The asymptotic of static isolated systems and a generalised uniqueness for Schwarzschild

0.16 MB·English

by  Martin Reiris

The asymptotic of static isolated systems and a
generalised uniqueness for Schwarzschild.
MARTIN REIRIS
[email protected]
MAX PLANCK INSTITUTEFU¨R GRAVITATIONSPHYSIK
Am M¨uhlenberg 1 D-14476 Golm, Germany
Itisprovedthatanystaticsystemthatisspacetime-geodesicallycompleteatinfinity,and
whosespacelike-topology outsideacompact setisthatofR3 minusaball, isasymp-
5
toticallyflat. Thematterisassumedcompactly supportedandnoenergyconditionis
1
required. Asimilar(thoughstronger)resultappliestoblackholestoo. Thisallowsus
0
2 tostatealargegeneralisationoftheuniqueness oftheSchwarzschildsolutionnotre-
quiringasymptoticflatness. TheKorotkin-Nicolaistaticblack-holeshowsthat,forthe
n
givengeneralisation,nofurtherflexibilityinthehypothesisispossible.
a
J
PACS:02.40.-k,04.20.-q.

6
Keywords:GeneralRelativity,staticsolutions,asymptotic.

]
c
q 1 Introduction
-
r
g AsymptoticflatnessisthebasicnotionusedinGeneralRelativity(GR)tomodelsystemsthatcan
[
be thoughtas “isolated” from the rest of the universe. It was used by Einstein himself at least

1 in heuristic form and is now a standard piece of differential geometry and of gravitational and
v theoreticalphysics.
0
The notion of asymptotic flatness is also epistemologically linked to the Newtonian theory
8
1 of gravitation. {1}In the 1916 manuscript The Foundation of the Generalised Theory of Rela-
1 tivity, Einsteinaddressedwhathecalledanepistemologicaldefect(butnotmistake)ofclassical
0 mechanics,whoseoriginhelinkedto E.Mach. Heimaginedtwobodies,A andB, madeofthe
.
1 samefluidmaterialandsufficientlyseparatedfromeachotherthatnoneofthepropertiesofone
0
couldbeattributedtotheexistenceoftheother. Observersatrestinonebodyseetheotherbody
5
rotatingataconstantangularvelocity,yetthesesameobserversmeasureaperfectroundsurface
1
: inonecaseandanellipsoidofrotationintheothercase. Hethenasked: “Whyisthisdifference
v
betweenthetwobodies?”. Necessarily,hecontinues,theanswercannotbefoundinsidethesys-
i
X temA+Bonly; Itmustlie in itsexterior: theouteremptyspace. Einsteinfoundthatthe source
r ofthepeculiardisparitywasomittingthattheemptyspaceshouldalsoobeyphysicallaws. These
a
laws,whichtreatthepartsAandBofthesystemA+B+EXTERIOREMPTYSPACEonanequal
footing, are the Einstein equations of GR. There is one point in Einstein’s elegant conclusions
thatis left slightly inconclusive. Itcan be arguedonthe base of GR, thatthe absolute space of
the18thand19thcenturieswasaninevitableconcept,as“corrections”totheNewtoniangravity
are simplytoo small. Thoughthisis unquestionable,it can also be demandedto GR to explain
too, whythis “backgroundsolution”, representingthe EXTERIOR EMPTY SPACE of the system
described earlier, is so distinguished in a theory that treats the geometry and the asymptotic of
space,essentiallyasavariable.
We findthenthataproblemofsometheoreticalimportanceistoanalyseasymptoticallyflat
(AF)solutionswithinthesetofsolutionsofGeneralRelativityandtofindcontextsinwhichthey
are indeed inevitable. Regardless of the “aesthetic” motivation just described, the study of the
{1}ThefollowingpassageismadeuponatextpreparedbymetoahighlightinCQG+.
1
1 INTRODUCTION
asymptoticofspacetimesisofcourseinterestinginitselfandcanproviderelevantinformationon
structureofsolutionstotheEinsteinequations.
Togiveourresultaframework,weredefineherestaticisolatedsystemsinthesimplestpos-
siblewaywithoutassumingasymptoticflatnessatinfinity. Weprovethenthatthesesystemsare
necessarilyAF.Thedefinitionofstaticisolatedsystemisasfollows.Theregionofthespacetime
outsidesomesetcontainingthesourcesshouldoftheform
M R R3 B3 , g N2dt2 g (1.1)
= ×( ∖ ) =− +
wherehereB3istheunitopenballinR3,N 0isthelapsefunction(thenormofthestaticKilling
field),andgisathree-metricinR3 B3. M>oreoverthisspacetimeregionshouldbegeodesically
∖
completeuntilitsboundary,namely,spacetimegeodesics(ofanyspacetimecharacter)eitherend
atitsboundaryoraredefinedforinfiniteparametrictime.
