Ihe space of solulions uf Einslein',; I:ljuations ncar it spac.:llnle with ...... denote the connected component of the identity of the Lie group. 11i/,.' and let .I1i1 .
ESSAYS IN U1iNI!ItAl Itl!lATIVITY
7 Symmetry Breaking in General Relativityt Arthur E. Fischer Departmenl of Mathematics University of California Santa Cruz. California
Jerrold E. Marsden Department of Mathematics Univllrsity of California Berkeley. California
and Vincent Moncrief Department of Physics Vale University New Haven. Connecticut
llifurcatiun Iheory is u~cd III 'lIIalyz.: Ihe space of solulions uf Einslein',; I:ljuations ncar it spac.:llnle with ~Ylllmelrics. The m.:thods dcvc:lopcd here allow one to describe: Jlrecisdy how Ihe sYllllllelry is broken as une brallche~ from a highl)' ~ymrnelric spacetime 10 nearby sra ,If lidlh .:ouplcd 10 gravity. sllch as gilllgc Ihenries. Since 11111,1 "I' Ih.: known 511huillilS Ilf Einslein', equalions have Killing lOynllllctrics. Ihe siudy of how Iliesc sYllullelrics ,III: brukclI hy SlIlilllllCrlllrbOlliulIS lakes lin cUIl~ldCfilble Iheorell.:al slllnilicancc.
t Itcscarch fllr Ihis work WIIS SUPllllrlCll ill pari by
NSF gnull PH Y7H-K21Sl.
79 (\)p),u.hl © 14180 b) A(4/.km..: ,',n... 10&:' All hllu, of ftptud\M.11t.NI til .uy lonu ,c.,e,wd ISIIN O·U·b¥IlSU-1
80
A. E. Fischer, J. E. Marsden. and V. Moncrief
7.
81
Symmetry Breaking in Generat Relativity
Bifurcation theory deals with the branching of solutions of nonlinear equations. Here we describe a new application of this theory to the determination of how the solution set of Einstein's equations branches near a spacetime with a one-parameter family of symllletries. The directions of these brunches arc not determined by the linearized theory of gravity alone, but arc completely characterized by the second-order terms. Thus the linearized theory of gravity ncar a spacetime with symmetry is not suflicient to capture the dominant clfccts of the nonlinear theory. These conclusions arc in accord with the earlier work of Fischer and Marsden [1-3], Moncrief [4-6J, and Arms and Marsden [7], which showed that for spacelimes with compact Cauchy surf.lces, a solution of Einstein's equations is linearization stahle if and only if it h.ls no Killing Helds. Our current work eXh:nds these results and describes precisely the geometry of the space of solutions of Einstein's equations in a neighborhood of.1 solution that has a single Killing vectollield. In particular, in a neighborhood of such a spacetime, ~he solutions cllluml be paramderizcd in .1 smooth wuy by elements of a hnear space, such as the space ofrour functions of three variables. (Sec Harrow and Tipler [X] for an application of this result.) We slmll begin by describing a theorem from bifurcation theory and give a simple example that is a prototype for whm is happening in relativity.
singularities of this type arc "conical" since they are determined by the second-order terms of their Taylor series. nle above result is proved by a method called" blowing up a singularity," wherein one considers the scaled equation
'n'eorL'm I i.el 4,: U~" -. U~ be II smooll, 1I1t/1'/'ill!1 .m( i.~/'yillll 1jJ(() = 0 (III(/ D4'(0) = () (i.e., (ill! mll/rix (Jl/l/vx J of purliul tlel'i"tlliIlI.'S of I/J pill/islu's tll (I'I! (}rigill). Lt!f
Silp/Jvse q, Illls a ncmc/eljl.'t/emle crifica/lllilllijtJld NcR ft fhrollgh 0; i.e., N is a su/ltlJat/ifultl 01/ wlJic1, DI/) = 0 tlllIl D 2 !/1(0) re.~(ric(etl to II space Iransverse 10 '/~I N (ils WIIOl!m SpilL'e illll,e origin) is ,/Omit/gillar. TJlfm (I'I! cut/ell/sium vf"'I! prl.'violls cl,eorem apply; ill partkular, (lJe Svlulions ol !/I(x) = 0 IIel" N Iltlve lire S"UClllre of u prodllCI ol a ('olle wi, II N.
= 0 tim/v'"
mltl .\'UPI'Wil! III(/( WIWIII!III:r Q(I,) .
cl J
Ims fllllli Ii, i.I!.,
(11(' mUJlIV 1-.
