recent results in mathematical gr
TRANSCRIPT
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International Journal of Modern Physics DVol. 22, No. 6 (2013) 1330012 (12 pages)c© World Scientific Publishing CompanyDOI: 10.1142/S0218271813300127
RECENT RESULTS IN MATHEMATICAL GR∗
SERGIU KLAINERMAN
Department of Mathematics, Princeton University,Princeton, NJ 08544
Received 9 January 2013Revised 23 January 2013Accepted 23 January 2013Published 23 April 2013
The current state of the vacuum evolution problem in general relativity is reviewed froma perspective of rigorous mathematical results.
Keywords: General relativity; evolution problem.
PACS Number(s): QC173.6, 04.20.−q
1. Introduction
For lack of any better alternative I will adapt the pragmatic and rather crude def-inition according to which mathematical general relativity (GR) encompasses thecircle of ideas and results in GR, developed within the mathematics community,using sophisticated geometric and analytic and, primarily, partial differential equa-tion (PDE) methods. Within this definition it is easy to distinguish two differentdirections: one which is primarily concerned with initial data sets and the problemof constraints and the second whose objective is to understand the problem of evo-lution. In this paper, I will restrict myself only to the latter. To simplify mattersI will only consider the vacuum evolution problem although many of the resultsdiscussed here can be extended to various matter models. We are thus interestedin solutions to the Einstein vacuum equations (EVE).
EVE : Ricαβ = 0, (1)
where Ricαβ denotes the Ricci curvature tensor of a four-dimensional Lorentzianspacetime (M,g). An initial data set for (1) consists of a three-dimensional
∗Based on a talk presented at the Thirteenth Marcel Grossmann Meeting on General Relativity,Stockholm, July 2012.
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3-surface Σ0 together with a Riemannian metric g and a symmetric 2-tensor kverifying the constraint equations,{∇jkij −∇i tr k = 0,
Rscal − |k|2 + (tr k)2 = 0,(2)
where the covariant derivative ∇ is defined with respect to the metric g, Rscal isthe scalar curvature of g and tr k is the trace of k with respect to the metric g. Inthis paper, I restrict myself to asymptotically flat initial data sets with one end. Iwill review some recent results in three different directions:
(1) The bounded L2 curvature conjecture (BCC).(2) Rigidity of Kerr.(3) Formation of trapped surfaces.
2. The Bounded L2 Curvature Conjecture
For a given initial data set, the Cauchy problem consists in finding a metric gsatisfying (1) and an embedding of Σ0 in M such that the metric induced byg on Σ0 coincides with g and the 2-tensor k is the second fundamental form ofthe hypersurface Σ0 ⊂ M. The first local existence and uniqueness result for theEVE was established by Bruhat, see Ref. 1, with the help of wave coordinateswhich allowed her to cast the equations in the form of a system of nonlinear waveequations to which one can applya the standard theory of nonlinear hyperbolicsystems. The optimal, classical result (based only on energy estimates and classicalSobolev inequalities) states the following.
Theorem 2.1. [Classical local existence3,4] Let (Σ0, g, k) be an initial data setfor the EVE (1). Assume that Σ0 can be covered by a locally finite system ofcoordinate charts, related to each other by C1 diffeomorphisms, such that (g, k) ∈Hs
loc(Σ0) ×Hs−1loc (Σ0) with s > 5
2 . Then there exists a uniqueb (up to an isometry)globally hyperbolic development (M,g), verifying (1), for which Σ0 is a Cauchyhypersurface.
2.1. Short history
The classical exponents s > 5/2 are clearly not optimal. By straightforward scalingconsiderations one might expect to make sense of the initial value problem fors ≥ sc = 3/2, with sc the natural scaling exponent for L2 based Sobolev norms.Note that for s = sc = 3/2 a local in-time existence result, for sufficiently small data,
aThe original proof in Ref. 1 relied on representation formulas, following an approach pioneeredby Sobolev (see Ref. 2).bThe original proof in Refs. 3 and 4 actually requires one more derivative for the uniqueness. Thefact that uniqueness holds at the same level of regularity than the existence has been obtained inRef. 5.
