v1 13 Mar 2002

arXiv:gr-qc/0203044v1 13 Mar 2002

CONSTRUCTING SOLUTIONS OF THE EINSTEIN CONSTRAINT EQUATIONS JAMES ISENBERG Department of Mathematics and Institute of Theoretical Science University of Oregon Eugene, OR 97403 USA E-mail: [email protected] The first step in the building of a spacetime solution of Einstein’s gravitational field equations via the initial value formulation is finding a solution of the Einstein constraint equations. We recall the conformal method for constructing solutions of the constraints and we recall what it tells us about the parameterization of the space of such solutions. One would like to know how to construct solutions which model particular physical phenomena. One useful step towards this goal is learning how to glue together known solutions of the constraint equations. We discuss recent results concerning such gluing.

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Introduction

The constraint equations of Einstein’s theory of the gravitational field are familiar to most of those who work in general relativity: If we choose a set of appropriate local spacetime coordinates (xa , t) and if we write Einstein’s field equations for the gravitational metric field g together with a coupled matter source fielda Ψ in the tensorial formb Gµν (g) = κTµν (g, Ψ),

(1)

Gtν (g) = κTtν (g, Ψ),

(2)

then the four equations

involve no time derivatives of higher than first order. These are the Einstein constraint equations. What makes the constraint equations important in the study of the physics of the gravitational field is their role in the initial value formulation of a We

presume that the matter source field is not derivative-coupled, and that the coupled field equations constitute a well-posed PDE system. b We use M T W 1 conventions on signs of the curvature and on indices: Greek letters run from 0 to 3 (spacetime) while Latin letters run from 1 to 3 (space). Here κ is a coupling constant.

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Einstein’s equations. We recall how the initial value formulation works: Say we want to construct a spacetime solution {g, Ψ} of the Einstein equations (1) on a manifold M 4 = Σ3 × R, where Σ3 is a smooth three-dimensional manifold. The first step in doing this is to find a set of initial data consisting of γab - a Riemannian metric on Σ3 cd K - a symmetric tensor field on Σ3 ψ - space covariant pieces of the matter field Ψ on Σ3 with {γ, K, ψ} satisfying the constraint equations (2). In terms of the data {γ, K, ψ} the constraint equations take the form R + (trK)2 − K cd Kcd = 2κρ(γ, ψ)

(3)

∇a Kba − ∇b (trK) = κJb (γ, ψ)

(4)

where ∇a is the Levi-Civitac covariant derivative corresponding to γ, R is its scalar curvature, ρ is a scalar function of γ and ψ, and Jb is a vector field function of γ and ψ.d Once we have obtained constraint-satisfying initial data {γ, K, ψ}, we may evolve this data in time by choosing everywhere on Σ3 a scalar field N (the “lapse function”) and a vector field M a (the “shift vector”) and then solving the Einstein evolution equations in the form d γab = −2N Kab + £M γab dt

(5)

d c K = N (Rdc + trKKdc ) − ∇c ∇d N + κTdc + £M Kdc dt d

(6)

and

where Tdc are the spatial components of the stress-energy tensor corresponding to the matter field Ψ. We note that the time evolution of the lapse and shift are not determined by the Einstein equations; they may be chosen freely, and they must be chosen for all t if one wishes to evolve γ and K in time. The role of N and M is to determine the coordinates relative to the evolving spacetime. c i.e.,

metric compatible and torsion-free for example, the “matter source field” is Maxwell’s electromagnetic field, then the spacetime covariant field Ψ is the spacetime one-form field Au , the space covariant fields ψ may be chosen to be the magnetic spatial one-form field Ba and the electric spatial vector field E a , and one calculates ρ = 21 (E a E b γab + B a B b γ ab ) and Jc = ηcmn E m B n , where ηcmn is the alternating symbol η123 = 1, etc. In addition, the data must satisfy the Maxwell constraint equations ∇a Ba = 0 and ∇a E a = 0. d If,

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After N (x, t) and M c (x, t) have been chosen and the evolved data γab (x, t) and K cd (x, t) and ψ(x, t) have been determinede , we may reconstruct the spacetime metric (on the manifold M 4 = Σ3 × I, where I is the interval for which the evolution can be carried out) via the formula g(x, t) = −N 2 dt2 + γab (dxa + M a dt)(dxb + M b dt).