Admittedly, the topologicalcondition, (whichas we will se below is fundamental),is moti-
vated mostly by historicalconsiderations, althoughof course, to modela system like a neutron
star, it is meaningless to make any other choice. On the other hand the geodesic completeness
until the boundary of (1.1) is the most basic condition that one can impose to ensure that the
spacetime is, roughlyspeaking, “endless”. From now on we will call it geodesic completeness
at infinity; This terminology is justified by the following fact: geodesic completeness until the
boundaryholdsiffeveryspacetimegeodesic,whoseprojectionintoR3 B3 leavesanycompact
∖
set,iscomplete.
Inthissetupweprove,
Theorem1.1. StaticisolatedsystemsareasymptoticallyflatwithSchwarzschildianfalloff.
ThistheoremisanexpressionoftheremarkableconsistenceofGeneralRelativityasaphysi-
caltheoryandshowstheinevitabilityofasymptoticflatnessincertaincontexts.
End AF
End not AF
Figure1:RepresentationofanAFendandanon-AFend.
Tounderstandtheimportanceandscopeoftheconditionsdefiningstaticisolatedsystems,let
usbringtwopurelyrelativisticexamplesintoconsideration. ThefirstistheSchwarzschildblack
hole. Itisastaticvacuumsolutionwithatopological-sphericalhole,itscurvaturedecaystozero
atinfinity,andthespacetimeisgeodesicallycompleteatinfinity.Yet,(thoughnotalwaysproperly
emphasised),Schwarzschilditisnottheonlystaticvacuumblackholesolutionin3+1dimensions
enjoying these attributes. The other solution we are referring to is the Korotkin-Nicolai static
2
1 INTRODUCTION
blackhole[7]. Itrepresentsatopologically-sphericalhole,notinsideanopen(infinite)three-ball
B3 asinSchwarzschild,butinsideanopen(infinite)solid-torusB2 S1. Itisaxiallysymmetric
×
andhastheasymptoticofastaticKasner[7]spacetime.Itsspaceisnotsimplyconnected;Forthis
reasonthehorizonisprolate,asitfeelstheinfluenceofitselfalonganaxisofsymmetryoffinite
length.TheparticularKasnerasymptoticisthesimultaneousresultofthepresenceoftheholeon
one side and of the non-trivialglobaltopologyon the other. Finite coversof the solution yield
staticspacetimeswithafinitenumberofblackholesinequilibrium.Fromthepointofviewofthe
Generaltheory of Relativity, the Korotkin-Nicolaiand the Schwarzschild solution are perfectly
acceptable,stilloneisAFandtheotherisnot. Thisshowsthat,inTheorem1.1,thetopological
assumptionrequiredforisolatedsystemscannotberemoved.
The proof of Theorem 1.1 is based on the results [8], [9] where AF was proved under the
extrahypothesisthat(outsideacompactset)N isboundedfrombelowawayfromzero{2}. This
hypothesis was used only to guarantee that the conformal metric N2g is metrically complete,
propertythatwasusedfundamentally. Ina sense, allthatwe dointhisarticle isto removethis
undesiredhypothesisonN butforstaticsolutions. WeshowthatthecompletenessofN2gholds
alwaysinstaticisolatedsystems,aswedefinedthemearlier.Thetechniquesofthisarticledonot
applydirectlyto strictlystationarysolutions(cf. Remark2.2). Thequestionofwhetherstrictly
stationaryisolatedsystemsarealwaysAFisstillopen,though,(asshownin[8],[9]),theyareAF
whenthenormoftheKillingfieldisboundedfrombelowawayfromzeroatinfinity.
AlongthesamelinesasinTheorem1.1, wecangeneralisethecelebrateduniquenessofthe
Schwarzschildsolution(Israel[6]{3},Robinson[10],Bunting-MasoodUmAlam[3])toaunique-
nessstatementamongan (a priori)muchlargerclass of static solutionsthan those AF. Accord-
ingly,weconsiderstaticsolutionsgivenbyavacuumstaticdata S ;g,N ,i.e. with
( )
NRic N, D N 0, (1.2)
=∇∇ =
andwithcompactbutnotnecessarilyconnectedhorizon¶ S N 0 .Asearlier,thesolutions
={ = }≠∅
aresaidtobegeodesicallycompeteatinfinityifspacetimegeodesics,ofanyspacetimecharacter,
eitherendatthehorizon(i.e. theboundary)oraredefinedforinfiniteparametrictime.