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would be equivalent to a global result. More precisely any smooth initial data, smallin the corresponding critical norm, would be globally smooth. Such a well-posedness(WP) result would be thus comparable with the so-called ε-regularity results fornonlinear elliptic and parabolic problems, which play such a fundamental role inthe global regularity properties of general solutions. It is believed that a critical WPresult must play an essential ingredient in the proof of Penrose’s Cosmic CensorshipConjectures.
For quasilinear hyperbolic problems critical WP results have only been estab-lished in the case of 1+1-dimensional systems, or spherically symmetric solutions ofhigher-dimensional problems, in which case the L2-Sobolev norms can be replacedby bounded variation (BV) type norms.c A particularly important example of thistype is the critical BV WP result established by Christodoulou for spherically sym-metric solutions of the Einstein equations coupled with a scalar field, see Ref. 7.The result played a crucial role in his famous Cosmic Censorship results for thesame model, see Ref. 8. As well known, unfortunately, the BV-norms are com-pletely inadequate in higher dimensions; the only norms which can propagate theregularity properties of the data are necessarily L2 based.
The quest for optimal WP in higher dimensions has been one of the majorthemes in nonlinear hyperbolic PDE’s in the past 20 years. Major advances havebeen made in the particular case of semi-linear wave equations. In the case ofgeometric wave equations such as Wave Maps and Yang–Mills, which possess awell understood null structure, WP holds true for all exponents larger than thecorresponding critical exponent. For example, in the case of Wave Maps definedfrom the Minkowski space R
n+1 to a complete Riemannian manifold, the criticalscaling exponents is sc = n/2 and WP is known to hold all exponents s ≥ sc for alldimensions n ≥ 2. This critical WP result, for s = n/2, plays a fundamental role inthe recent, large data, global results of Refs. 9–12 for 2+1-dimensional wave maps.
The role played by critical exponents for quasilinear equations is much lessunderstood. The first WP results, on any (higher-dimensional) quasilinear hyper-bolic system, which go beyond the classical Sobolev exponents, obtained in Refs. 13–17, do not take into account the specific (null) structure of the equations. Yet thepresence of such structure was crucial in the derivation of the optimal results men-tioned above, for geometric semilinear equations. In the case of the Einstein equa-tions (in 1 + 3 dimensions) it is not at all clear what such structure should be, ifthere is one at all. Indeed, the only specific structural condition, known for EVE,discovered in Ref. 18 under the name of the weak null condition, is not at all ade-quate for improved WP results. It is known however, see Ref. 19, that without sucha structure one cannot have WP for exponents s ≤ 2. Yet EVE is of fundamental
cRecall that the entire theory of shock waves for 1+1 systems of conservation laws is based on BVnorms, which are critical with respect to the scaling of the equations. Note also that these BVnorms are not, typically, conserved and that Glimm’s famous existence result6 can be interpretedas a global WP result for initial data with small BV norms.
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importance and as such it is not unreasonable to expect that such a structure mustexist.
Even assuming such a structure, a result of WP for the Einstein equations at,or near, the critical regularity sc = 3/2 is not only completely out of reach but mayin fact be wrong. This is due to the presence of a different scaling connected to thegeometry of boundaries of causal domains. It is because of this more subtle scalingthat we need at least L2-bounds for the curvature to derive a lower bound on theradius of injectivity of null hypersurfaces and thus control their local regularityproperties. This imposes a crucial obstacle to WP below s = 2. Indeed, any suchresult would require, crucially, bilinear and even trilinear estimates for solutions towave equations of the form �gφ = F . Such estimates, however, depend on Fourierintegral representations, with a phase function u which solves the eikonal equationgαβ∂αu∂βu = 0. Thus the much needed bilinear estimates depend, ultimately, onthe regularity properties of the level hypersurfaces of the phase u which are, ofcourse, null. The catastrophic breakdown of the regularity of these null hypersur-faces, in the absence of a lower bound for the injectivity radius, would make theseFourier integral representations entirely useless.
