say Ck, would only be possible if the
initialdataisstationarytoagivenorder—whichdepends onk. Thekindofinitialdatarequired
can be constructed by means of the gluing techniques of [9, 10, 7]. Intuitively, the absence of
gravitationalradiationnearspatialinfinitywhichtherigidityconjectureasserts,suggeststhatthe
incoming radiation has to die off in the infinite future, and at the same time the system cannot
have emitted gravitationalwaves for all times in the infinite past.
Now, the calculation of the asymptotic expansions which arise from the construction of the
cylinder at infinity is performed in a gauge2 which is adapted to a Cauchy initial value problem
—remarkably,thegaugealsoallowstoreadoffthestructureandlocationofinfinitydirectlyfrom
the initial data. On the other hand, the discussion of the gravitational field near null infinity is
usually done using a gauge which is based on geometric structures of null hypersurfaces. Having
two different gauges, hampers, to a certain extent, extracting the physics of the system. It also
difficults the attempts to assess the relevance of the presence of obstructions to the smoothness
tonullinfinityintheasymptoticexpansions. Oneoftheaspectsoftheanalysisoftheasymptotic
behaviour of the gravitational field via the definition of asymptotic simplicity and related con-
structions is that it allows to rephrase the decay of the fields in terms of differentiability at the
conformalboundary. Thus,itisnaturaltoexpectthatthepresenceofobstructionstosmoothness
in the asymptotic expansions may result in a modified peeling behaviour —this idea has already
been explored in [51].
A first study of the transformation between the gauge used in Friedrich’s analysis of spatial
infinity andthegaugegenerallyusedtodiscussnullinfinityhasbeengivenin[24]. Inthe present
article their analysis will be extended to the developments of a class of initial data sets which
are not time symmetric. The consideration of initial data sets with a non-vanishing second
fundamental form will allow us to examine some time asymmetric aspects of the structure of
the conformal boundary of asymptotically flat spacetimes which up to now have remained terra
incognita. On the other hand, the discussion shall be restricted to conformally flat initial data
1Hereandinthesequel,smooth willalwaysmeanC∞.
2Inthisarticle,agauge choice isunderstoodas acertainchoiceofcoordinates, vector orspinorialframe,and
alsoachoiceofrepresentativeintheconformalclassofthe physicalspacetime.
2
sets. This is done for conciseness and for the ease of calculations and presentation. Previous
analysis with conformally flat initial data sets —see e.g. [50, 52]— have shown that most of the
crucial phenomena and structure observed in this setting admit a counterpart when discussing
non-conformally flat data sets —like in [49, 53]. Furthermore, the detailed understanding of
conformally flat initial data sets is of relevance in view of their use in the numerical simulation
of black hole spacetimes —see e.g. [27] for a recent discussion on this. A cautionary note is,
however,due: mostoftheconformallyflatinitialsetsconsideredinthenumericalsimulationsare
boosted. This is done on physical grounds. Here, initial data sets with linear momentum —i.e.
boosted— will not be considered. The reason being that, as it will be discussed in the sequel,
they are more technically involved and less smooth. The effects of this lower regularity are of
interest in themselves and will be discussed elsewhere.
Thearticleisstructuredasfollows: section2discussessomeideasofthebasissetupandfixes
some notation; section 3 describes some aspects of the “cylinder at spatial infinity”-formalism;
section4considersrelevantaspectsoftheclassofinitialdatasetstobeused;section5musesover
the asymptotic expansions that can be calculated using the cylinder at spatial infinity; section 6
goes briefly about the construction of a frame adapted to null infinity —the NP gauge— while
section 7 discusses how it is related to the gauge used in the expansions of section 5. This
article revisits the analysis carried out in [24] of “Bondi type” systems —what here is called the
Newman-Penrose gauge— and combines it with the asymptotic expansions for the development
of conformally flat initial data sets which have been calculated in [52]. In section 8 analysis
of the implications of these expansions and the related obstructions to the smoothness of null
infinity is done —under some existence and regularity assumptions. In section 9 the machinery
is used to calculate expansions of the spin coefficient σ on null infinity in the NP gauge. The
behaviourofthiscoefficientisrelatedtovariousphysicalaspectsofthetheoryofisolatedsystems.
