The Initial V nIue Problem

and the Dynamics of Gravitational Fields l)

J. E. Marsden (Berkeley)

This lecture will survey some of the recent advances that have been made in the dyna­mics of general relativity and other classical relativistic field theories. In addition, we shall indicate a few open problems that appear to be of basic interest.

1. Existence and Uniqueness Theorems for Geomctrodynamics

The basic existence and uniqueness theorem states that Cauchy data on a spaceIike hypersurface determines uniquely (up to spacetime diffeomorphisms) a piece of spacetime (filled with whatever matter or other fields arc under consideration) containing the hypersurface. Moreover, it makes sense to look at the maximal devel­opment of such Cauchy data, just as it makes sense to look at maximal integral curves of ordinary differcntiai equations.

The rigorous theory developing results of this type begins with Choquet-Bruhat [1948}, [1952]. The subject 8S it existed up until about 1972 is adequately presented in Hawking and Ellis [1973]. Some of the key developments since then are as follows, in more or less chronological order:

(a) Fischer and Marsden (1972a] show how to write the evolution equations as a first-order symmetric hyperbolic system.

(b) Miiller-zum-Hagen and Seifert [1977] study the characteristic initial value problem.

(e) Hughes, Kato and Marsden [1977] prove a. conjecture of Hawking and Ellis, showing that the equations are well posed, with the metric in HS, 8> 2.5. (See also Fischer nnd Marsden [1979a].)

(d) For asymptotically flat spaeetimes, Choquet-Bruhn.t and Christodoulou [1980] and Christodoulou [1080] prove well-posedness in the weighted Sobolev spaces of Nirenberg-Walker Ilnd Cantor ("SNWC spatcs"; sec Cantor [1979]). The crucial point here is to allow Hilbert spaces (el. McOwen [1979]).

(e) Christodoulou and O'llurchadha [1980] solve the boost problem in S~'WC spaces; i.e. they show that the piece of spacetime generated by the initial data is large enough at spatial infinity to include boosts. The methods may allow also for capturing a piece of Q .

I) Research partinlly supported by the National Science }'ouJI(llltion.

IIIj J. E. M aT~dC 1\ (Berkeley)

SO IllO olrclllrrohlclll ll for I.

(i) Gnuge. Problem: Current ex istencc theory i;; bmwd on the harmonic gauge of Choquet-Brllhal. l~ t here a direct. proof valid in nlly gauge?

(ii) Global Problem: }'ind a gauge (such us the constilll t mean cur'Vlllurc gauge, i,c, x-gauge) in which globa l exis tence hold;;. B,oc:en! work of Ch ristodoulou, Eardley Rnd Moncrief on Yang-Mills fields Rlld iIIax\\'c ll K lei n-Gordon fie lds g ives one hOJle thal the gravitnt i01l1l1 problem mil)' be &lIvable. (Scc Sega l {1 97!)], Moncrief [1!lSOn, b, c] and E Rrd lcy and Moncrief lIn80]), :Fo r globn lly hyper­bolic s pacctirncs with a com pact Ca uchy sur' fncc (coslllologiCII I cUBe) global cxistcnce in u constant IlWUIl curvature g,l llgO impl ies tlmt thc evo lu tion in that gauge caplurcslhc en lire spaeet il1l!J; sec below lind i'l larsdcn and Tipler [1!J80J. For non-campa!;! Ca nehy surfu ces, th is need not. he true: sec Eardley and Smarr [ 1979].

(iii) BU1wdarie8 (11Hl Om lI i/aliou(ll Shock..'l. Try to lowcl' 8 liVen below ~ ,5 ill the Cauchy problem to al low for jlllnp discontinu ities in t he second ueri valivcs of the metric (8 = 2.5 is the cJ'!wial vnlue, Lelow which jumps lire allowed). This wou ld a llow gravitational shocks. 'I'he sol utions lire preslImubly lIoll-unique if II < 2,5 find the physically corrcct ones arc picked oul by !lome kind o f entropy condition, as o ne does in gns dynamics. Call rct:cnt advances in g(;oll1etrie optics unci Fourier integra l o]Jcrntors (G uillcmin ami Sternberg [Hlii ]) he used ill the s tudy of grav itationa l \\'I\ I'e); und shoc\;:s~

2. Hamiltoniall Structures

The H amiltonian fOI'll1U lism in genera l relativity goes buck to Choquel -13ruhat, Dirac, Bergmann and Arnowitt·Deser find Misner'. Th is formalism (referred to com­mo nly ns the AD,\I formalism) is fou lld in , f{lrcxnmpl c, Misner, TlronlC lInd Wheeler [ Hli3].