(7)

We similarly reconstruct Ψ. It then follows (from the Gauss-Codazzi-Mainardi 3+1 decomposition equations for the spacetime curvature) that g and Ψ satisfy the Einstein equation (1) on M 4 . Does the initial value formulation work? This general question is best considered by breaking it up into a number of more specific questions: 1. Does every choice of initial data {γ, K, ψ} which satisfies the constraint equations evolve into a spacetime solution {g, Ψ}? 2. How do we find data {γ, K, ψ} which satisfy the constraint equations (3)-(4)? 3. What is the space of solutions of the constraint equations? 4. If we wish to use Einstein’s equations to model the physical behavior of a particular physical situation (e.g., a black hole collision), how do we build that physical situation into the choice of initial data? 5. What criteria should we use in making a choice of the lapse and shift? 6. What do we know about the long time behavior of solutions? 7. Is the initial value formulation a practical way to construct and study solutions of Einstein’s equations? In this essay, we shall focus our attention on questions 2, 3, and 4, which concern finding and understanding solutions to the constraint equations. We start in Section 2 by discussing the conformal method for constructing solutions to the constraints. After recalling the basic steps of the conformal method, we state the results which tell us the extent to which the method is known to work. Based on these results, we discuss (also in Section 2) what is known about the function spaces which parameterize the set of solutions. Knowing how to construct constraint-satisfying initial data sets from conformal data sets, and knowing how to parameterize the space of all such initial data sets, does not tell us much about how to use solutions constructed via the e The

coupled Einstein-matter source field equations determine evolution equations for ψ.

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initial value formulation for studying physical questions. With observational data from LIGO and other gravitational radiation detectors expected to start arriving within the next few years, it is crucial that we learn how to do this. Although progress to date on this front has been fairly limited, recent results on the gluing of initial data sets could be useful. We discuss some of these gluing results and the ideas involved in Section 3. Only if initial data can be evolved is it worthwhile to construct it. We comment briefly on evolution in Section 4, touching on questions 1, 5, 6, and 7. We make some concluding remarks in Section 5. 2

The Conformal Method and the Space of Solutions of the Constraint Equations

For the past thirty years, the conformal method has been the most successful procedure for producing and studying solutions of the Einstein constraint equations, both numerically and analytically. The key idea of this method, which was developed primarily by Lichnerowicz, Choquet-Bruhat and York2 is to split the initial data {γ, K, ψ} into two parts: The first part is to be chosen freely, while the second part is to be determined by the constraints. As a consequence of the split, the constraint equations take the form of a determined PDE system of four equations for four unknowns (equations (12)-(13) below), rather than the form of an underdetermined system of four equations for twelve unknowns (equations (3)-(4)). To illustrate how this method works explicitly with matter source fields as well as gravitational fields, let us consider the Einstein-Maxwell equations, for which the initial data consists of {γab , K cd , Ba , E a } and for which the full set of constraint equations takes the form (presuming no charges present) R + (trK)2 − K cd Kcd = κ(B a Ba + E a Ea )

(8)

∇a Kba − ∇b (trK) = κηbmn B m E n

(9)

∇a Ba = 0

(10)

∇a E a = 0

(11)

For this system, the split of the data is as follows: Free (“Conformal”) Data: λab - a Riemannian metric (specified up to conformal factor)

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σ cd - a divergence-freef and trace-free symmetric tensor field (∇c σ cd = 0) (λcd cσ cd = 0) τ - a scalar field β a - a divergence-free vector field εa - a divergence-free vector field Determined Data φ - a positive definite scalar field W a - a vector field Determining Equations ∇a (LW )ab = ∇2 φ =

2 6 φ ∇b τ + κηbmn β m εn 3

1 1 Rφ − (σ ab + LW ab )(σab + LWab )φ−7 8 8 1 κ a − (β βa + εa εa )φ−3 + τ 2 φ5 8 12

(12)

(13)

Here the covariant derivative, the Laplacian, the scalar curvature, and the index manipulations all correspond to the metric λab . Note that L is the conformal Killing operator, defined by 2 LWab := ∇a Wb + ∇b Wa − λab ∇c W c . 3

(14)

The determining equations (12)-(13) are to be solved for φ and W a . If, for a given set of conformal data {λab , σ cd , τ, β a , εa } one can indeed solve (12)-(13), then the reconstructed data γab = φ4 λab

(15)

1 K cd = φ−10 (σ cd + LW cd ) + φ−4 λcd τ 3

(16)

Ba = φ−2 λab β b

(17)

E a = φ−6 εa

(18)

f In

the conformal data, the divergence-free condition is defined using the Levi-Civita covariant derivative compatible with the conformal metric λcd .

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constitute a solution of the constraint equations (8)-(11). Does the conformal method always work in the sense that, for any choice of conformal data {λab , σ cd , τ, β a , ǫa } one can always solve (12)-(13) for {φ, W a } and thereby obtain a solution {λab , K cd , Ba , E a } for the constraints (8)-(11)? It is easy to see that this is not the case, as illustrated by the following simple example: Let us choose the manifold Σ3 to be the three-dimensional sphere S 3 , and let us choose the conformal data to consist of the round sphere metric λ = g(round) with R = 1, together with σ cd = 0, εa = 0, β a = 0, and τ = 1. With τ constant and with ηbmn εm β n equal to zero, the right hand side of equation (12) vanishes. Since the operator ∇ · L is self-adjoint and injective (up to conformal Killing vector fields), it readily follows that the solutions W a to (12) for this choice of conformal data all satisfy the condition LWab = 0 (which is the conformal Killing equation). Using this result, we find that the remaining equation (13) takes the form 1 1 φ + φ5 . (19) 8 12 Since we seek solutions φ which are positive definite, for any such solution the right hand side of equation (19) is positive define. But the maximum principle tells us that there are no functions φ on a compact manifold for which ∇2 φ > 0 (or < 0). Hence, for this choice of conformal data, there is no solution {φ, W a } to equations (12)-(13), and therefore no corresponding solution to the Einstein-Maxwell constraint equations. This example shows that there are choices of conformal data which are not mapped to solutions of the constraint equations by the conformal method. What happens generally? This question is difficult to address in any complete sense because there is a very wide variety of cases of physical and mathematical interest, and each must be handled separately. The various cases may be classified using the following criteria: ∇2 φ =