2.2. Main result
These considerations lead one to conclude that, the following conjecture, proposedin Ref. 20, is most probably sharp.
Conjecture [BCC]. The Einstein-vacuum equations admit local Cauchy develop-ments for initial data sets (Σ0, g, k) with locally finite L2 curvature and locally finiteL2 norm of the first covariant derivatives of k.d
Remark 2.1. It is important to emphasize here that the conjecture should be pri-marily interpreted as a continuation argument for the Einstein equations; that isthe spacetime constructed by evolution from smooth data can be smoothly contin-ued, together with a time foliation, as long as the curvature of the foliation and thefirst covariant derivatives of its second fundamental form remain L2-bounded onthe leaves of the foliation. Note that the conjecture implies the break-down criterionpreviously obtained in Ref. 21 and improved in Ref. 22. According to that criteriona vacuum spacetime, endowed with a constant mean curvature (CMC) foliation Σt,can be extended, together with the foliation, as long as the L1
tL∞(Σt) norm of the
deformation tensor of the future unit normal to the foliation remains bounded.
To prove the conjecture one has to deal with the following crucial ingredients:
A. Provide a coordinate condition, relative to which EVE verifies an appropriateversion of the null condition.
dThe precise theorem requires other weaker conditions, such as a lower bound on the volumeradius.
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B. Provide an appropriate geometric framework for deriving bilinear estimates forthe null quadratic terms appearing in the previous step.
C. Construct an effective progressive wave representation ΦF (parametrix) for solu-tions to the scalar linear wave equation �gφ = F, derive appropriate bounds forboth the parametrix and the corresponding error term E = F − �gΦF and usethem to derive the desired bilinear estimates.
As it turns out, the proof of several bilinear estimates of Step B reduces to the proofof a sharp L4(M) Strichartz estimate for a localized version of the parametrix ofStep C. Thus we will also need the following forth ingredient.
D. Prove of sharp L4(M) Strichartz estimates for a localized version of theparametrix of Step C.
Note that the last three steps are to be implemented using only hypothetical L2-bounds for the spacetime curvature tensor, consistent with the conjectured result.
Remark 2.2. As mentioned above, the only known structural condition related tothe classical null condition, called the weak null condition, tied to wave coordinates,fails the test. Indeed, the following simple system in Minkowski space verifies theweak null condition and yet, according to Ref. 19, it is ill posed for s = 2.
�φ = 0, �ψ = φ · ∆φ.Coordinate conditions, such as spatial harmonic,e also do not seem to work.
The conjecture has been recently proved in collaboration with Rodnianski andSzeftel in the sequence of Refs. 23–28 and summarized in 29. In dealing with Awe need to abandon coordinate conditions and rely, instead on a Coulomb typecondition, for orthonormal frames, adapted to a maximal foliation. Such a gaugecondition appears naturally if we adopt a Yang–Mills description of the Einsteinfield equations using Cartan’s formalism of moving frames, see Ref. 30. It is impor-tant to note nevertheless that it is not at all a priori clear that such a choicewould do the job. Indeed, the null form nature of the Yang–Mills equations in theCoulomb gauge is only revealed once we commute the resulting equations withthe projection operator P on the divergence free vectorfields. Such an operationis natural in the case of the Yang–Mills equation on the flat Minkowski space, seeRef. 31, since P commutes with the flat d’Alembertian. In the case of the Ein-stein equations, however, the corresponding commutator term [�g,P ] generatesf
a whole host of new terms and it is quite a miracle that they can all be treatedby an extended version of bilinear estimates. At an even more fundamental level,the flat Yang–Mills equations possess natural energy estimates based on the time
eMaximal foliation together with spatial harmonic coordinates on the leaves of the foliation wouldbe the coordinate condition closest in spirit to the Coulomb gauge.fNote also that additional error terms are generated by projecting the equations on the componentsof the frame.