The implications of these expansions are analysed. It is shown that if the spacetime peels, then
the behaviour of σ is such that it is always possible to single out the Poincar´e group out of
the asymptotic symmetry group in a canonical way. Expansions of the Bondi mass are also
considered. In section 10 the formalism is used to write down the Newman-Penrose constants of
thedevelopmentsoftheclassofinitialdataunderconsiderationintermsofinitialdataquantities.
These expressions allow to show that the constants at future null infinity are equal to those at
past null infinity.
There is much more to the aforementioned analysis than what is presented in this article.
In particular, most of the calculations involved in the analysis have been carried out using ex
profeso scripts in the computer algebra system Maple V. A first subset of the scripts constructs
the initial data; anothersubset determines the coefficients ofthe asymptotic expansions in terms
of initialdata quantities; finally a thirdsubset calculatesthe transformationbetween gaugesand
allows to calculate the asymptotic expansions of NP objects. In order to keep the presentation
at a reasonable length, the explicit expressions of intermediate steps will be omitted, and more
emphasis will be put on to the qualitative aspects of the results. Some familiarity with the use
of spinors —in particular the space-spinor formalism of [41]— will be assumed. Details of the
diverse other formalisms used—the cylinder at spatialinfinity, the formalismofnull infinity, the
construction of gauges, and how to relate them— is kept to the minimum necessary to motivate
the analysis and to point out the many subtleties arising. In any case, the reader is referred to
the various references cited for more complete details. Finally, are some concluding remarks and
a brief appendix containing some useful spinorial expressions.
2 Basic setup
Let ( ,g ) be a vacuum spacetime arising as the development of asymptotically Euclidean
µν
M
Cauchy initial data (S,h ,χ ). Let Cµ be the conformally invariant Weyl tensor of the
e αβ αβ νλρ
metric g˜ . Tilded quantities will refer to quantities in the physical spacetime, whereas untilded
ones willµdνenote,genereiceally,qeuantitiesin anunphysical —i.e. conformallyrescaled—spacetime.
Indices µ,ν,λ,... (second half of the Greek alphabet) are spacetime indices taking the values
0,...,3; the indices α,β,γ,... are spatial ones with range 1,2,3. The Latin indices a,b,c,...
will be used in spatial expressions which are valid for a particular coordinate system (usually a
3
Cartesian normal one). The indices i,j,k,... are spatial frame indices, while A,B,C,... will be
spinorial indices taking the values 0,1.
The initial data sets are taken to be maximal —that is, χ = χα = 0, and the metric to be
α
h negative definite —the latter in order to ease the later use of spinors. For simplicity and
αβ
e e
definiteness, it shall be assumed that the initial hypersurface contains only one asymptotic end
e
—that is, a region which is diffeomorphic to R3 minus a ball.
The analysis shall be concentrated in the region of spacetime near null and spatial infinity.
In order to discuss the behaviour of the development of the initial data sets in the asymptotic
region use will be made of the so-called conformal picture. Let i denote the point at infinity
correspondingtotheasymptoticendunderconsideration. Thepointiisobtainedbyconformally
compactifying the hypersurface with a conformal factor Ω which can be obtained by solving
S
theEinsteinconstraintequations—seesection(4). Let betheresultingcompact3-dimensional
S
manifold. Themanifoldisidentifiedinastandardwaywith i . Mostofthediscussionwillbe
S\{ }
carriedoutinasufficientsmallneighbourhood (i)ofradiusacentredoniandonthe resulting
a
B
associated domains of influence and dependence =J+( (i)) J−( (i)).