Manifold and Asymptotic Geometry • Closed Σ3 • Asymptotically Euclidean • Asymptotically hyperbolic • Compact Σ3 with boundary conditions Mean Curvature • Maximal

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• Constant non maximal (CMC) • Non Constant (non CMC) Fields • Einstein vacuum • Einstein - standard matter source fieldg Function Spaces • C α (analytic) • C ∞ (smooth) • C k+β (H¨older) • Hkp (Sobolev) p • Hk,δ (weighted Sobolev)

For a number of cases (labeled using these criteria), we have a fairly complete understanding of which sets of conformal data map to solutions and and which do not. This is true for the following: 1. Closed Σ3 ; CMC; Einstein vacuum or Einstein-standard matter source field; λ ∈ C 3 and σ, ε, β ∈ H2p (p > 3). 2. Asymptotically Euclidean; Maximal; Einstein vacuum or Einsteinstandard matter source field; Data in weighted Sobolev spaces. 3. Asymptotically hyperbolic: CMC; Einstein vacuum or Einstein-standard matter source field; Data in weighted Sobolev spaces. For these three cases, one finds that all sets of conformal data which are not readily disqualified by simple sign criteria or by the maximal principle are mapped to solutions by the conformal method. To explain the sense in which this holds, let us focus for the moment on the first of these cases; that with Σ3 closed: The Yamabe classification of metric conformal classes is a key tool for determining which sets of conformal data on a closed manifold Σ3 are mapped g We label as “standard matter source field” any field theory which couples to Einstein’s theory without derivative coupling. Included are Einstein-Maxwell, Einstein-Yang-Mills, Einstein-Dirac, Einstein-Klein-Gordon, Einstein-Cartan, and Einstein fluids. For all these theories, there are conformal splittings such that, if ∇τ = 0, the constraint equations semi-decouple. See Isenberg and Nester4 .

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to solutions of the constraints. This classification is based on the Yamabe Theorem3 which says that every metric λab on a closed manifold Σ3 is conformally related to another which has constant scalar curvature either +1, 0, or −1; and moreover, the sign of R for constant scalar curvature metrics conformally related to a given λab is unique. Thus the set of all metrics (and metric conformal classes) on a closed Σ3 is partitioned into the three Yamabe classes Y + (Σ3 ), Y 0 (Σ3 ), and Y − (Σ3 ) depending on this sign. The Yamabe theorem and Yamabe classification are important for CMC conformal data on a closed Σ3 because solubility of (12)-(13) is invariant under the conformal transformation {λab , σ cd , τ, β a , εa } → {θ4 λab , θ−10 σ cd , τ, θ−6 β a , θ−6 εa } for any positive θ. (This is not true if ∇τ 6= 0). Thus, to determine for which sets of conformal data (12)-(13) admit a solution, one may without loss of generality restrict attention to conformal metrics with R(λ) = −1, 0, or +1. As a consequence of this conformal invariance and the resulting simplification, one immediately sees from (13) and the maximum principle that the following categories of conformal data do not lead to solutionsh {Y + ; (σ, ρ) ≡ 0; τ = 0}, {Y − ; (σ, ρ) ≡ 0; τ = 0}, {Y + ; (σ, ρ) ≡ 0; τ 6= 0}, {Y 0 ; (σ, ρ) ≡ 0; τ 6= 0}, {Y 0 ; (σ, ρ) 6≡ 0; τ = 0}, {Y − ; (σ, ρ) 6≡ 0; τ = 0}. Much less evident is the fact that for all other sets of conformal data on closed Σ3 (12)-(13) can be solved, and we obtain a solution of the constraints. The proof of this result (with one case left out) is discussed in the review paper of Choquet-Bruhat and York2 ; the complete proof appears in the work of the author5 . Analogous results, with a few important modifications, hold for the other two cases mentioned above. Specifically, for the asymptotically Euclidean case with τ = 0, we first note that there is a Yamabe-type partition of asymptotically Euclidean metrics into two classes Z + and Z − , with the Z + conformal geometries admitting a conformal transformation to metrics with R = 0, and the Z − conformal geometries not admitting such a transformation. It follows from work of Brill and Cantor6 that asymptotically Euclidean metrics with non-negative scalar curvature, as well as those satisfying a certain integral condition, are contained in Z + . The work of Cantor7 (See also8 ) shows that a set of asymptotically Euclidean conformal data {λab , σ cd , τ, β a , εa } (in appropriate function spaces) is mapped to a solution by the conformal method if and only if λ ∈ Z + (The behavior of the other fields is irrelevant.). For the asymptotically hyperbolic case (with τ 6= 0), there are important h Here

(σ, ρ) ≡ 0 means that σcd and ρ are identically zero everywhere on Σ3 , while (σ, ρ) 6≡ 0 means that there exists at least one point on Σ3 at which σcd or ρ is not zero.