This form ulism CUll bc cxh ibited liS folloll'8 (Fischct<\ [lIrsden llDiG, Ini!)}), Le t a s licing of spacet imc (\', (~ )y) be gi l'cn , bnSlxl on a 3-111!\nifold :'II, 'fIJi .'; s1i!;ing determines II eurvo y(i.) of Riernu nnill ll metrics on J'. l lind !l curve of sy mmetric tenoor densities n(),) (the conj ugate rnonu:nturn), Let the fl 1icing have!t la pse function Nand [( shift I'ector f ield X. E insl.ein·s 1'1ICII ll1ll eqtllllions Bin {(.Ily) = () (the Einstoin tensor formed from (41g) arc then t:q ui\'!dent to the !;volution cqualiorm in udjoint forrll

-. ..:... = Jl 0 Drp(g, :1)* .. ; a ( " ) (X) aI, :-t j\ ~ ~ ( 0 ') - I 0

together with tho cO Il ~lru i Bts

(IJ(y,:'f) = 0

wll erc (/J(y,:'f) = (Jf'(y, ;J). 5 (y, :1)) is the s uper energy-momentum.

Thl} Initi:d VailiC ProblCIil and t110 Dynnmics of Gravitational Fields 11 7

The choice of slicing is a gauge choice, and one rnrty wish to determine it a long with l hc dYIlIlIll ics. rn pR,-!icnlar, the conHtant mean curvature gauge is especially inter­cl'l ting, RS we 11IIve a lready noted. As was indicated illl)rofcssor Wheeler's lecture and in Qadir nnd Wheeler [19801, this ga uge has the property thnt its spaeelike hyper­!<urfaces tend to Ilvoid s ingulll,- ities_ I'f one Cl\ 1l show t hat ti le mean curvutllre Y. runs fro m -00 to 00 and ]\1 is compact (closed uni\' crscs) t hen the foliat ion fi1l5 out the whole Cnuehy developmen t nnd in fnct this devcloJlmeut is it " Wheeler universe" (see Tipler and Marsden [19801).

Some ot her developments of interest in th e Hamiltonian forlluilism a n; :

(a) The H am iltonian and sy mplectic struc tufes are in\'cs ligatL'l.i direelly [rom the four dimensiona l point of ... iew in Kijowski and Sze~.yrbR 11 07(;1, and Kijowski ami Tu lezyjow [J!J7!.l].

(b ) There has Ucen dC\'c1opmc nt of the idea that the I;onstrnint (/J = 0 is the salllens the vanishing of the Noether CUlTent ge nerat ed by tlllJ gauge grOllp of 1-e1ativity i. c_ all diffeomorphisms of V ((-quailing the identil)" at i11finity for open uni­verses); for relativity, sec l'i ~ ehor and Marsden [ HJ72hl, for gauge theories, !lee Cor-doro and Teit elboim 1197(;], Moncrief [19771 a nd Arms [1078].

(c) The Poincnr6 group at infinity or the R\l S group have Kocther current s of interest ns well, (nlthoug h we do not Rot them zero) HITch as the ADM cnergy­momcnllllll tenso,- or the B!\IS energy-momentum tensor ; sec llegge-Teitdboi m [1!)7-1] and AshtcJwr and Strellblc P9S0}_

(II) How r..s licings fit together with tho 13:\IS gro up and gl'Rv ilat iona l radiation has been invcstignted by Stlllllhiefl (l980l- (.For relnted information 011 x-slieill r,'S_ sec ChOqllct-BrTlllllt, }'i.~ eher Rnd !\Iarsdcn [Hl7D], Eunlie), lind Smurr lI!)7D), _'I Arsdon a nd Tipler [19S0 ] and 'l'reihcrgil {19S01 and references there in ).

(e) "l'eitcl boim L1 9771 a nd l)i ln ti [I !J78] huve investigated the gcomclrodynrullie_'! o f sllpergmvity .13uo [lD81] Ims pu t it onto the adjo int form ilhoye_

SOllie 0lu)/I Ilrohlc ms rol' 2.