a a
N B ∪ B
Asit iscustomary,the discussionofthe initialdatawillbecarriedoutinterms ofthe tensors
h =Ω2h˜ =ϑ−4˜h , χ =Ωχ =ϑ−2χ˜ , ψ =Ω−1χ˜ =ϑ2χ˜ (1)
αβ αβ αβ αβ αβ αβ αβ αβ αβ
on the conformally rescaled manifold . In this work attention will be restricted to conformally
S
flat initial data sets and their developments. This assumption will prove fruitful and will allow
to infer features of the development of more general types of initial data sets. Under these
assumptions coordinates xa can be chosen such that:
h= δ dxadxb, in (i). (2)
ab a
− B
Unless otherwise stated, xa denote some normal coordinates with origin at i based on a h-
{ }
orthonormal frame e .
j
3 The Friedrich-gauge
In [20] a certain representation of the region of spacetime = J+( (i)) J−( (i)) —which
a a
canbe thoughtofasthatclosetonullinfinity, I =I− IN+,andspBatialin∪finityB, i0—hasbeen
∪
introduced. The standard representation of this region of spacetime depicts i0 as a point. In
contrast, the representation of [20] depicts spatial infinity as a cylinder —the cylinder at spatial
infinity. The technicalandpracticalgroundsforintroducing this descriptionhavebeen discussed
at length in the seminal reference. The original construction in [20] was carried out for the class
of time symmetric metrics with analytic conformal metric h . However, as seen in [52, 53], the
ab
construction can be adapted to settings without a vanishing second fundamental form.
Friedrich’s construction makes use of a blow up of the infinity of the initial hypersurface
to S2. The blow up of i to S2 requires the introduction of a particular bundle of spin-frames
over a(i). Let τAA′ be the spinorial version of a timelike vector τµ. Assume the normalisation
τAA′τBAA′ = 2. Let SU( a(i)) denote the fibre bundle of spinorial dyads δA A=0,1 = oA,ιA
B { } { }
over a(i)withprojectionπ,suchthatτAA′ =oAo¯A′+ιA¯ιA′. ThefibresofSU( a(i))areoftype
SU(2B,C)={tAB ∈SL(2,C)|τAA′tABt¯A′B′ =τBB′}. With this notation, definBe the set
= δ(ρ,tA ) SU( (i)) ρ <a, tA SU(2,C) , (3)
Ca B ∈ Ba | | B ∈
(cid:8) (cid:12) (cid:9)
(cid:12)
where ρ is an affine parameter of horizontal curves on SU( (i)) which project to geodesics on
a
(i) starting at i. The matrices tA SU(2,C) can beBregarded as “angular coordinates”.
Ba B ∈
Crucial for the applications is that, 0 = π−1(i) = ρ = 0 , the fibre over i, is such that
0 SU(2,C). Thus, working on CI(i) provides a re{present}ation of (i) where i has been
a a
I ≃ B
“blown up” to a sphere. Note, however, that while (i) is 3-dimensional, is 4-dimensional.
a a
B C
This extra dimension can be interpreted as giving rise to the spin-weight of the diverse spinorial
objects when lifted from (i) to . This issue will arise again in section 7.
a a
B C
4
I+
I+
I
τ
I0 S
ρ
I−
I−
Figure 1:
Figure 2: The region of the —unphysical spacetime— near spatial and null infinities. The con-
formal factor leading to this representation is given by Θ=ωΩ−2(1 τ2).
−
Theconformalstructureofthespacetime( ,g)isexploitedbytheintroductionofconformal
M
Gaussian coordinates based on a certain type of conformal invariants: conformal geodesics —
see e.g. [21]. Accordingly, the spinor field τAA′ will be thought of as being tangent to timelike
conformal geodesics which are orthogonal to (i). For a gauge based on conformal Gaussian
a
B
coordinates, there exists a canonical conformal factor, Θ —i.e. a choice of representative of the
conformal class of the metric g . It is given by
µν
κ2τ2 2Ω
Θ=κ−1Ω 1 , with ω = =ρ+ (ρ), (4)
(cid:18) − ω2 (cid:19) D ΩDαΩ O
α
| |
p
where Ω = (ρ2) is the conformal factor of the initial hypersurface —usually obtained from
O
solvingthe constraintequations—,κ=κ′ρ, withκ′ smooth,κ′(i)=1,andτ anaffine parameter
of the conformal geodesics. The function κ expresses the remaining conformal freedom in the
construction. For the ease of future calculations the choice
κ=ω, (5)
willbemade. Asitwillbeseen,withthischoice,thelocus ofnullinfinitycorrespondstoτ = 1.