Thetheoremisthefollowing.
Theorem1.2. Let S ;g,N bethedatasetofastaticvacuumspacetimewithcompacthorizonand
( )
geodesicallycompleteatinfinity.Then,thespacetimeisSchwarzschildiffaconnectedcomponent
ofthecomplementofacompactsetinS isdiffeomorphictoR3 B3.
∖
Observethatinthisstatementnothingissaidaboutthe(ifany)otherconnectedcomponentsof
thecomplementofthecompactset. Inprincipletherecouldbemanyotherunboundedconnected
components. That this cannot happen must be discerned after some analysis. This is in spirit
similarto“topologicalcensorship”-typeoftheoremsasin[4],althoughourtechniqueisdifferent
aswecannotrelyonanygivenstructureatinfinity.
Inparalleltothediscussiongivenatthebeginningoftheintroduction,itisworthnotingthat
Theorem1.1canbeinterpretedasaresulton“asymptoticuniqueness”,(hereasymptoticflatness),
and that, in this sense, it is a close relative of the uniqueness of the flat Minkowski spacetime
amongcomplete(simplyconnected)vacuumstaticspacetimesprovedbyM.T.Andersonin[1].
Anderson’s result is a direct consequence of a curvature decay that we will explain in Section
{2}Thedefinitionofstaticisolatedsystemin[8]isthesameastheonehere,exceptthatitincludesthehypothesisthatN
isboundedfrombelowawayfromzero,seetheremarkinsidetheproofofTheorem1.1.
{3}Israelbreakthoughin1967,wasthefirstuniquenesstheoremforSchwarschildandrequiredthatNcouldbechosen
asaglobalradialcoordinate.
3
2 BACKGROUND MATERIAL.
2.1. Westresshoweverthatsuchdecayisnotnearlysufficienttodeduceasymptoticflatness. The
Korotkin-NicolaisolutionsatisfiesthiscurvaturedecayandisnotAF.
The rest of the article is roughly organised as follows. Sections 2.1, 2.2 and 2.3 deal with
someimportantfactsabouttheglobalstructureofthevacuumstaticsolutions.Section3contains
theproofsofTheorems1.1and1.2. Proposition3.1showstheexistenceofanaturalpartitionof
staticendsoftheformR3 B3. Proposition3.2thenprovesthatthelapseN canhaveonlythree
types of behavioursat infi∖nity and Proposition 3.3 provesthe completenessof N2g on the end.
TheproofofTheorems1.1and1.2aregivenafterwards.
Acknowledgments. IamgratefultoMarcMarsforinterestingdiscussionsonrelatedtopics.
2 Background material.
A smooth Riemannian metric g on a smooth connected manifold S (with or without boundary,
compactornot)inducesthemetric
dist p,q inf length g g smoothcurvejoining ptoq . (2.1)
pq pq
( )= { ( )∶ }
Thespace S ;g issaidmetricallycompleteif S ;dist iscomplete. IfS hascompactboundary
thenmetric(com)pletenessisequivalenttothege(odesic)completenessuntiltheboundaryof S ;g ,
(by Hopf-Rinow). On the other hand, geodesics in S ;g lift to geodesics perpendicular(to th)e
( )
staticKillingfieldintheassociatedspacetime.i.e. in
M R S , g N2dt2 g (2.2)
= × =− +
Hence, if ¶ S is compact, geodesic completeness until the boundary of M;g implies metric
completenessof S ;g . ThisisusedinProposition3.3. ( )
( )
Geodesic completeness until the boundaryof M;g is a basic assumption in the two main
( )
theoremsin thisarticle. However,regardingpossible mathematicalapplications, it is important
whenpossibletoassumeonlythemetriccompletenessofthedata. Wewillmakesomeremarks
inthisrespect.
If¶ S ,wedefinethemetricannulusA a,b ofradii0 a bby
≠∅ ( ) < <
A a,b p S a dist p,¶ S b (2.3)
( )={ ∈ ∶ < ( )< }
wheredist p,¶ S inf dist p,q q ¶ S .
( )= { ( )∶ ∈ }
2.1 Anderson’s curvature decay.
Anderson’scurvaturedecay[1]isanimportantpropertyofstaticsolutions. Itsaysthatthereisa
universalconstanth 0suchthatforanystaticdata S ;g,N wehave
> ( )
h N 2 h
Ric p , and ∇ p (2.4)
∣ ∣( )≤ dist2 p,¶ S ∣ N ∣ ( )≤ dist2 p,¶ S
g g
( ) ( )
Theoptimalconstanth canbeseentobegreaterorequalthanone,butitisnotknowifitisone.