(i) Find sufficient. condit ions 011 a relativistic field theory \\' ith a g iven e:nuge g roup to clIs urc tha t th e (-onstrai nt s in i~ Dirac nnalysis will be the zero level of the cor­rcsponding Noethcr current. (Th is is true fOI' al l lhc exnmplcs mentiOllctl above).

(Ii) How is tho classical Not:ther constmint "color charge = 0" for Yang-M ills field s on _\I inkowsk i SpA ce rclah.:<1 10 quark confineme nt? (Sec A fllI ~ , ,\Inrsdell and " Io nerief [ Hl80] for some disellssioll) _

(ii i) I s it truc Ili llt the long time dynam ics fo!' a typ ica l rela tivist ic field theory is chnOlid I s the K olmogorov-.-\rno ld-]\[oser theory rele .... ant? (Recen tly, Ba rrow has embarked 011 it very clllig ht enillg ill vestigatioll of MisTlcr 's llliXlIlIU, ler lIlodel fronl the point of dew of chaoti c dynaIII ical syste ms) _

118 J. E. :\larsden (Berkeloy)

3. Spaces of Solutions and Linearization Stability

Let V be a fixed four manifold and let {f be the set of nil globally hyperbolic Lorentz metrics g = (4)g that satisfy the vacuum Einstein equations Ein (g) = 0 on V (plus some additional technical smoothness conditions). Let flo E 8 he a given solution. We ask: what is the structure of C in the neighborhood of go? . There are two basic reasons why this question is asked. First of all, it is relevant to the problem of finding solutions to the Einstein equations in the form of a per­tnrbation series:

).2 g().) = Yo + )J~l + - It2 + ...

2

where;' is a small parameter. U y(l) is to solve Ein (y(i.)) = 0 identically in ). then clearly hi must satisfy the linearized Einstein equation/J:

D Ein (g) • 1~1 = 0

where D Ein (g) is the derivative of the mapping g I-+' Ein (g). For such a perturbation series to be possible, is it sufficient that hi satisfy the linearized Einstein equations, i.e. is hi necessarily a direction of linearized atability? We shall see that in general t.he answer is no, unless drastic additional conditions hold. The second reason why the structure oC tJ is of interest is in the problcm of quantizlltion of the Einstein equations. Whether one quantizes by means of direct phase space techniques (due to Dirac, Segal, Souriau and Kosbmt in vll,rious Corms) or by Foynman path integrals, there l will be difficulties near places where the space of cla8!~ical solutions is such that the linearized theory is not a good approxilllntion to the nonlinear theory.

The dynamiclll formulation mentioned in § 2 is crucial to the analysis of this problem. Indeed, the essence of the problem reduces to the study of structure of the space of solutions of the constraint equations $(g, n) = O.

As we shall see, the answer to these questions is this: Ii' has a conical or quadratic singularity at go if and only if there is Il nontrivial Killing field for Yo that belongs to the gauge group generating $ = 0 (thUll, the flat metric on ']'3 X IR has such Killing fields, but the )Iinkowski metric has none.) When 6' hns such a singularity, we spenk of a bifurcation in the space of solutions.

(a) Brill and Deser [1973] considered perturbations of the flat metric on '1'3 X IR and discovered tho first example of trouble in perturbation theory. They found, by going to a second order pcrturbatioll analysis, that they hud to readjust the first order perturbations in order to avoid inconsistencies at second order. This was the first hint of a conical structure for t' near solutions with symmetry.

(b) Fiseher and ~Iarsden [197:{] found general sufficient conditions for 0' to be a manifold in terms of the Cauchy data for vacuum spacetimcs.

(c) Choquet-Bruhat and Doser [1973] proved a version of the theorem that C is It

manifold near l\1inkowski space, which WIlS hiter improved by Choquet-Bruhat, Fischer and Marsden [1979].

The Initial Val.ue Problem and the Dynamics of Gravitational Fields 119

(d) :\Ioncrief [1975aJ showed that the sufficient conditions derived by Fischer and .i\clarsden for the compact case where equivalent to the requirement that (V, go) have no Killing fields. This then led to the link between symmetries and bi­furcations.

(e) Moncrief [1975b] discovered the general splitting of gravitational perturbutions generalizing Deser's [1967] decomposition. The further generalization to momen­·hun maps (general Noether currents) was found by Arms, Fischer and Marsden [1975]. 'l'his then applies to other examples such as gauge theory and also gives York's decomposition (York [1974]) as special cases.