±
Other choices are possible, and indeed, the calculations in [24] use the apparently simpler κ=ρ
which renders a more complicated expression for the locus of null infinity.
Inordertocompletetheconstructionofarepresentationof =J+( (i)) J−( (i))—the
a a
N B ∪ B
region of spacetime close to null and spatial infinity— consider the set = κ1/2 of scaled
a,κ a
C C
spinor dyads κ1/2o ,κ1/2ι . Define the bundle manifold
A A
{ }
= (τ,q) q , 1 τ 1 , (6)
a,ω a,ω
M ∈C − ≤ ≤
(cid:8) (cid:12) (cid:9)
see figure 1. The conformal factor Θ given(cid:12)by (4) can be used to define the following relevant
sets:
= (τ,q) ρ(q)=0, τ <1 ,
a,ω
I ∈M | |
± =(cid:8) (τ,q) a,ω(cid:12)(cid:12) ρ(q)=0, τ = (cid:9)1 ,
I ∈M ±
I± =(cid:8) (τ,q) a,ω(cid:12)(cid:12) q a,ω 0, τ =(cid:9) 1 ,
∈M ∈C \I ±
(cid:8) (cid:12) (cid:9)
which will be referred to as, respectively, the cyli(cid:12)nder at spatial infinity, the critical sets where
null infinity touches spatial infinity and future and past null infinity.
Consistent with the structure of the bundle space it is possible to introduce a calculus
a,ω
based on operators X , X and X —related to a baMsis of the Lie algebra su(2,C)— and the
+ −
derivatives ∂ρ, ∂τ. For this, use will be made of a frame cAA′ and the associatedspin connection
coefficients ΓAA′BC of . Conforming with the use of space-spinors one writes
∇
cAA′ =τAA′τEE′cEE′ −τBA′cAB, cAB =τ(AB′cB)B′, (8)
5
with
τEE′cEE′ = √12∂τ, cAB =c0AB∂τ +c1AB∂ρ+c+ABX++c−ABX−. (9)
It is also convenient to introduce an unprimed version of ΓAA′BC,
ΓABCD =τBB′ΓAB′CD, ΓAA′CD =ΓABCDτBA′, (10)
which will be decomposed as
1 1
Γ = (ξ χ ) ǫ f , (11)
ABCD √2 ABCD− (AB)CD − 2 AB CD
where χ agrees on (i) with the second fundamental form of the initial hypersurface.
ABCD a
B
The discussion of the curvature will make use of the spinor ΘAA′BB′, related to the Ricci
tensor of the Weyl connection which can be constructed from and a spinor fAA′ with spatial
∇
part given by fAB —fAA′ is related to the acceleration of τAA′. For later use, define
ΘABCD =τCA′τDB′ΘAA′BB′. (12)
TheremainingpartofthecurvatureisencodedintherescaledWeylspinorφ —thespinorial
ABCD
counterpart of the rescaled Weyl tensor, dµ =Θ−1Cµ .
νλρ νλρ
Proposition 1. There is a frame cAA′ on a,ω for which
M
τAA′cAA′ =√2∂τ, (13a)
τAA′fAA′ =0, fAB = τCC′ΓCC′AB, (13b)
−
τBB′ΘAA′BB′ =0, ΘABCC =0. (13c)
The description of in terms of the manifold , with the conformal factor Θ given by
a,ω
N M
(4), and the frame cAA′ of proposition 1, will be called the Friedrich gauge —F-gauge for short.