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questions regarding the asymptotic behavior of solutions; but ignoring those, the results are very simple: For every set of asymptotically hyperbolic conformal data, equations (12)-(13) can be solved, and we obtain a solution to the constraints9 10 . These results together tell us that if we are interested in constant mean curvature (or maximal) initial data, the conformal method works, apart from a few readily identified special cases. This is useful for two reasons. The first is that since equations of the form of the determining equations (12)-(13) can be handled by modern computer algorithms (at least when they are semidecoupled, which holds so long as we restrict to conformal data with constant τ ), the conformal method is an effective way to produce CMC initial data sets for numerical relativity. The second is that, as a consequence of uniqueness theorems relating conformal data and solutions of the constraints5, together with the results noted above, certain function spaces of conformal data provide an effective parameterization of the set of solutions of the constraints11. In light of the uniqueness of CMC foliations of spacetime solutions of Einstein’s equations12 , one may use these function spaces of conformal data to parameterize the set of spacetime solutions as well (ignoring those solutions which do not admit CMC or maximal foliations30 14 ). These results are very useful, but they are limited in a number of ways, and so there is a lot more that needs to be done. First, one would like to know which non CMC conformal data sets lead to solutions. While there are some results concerning this question, they all presume that |∇τ | is small. Indeed, for small |∇τ |(with certain restrictions on the zeros of τ ) our results show that solutions to the coupled system (12)-(13) always exist15 16 17 8 18 . Unfortunately, the iteration methods we have used to prove the results just cited do not appear to be able to handle general |∇τ |, so new ideas are likely needed. Next, we would like to be able to generate solutions to the constraint equations on manifolds with boundaries. The initial value-boundary value problem has been studied for Einstein’s theory in Friedrich’s formulation19 , but there has not been much mathematical work on it in the standard {γ, K} formulation discussed here. In view of the recent numerical work on spacetimes with excised black holes, a mathematical treatment, in terms of the {γ, K} formulation, of solutions with boundaries would be useful. A third issue regarding the construction of solutions of the constraint equations which should be addressed further is the loosening of differentiability requirements for solutions, including weak solutions. The motivation for doing this comes from the progress that has recently been made by Klainerman and Rodnianski20 and by Smith and Tataru21 in proving local well-posedness

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for the Einstein equations for data of lower differentiability. The goal (based 2 2 on energy considerations) is H3/2 ; so far H2+ǫ has been obtained. This leads 2 2 one to try to construct initial data sets in H2+ǫ or even H3/2 . The results 5 10 8 to date produce initial data sets which are considerably more restricted (i.e., smoother) than this, but work is proceeding toward obtaining solutions of the constraint equations with this low differentiability. The question that remains is how to construct initial data sets which can be used to model physical phenomena of interest. We discuss this in the next section.

3

Physical Modeling and the Gluing of Solutions of the Constraint Equations

Say we want to study the gravitational radiation produced by the inspiral collision of a neutron star and a black hole. The first step of a numerical modeling of this phenomenon is to find initial data {γ, K, ψ} which constitutes a snapshot of the two objects and their ambient spacetime at some moment prior to the collision. It is important that the initial data accurately represent the two objects and their surroundings in a relatively quiet pre-collision state, without any extraneous unrelated radiation, or the modeling will be of little use. The electromagnetic analogue of this modeling problem is familiar and easily handled, since the background space is fixed and independent of the electromagnetic fields in Newtonian-Maxwell theory, and since the representation of electromagnetic radiation is well understood in this theory as well. So, too, if we work with a post-Newtonian approximation to Einstein’s theory, the choice of initial data for phenomena like the inspiral collision is understood to some extent22 23 24 . However, attempts to find physically accurate initial data sets for phenomenon such as this in terms of the full Einstein theory have been stymied, despite many years of significant effort, both numerical and analytical. The conformal method can of course be used to generate candidate sets of data, and such sets are used in most numerical modeling studies. The lack of a priori control of the conformal factor and therefore of the ambient space, together with the difficulty in representing and recognizing gravitational radiation in the gravitational radiation in the conformal data {λ, σ, τ, ψ} for massive relativistic objects renders these sets somewhat suspect, however. One approach that has been discussed to try to control the choice of initial data is to work with sets of post-Newtonian initial data in the very