(f) D'Eath [1976] obtained the basic linearization stability results for Robertson­Walker universes.

(g) Moncrici [1976] discovered the spacetime significance of the second order condi­tions that ariso when one has a Killing field and identified them with conserved quanti tics of Tn 11 b [1970]. Arms and Marsden L 19791 showed tha t the second order conditions for compact spacelike hypersurfaces are nontrivial conditions.

(h) The description of the conical singularity in If ncar a spacetime with symmetries is due to Fischer, Marsden and Moncrief [1980J for one Killing field and to Arms, Fischer, Marsden and Moncrie.f [1981] in the general casco

(i) :Moncrief [1977], Coll [1975] and Arms [1977, 1979] obtained the basic results for pure gauge theories and eleetromagnetislll and gauge theories coupled to gravity.

r (j) An abstract theory valid for arbitrary momentulU maps was developed by Arms, ~farsden and Moncrief {1980].

(k) Moncrief [1978] investigated the quantum analogues of linearization stabilities. Using 'f'3 X JR., he shows that, unless such conditions are imposed, the correspond­ence principle is violated.

For vacuum gravity, let usstnle one of the main results in the cosmological case: suppose Yo has It compact spacclike hypersurface M c:: V. (Actually we require the existence of at least one of constant mean curvature for teclmical reasons). Let Suo he the Lie group of isometries of go and let k be its dimension.

'fhcoreUl.

1. k = 0, then tS is n smooth manifold in a neighborhood of (/0 with tangent space at go given by the Rolutions of the linearized Einstein equations.

2. If k > 0 then If is not It smooth manifold at Yo. A solution hl of the linearized equa­tions is tangent to a curve in e if and only if 1~1 is Rllch that Taub conserved quanti­ties "anish; i.e. for every Killing field X for go.

J X . {D2 Ein (Yo) • (hl, "1)] . Z dPM = 0 .11

where Z is the unit normal to the bypcrsurfnce M and "." denotes contraction with respect to the metric go.

120 J . £. ~l[\Tildcn ( llcrk{·\ey )

,·\11 explicitly known solutions vosscss symmetries. so while I. is "generic" , 2. i~

what OCClI!"S in exa1l1ples. This I heol"c1l1 gi\'C5 a c01l1pletc answer to the pertu rbation qucstion: a perturba tion series is possible i f Rnd only if nil t he Tau b qua nt ilies vanish.

Let liS g ivc n brief 1Ihstract ind icalio n of why su ch sccond order conditions shou ld c01l1e in. Suppose X Rnd Y Il rc BRnnch spaccs und :F: X _ Y is n s mooth map. Sup­pOHC F(Xu) = 0 find x(i. ) is II cur·ve with z(O) = Xo flnd Ji( xU.») E"I O. l .(Jt hi = x '(O) so by the chnin rule ])F(xo) . hi = O. Ko\\" suppose nF(x,,) is not surjective and ill

" fnel supposc there is II lincnrfullctionrti' E Y· orthogona l to its range : (I, nF(xo) . II) = 0 for nil ·11. E X. By d iHerclltintillg F( x(i.») = 0 twice at i. = 0, wc ge l

D~F(xo) · (hlf ltd + Dl~(xo)' x"(O ) = O.

Applying I gh'cs

(I, D~F{xo) ' (hI> li d) = 0

whi<:!1 nrc nece:::s:l r)" second o rdcr conditions thnt rHllst be sa tisfi cd by hi. lt is by this genera l 1I1ethod that one nrrivcs fl.t lhe TRUh conditions. The illsufl

of whe t/rer or not [hellc conditions It rc s rrfficien t is 1Jl1IO": !! deeper requiring extens h'c ana lysis and bifu rcation theory (for J: = 1 th e i\ lo!"se lellllJla is USl.-o, while fo r f: > 1 the Kunlnishi dcforlllRI ion theory is need ed sec 1\ urnnish i [19051, A tiyalL , Hit chin lind Singer ]19i8]nlld § 'I below ).

SO IllO 0llen prohh:ms rur :J .