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symmetry of the Minkowski space. There are no such timelike Killing vectorfield incurved space. We have to rely instead on the future unit normal to the maximalfoliation Σt whose deformation tensor is nontrivial. This leads to another class ofnonlinear terms which have to be treated by a novel trilinear estimate.
3. Rigidity of Black Holes
It is widely expectedg that the domains of outer communication of regular, station-ary, four-dimensional, vacuum black hole solutions are isometrically diffeomorphicto those of the Kerr black holes. Due to gravitational radiation, general, asymp-totically flat, dynamic, solutions of the Einstein-vacuum equations ought to settledown, asymptotically, into a stationary regime. Thus the conjecture, if true, wouldcharacterize all possible asymptotic states of the general vacuum evolution. A sim-ilar scenario is supposed to hold true in the presence of matter.
So far the conjecture is known to be trueh if, besides reasonable geometric andphysical conditions, one assumes that the spacetime metric is real analytic in thedomain of outer communication. This last assumption is particularly restrictive,since there is a priori no reason that general stationary solutions of the Einsteinfield equations should be analytic in the ergoregion, i.e. the region where the station-ary Killing vector-field becomes space-like. Hawking’s proof of his famous rigidityresult starts with the observation that the event horizon of a general stationarymetric is nonexpanding and the stationary Killing field must be tangent to it. Spe-cializing to the future event horizon H+, Hawkingi (Ref. 34) proved the existenceof a nonvanishing vector-field K tangent to the null generators of H+ and Killingto any order along H+. Under the assumption of real analyticity of the spacetimemetric one can prove, by a Cauchy–Kowalewski type argument (see Ref. 34 andthe rigorous argument in Ref. 38), that the Hawking Killing vector-field K can beextended to a neighborhood of the entire domain of outer communication. Thus,it follows, that the spacetime (M,g) is not just stationary but also axi-symmetric.To derive uniqueness, one then appeals to the theorem of Carter and Robinsonwhich shows that the exterior region of a regular, stationary, axi-symmetric vac-uum black hole must be isometrically diffeomorphic to a Kerr exterior of mass Mand angular momentum a < M . The proof of this result originally obtained byCarter35 and Robinson,36 has been strengthened and extended by many authors,notably Mazur,39 Bunting,40 Weinstein;41 the most recent and complete account,which fills in various gaps in the previous literature is the recent paper of Chruscieland Costa.37 A clear and complete exposition of the ideas that come into the proof
gSee reviews by Carter32 and Chrusciel33 for a history and review of the current status of theconjecture.hBy combining results of Hawking,34 Carter35 and Robinson,36 see also the recent work ofChrusciel–Costa.37iThe real achievement of Hawking was to prove the existence of such a vectorfield in the degeneratecase; in the case of a nondegenerate horizon his observation is trivial.
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can be found in Heusler’s book.42 We remark that the Carter–Robinson theoremdoes not require analyticity.
As mentioned earlier the assumption of real analyticity made by Hawking ishowever very problematic; there is simply no physical or mathematical justificationfor it. Moreover it completely trivializes the extension problem; indeed to performit one does not need anymore the Einstein equations. (It is as if one replaces themarbitrarily with the linear Cauchy–Riemann equations!)
During the past six years, I have developed, in collaboration with Ionescu andAlaxakis (see Refs. 43–45), a strategy to dispense of analyticity. In Ref. 43, based ona characterization of the Kerr solution (see Ref. 46) by the vanishing of four covari-ant complex valued tensorj S, we were able to remove the analyticity assumption byreplacing it with a complex scalar condition to be satisfied on the bifurcate sphereof the horizon. The main idea of the proof was to derive a covariant wave equa-tion for S, show that S vanishes on the bifurcate event horizon of the stationarysolution, and then use Carleman estimates to deduce that S must vanish in theentire domain of outer communication of the spacetime. In Ref. 45, we also removethe additional complex scalar condition but require instead that the Mars–Simontensor is sufficiently small. In other words we prove that any regular, nondegener-ate stationary, asymptotically flat solution which is a small perturbation of a givenKerr solution K(a,m), 0 ≤ a < m, is in fact a Kerr solution.