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early pre-radiation stages of the collision and then try to evolve these into the later, more relativistic stage. This approach has some promise, but can be very expensive in numerical evolution time. Another possible approach that could be tried relies on “gluing.” The idea here is to start with well-understood initial data sets for separate portions of the spacetime–say, one for the black hole and one for the neutron star–and develop a procedure for joining these together into one set. Mathematically, the idea is to join two or more solutions of the constraint equations into a single solution in such a way that, away from the joining region, the data is largely unchanged. Physically, the challenge is to understand and control the physical effects of the joining region. There has been substantial progress during the past couple of years on the mathematical aspects of gluing solutions of the Einstein constraint equations. Mazzeo, Pollack, and the author (IMP) have proven theorems which prescribe a procedure for gluing connected sums of sets of initial data25 , while Corvino and Schoen26 27 have shown that rather general asymptotically flat initial data sets can be glued to exact Schwarzschild or exact Kerr exteriors outside a transition region. We first discuss the IMP work, and then briefly describe the results of Corvino and Schoen. 3.1

Connected Sum Gluing

We start with a pair of constant mean curvature solutions {Σ31 , γ1 , K1 } and {Σ32 , γ2 , K2 } of the vacuum constraint equations (3)-(4). (We presume for now that there are no matter source fields ψ, and so ρ = 0 and J = 0). Each of these solutions may be asymptotically Euclidean, asymptotically hyperbolic, or one may have either or both of Σ31 and Σ32 closed. The two solutions need not have matching asymptotic properties, but they do need to have the same constant mean curvature τ . If we now pick a pair of points p1 ∈ Σ31 , p2 ∈ Σ32 on each manifold, then the ˜ 3 , γ˜s , K ˜ s} gluing procedure produces a one-parameter family of solutions {Σ 3 3 3 3 3 ˜ with the following properties: (i) Σ = Σ1 #Σ2 , with Σ1 joined to Σ2 by a “neck” (S 2 × R) connecting a neighborhood of p1 to a neighborhood of p2 ˜ s } can be made to be arbitrarily close (both now excised); (ii) the data {˜ γs , K ˜ 3 away to the original data {γ1 , K1 } and {γ2 , K2 } on appropriate regions of Σ ˜ from the neck by choosing s sufficiently large; (iii) trK = τ ; and (iv) the geometry on the neck (i.e., lengths, curvature) can degenerate for s → ∞, but its behavior is controlled by exponential bounds in s. This gluing procedure works for quite general classes of data. The only restrictions on {Σ31 , γ1 , K1 } and {Σ32 , γ2 , K2 } besides the CMC matching con-

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dition trK1 = trK2 and certain regularity restrictions (stated in terms of weighted H¨ older spaces 25 ) are that if either of the sets of data has Σ3 closed, that set of data must have K cd not vanishing identically, and must be free of nontrivial conformal Killing vector fieldsi . Neither of these latter conditions are needed for asymptotically Euclidean or asymptotically hyperbolic data, and we note that if a set of data {Σ3 , γ, K} fails to satisfy either condition, it will generally satisfy them both after a small perturbation is made. The procedure allows a number of interesting sets of initial data to be produced. One can, for example, add a sequence of small black holes to any asymptotically flat spacetime {M 4 , g} by gluing a sequence of copies of Minkowski data to asymptotically Euclidean data for {M 4 , g} and then evolving the glued data. These spacetimes (somewhat reminiscent of the Misner multi-black hole solutions28 , but far more general) may be constructed with the interiors either independent or connected. Another set of spacetimes one obtains via this gluing procedure allows one to study the question of black holes in cosmological spacetimes: One can glue asymptotically Euclidean or asymptotically hyperbolic data to cosmological data on closed Σ3 and then see if black hole-like physics developsj . This gluing procedure also permits us to add one or more wormholes to most solutions of Einstein’s equations: If one chooses the pair of points p1 and p2 on a single connected initial data set {Σ3 , γ, K} and carries out the gluing procedure, one obtains initial data much like {Σ3 , γ, K} except with an S 2 × R wormhole glued on. To what extent can one carry out these gluings in practice, say, numerically? To answer this question, we shall now describe the basic steps of the connected sum gluing procedure: Step 1: One starts by performing a conformal transformation of the data γab → ψ −4 γab , σ cd → ψ 10 σ cd , (Here σ cd is the trace-free part of K cd.) which is trivial (ψ = 1) away from a neighborhood of each of the points p1 and p2 , and which is singular at each of the points (with a specified type of singular behavior). The effect of this conformal blowup is to replace a ball surrounding p1 with an S 2 × R infinite length half-cylinder which is asymptotically a standard round half-cylinder, and to do the same in a neighborhood of p2 . Note that there are standard explicit formulas one can choose for ψ to carry out this first step. Step 2: Each of the newly added half cylinders extends out infinitely far. If one cuts each at a distance s/2 (measured using the local conformally i Actually, a Killing vector field can be present, so long as it does not vanish at the chosen points p1 or p2 . j Note that there is no accepted definition of a black hole in a spacetime without some sort ”asymptotic infinity” structure.