(i) Is the nhove phellolllcnon a p!)clI!inril)" nboll l vnelHlIll gra\·ity or is there an ahs\rucl Iheorell! nppli vnb[c to fI brond class of I"eJa ti" l;; tie field theories? T he exnmples wh id l have hevll a lld nrc heing \\"orkcIJ ou t l'> uggest t hat the lnller is lhe cnsc. Good exa mples !l rc lI re Yllng-~Jill s Nlrra t ions for gnuge theory (i\lonerief [Olii], Arms [WiU \) the 1':ills1eill-Dil'll(: equll iions (eL \'elson nnd 1'cilelho illl IWi8)}, thc Eins teill-E:ulc,· equations (Bao a nd ~' l nrSl l en 11981]) and Sllpcr­gravity (Pillili [1\IiS], Ib o \ 19811 ). In e:leh of these ex lt lllplct; thc!·c is It gll ugc group plnring ihe role o f the rliffcomorphisrrr group of s pacetime for \'a eUIIIll gravits . ThiH gauge group 1I<.:ls 01' Ihe rickls: whcn it fixe" a field, it is a symmetry fo!· Ihat field. The I·ela tion:;hip hel ll·cell sY!llIlietri lJ!' o f it fie ld fIIld singularitic;: in the RJlll ce of solutions of the classical eq uat ions is the n liS il is for VU CUIlIlI

gra\'i t.v. For· this progra m to ta rry t hrough, olle firsl writes tbe fOI\I' dilllcnsiorUiI equa­tions 11 8 Hllilriltorrian e \'o!lItion ef]lIlltioll fi p!lIS constrnintequfltions by mcans of the 3 + 1 procedures of Diral' . The (·onstrain l. eqlln tiolls then IIJUSl

I. oe Ihe N oel/a: r cOJ/serall q!III/!/ilic8 l or Ihe {j(WYC group /1 m! 2. sol lsly smne technical ell i1!/icil!l cOlldi/ioIlS : ( [)Ib )· musl OIl (In elfiptic opera/or . .'\s is alreudy been IIIClition('<1. for I, it lIlay be necessary to sir rink rh e gRlIge group so lllewha t, cspecially fOJ· s pn vclimcs lhat lire !lot sp:! tiaHy cornJJ~te l. For eXlllllple Ihe iso­rlll:lrie.'l of !\tinlw\\'ski t; pace do not. helollg to thc ga uge grOllp genernling the <.:ollslraints bllt rulhcr thcy geIH,r·at c the 10Ial ellcrgy-momell l ll lll veclor of the

The Initial Value Problem and the Dynamics of Gravitational Fields 121

spacetime ..• that this vector is time-like is the now famous positive energy problem ... sec Brill and Deser [1978], Choquet/Bruhat, Fischer and Marsden [1979], Deser and Teitelboim [1976] and Schoen and Yau [1979, 1980].

(ii) In the space of solutions, the kernel of the symplectic form coincides with the infinitesimal gauge transformations (this follows from Moncrief's decomposi­tion). Therefore, one can construct the space of true degrees of freedom, the quotient of C by the gauge group. Using Marsden-Weinstein [1974], one proves that this quotient is a smooth symplectic manifold near points where 6' is smooth. This leaves open the question: what is &'/(gauge group) like near points of sym­metry, where & is singularl

(iii) How should one treat the Schwarzschild solution in the context of linearization stability? Do singularities in the space of solutions affect spacetime singulari­ties in the sense of Hawking and Penrose? Do they affect Cauchy horizons1

~

4. Bifurcations of ~Iomentum Maps

The role of the constraint eqUlltions as the zero set of the Noether conserved quantity of the gauge group leads one to investigate zero sets of the conserved quantities asso­ciated. with symmetry groups rather generally. One goal is to begin answering qucs­tion (i) in the previous section. This topic is of interest not only in relativistic field theories, but in classical mechanics too. For example the sct of points in the phase space for n particles in .lR.3 corresponding to zero total angular momentum is an inter­esting and complicated. set, even for 11 = 2!

We shall present just a hint of the relationship between singularities and symme­tries. The full story is a long one; one finally ends up with an answer similar to thnt in vacuum relativity. We refer to Arms,l\Iarsden and Moncrief [1980] for additional details.

First we need a bit of notation (see Abraham and Marsden [1978], Chapter 4). Let 1\1 be a. manifold and let a. Lie group G act on M. Associated to each element ~ in, the Lie algebra ru of G, wc have a vector field ~l[ naturally'induced on ~I. We shall dcnote the action by 0 : G X M - M and we shall write 0 q : 1\1 - M for the transformation

of M associ~ted. with the group element g E G. This 'M(X) = ~ 0 expUl !(x)!t'-o' dt -

Now let (P, (.0) be a sympledic manifold, so w is n. c109Cd (weakly) non-degenerate two-form on P and let 0 be an action of a Lie group G on P. Assullle the action is symplectic: i.e. tP;w = lJJ for all g E G. A momentum mapping is a smooth mapping .J: P _ &* such that

(dJ(x) . f'z, ~> = w(,p(x), f'z)

for all ~ E (M, Vz E TzP where dJ(x) is the derivative of J at x, regarded lUI a linear map of TzP to OJ'" and (,) is the natural pairing between GJ nnd GJ*.