Upperboundscanbegivenbutfarfromone.
As anapplicationof the curvaturedecayletusproveherea propositionthatwill beused in
the proof of Theorem 1.2 to rule multiple ends when it is known that there is one that is AF.
In the statement we use S d to denote the manifold resulting from removingfrom S the tubular
4
2 BACKGROUND MATERIAL. 2.1 Anderson’scurvaturedecay.
neighbourhoodof ¶ S andradiusd , i.e. S d S p distg p,¶ S d . We assume thatd d 0
withd 0 smallenoughthat¶ S d isalwayssmo=oth∖. { ∶ ( )< } <
Proposition2.1. Let S ;g,N beastaticvacuuminitialdatasetwithcompacthorizon(¶ S N
0 )and S ;g me(trically)compete. Thenthereis0 e 1suchthatforeverye e th=e{rei=s
0 0
d }≠d∅0suchth(at )S d ;N−2e g ismetricallycompleteand<¶ S d<isstrictlyconvex(withre<specttothe
< ( )
inwardnormal).
Proof. Given0 e 1,theconvexityof¶ S d ford d 0smallenoughisdirect(andweleaveitto
thereader)asth<efa<ctorN−2e “blowsup”theboun≤dary¶ S uniformly,(observehoweverthat,as
e 1,¶ S “remains”atafinitedistancefromthebulkofS ).
<Soletusprovethat,ifwechosee smallenough,thespace S d ,N−2e g ismetricallycomplete.
As we are assuming d d 0, it is enough to prove that, if e(is small e)nough, S d ,N−2e g is
metricallycomplete.We<willdothatbelow,theargumentisthusindependentofd(. 0 )
Itisenoughtoprovethat,(ife smallenough),thefollowingholds:foranysequenceofpoints
pi whoseg-distanceto¶ S d diverges,the N−2e g -distanceto¶ S d alsodiverges. Equivalently,
itisenoughtoprovethatfo0ranysequence(ofcurv)esgistartingat¶ S0d andendingat piwehave
0
si 1
∫0 Ne gi s dsÐ→∞ (2.5)
( ( ))
wheresistheg-arclengthofgi startingfrom¶ S d . Thecurvaturedecay(2.4)impliesrightaway
0
theestimate,
N p c 1 distg p,¶ S d h (2.6)
( )≤ ( + ( 0))
foranyp S andwhereh 0isuniversalbutcdependson S ,g andd 0.Asdistg gi s ,¶ S d s,
∈ > ( ) ( ( ) 0)≤
thenwehave
N g s c 1 s h (2.7)
i
( ( ))≤ ( + )
Thus,ife 1 h then,
< /
si 1 ds si 1 ds 1 1 s 1−eh 1 (2.8)
∫0 Ne gi s ≥∫0 ce 1 s he = ce 1 eh (( + i) − )
( ( )) ( + ) ( − )
≥ ce 11eh ((1+distg(pi,¶ S d0))1−eh −1)Ð→∞ (2.9)
( − )
aswished.
∎
The importanceof Proposition 2.1 roots in that the Ricci curvature of the metric g˜ N−2e g
hastheexpression{4} =
1
R˜ic ˜ ˜ f ˜ f ˜ f (2.10)
=−∇∇ +c∇ ∇
where f andcdependone andaregivenby
1 1 2e e 2
f 1 e lnN, and ( − − ) (2.11)
=−( + ) c = 1 e 2
( + )
Inparticular,if0 e √2 1thenc 0. Thismeansthatc-Bakry-EmeryRiccitensorR˜icc given
f
< < − >
by
R˜icc R˜ic ˜ ˜ f 1 ˜ f ˜ f, (2.12)
f = +∇∇ −c∇ ∇
{4}Useforthisthat ifg˜=e2f gthenR˜ic=Ric−(∇∇f −∇f ∇f )−(D f +∣∇f ∣2)gandthat∇˜iVj=∇iVj−(Vj∇if +
Vi∇jf −(Vk∇kf )gij).
5
2 BACKGROUND MATERIAL. 2.2 TheBallCoveringProperty.
iszero.Wewillusethisfundamentallylater.
2.2 The BallCovering Property.
Asobservedin [2], Liu’sballcoveringpropertyholdsfor(metricallycomplete)static solutions
S ;g withcompactboundary.Namely,forany0 a bthereisr andn suchthatforanyr r
0 0 0
( ) < < ≥
thereisasetofballs B pi,ar 2 ,pi A ar,br ,i 1,...,nr n0 ,coveringA ar,br . Hereand
{ ( / ) ∈ ( ) = ≤ } ( )
belowAistheclosureofA.