(12lji
(I) jilr .\ IIC'Clr 0 is Iwml!omol'plric 1(1 lire COllI! tJf SO/III iOlls of
(2)
Notice th:1l Ell· (2) is a set of Ii simultuneotls homogeneous qU&ldmtic eqll'2
X
SJ2'J
be the symplectic matrix on ·P.II. llere AJ is the space of olle-form densities on M; Sz is the space of symmetric two-covariant tensurficlds on M; sj is the space of symmetric two-contmvarinnt tensor d\!nsities 011 M; and ;'I' is the space of vector fields on M. The derivatives of I/I«fI,'" .'I, and their natural 1-2 adjoints arc related as follows: Letllllltl
(kcr('li"I/I,P')*)*
nW® (ker(D.1(g. n) J»*
( )*: SJ x S 2
.....
.
= 1),t(U. n) ,. J* = - 1),I(y, n)" J: SJ -. AJ;
(Ill. II) t-> D,I(y, n) ,( ~,~:).
(D.t(g.
0
u
S2 x SJ; «(0,11).-. «(Ill'. II "till (I)'
Proof (Here w' is the covariant form of (d and I,' the contravariant form of II.) D.I(g, n)*· X = (- LxX, Lxg). Hence J 0 D.I(g. n)*· X = (Lxg, Lxn) = (T;.t'/'«u .•,). X, and thus (T;.. "',~ .•,). = D.t(g, n) J* = - D.I(g. n)" J since J* = -J, The! splitting follows from injectivily of the symbol of D.I(g, n)* (sec Fischer and Mnrsden [17]). • 0
Lemma JJ Tile orllit f!J'fP' through (y, n) is u dosed su/mltJllifold of T*.II wilh wnyent space tit (g, n) given by 7;~,,,)(!JIIP' = range
J .. D.t(y, x)·
= {(LxY. Lx n)1 X is a vector field on M}. Proof
Let '!l: !iJ t-> T*. 1/; '11-> ('1*9, 'I*n). We haVe!
'r" '!leX) = (,,*L x !!, 'I*Lxn), Since X t-> txu is elliptic, 'l~ 'JI hus closed range! and linite-dimensional kernel. By the arguments of Ebin and Marsden «(23]. Appendix B), ker ~ 'f' is a subbundle of T!tI. It follows from the implicit function theorem that the range of ~p is an immersed submanifuld. From Ebin ([22], Proposition 6.13), it follows that ~I' is an open Illap onto its range and that the range is closed. The lemma then follows. II The proof of our results depends on it carefully constructed slice for the action of !iJ 011 T* .11, For the action of f;.1 on 4/. the Ebin-Palais slice theorem (sec Ebin [22]) asserts the existence of a slice. To avoid unnecessary technicalities. our slice will be an alline one. L2 orthogonal to the orbit of ~. Let Ifl = U E ~(M)lry = g},
1" =
'I;JI/I«p,=J.,(I),/(y.n»*:.'1'->sl x SJ;X. • (I'xy,l,x n ).
)
= range! 'f;J"'III .• ll)
= mnge! J
where ( )* dellotes tile L2 utljoilll map
J(I
O;J"'«U .• ,)*
(1.\'
We shall also need the following result:
where f*n = (f - I )~n is the pull-back of contravariant tensor densities. For (y, n) E T* .11, let
J
SJ Sl,/its ['2 ortlwyo/lully
Sz x S~
,lIld
An important ingredie:t in the analysis of the singularity that occurs in (8) about such a (y, x) is the Ebill' Palais slice theorcm (sec Ebin (22]) extended to a contangent bundle action. We now describe this analysis. Let II = Ricm(M) denote the space of Riemauuian mctrics on a compact manifold M,and let 9 = DilT(M)denotc the group of dille om or ph isms of M. Then !iJ ncts on by pull-b:tck. &IIld !iJ lifts natuntlly to n symplectic nction on the L1"'Cotangcnt bundle '1'* ,II;
89
Symmetry Breaking in General Relativity
If E !»(M)ll*n
= n},
and 1«11 •• , = I, () I" . Since the isometry group luis a compacl finite-dimensional Lie group. and since 1rr is a closed subgroup of fIJ(M). I «~ •• ' = I II () I. is also a compact
)
A. E. Fischer, J. E. Marsden, and V. Moncrief
90
finite-dimensional Lie group, I,".", is the isotropy group at (g. n) for the action of !» on 'f* .. II.
l'/wtJrem J2 Tire
Symmetry Breaking in General Relativity
Curull"ry 13 Let (0, x) E "'*.11. lIlId leI (g', x') e SI,. ""
IICl itm
be a ..,lice at (g, n).