In many parts of the sequel it will be necessary to lift objects defined in the base manifold
to the manifold . Scalar and space spinorial objects can be lifted in a natural way, with
a,ω
Ntheir “angular depeMndence” expressed in terms of the matrices tA SU(2,C). Functions from
SU(2,C) to C canbe expandedinterms ofcertainfunctions T l wBh∈ichcanbe thoughtofas the
nk
lifts of spin-weighted spherical harmonics —see [20, 23] for a more extended discussion:
2n+1
Y ( i)s+2n−m T n−m . (14)
s nm 7→ − r 4π 2n n−s
TheoperatorsX ,X andX canbedefinedbytheiractiononT l . TheT l ’sformacomplete
+ − nk nk
set of orthonormal functions with respect to the Haar measure of SU(2,C) —this fact will be
used severaltimes.
4 Conformally flat data without logarithmic singularities
A discussion of conformally flat initial data which is well suited for the purposes here presented
has been given in [13]. Its translation into the language of space-spinors has been carried out in
[53, 54]. Here, the main results required are presented.
The space-spinorial versions of the momentum and Hamiltonian constraints are given by
1
DABψ =0, DABD ϑ= ψ ψABCDϑ−7. (15)
ABCD AB ABCD
8
The spinorial translation of the approach adopted in [13] —see e.g. [50]— suggests a solution of
the form —see the appendix for a definition of the irreducible spinors involved—:
ξ η¯ η
1 1
ψ = (3x x +h )+ (x y +x y )+ (x z +x y )
ABCD ρ3 AB CD ABCD ρ3 AB CD CD AB ρ3 AB CD CD AB
µ¯ µ
2 2
+2 (y y )+2 (z z ), (16)
ρ3 AB CD ρ3 AB CD
6
where
ξ =X2λR+A+ρQ, (17a)
− 2
η = 2ρ∂ X λR+X λI +ρX Q+iX J, (17b)
1 − ρ − 2 − 2 + +
µ =2ρ∂ (ρ∂ λR)+X X λR 2λR 2λR ρ∂ λI, (17c)
2 ρ ρ 2 + − 2 − 2 − 2 − ρ 2
with3
λ =X2λ=X2λR+X2λI =λR+λI, (18)
2 + + + 2 2
andλistheliftofaC∞functionon (i) i . Notethatanoverbardenotescomplexconjugation.
a
B \{ }
Finally,
J =3(iJ J )T 0 +3iJ T 1 3(iJ +J )T 2 , (19a)
1− 2 2 1 3 2 1− 1 2 2 1
3 3 3
Q= (iQ +Q )T 0 + (Q )T 1 + (iQ Q )T 2 . (19b)
2 2 1 2 1 2 3 2 1 2 2− 1 2 1
Intheaboveexpressions,A,denotesaconstantrelatedtotheexpansionsconformalKillingvector.
The term J expresses the angular momentum of the initial data, while Q is associated with the
so-calledconformalboosts. Notethatboostedinitialdatasets—i.e. withlinearmomentum—are
excludedfromtheclassunderconsideration. Iftheinitialdataissuchthatithasanon-vanishing
linear momentum, then it can be shown that the asymptotic expansion of ϑ around infinity will
contain logarithmic terms already at order (ρ2). As firstly noted in [20, 50], even completely
O
regular initial data sets —e.g. time symmetric, conformally flat— may give rise to obstructions
to the smoothness of null infinity. It is expected that initial data with logarithms will render
further obstructions. This kind of obstructions arising from particular non-smooth features of
the data have not yet been studied. Thus, attention is restricted to initial data sets which are as
regular as possible.