The proof starts with Hawking’s vectofield K, mentioned above, which can beeasily constructed along the event horizon. In the absence of analyticity one wouldlike to extend this Hawking vectorfield K by solving a covariant wave equation�gK = 0 with prescribed data on the horizon. This problem is however ill-posed,i.e. one cannot prove existence in the nonanalytic, smooth, category.
Our work in Ref. 45, circumvents this difficulty by starting with some natural,geometric extension, of K consider its associated flow Ψt, and show that, for small|t|, the pull back metric g′ = Ψ∗
t g must coincide with g. This is based on the factthat both metrics verify the same Einstein equation and coincided on the horizon.We are then reduced to prove a uniqueness result for two such solutions. Thoughthe problem is ill-posed, just as we have remarked above, one can neverthelessprove uniqueness. This strategy is illustrated in Ref. 44 where we prove a localversion of Hawking’s local rigidity result which does not rely on analyticity. Theresult has recently been generalized in Ref. 47. I should note here that these resultsare unconditional (i.e. do not require the smallness of the Mars–Simon tensor). InRef. 47, we also show that one can find local, stationary, vacuum extensions of aKerr solution K(m, a), 0 < a < m, in a future neighborhood of a point p of thepast horizon (p not on the bifurcation sphere!), which admits no extension of theHawking vector-field of K(m, a). This result illustrates one of the major difficultiesone faces in trying to extend Hawking’s rigidity result to the more realistic setting
jWhich is traceless and possesses all the symmetries of the Riemann curvature tensor.
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of smooth stationary solutions of EVE; unlike in the analytic situation, one cannothope to construct an additional symmetry by relying only on local information.
It remains, of course, to remove the smallness assumption made in Ref. 45. It isclear that the local, unconditional result in Ref. 44 has to be the starting point ofany strategy for achieving this goal. So far we have been able to refine the smallnesscondition in Ref. 45 in various directions but the general case still eludes us.
4. Formation of Trapped Surfaces
The best well known results in GR which provide important global information forsolutions of the general initial value problem, without any symmetry assumption,are the global stability of Minkowski space, see Refs. 48 and 49 and the singularitytheorem of Penrose.50
Theorem 4.1. [Global Stability of Minkowski] Any asymptotically flat initial dataset which is sufficiently close to the trivial one has a complete maximal futuredevelopment. Moreover the curvature of the development is globally small and tendsto zero at infinity, along any direction. The precise rates of decay are called peelingproperties.
At the opposite end of this result, when the initial data set is very far from flat,we have Penrose’s singularity theorem.50
Theorem 4.2. [Penrose] If the manifold support of an initial data set is noncom-pact and contains a closed trapped surface the corresponding maximal developmentis incomplete. The results holds true in the presence of any matterfields which verifythe positive energy conditions, i.e. if for any null vector L,
Ric(L,L) ≥ 0. (3)
The result leaves open the fundamental question concerning how trappedsurfaces can form, in the first place, from regular initial conditions. RecentlyChristodoulou51 has proved such a result by combining the global methods used inthe global stability of Minkowski with a novel ansatz on the form of initial data,prescribed on an outgoing null hypersurface (and trivial on a transversal one) or atpast null infinity.
The ansatz (Christodoulou calls it “short pulse”), which allows certain compo-nents of the data to be large with respect to a parameter δ−1, with δ > 0 small, isdynamically stablek and thus, through a simple mechanism of amplification, leadsto a trapped surface. He observes that, once one can prove the dynamic stabilityof the ansatz (in other words the persistence of the ansatz for a sufficiently largeregion of the spacetime), the formation of the trapped surface is a straightforwardconsequence. More that 500 pages of the work is taken by the proof of the stability
kThe stability of the ansatz, whose proof takes by far most of the work, is based on a variation ofthe methods of Refs. 48 and 49.
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of the ansatz. In a subsequent work written in collaboration with Rodnianski,52 wegive both a significant extension as well as a large simplification of Christodoulou’sproof.