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transformed metric) out from the neighborhoods of p1 and p2 , and identifies the two cut half cylinders on the joining sphere, then one has the manifold ˜ 3 = Σ3 #Σ3 , with its connecting s-parametrized ”neck”. Then, using a pair Σ 1 2 of cut-off functions χ1 and χ2 , one may patch together the metric γˆs = γ1 χ1 + γ2 χ2 and the traceless apart of the second fundamental form σ ˆs = σ1 χ1 +σ2 χ2 . ˜ 3 , and both depend on the gluing distance Both γˆs and σ ˆs are well-defined on Σ parameter s. One also glues together the conformal function ψˆs = ψ1 ξ1 +ψ2 ξ2 , using a different set of cut-off functions. (For details, see IMP25 ). This step, like the first, can be done using standard explicit formulas for χ1 , χ2 , ξ1 , and ξ2 . Step 3: A careful choice of the cut-off functions results in σ ˆs being everywhere traceless with respect to γˆs . The conformal transformation σ cd → ψ 10 σ cd guarantees that away from the neck, σ ˆs is divergence-free with ˆ cσ respect to γˆs . However, in the neck, ∇ ˆscd 6= 0. So one replaces σ ˆscd by µ ˆcd s , which is obtained by solving ˆ c (LY ˆ s )cd = ∇ ˆ cσ ∇ ˆ cd (20) s

for the vector field

Ysc ,

and setting ˆ s )cd . µ ˆcd ˆscd − (LY s = σ

(21)

˜3 The tensor field µ ˆcd ˆs . s is divergence-free everywhere on Σ with respect to γ This step is straightforward to carry out in practice. However, to be able to proceed further with the gluing procedure, one needs to know not only that the divergence-free field µ ˆcd ˆs − σ ˆs kC k+β is very small s exists, but also that k µ for sufficiently large s. It is shown in IMP that this estimate always holds. ˜ 3 . So one Step 4: {ˆ γs , µ ˆs , τ } is a standard set of conformal data on Σ may set up and attempt to solve 1 2 5 −7 ˆ s φs − 1 µ ˆ 2 φs = 1 R ˆscd µ ˆcd τ φs (22) ∇ s φs + 8 8 12 for φs . Although the Yamabe class of the metric γˆs constructed in Step 2 is not evident, the estimates obtained in Step 3 allow one to show that (22) has a unique solution. Further, with a considerable amount of work, one shows that k φs − ψs kC k+β is very small for sufficiently large s. As noted above, once one knows that a solution to (22) exists, one can readily obtain that solution numerically. Step 5: Using formulas analogous to (15)-(16), one constructs γ˜s and ˜ s from φs and {ˆ K γs , µ ˆs , τ }. One immediately verifies that for each value of ˜ 3 , γ˜s , K ˜ s } solves the vacuum constraint equations. As a the parameter s, {Σ consequence of the estimates of Steps 3 and 4, one verifies that away from the ˆ s approach the original data for sufficently large s. neck, γˆs and K

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We see from the description of these five steps that in practice, this connected sum gluing procedure involves making a few choices of standard cut off functions, plus solving familiar elliptic equations. It should be a useful procedure for studying solutions of Einstein’s equations numerically. We note that, to date, the IMP connected sum gluing procedure has been proven to work for the vacuum Einstein case only. It is expected that it can be implemented for Einstein-Maxwell, Einstein-Yang-Mills, and the other Einstein-standard matter source fields as well; work is under way to show this. It would be nice to also extend the procedure to non CMC data, but this is likely to be difficult. 3.2

Exterior Schwarzschild and Kerr Gluing

The starting point for the exterior Schwarzschild gluing procedure is any asymptotically Euclidean, time symmetric (K cd = 0) initial data set {Σ3 , γ}. One chooses a compact region Λ3 in Σ3 . One then shows 26 that there is an asymptotically Euclidean time symmetric solution {Σ3 , γ˜} such that (i) inside a ball Br which contains Λ3 as a subset, γ˜ |Br = γ; and (ii) outside the ball B2r , we have γ˜ |Σ3 \B2r = γSch(m) , where γSch(m) is the spatial Schwarzschild metric for some positive mass m. We note three important features of this type of gluing: First, when the gluing is completed, the region inside Br as well as the exterior region outside B2r are unchanged. This is not the case for the connected sum gluing, in which the original solutions are slightly changed (away from the neck) by the gluing procedure. Second, the analysis used to prove this result does not rely in any way on the conformal method. That is, one does not construct the solution {Σ3 , γ˜} by first specifying a conformal metric everywhere and then solving for the conformal factor. Indeed, one can carry out exterior Schwarzschild gluing only if one works with the constraints as an underdetermined system to be solved for γ˜, rather than as a determined system to be solved for φ. Third, the proof that this gluing works does not readily translate into a step by step procedure that one can carry out numerically. The proof guarantees existence, but does not prescribe a procedure for constructing {Σ3 , γ˜ } The result is quite surprising and remarkable. It says that for any compact portion of a time symmetric asymptotically Euclidean solution, one can arrange the gravitational fields in an annular region around it so that outside this region, the gravitational field is exactly Schwarzschild. Note that the analogous result does not hold true for Newtonian gravity. Note also that, combining this result with the work of Friedrich29 , one may be able to produce a very large family of solutions of Einstein’s equations with a complete