Even if the initial data sets are non-boosted, there is still the possibility of having solutions
to the Hamiltonian constraint whose asymptotic expansions contain logarithms. The conditions
under which the solutions to the Euclidean momentum constraint give rise to logarithm-free
solutions of the Hamiltonian constraint have been spelled in [13]. From their results it follows
that if
λ♭
λ= +λ♮, (20)
ρ
where λ♭ and λ♮ are the lifts of C∞ functions on (i), then the asymptotic expansion of the
a
B
conformalfactor ϑ contains no logarithms. Recalling that the lift of an analytic function f to
a
C
admits an expansion of the form
∞ 2q
f = f T k ρp, (21)
p;2q,k 2q q
Xp=0q∈XQ(p)Xk=0
whereQ(p)= 0,2,...,p ifpisevenandQ(p)= 1,3,...,p ifpisodd,thefollowingassumption
{ } { }
is made:
Assumption 1. The class of initial data ( ,h ,χ ) under consideration is conformally flat
αβ αβ
S
and with a solution to the momentum constraint given by (16) where the lift of the complex
function λ to is of the form
a
C
∞ p 2q
λ= λ T k ρp. (22)
p;2q,k 2q q
Xp=2Xq=2kX=0
For the class of initial data comprised by assumption 1, the lift of the conformal factor ϑ to
admits the parametrisation
a,ω
M
ϑ=1/ρ+W, (23)
3Inolderaccounts—seee.g. [33]—,λR andλI arereferredtoas,respectively,theelectric andmagnetic parts
2 2
ofλ2.
7
where
∞ p 2p
m 1
W = + W , W = w T k , (24)
2 p! p p p;2q,k 2q q
Xp=1 Xq=0Xk=0
with m the ADM mass, and w C satisfying reality conditions so that W =W.
p;2q,k
∈
4.1 A further regularity condition
Attention will be now given to a different type of regularity condition on the initial data. In
[20] it has been shown that, generically, a specific class of obstructions to the smoothness of null
infinity will be present unless certain coefficients in the expansion of the spinor φ on the
ABCD
initial hypersurface, , satisfy a particular relation. In what follows, objects expressible in terms
S
ofinitialdatawhichconroltheregularityofthedevelopmentatnullinfinitywillbereferredtoas
obstructions. The obstructions described in [20] are relatedto the structure of the principal part
of the evolution equations (30b), and are shared by zero-rest-mass fields —see [46]. For the case
of time symmetric data sets these conditions on the Weyl spinor can be rewritten as a condition
on the Cotton-Bach tensor. Namely:
D D b (i)=0, p=0,1,..., (25)
(ApBp··· A1B1 ABCD)
where b is the spinorialversionof the Cotton-Bachtensor. The way the above formula has
ABCD
to be modified in order to accommodate initial data sets which are not time symmetric is not
known. However, the conditions on the coefficients of the expansion of the Weyl tensor given in
[20]canbe calculatedexplicitly uptoanydesiredorderforconformallyflatinitialdatasets. One
finds that —cfr. with [53]—:
λ +λ¯ =0, (26)
p;2p,k p;2p,k
with p=2,3, , k =0,...,2p.
···
An explicit calculation shows that at least for p= 2,...,6 the conditions (26) are equivalent
to
D D Cℜ (i)=0, p=1,...,6, (27)
(Ap−1Bp−1··· A1B1 ABCD)
where
Cℜ =√2DE χℜ , (28)
ABCD (A BCD)E
χℜ = Ω2ψℜ , with ψℜ being the part of the general solution to the Euclidean mo-
ABCD ABCD ABCD
mentum constraint which is obtained arising from the real part of λ, that is, λR 4. It is noted
that for the class of data under consideration, Cℜ (i) = 0, hence this particular condition
ABCD
poses no restriction on λ. The equivalence of the conditions (26) and (27) to all orders p will be
discussed elsewhere.