Asadirectcorollarywehavethat,forany0 a bandr r ,asintheballcoveringproperty,
0
< < ≥
anytwopointsinthesameconnectedcomponentofA ar,br canbejoinedbyacurveoflength
( )
lessorequalthann0arentirelycontainedinA ar 3,3br .
( / )
LetAc ar,br beaconnectedcomponentofA ar,br . Bythecurvaturedecay(2.4)wehave
N N 3(h ara)lloverAc ar 3,3br .Integrating(thisin)equalityalongcurvesasintheprevious
∣∇ / ∣≤ / ( / )
paragraphweobtain
max N p p Ac ar,br
{ ( )∶ ∈ ( )} C a,b (2.13)
min N p p Ac ar,br ≤ ( )
{ ( )∶ ∈ ( )}
ThisisatypeofHarnackinequalityforN andisfundamental.
Remark2.2. Itisnotknownatthemomentifasimilarballcoveringpropertyholdsforstrictly
stationarysolutions.ThisisamainobstacletoextendTheorem1.1tostationaryisolatedsystems.
2.3 Spacetime geodesics instaticspacetimes.
Let S ;g,N beastaticvacuumdataandlet M,g beitsassociatedspacetime. Werecallherea
usef(ulway)todescribespacetimegeodesicsG( t i)ntermsofcertainmetricsconformaltoginS .
( )
ThisgoesbackatleasttotheworkofH.Weyl[12]from1917.
Letg P G betheprojectionofG intoS . Thenitisdirecttoseethatg satisfiestheequation
= ( )
g′g ′ a2∇N (2.14)
∇ = N3
whereg ′ dg dt andwhereaistheconstanta g G ′,¶ . Moreoverwehave
t
= / = ( )
g ′2 e a2 (2.15)
∣ ∣ = +N2
wherethenormonthel.h.siswithrespecttogandwheree 1,0,1accordingtothecharacter
=−
typeofthegeodesic.
Thendefinee2f by
e2f e a2 (2.16)
=( +N2)
wherever the right hand side is positive (this includes the projection of the geodesic). Finally
considertheconformalmetrics
gˆ e2f g, gˇ e−2f g. (2.17)
= =
anddenotebyds,dsˆ efds,anddsˇ e−fdstheelementsoflengthofg withrespecttog,gˆandgˇ
= =
respectively.
Inthissetupwehavethefollowingcharacterisation:IfG t isaspacetimegeodesictheng sˆ
isageodesicofgˆanddt dsˇ. Converselyifg sˆ isageode(sic)ofgˆthenthecurve ( )
= ( )
G sˇ sˇ a dsˇ′,g sˇ R S M (2.18)
( )=(∫ N g sˇ′ ( ))⊂ × =
( ( ))
6
3 THE PROOFS
isaspacetimegeodesicwithg G ′,G ′ e ,hencewitht sˇ.
( )= =
Two pointsare particularlyimportantaboutthis characterisationof spacetime geodesics, (i)
spacetime geodesics can be constructedout of the projectedcurveswhich in turn can be easily
foundthroughlength-minimisation,(ii)astheaffineparameterofspacetimegeodesicsisthegˇ-arc
lengthoftheprojectedcurve,awayisopenedtolinkspacetimegeodesiccompletenessatinfinity
tothemetriccompletenessofgˇ N2g. Wewillexploitthesetwoobservationsduringtheproofof
Proposition3.3. Wewilluseonl=ythecharacterisationofnullgeodesics,i.e. e 0,althoughother
=
typesofgeodesicscanbeusefulinsimilarcontexts.
3 The proofs
Every, smooth, connected, compact, boundaryless and orientable surface F embedded in R3
divides R3 in two connected components. Below we will work with such surfaces F embed-
ded in R3 B3 and will denote by M F to the closure of the bounded connected component
of R3 B∖3 F. Two facts are dire(ct)to check. First, for any disjoint F and F such that
1 2
¶ B(3 M∖ F)∖for i 1,2, then either F M○ F or F M○ F , (here ○ Interior). Second,
i 1 2 2 1
ifase⊂t F(,i) 1,...=,n 1 ofsuchsurfac⊂esis(suc)hthat¶ ⊂R3 be(lon)gstoabou=ndedcomponentof
i
S i=n{F th=enthereis≥at}leastoneF suchthat¶ B3 M F . Wewillusethesefactsintheproof
⋃i=1 i i i
∖ ⊂ ( )
ofthefollowingproposition.