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I (null asymptotic infinity). One large but unsurprising restriction on the type of data which can be glued to an exterior Schwarzschild spatial slice is that the data have K cd = 0. More recent results of Corvino and Schoen allow one to remove this restriction provided that one replaces the Schwarzschild exterior by a slice of the more general Kerr (rotating black hole) They do impose restrictions on the data {Σ3 , γ, K} which can be patched to an exterior Kerr slice: They require certain decay behavior in γ and K which are essentially just enough so that the momentum and angular momentum integrals at infinity are welldefined. These restrictions appear to be minor, in view of the result. Again, one consequence of these results might be an even larger family of solutions with complete I.

4

Some Comments on Evolution

Of the seven questions raised in the introduction, we have focussed in this essay on those three – 2,3, and 4 – which deal with the constraint equations. We shall now briefly comment on questions 1,5, and 6, which mostly concern evolving sets of initial data into spacetime solutions. The first question addresses the issue of local existence and well-posedness of the initial value formulation of the Einstein equations. To a large extent, this issue was settled fifty years ago by Choquet-Bruhat30 who proved (using harmonic coordinates) that for smooth initial data (C l , with large l) the Einstein system is well-posed. While this result is sufficient for most purposes, there have been strong attempts in recent years to prove well-posedness for more general classes of data. The goal is to prove it for data in the Sobolev space H 23 . The recent work of Klainerman and Rodnianski20 , and of Smith 2

2 and Tataru21 achieves it for H2+ǫ (ǫ > 0) data. Well-posedness theorems show (among other things) that data in the specified function space can always be evolved to a spacetime solution for a sufficiently small time duration into the future and into the past. They say little about how long a spacetime solution will last, or what its behavior will be far into the future and far into the past. While these issues are very difficult and far from resolved, there has been substantial progress on them in recent years. We note in particular the work of Christodoulou and Klainerman31, which verifies the stability of Minkowski spacetime (and thereby proves that there is a nontrivial family of asymptotically flat spacetimes which extend for infinite proper time into the future and into the past); that of Andersson and Moncrief32 which does the same for the stability of the expanding

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k = −1 vacuum Friedman-Robertson-Walker cosmological spacetimes; and the work of Christodoulou33 which verifies weak cosmic censorshipk for spherical collapsing Einstein-scalar field spacetimes. There has also been a collection of works by Berger, Chrusciel, Garfinkle, Kichenassamy, Moncrief, Rendall, Ringstrom, Wainwright, Weaver, and the author which provide increasing evidence for the presence of asymptotically velocity dominated behavior and mixmaster behavior in cosmological spacetimes, and further tends to support the validity of the strong cosmic censorship conjecturel in these spacetimes. (See, for example, the review papers by Berger34 and Rendall35 .) The construction or study of spacetime solutions of Einstein’s equations via the initial value formulation requires one to make a choice of spacetime foliation (i.e., a specification of the t = constant hypersurfaces which fill the spacetime) and of spacetime threading (i.e., a specification of the xa = constant observer world lines which are everywhere transverse to the foliation, and fill the spacetime). The choice of foliation is controlled infinitesmally by the lapse function N , and the choice of threading is similarly controlled by the shift vector M a . In constructing a spacetime from specified initial data, one may always make the simple choice N = 1 and M a = 0 (“Gaussian normal coordinates”). However, this choice generally leads to a non physical and premature breakdown in the evolution (a coordinate singularity). Consequently, the maximal development37 of the initial data may not be obtained. So, one criterion for a good choice of the lapse and shift is that the choice effectively avoids such coordinate singularities. Also important is that the choice be relatively easy to implement in practice, and that it not obscure the gravitational physics of the spacetime (e.g., by simulating gravitational radiation which is not physically present). Constant mean curvature or maximal slicing, coupled with some sort of coordinate shear minimizer, is often cited as a choice of foliation and threading which avoids coordinate singularities and clarifies the physics. However, implementing this choice requires that one solve a set of elliptic equations on each time slice. This is a large expense in computer time for numerical constructions, and it precludes explicit forms for the lapse and shift in analytical studies. For special families of solutions such as the Gowdy spacek The

weak cosmic censorship conjecture says that in asymptotically Euclidean spacetime solutions, the singularities which develop during gravitational collapse are generically shielded from the view of observers at infinity by the development of an event horizon. l The strong cosmic censorship conjecture36 says that in generic solutions developed from Cauchy data on a compact Cauchy surface, a Cauchy horizon (with its attendant causality difficulties) does not form.