5 Expansions near spatial infinity
Theevolutionofthespinorialquantitiescµ ,Γ ,Θ andφ describingthegeome-
AB ABCD ABCD ABCD
try ofthe manifold willbe givenby means ofthe conformal propagation equations [20,23].
a,ω
M
The procedure described in section 3, by which the point i0 is replacedby the cylinder at spatial
infinity , rendersregulardataandequationsfor the unphysicalspacetime. Moreover,itleadsto
I
an unfolding of the evolution mechanism which allows to analyse the process at arbitrary order
and in all details. This kind of analysis is made possible by the total characteristic character of
the cylinder at spatial infinity with regard to the conformal propagation equations — can be
I
understood as a limit set of incoming and outgoing light cones.
4ThetensorialequivalentofCABCD isgivenby
Ckl=Diχj(kǫijl).
8
In order to briefly discuss the equations implied by the extended conformal equations it is
convenient to write:
υ = cµ ,Γ ,Θ , (29a)
AB ABCD ABCD
φ=(cid:0)(φ ,φ ,φ ,φ ,φ ), (cid:1) (29b)
0 1 2 3 4
In this manner, one can write the propagationequations concisely in the form
∂ υ =Kυ+G(υ,υ)+Lφ, (30a)
τ
√2E∂ φ+AABcµ ∂ =B(Γ )φ, (30b)
τ AB µ ABCD
where K, G denote, respectively, linear and quadratic functions with constant coefficients, L de-
notes a linear function with coefficients depending on the coordinates via Θ and related objects,
while E denotes the (5 5) unit matrix, AABcµ are (5 5) matrices depending on the coordi-
× AB ×
nates and B(Γ ) is a linear matrix-valued function of the connection Γ with constant
ABCD ABCD
coefficients acting on φ. The reader is referred to [20] for a full list of the equations.
Theevolutionequations(30a)and(30b)aretobesupplementedwithinitialdatawhichcanbe
readilycalculatedonceaninitialdataset( ,h ,χ )satisfyingthevacuumconstraintequations
αβ αβ
S
is known —see [52] for the details.
Theunfoldingoftheevolutionprocessreferredtointhepreviousparagraphscanbethoughtas
the calculationofa certaintype ofasymptoticexpansionsforthe diversefieldquantities grouped
in the vectors υ and φ. Consistent with the class of initial data under consideration —which is
expandable in terms of powers of ρ— one can put forwardthe following expansion Ansatz:
1
υ υ(p)ρp, (31a)
∼ p!
Xp≥0
1
φ φ(p)ρp. (31b)
∼ p!
Xp≥0
The upper limits of the above summations have been intentionally left open to emphasise some
convergenceissueswhichwillbediscussedlater. Transportequationsfortheaboveequationscan
be calculated by differentiating the propagation equations (30a)-(30b) and then evaluating at
I
—that is, ρ=0. These equations are schematically of the form
∂ v(p) =Kv(p)+G(v(0),v(p))+G(v(p),v(0))
τ
p−1
+ G(v(j),v(p−j))+L(j)φ(p−j) +L(p)φ(0), (32a)
Xj=1(cid:16) (cid:17)
√2E+AAB(c0 )(0) φ(p)+AAB(cC )(0)∂ φ(p) =B(Γ(0) )φ(p)
AB AB C ABCD
(cid:0) (cid:1)p
p
+ B(Γ(j) )φ(p−j) AAB(cµ )(j)∂ φ(p−j) , (32b)
Xj=1(cid:18)j(cid:19)(cid:16) ABCD − AB µ (cid:17)
These equations will be referred to as the transport equations of order p. A crucial feature
of the transport equations is that their principal part is universal. The solution for p = 0
is also universal —that is independent of the initial data— and agrees with the solution for
Minkowski spacetime. For p 1 they are linear differential equations for the unknowns of
≥
order p. Furthermore, note that the subsystem (32a) consists only of differential equations. The
transport equations (32a)-(32b) are decoupled in the following sense: knowledge of φ(j), φ(j),
j = 0,...,p 1, together with the initial data v(p) allows us to solve the subsystem (32a) to
I0
− |
obtain the quantities v(p). With v(k), φ(l), k = 0,...,p and l = 0,...,p 1 and the initial data
−
φ(p) at hand one could, in principle, solve the equations (32b) to get φ(p).