Both the original proof in Ref. 51 as well as that of Ref. 52 rest on two mainingredients:
(1) A semi-global existence result for the characteristic initial value problem withinitial data which (on the outgoing null hypersurface) is allowed to be large,in an appropriately measured way, relative to small parameter δ > 0. Theprecise dependence on δ, which Christodoulou calls the short pulse method,was subsequently relaxed in Refs. 52 and 53, (see also Ref. 54). In these resultsthe data on the incoming null hypersurface is assumed to be flat. This restrictionhas been recently removed in Ref. 55.
The semi-global result allows one to construct the future development of theinitial data together with a double null foliationl (u, u), 0 ≤ u ≤ u∗, 0 ≤ u ≤ δ,and full control on all the geometric quantities associated to them.
(2) A uniform lower bound on the integrals of the outgoing null shear χ alongall short segments (of size δ!) of the outgoing null generators of the initialconfiguration. Using the estimates obtained in the constructive step (1) onecan then deduce, by a strikingly simple ODE argument, that a trapped surfacemust form for some later two-sphere of the double null foliation. More precisely,if we denote by M0 be a real valued function on the initial sphere S0,0 = {u =0} ∩ {u = 0} defined by
M0(ω) =∫ δ
0
|χ|2(u′, ω)du′, (4)
a necessary and sufficient condition that the surface S(u∗, δ) = {u = u∗}∩{u =δ} is that the lower bound,m
2(r0 − u)r20
<
∫ δ
0
|χ0|2(u′)du′ −O(δ0), (5)
holds uniformly for every outgoing null generator emanating from S0,0.
Condition (5) has been recently dramatically weakened in Ref. 56. The new resultrelies heavily on the hard part of the above results, i.e. the construction of the space-time in (1). We modify however part (2) by combining Christodoulou’s argumentwith a deformation argument along the incoming null hypersurfaces u = const. Thisallows us to replace his uniform condition with a localized condition in a neighbor-hood of a null geodesic of {u = 0}. Thus the trapped surface we find does not
lSuch that the initial configuration is given by the incoming {u = 0} and outgoing {u = 0} initialnull hypersurfaces.mHere r0 is such that |S0,0| = 4πr2
0.
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belong to the double null foliation (u, u) as in the previous results but rather on anappropriate deformation. Here is a somewhat precise formulation of our result.
Theorem 4.3. [Main Theorem] Let Bp(ε) be a geodesic ball of radius ε in S0,0.
Suppose given initial data, verifying some variation of Christodoulou’s δ-pulseconditions, such that M0 satisfies
infω∈Bp(ε)
M0(ω) ≥Mε. (6)
Then, there exists a constant c > 0, depending only on the metric on S0,0, suchthat if
r20Mεε2
≤ c(1 − u∗), (7)
then, for δ > 0 sufficiently small the future development D(u∗, u∗ = δ) of thecorresponding characteristic data contains a trapped surface.
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Submitted to Inventiones, arXiv:1204.1767.24. J. Szeftel, Parametrix for wave equations on a rough background I: Regularity of the
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26. J. Szeftel, Parametrix for wave equations on a rough background III: Space-timeregularity of the phase, Submitted to Asterisque, arXiv:1204.1770.
27. J. Szeftel, Parametrix for wave equations on a rough background IV: Control of theerror term, Submitted to Asterisque, arXiv:1204.1771.
28. J. Szeftel, Sharp Strichartz estimates for the wave equation on a rough background,Submitted to Asterisque, arXiv:1301.0112.
29. S. Klainerman, I. Rodnianski and J. Szeftel, Overview of the proof of the boundedL2 curvature conjecture, arXiv:1204.1772.
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54. M. Reiterer and E. Trubowitz, Commun. Math. Phys. 307 (2011) 275.55. J. Luk and I. Rodnianski, Nonlinear interaction of impulsive gravitational waves for
the vacuum Einstein equations, Preprint (2012), arXiv:1209.1130.56. S. Klainerman, J. Luk and I. Rodnianski, A fully anisotropic mechanism for formation
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