Proof The inclusion follows from properly (ii) of a slice.
1'*,/1- '1'*,//;
litis a .~Ii(.'e Sell.• , c 1'*.11 til ellt:/1 (g, n) E 1'*. /I; i.e., SeQ .• ' is a slIbmwlijoltl 01 /I ctmwillillg (g, n) Sl/clr tllat
"'*.
(i) if I E lell.• ,. tlrell I*(S'II .• ,) = Sell •• ,; (ii) if IE fJ tlllIl 1*(SIi/••') ("\ S'II .• ' :j. ,p, t"ell IE 11110"';
and
I/u!re is a local cross sccliml X: !:pII !!I. n ,
--
!lJ.
defilled illlllleiglrbor/wOtI U ofl Ire illem;t y ,'oset sl/cll ,lIat lire map (,p. (g'. n'» t-t (x(,p»*Cg', n') is lI/wmeomorp/ristti of U X S'II.", 01110 lllwiylJlwr/wlJIl of (g. n) ill T·./I. JIII'tlrticular lire slice Sl" .• ,.~",eeps out U lIt'iyIl/JOr/wml of (g. n) IIlIller tire !Jroup tlCI iOIl. Tire Illllljl!tII slJCIee W till! slice at (Y. n) is give" by
= (kerO;".p'II.",)*)· = (ker(lJ.; I. the set IG,ASI,._I will have the structure of .. coneS on cones:' The result of Theorem 15 can be generalized to give the structure of the constraint equations
cJ)(Y. x)
= (.f{'Cg. x)• .I(g. x» =
in a neighborhood of ({I. n) e rtf
= ((f./f' A
l(f,..
which satisfies the hypotheses
Skelch of Proof of 1'!Jeorem 9 We have already remarked on the nece~sity of the second-order condition for integrability of first-order deformations 14111. To prove sufliciency, we lirst consider the c:lse that HI X is sp:lcelike. From the f~lct that I: h1ls constant mean curvature, one can show that 141 X is pllwllel to I:. so that the perpcmlicular - parallel decomposition of 141 X along I: is (141 X ~. 141 X tI) = (0, X). where X is now it vector field on the submanifold I: (identified with M). l.et (g. n) be the Cauchy data induced 011 E by 141g. In Moncrier (4J, it is proven that
= {(loll X 1,141 X 11)1141 X e.i ,••.,1.
where .$1'.'11 is the space of Killing vcctor fields for 14•g. In the case at hand. 14 1g has only a single spacclike Killing vector lield. so ker(l)(J)(y. nW is the one-dimensional space: spanned by (0. X). Moreover, Dt1l(y. n)· . (0, X) = 1>.1(0. n)· . X = (- Lx n. I.xy).
so that if (0. X) e ker Dcll(g. n)·. then L,d/ = 0 :lnd I. x x = O. Le:t 141yo e 8 ..... be" maximal dcvclopment of (I:. !/. n). su th;1I (14Igll • V ). Let 1"'11.. dcnote the isometry group OfHIYII and let 4
8 4"."..
=
{(4Ig'E~'......!J II .., = F-' .. /'.'d"/-" 111'41
4 (1 10,.
Vol)
£;
forsumcl:EDifJlV4 )}.
e II'''Itlu is the set of soIUli''IIs to the lield clluatinns with the same sJ'/11111t'try .. -.. v 4 I }'I'" as l·"!!o' Thus lJ"'II11 includes all solutions c 'O' related to HI!}'I hy a transformation P; lolly' = F· 1411111, hut it 1I1so includes other non isometric solutions as well. -I
95
Symmetry Breaking in General Relativitv
Let P be the projection to range D!J)(g, n), and define the function
v: p = Wi,
n) IPflJ(ii, ii) ::; O}
by
f(ll,w) =
L«(4IX.1,(4)XII)·~(g + ",n + w)
0
(i) dim .1 111 • _I = I: and (ii) tr x' = const. The conclusion' is identical to (a)-(c) of Theorem 15 with fC replacing {C,,; see Fig. 2. Thus the conical singularities in fCJ are carried over to. {t. Using this result on the structure of lC, we can now sketch a proof of 1 heorem 9 (see [25J for details).
ker(D
97 (",.,I),II.hl © ,",!So by A,·.d