I0
|
Inpractise,inordertocarryouttheabovedescribedalgorithmonemakesuseofthefactthat
allthefieldquantitiesappearingintheevolutionequations(30a)-(30b)haveadefinitespin-weight
and expansiontype. Thus, the coefficients v(j), φ(l), 0 j,l p can be rewritten in terms of the
≤ ≤
9
functions T m and τ-dependent coefficients. For example, for the coefficients appearing in the
n k
components of the rescaled Weyl tensor one writes:
p 2q
1
φ = a (τ)T k ρp. (33)
j p! j,p;2q,k 2q q−2+j
p≥X|2−j|q=X|2−j|kX=0
The products of functions T m which naturally arise in (32a)-(32b) are linearised by means of
n k
the use ofthe Clebsch-Gordancoefficients ofSU(2,C) —note thatthe products appearingin the
principal part of the equations are somehow trivial as they involve the function T 0. The above
0 0
“decomposition in terms of spin-weighted spherical harmonics” allows for an implementation
in a computer algebra system5 whereby the diverse transport equations in the hierarchy and
their solutions are explicitly calculated to a given order p. The order p is restricted by the
computer resources available —generally up to p = 7 for a spacetime without axial symmetry.
Theassumptionofaxialsymmetryallowstocarrythe calculationsuptoacoupleofordersmore.
Animportantquestionintheprogrammeisthesense—ifany—inwhichtheexpansions(31a)
and (31b) approximate the solutions of the equations (30a)-(30b). In particular, one would like
to be able to estimate the residues
p
R (υ)=υ 1 υ(p′)ρp′, (34a)
p+1 − p′!
pX′=0
p
R (φ)=φ 1 φ(p′)ρp′. (34b)
p+1 − p′!
pX′=0
The estimation of the above residues and proving that the asymptotic expansions actually do
approximate the solution to the conformal field equations is one of the remaining outstanding
problems in the analysis of the structure of spatial infinity. It is expected that some suitable
extension of the estimates discussed in [22] in the case of the spin-2 massless field (linearised
gravity) will permit to remove this hurdle. Some of the technical complications of the problem
lie in the degeneracy of the principal part of the equations (30b) and (32b) at the critical sets
±, which precludes the use of standard methods of the theory of partial differential equation.
I
Further,asdiscussedin[20,23,50,52,53],this degeneracytranslatesinanon-smoothbehaviour
of the solutions of the transport equations —which presumably will also extend to the actual
solutions of the field equations. It can be seen that for the data of assumption 1 and if the
further regularity conditions (26) hold, then up to p=4 the coefficients in (31a) and (31b) have
all polynomial dependence in τ. Starting with p = 5, logarithmic terms which are singular at
τ = 1 appear. This adds further complexity to the analysis of the asymptotic expansions.
±
Hence, most of the calculations of expansions are stopped once the first logarithmic divergences
have appeared.
In order to proceed with the analysis of this article, the following assumption will be made:
Assumption 2. For the class of initial data under consideration, the propagation equations
(30a)-(30b) have a uniquesolution in . Further, the expansions (31a) and (31b) where their
a,ω
M
coefficients are given by the solutions of equations (32a) and (32b) approximate these solutions in
the sense that the residues (34a) and (34b) can be suitably estimated and are finite in the region
of interest. The regularity of the solutions is at least that of the τ-dependence of the expansions.
6 The Newman-Penrose gauge
This section discusses a certain class of gauge conditions on the spacetime ( ,g ) near each
µν
component of null infinity —i.e I+ and I−— known as the Newman-PenMrose gauge —NP
gauge. This class of gauge conditions is convenient for the analysis of radiative properties of
the spacetime, like its peeling behaviour and its mass loss. The NP gauge consists of certain
requirements on the conformal gauge of the spacetime, together with adapted coordinates and
5Thecomputer algebrasystemmaple Vhasbeenusedforthispurpose.
10