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times or the U (1) Symmetric solutions , there are certain choices of lapse and shift picked out by the geometry (areal for Gowdy38 , harmonic time for U (1) Symmetric39 ); but these are special cases. More generally, the choice of lapse and shift remains a difficult issue. 5

Concluding Remarks

We have discussed a number of the challenges that one encounters in constructing and studying spacetimes via the initial value formulation. These occur both in finding sets of initial data which satisfy the Einstein constraint equation, and in evolving these sets of data. Some are fairly difficult. However, paraphrasing Winston Churchill’s description of democracy, one finds that the initial value formulation is the most impractical way to work with solutions of Einstein’s equations, except all of those other ways which have been tried from time to time40 . The fact that most numerical studies of solutions are carried out using the initial value formulation attests to the relative practicality of this approach. 6

Acknowledgments

I thank the Scientific Committee of GR16 for inviting me to speak, and I thank the local Organizing Committee and the local South African hosts for running a very fine conference. Portions of the work discussed have been supported by NSF grant PHY 0099373. References 1. C. Misner, K. Thorne and J. Wheeler, Gravitation (Freeman, San Franciso, 1973). 2. Y. Choquet-Bruhat and J. York in General Relativity and Gravitation, ed. A. Held (Plenum, New York, 1980). 3. R. Schoen, J. Diff. Geom. 20, 479 (1984). 4. J. Isenberg and J. Nester, Ann. Phys. 107, 368 (1977). 5. J. Isenberg, Class. Qtm. Grav. 12, 2249 (1995). 6. D. Brill and M. Cantor, Comput. Math. 43, 317 (1981). 7. M. Cantor, Comm. Math. Phys. 57, 83 (1977). 8. Y. Choquet-Bruhat, J. Isenberg and J. York, Phys. Rev. D 61, 084034 (2000). 9. L. Andersson, P. Chrusciel and H. Friedrich, Comm. Math. Phys. 149, 587 (1992).

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10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40.

L. Andersson and P. Chrusciel, Dissert. Math. 355, 1 (1996). J. Isenberg, Phys. Rev. Lett. 59, 2389 (1987). D. Brill and F. Flaherty, Ann. Inst. H. Poincare 28, 335 (1978). D. Brill in Proc Third Marcel Grossman Meeting, ed. H. Ning (NorthHolland, Amsterdam, 1982). R. Bartnik, Comm. Math. Phys. 117, 615 (1998). Y. Choquet-Bruhat, J. Isenberg and V. Moncrief, C. R. Acad. Sci. Paris 315, 349 (1992). J. Isenberg and V. Moncrief, Class. Qtm. Grav. 13, 1819 (1996). J. Isenberg, Fields Inst. Comm. 15, 59 (1997). J. Isenberg and J. Park, Class. Qtm. Grav. 14, A189 (1997). H. Friedrich and G. Nagy, Comm. Math. Phys. 210, 619 (1999). S. Klainerman and I Rodnianski, preprint: math.AP/0109173. H. Smith and D. Tataru, preprint: Sharp local well-posedness results for nonlinear wave equation. T. Damour in 300 Years of Gravitation, eds. S. Hawking and W. Israel (Cambridge Univ Press, New York, 1987). A. Buonanno and T. Damour, Phys. Rev. D 62, 06401 (2000). K. Alvi, Phys. Rev. D 64, 104020 (2001). J. Isenberg, R. Mazzeo, and D. Pollack, preprint: gr-qc/0109045. J. Corvino, Comm. Math. Phys. 214, 137 (2000). J. Corvino and R. Schoen, unpublished. C. Misner, Ann. Phys. 24, 102 (1963). H. Friedrich in Gravitation and Relativity at the Turn of the Millennium, eds. N. Dadhich and J. Narlikar (Inter-University Centre, Pune, 1998). Y. Bruhat, Acta Math 88, 141 (1952). D. Christodoulou and S. Klainerman, The Global Nonlinear Stability of the Minkowski Space (Princeton Univ Press, Princeton, 1993). L. Andersson and V.Moncrief, unpublished. D. Christodoulou, Ann. Math. 149, 183 (1999). B. Berger, Living Reviews 5, 2002-1 (2002). A.Rendall, Living Reviews 3, 2000-1 (2000). R. Penrose in General Relativity-An Einstein Centenary Survey, eds. S. Hawking and W. Israel (Cambridge Univ Press, New York, 1979). Y. Choqet-Bruhat and R. Geroch, Comm. Math. Phys. 14, 329 (1969). R. Gowdy, Ann. Phys. 83, 203 (1974). V. Moncrief , Ann. Phys. 167, 118 (1986) W. Churchill, House of Commons, 11 Nov., 1947.

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