Proposition 3.1. Let S ;g,N be a metrically complete vacuum static data set with S R3
B3. Then,thereis ase(tof(sm)ooth,connected,compact,boundarylessandorientable)su≈rface∖s
S ;j 0,1,2,3,... ,suchthatthefollowingholdsforevery j,
j
{ = }
1. Sj isembeddedinA 21+2j,22+2j ,
( )
2. ¶ S M S ,
j
⊂ ( )
3. M Sj M Sj+1 ,
( )⊂ ( )
ThesurfacesS willbeusedonlyasreferencesinsidethemanifoldS ,theirgeometriesplayno
i
¶rolMe.OSbj+s1erveMth○atSS j∖M. T(Shkis)l=as⋃tojj==bk∞seMrv(aStijo+n1)w∖ilMlb(eSuj)sewditwhhtheenuwneioanppdliysjothinetmanaxditmhautmSpj+r1in∪cSipjl=e
( ( )∖ ( )○)
toN onM Sj+1 M Sj .
( )∖ ( )
Proof. IntheargumentthatfollowswetreatS andR3 B3 indistinctly. Theconstructionofthe
surfacesSj, j 0,1,2,...isasfollows. Let f S → 0,∖ bea(any)smoothfunctionsuchthat
f 1on p di=st p,¶ S 21+2j and f 0on ∶p dis[t p∞,¶)S 22+2j .Letxbeanyregularvalue
of≡f in {0,1∶ . T(henwe)≤canwri}te f−1 ≡x F{ .∶.. F( whe)re≥eachF} isa(connected,compact,
1 n i
boundar(yles)sandorientable)surfaceem(b)e=dded∪inA∪21+2j,22+2j .Now,asS isthedisjointunion
of the sets f−1 x, , f−1 x i=∞F and f−1 ( ,x , an)d as p dist p,¶ S 22+2j
⋃i=1 i
f−1 ,x (w(e c∞on)c)lude t(ha)t=¶ S , which lies ins(i(d−e∞f−1))x, , m{us∶t bel(ong to)a≥bound}e⊂d
com(p(o−n∞ent)o)fS ⋃ii==1nFi. Hence¶ S M Fi∗ forsomeFi∗(,((se∞e)th)ebeginningofthissection).
WesetSj Fi∗. ∖ ⊂ ( )
=
WeverifynowthatthesurfacesS satisfytheproperties1-3. ByconstructiontheS ’ssatisfy
j j
○ ○ ○
already 1 and 2. Now, either M Sj M Sj+1 or M Sj+1 M Sj . If M Sj+1 M Sj
thenSj+1 p dist p,¶ S 22+(2j )w⊂hich(is im)possibl(ebec)au⊂se Sj(+1 )A 23+(2j,24+)2⊂j . T(hus),
M Sj M⊂○{Sj∶+1 ,(showin)g<proper}ty3. ⊂ ( )
( )⊂ ( ) ∎
We claim that, for any j 0, the surfacesSj+1 and Sj lie in the same connectedcomponent
oftheannuliA 21+2j,24+2j ≥. Toseethis,considerarayg s ,s 0,startingat¶ S ats 0,(i.e.
( ) ( ) ≥ =
7
3 THE PROOFS
dist g s ,¶ S sforalls 0; sisarc-length). Letsj bethelasttimethatg s Sj andletsj+1
bet(he(fi)rsttim)=e thatg s ≥Sj+1. Then,sj 21+2j becauseSj A 21+2j,22+(2j)∈andsj+1 24+2j
becauseSj+1 A 23+2(j,2)4∈+2j .Hencethea≥rc g s s sj,sj+⊂1 (mustlieinsid)eA 21+2j,≤24+2j
because dist⊂g s(,¶ S s fo)r all s. We con{clu(d)e∶th∈en[ that S]j}and Sj+1 must lie(in the sam)e
connectedco(m(po)nent)o=fA 21+2j,24+2j .
ThisclaimandProposit(ion3.1willb)eusedintheproofofthefollowingproposition.
Proposition3.2. Let S ;g,N beametricallycompletevacuumstaticdatasetwithS R3 B3
andN 0. Then,oneo(fthefo)llowingholds, ≈ ∖
>
1. N convergesuniformlytozeroovertheendofS ,
2. N convergesuniformlytoinfinityovertheendofS ,
3. C N C forconstants0 C C .
1 2 1 2
< < < < <∞
Proof. Toshortennotationwewillwritemax N;W max N p p W whereW arecompact
sets(samenotationformin N;W ). { }∶= { ( )∶ ∈ }
Supposethatthereisa{diverg}entsequence pi forwhichN pi →0asi→ . Weclaimthat,
inthiscase,N tendsuniformlytozeroovertheend. ( ) ∞
○
Foreveryilet jibesuchthat pi∈M(Sji)∖M (Sji−1). Supposefirstthat
max N;Sji →0 (3.1)
{ }
′
Then,foranyi ithemaximumprinciplegives
>
○
max N;M S M S max max N;S ,max N;S (3.2)
{ ( ji′)∖ ( ji)}≤ { { ji′} { ji}}
′
Lettingi → andusing(3.1)weobtain
∞
sup N p p S M○ S max N;S (3.3)
{ ( )∶ ∈ ∖ ( ji)}≤ { ji}
wherether.h.stendstozeroasitendstoinfinity. ThisprovesthatN tendsuniformlytozeroas
claimed.