122 J. E. Marsdon (Berkeley)

A momentum map is Ad*-equh:ariant when the following diagram L'Ommutes for each g E G:

p CPg ---.... P

Jl 1"

<»* Ad;_1

, Gl*

where Ad;_l denotes the co-adjoint action of G on OJ"'. If J is Ad* equivariant, we eall (P, (I), G, J) a Ilamiltonian G-8pace.

Momentum maps represent the (Noether) conserved quantities associated with symmetry groups acting on phase space. This topic is of course a very old one, but it is only with more recent work of Sauriau and Kostant that a deeper understanding has been achieved.

See Fischer and Marsden [19i9] for the sense in whieh the map fj) described in § 2 is the momentum map associated with the group of diffeomorphisms of spacetime. See Moncrief [197i] and Arms [1979] for the corresponding result for gauge theory.

Let SZo = (the component of the identity of) {g E G I gxo = xol. called the symmc­try group of Xo. Its Lie algebra is denoted sz, so

Let (P. w, G. J) be a Hamiltonian G-space. If Xo E P, Po = ,J(xo) and if

dJ(xo) :TzP --)0 QJ*

is surjcctive (with split kernel), then locally J-l(po} is a manifold and (J-l(p) I p E ~*} forms It regular local foliation of a neighbourhood of Xo. Thus, when dJ(xo) fails to be surjective, the set of solutions of J(x} = 0 could faj) to be a lIlanifold.

Theorem. dJ(xo) is surjective if and only if dim S".. = 0; i.e. ",r. = (OJ.

Proof. dJ(xo) fails to bc surjective if there is It ~ =1= 0 such that (dJ(xol . vZ.' ,) = 0 for all to".. E T.r,P. From the definition of momentum map, this is equivalent to wzo(;p(xo), v"..} = 0 for all V.l'o. Since (I) .... is non-degenerate, this is, ill turn equivnlent to ,p(xo) = 0; i.e. "r. 9= (01:

One then goes on to study the structure of .J-l(pO) when Xo hus symmetries, by investigating second order conditions and using methods of bifurcation theory. It turn.':! out that, as in relativistic field theories, J-l(pO) has quadratic singularities characterized by the vanishing of second ordcr conditions. The connection is not an accident since the structure of the space of solutions of a relativistic field theory is determined by the vanishing of the momentum map associated with the gauge group of that theory.

The Initial Value Problem and the Dynamics of Gra"itational Fields 12:l

Some open problems for 4.

{i) All the results obtained so far on spaces of solutions arc local. What is the global stru(,ture? Is there a global Morsetype theory for momentum mapsl

(ii) ~Iuch current work on Yang-Mills fields and the twistor program for gravity utilize a Euclideanized viewpoint. Some routine calculations show that in such a context the connection between symmetries and bifurcations is lost. (In parti­cular, the symmetries discussed by Rebbi and Jackiw [1976] are not related to Euclidean linearization instabilities.) What has become of the difficulties with perturbation series and quantization encountered in the Lorentz context?

(iii) Bifurcation theory exploits connections between symmetry and bifUrcation to study phenomena like pattern formation. See for example, Golubitsky and Schaeffer [1979] and Sattinger [1980]. Can one use this theory in relativity to study physical consequences of breaking the symmetry of a solution of a rela­tivistic field theory?

References

R. Abmham and J. l\[arsdoIl [1978]. Foundations of Mechanics. Seeond Edition. Addison­Wesley.

J. Arms [1977]. Linearizution stability of the Einstcin-lfaxweJl system, J. math. Phys .• 18, 830-3.

- (1979J. Linearization stability of gravitational and gauge fields, J. math. Phys. 20, 443-453.