To prove (3.1) we recall first that Sji and Sji−1 lie in the same connected component of
A 22j−1,22j+2 . Therefore,ascommentedinSection2.2,wehave
( )
max{N;Sji}≤cmin{N;Sji∪Sji−1} (3.4)
wheretheconstantcisindependentofi. Ontheotherhand,bythemaximumprinciplewehave
○
min{N;Sji∪Sji−1}≤min{N;M(Sji)∖M (Sji−1)}≤N(pi) (3.5)
Combining(3.4)and(3.5)weobtain
max N;S N p (3.6)
{ ji}≤ ( i)
wherether.h.stendstozero.Thisimplies(3.1)asdesired.
Inthesamemanneroneprovesthatifthereisadivergentsequence pi suchthatN pi →
asi→ thenN tendsuniformlytoinfinityovertheend. ( ) ∞
∞
Ifnoneofthesituationsconsideredaboveoccursthen0 C N C forconstantsC ,C .
1 2 1 2
< < < ∎
To show asymptotic flatness for isolated systems using [8], [9], we need only to prove the
completenessofN2gusingthatthestatic spacetimeisgeodesicallycompleteatinfinity. Thisis
doneinthenextproposition.
8
3 THE PROOFS
Proposition 3.3. Let S ;g,N be a static vacuum data set, with S R3 B3 and N 0 on S .
Assumethattheassoci(atedspa)cetime ≈ ∖ >
M R S , g N2dt2 g (3.7)
= × =− +
isgeodesicallycompleteatinfinity.Thenthespace S ;N2g ismetricallycomplete.
( )
Proof. Theproofisbycontradiction. Soletusassumethat S ;N2g isnotmetricallycomplete.
Wewillexplainlaterhowthiscontradictsthegeodesiccomp(leteness)atinfinity. Duringtheproof
weusethesamenotationasinProposition3.2. Wewillalsouse,aswasexplainedinSection2,
thatunderthehypothesisoftheproposition,thespace S ;g ismetricallycomplete.
Webeginbyprovingthat ( )
j=∞
max N;S 22j (3.8)
∑ j
j=1 { } <∞
Letb sj,sj+1 →M Sj+1 M○ Sj beanycurvewithb sj Sj andb sj+1 Sj+1. Weclaim
thatth∶e[n ] ( )∖ ( ) ( )∈ ( )∈
sj+1N b s ds c max N;S 22j (3.9)
∫ 1 j
sj ( ( )) ≥ { }
wheretheconstantc isindependentof j. Toseethiswrite
1
∫ sj+1N b s ds min N;M Sj+1 M○ Sj length b (3.10)
sj ( ( )) ≥ { ( )∖ ( )} ( )
andnotethat,
1. length b 23+2j 21+2j 622j,becauseSj A 21+2j,22+2j andSj+1 A 23+2j,24+2j ,
and, ( )≥( − )= ⊂ ( ) ⊂ ( )
○
2. min N;M Sj+1 M Sj max N;Sj ,because
{ ( )∖ ( )}≥ { }
○
min N;M Sj+1 M Sj min N;Sj+1 Sj (3.11)
{ ( )∖ ( )}≥ { ∪ }
bythemaximumprinciple,andbecause
min N;Sj+1 Sj c2max N;Sj , (3.12)
{ ∪ }≥ { }
where c is independentof j, by whatwas explainedin Section2.2, (see also the remark
2
aftertheproofofProp.3.1).
Theformula(3.9)isthenobtainedmakingc 6c .