- [1980]. The structure of the solution sot for the Yang-Mills equations (preprint). J. Arms, A. Fisher and J. Marsden [1975]. Uno approche symplcctique pour des theoremes

de decomposition en geometrie ou relativite glm6rale, C.R. Acad. Sci. Paris, 281, 517 -520. J. Arms, A. Fisher, J. Marsden and V. lloncrief [1980]. The structure of the space of solutions

of Einstein's equations; II Many Killing fields (in preparation). J. Arms and J. Marsdon [1979J. The absence of Killing fields is necessary for linearization

stability of Einstein's equations, Ind. Univ. Math. J. 28, 110-125. J. Arms, J. Marsden and V. ![oncrief [1980]. Bifurcations of momentum mllppings Commun.

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Phys. liO, 548-570. AI. Cantor [1979J. Some problems of global analysis on asymptotically simple manifolds.

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Sci Paris 226, 1071. (Sec also J. Ration. Alecll. Anal., 0 (1956) 951-96.) [1952]. ThCoremc d'existcnce pour certains systems d'equations uux dcrivecs partielles non lineaires, Acta math. 88, 141-225.

12-1 J. E. MeBClell (Berkeley)

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Y. Choquct.-Uruhat..8lld S. Oeser [ IUiZ}. St nhi litc init io Ie de i'cspncotonlp8 tic ~ l inkow8ki. C. n . ' \Clld. Sci. l'nris 275, 101U- Jfl2i .

- [1973J. On the ~t!l bilit.y of fl nt. SJI!ll:ll. AnTI, rh~a , Sl . 105- 1 OS. Y. Choquot..Brllha L. ,\ . 1-'is!\t:,r u nd ,,- :\ l lLr~tlen [ 1979]. :\In)(i l11nl H YI)Crllllrfn (.'Cij nnd Pos it i \"it~'

o f ~lnsa . in h olotcd Grn\'itot in~ Sy~tc/l1q nnd GCllcrol Bc lritil'il ? , J , Ehlc/"!!. cd .. Ill. liM' Phy"iclI l Soeicty. :122/:19",

D. Christonou lou [l9S0}. The boost. problem for wellkly l:ouJlled quusitinco r hyperholic syste lll t5 of thclICl'Onu order (preJlr illt.).

- D. Chrilltodoul u aud N.O':\ lurcluulhn [ 1080J. 1'I11l bOO!J,t. Jlrohlem ill ~etlOru l rc lnth' ity (JlI'C' prin t).

B. Coil [l1)ir.l. Slir b dcterm illlition, jlll r des d onn&! de CII nchy. d es ehn mps de K illing adll1 ffl Jla r un cspa(.'e·tenl IJ8, tI ' ~:iIlRle i n·~laxwell. C. R ... \ ead. Sci. P a ris !:!S I, II OU- 12. [ WiG]. Sur I:. iltahi1it6, H!u!a irc dCl! Cljllfitio liR d' Einteiu du "ide. C. n. ~\cnd. Sci. 1):I ris 2S:? 2~7- 2!iO.

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Ein lilcin· Y nng.Mills HyHtc ll1, ' \ 1111. I'Ii )"!!. (N .Y.) . 100, On? - :\ I.

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S. DCIlCr and C. Teitl! lboim [ !!IiG). SIIJ~rgm"it.r has JlOI!:itil'c cnergy. I'hYIi. H ('\·. Lotte/"!! :1lI. 2~9-fj2.

D. Dlnllcy II lId , ' . :U oncrief [ 1050). C lohlll cxi ~ 1.(! llcc for til!" " ling-M ills cqulltioll!l. D. Eardley li nd L. Smllrr P Il7!)). T ime fUllc lion~ in gCller,,1 ro1uti\'it y r. IlI l1rgiull l hound

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The l nitial Value 1"1'01.110111 a nd the DYIlum ic8 of Om\'il lltiollnll~' jcld8 125

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:\1. Kurauishi {1075]. Now proof for th., cxis~nt'i'l of loca lly cOmplete famUiel! of complex strlll'­t.ures. ProD. of Conf. Oil Complex }\rlltly.!! is, ("- Ac ppJi et nl. , (:ds), S pringer .

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D. H. Sntlinger [ ID7111. Group thooret ie met hods in hifureulion theory. Springer Lcetur.~ ~otes in Mnth. lliO'!. .

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126 J . "E. !\laT1jden (Berkeley)

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AddreM : Prof. Dr. J . E . .\Ja rsdclI

Depnrtment of Mathematic!:! Univcrsity of CUtifOflliu Berkeley, Ca lifornia 9H 20 USA