1 2
Now, if S ;N2g isnotmetricallycomple=te,thenonecanfindasequenceofpoints p, with
i
distg pi,¶ S (→ b)utwithdistN2g pi,¶ S uniformlybounded. Fromthedefinitionofdist, this
impli(es that)ther∞e is a sequence of(curves)a s ; s 0,s starting at ¶ S and ending at p, for
i i i
which ( ) ∈[ ]
s=siN a s ds K (3.13)
∫s=0 ( ( )) ≤ <∞
whereK isindependentof j. Foreveryilet j bethegreatest j suchthat p M S . Then,for
i i j
every j ji 1onecanfindaninterval sj,i,sj+1,i suchthatthecurveb jdefine∉dby(b )j s a i s ,
s sj,i,≤sj+1−,i , has range in M Sj+1 [M○ Sj a]nd moreoverwith b j sj,i Sj and(b j)s=j+1,(i )
Sj∈+[1. Using(3].9)wewrite ( )∖ ( ) ( )∈ ( )∈
K s=siN a ds j=ji−1 sj+1,iN b ds j=ji−1c max N;S 22j (3.14)
≥∫s=0 ( ) ≥ ∑j=1 ∫sj,i ( j) ≥ ∑j=1 1 { j}
9
3 THE PROOFS
Takingthelimiti→ gives(3.8)aswished.
∞
Weproceednowwiththeproof.ByProposition3.2weknowthatNmustgouniformlytozero
atinfinityotherwiseN wouldbeboundedfrombelowawayfromzeroandthemetricN2gwould
beautomaticallycomplete.IfN→0uniformlyatinfinity,then S ;N−2g)ismetricallycomplete.
AswasexplainedinSection2.3,null-spacetimegeodesicsp(rojectinto N−2g -geodesicsand
theaffineparameteristhe N2g -arclength. Wewillseebelowthatif S ;(N2g i)snotmetrically
completethenthereisani(nfinit)e N−2g -geodesicwhose N2g -lengt(hisfini)te. Thiswouldbe
againstthehypothesisthatthespac(etime)isgeodesicallycom(plet)eatinfinityandtheproofwillbe
finished.
LetG s ,s 0bearayforthemetrictoN−2gandstartingat¶ S . Foreach j 1lets bethe
j
lasttimet(ha)tG ≥s Sj. LetG j betherestrictionofG to sj,sj+1 . ThenG j S ≥M○ Sj andG
istheconcatena(ti)on∈ofthecurvesG , j 1. Now, [ ] ⊂( ∖ ( ))
j
≥
s=∞N G s ds j=∞ sj+1N G s ds j=∞max N;S length G (3.15)
∫s=s1 ( ( )) = ∑j=1∫sj ( j( )) ≤ ∑j=1 { j} ( j)
wheretoobtaintheinequalityweusethat,
sup N G j s s sj,sj+1 sup N p p S M○ Sj max N;Sj . (3.16)
{ ( ( ))∶ ∈[ ]}≤ { ( )∶ ∈ ∖ ( )}≤ { }
which is obtained from the inclusion G S M○ S (for the first inequality), and from the
j j
maximumprinciple(forthesecond).Thus⊂,(ifw∖epro(ve)t)hatforaconstantc independentof jwe
3
have
length G c 22j (3.17)
j 3
( )≤
thenwecanuse(3.8)inconjunctionto(3.15)toconcludethat
N G s ds (3.18)
∫
( ( )) <∞
whichwouldimplythatthereisanincompletenullgeodesicinthespacetime.
Letusprovethentheinequality(3.17). WewillplaywiththefactthatG isarayforN−2g.
Firstnote
sj+1 1 length G j length G j
∫sj N G j s ds≥ max N(;G j) ≥ max N(;Sj) (3.19)
( ( )) { } { }
wherethesecondinequalityisobtainedfromtheinclusionG S M○ S andbecausemax N;S
j j
M S max N;S bythemaximumprinciple. ⊂ ∖ ( ) { ∖
j j
(Th)e}n≤recal{l from}the discussion after Proposition 3.1, that Sj and Sj+1 lie in the same con-
nectedcomponentAc 21+2j,24+2j ofA 21+2j,24+2j . Hence,G sj Sj andG sj+1 Sj+1
lie also in Ac 21+2j,2(4+2j . Then), as in(Section 2.2), we can joi(nt)G (∈sj )to G sj(+1 t)h(r∈ough )a
curveG ′ ofle(ngthlessor)equalthanc22j, (c isa constantindependen(to)f j), en(tirel)ycontained
j
inaconnectedcomponentAc 21+2j 3,324+2j ofA 21+2j 3,324+2j . ThiscurveG ′j musthave
N−2g)-lengthgreaterorequal(thant/he N−2g))-length(ofG /becauseG) ,(beingaray),minimises
j j
(the N−2g)-lengthbetweenanytwoofit(spoints.Thuswecanthewrite
(
sj+1 1 s′j+1 1 ′ c22j
ds ds (3.20)
∫sj N G s ≤∫s′j N G ′j s′ ≤ min N;Ac 21+2j 3,324+2j
( ( )) ( ( )) { ( / )}
10