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Solutions of the vacuum Einstein equations with initial data on past null infinity

We prove existence of vacuum space-times with freely prescribable cone-smooth initial data on past null infinity.


Introduction
A question of interest in general relativity is the construction of large classes of space-times with controlled global properties. A flagship example of this line of enquiries is the Christodoulou-Klainerman theorem [3] of nonlinear stability of Minkowski space-time. Because this theorem carries only limited information on the asymptotic behaviour of the resulting gravitational fields, and applies only to near-Minkowskian configurations in any case, it is of interest to construct space-times with better understood global properties. One way of doing this is to carry out the construction starting from initial data at the future null cone, I − , of past timelike infinity i − . An approach to this has been presented in [10], but an existence theorem for the problem is still lacking. The purpose of this work is to fill this gap.
In order to present our result some terminology and notation is needed: Let C O denote the (future) light-cone of the origin O in Minkowski space-time (throughout this work, by "light-cone of a point O" we mean the subset of a spacetime M covered by future directed null geodesics issued from O). Let, in manifestly flat coordinates y µ , ℓ = ∂ 0 + (y i /| y|)∂ i denote the field of null tangents to C O . Letd αβγδ be a tensor with algebraic symmetries of the Weyl tensor and with vanishing η-traces, where η denotes the Minkowski metric. Let ς be the pull-back ofd αβγδ ℓ α ℓ γ to C O \ {O}. Let, finally, ς ab denote the components of ς in a frame parallelpropagated along the generators of C O . We prove the following: where C αβγδ is the Weyl tensor of g, extends smoothly across {Θ = 0}, and ς ab are the frame components, in a g-parallel-propagated frame, of the pull-back to C O of d αβγδ ℓ α ℓ γ . The solution is unique up to isometry.

Strategy of the proof
The starting point of our analysis are the conformal field equations of Friedrich. The task consists of constructing initial data, for those equations, which arise as the restriction to the future light-cone I − of past timelike infinity i − of tensors which are smooth in the unphysical space-time. We then use a system of conformally invariant wave equations of [13] to obtain a space-time with a metric solving the vacuum Einstein equations to the future of i − . Now, some of Friedrich's conformal equations involve only derivatives tangent to I − , and have therefore the character of constraint equations. Those equations form a set of PDEs with a specific hierarchical structure, so that solutions can be obtained by integrating ODEs along the generators of I − . This implies that the constraint equations can be solved in a straightforward way in coordinates adapted to I − in terms of a subset of the fields on the light-cone. However, there arise serious difficulties when attempting to show that solutions of the conformal constraint equations can be realized by smooth space-time tensors. These difficulties lie at the heart of the problem at hand. To be able to handle this issue, we note that ς determines the null data of [12]. These null data are used in [12] to construct smooth tensor fields satisfying Friedrich's equations up to terms which decay faster than any power of the Euclidean coordinate distance from i − , similarly for their derivatives of any order; such error terms are said to be O(|y| ∞ ). For fields on the light-cone, the notation O(r ∞ ) is defined similarly, where r is an affine distance from the vertex along the generators, with derivatives only in directions tangent to the light-cone. In particular the approximate solution so obtained solves the constraint equations up to error terms of order O(r ∞ ). Using a comparison argument, we show that the approximate fields differ, on C O , from the exact solution of the constraints by terms which are O(r ∞ ). But tensor fields on the light-cone which decay to infinite order in adapted coordinates arise from smooth tensors in space-time, which implies that the solution of the constraint equations arises indeed from a smooth tensor in space-time. As already indicated, this is what is needed to be able to apply the existence theorems for systems of wave equations in [7], provided such a system is at disposal. This last element of our proof is provided by the system of wave equations of [13], and the results on propagation of constraints for this system established there. 1

From approximate solutions to solutions
Recall Friedrich's system of conformally-regular equations (see [11] and references therein) Here Θ is the conformal factor relating the physical metricg µν with the unphysical metric g µν = Θ −2g µν , the fields d µνσ κ and L αβ encode the information about the unphysical Riemann tensor as made explicit in (2.6), while the trace of (2.3) can be viewed as the definition of s.
We wish to construct solutions of (2.1)-(2.6) with initial data on a light-cone C i − , emanating from a point i − , with Θ vanishing on C i − and with s(i − ) = 0. (The actual value of s(i − ) can be changed by constant rescalings of the conformal factor Θ and of the field d αβγ δ . For definiteness we will choose s(i − ) = −2.) As explained in [9], such solutions lead to vacuum space-times, where past timelike infinity is the point i − and where past null infinity We will present two methods of doing this: while the second one is closely related to the classical one in [2], the advantage of the first one is that it allows in principle a larger class of initial data, see Remark 2.6 below.
Let, then, a "target metric"ĝ be given and let the operator∇ denote its covariant derivative with associated Christoffel symbolsΓ σ αβ . Set (2.7) Consider the system of wave equations which [13, Section 6.1] follows from (2.1)-(2.6) when H σ vanishes: (2.14) Further, the field ξ µνσ above will, in the final space-time, be the Cotton tensor, related to the Schouten tensor L µν as Finally, the operator ✷ (H) g is defined as = ✷ g , so one may wonder why we are not simply using ✷ g . The issue is that the operator ✷ g on tensor fields of nonzero valence contains second-order derivatives of the metric, so that the principal part of a system of equations obtained by replacing ✷ (H) g by ✷ g in (2.8)-(2.13) will not be diagonal. This could be cured by adding equations obtained by differentiating (2.13), which is not convenient as it leads to further constraints. Instead, one observes [13, Section 3.1] that the second derivatives of the metric appearing in ✷ g can be eliminated in terms of the remaining fields above. For example, for a covector field v, with f λ changing from line to line. This leads to the definition consistently with (2.15). An identical calculation shows that the operator (2.15) has the properties just described for higher-valence covariant tensor fields.
It follows from the above that the principal part of ✷ This implies that the principal part of (2.8)-(2.13) is diagonal, with principal symbol equal to g µν p µ p ν times the identity matrix. In particular, we can use [7] to find solutions of our equations whenever suitably regular initial data are at disposal.
Let (x 0 , x 1 ≡ r, x A ) be coordinates adapted to the light-cone C i − of i − as in [1,Section 4], and let κ measure how the coordinate x 1 differs from an affine parameter along the generators of the light-cone of i − : There are various gauge freedoms in the equations above. To get rid of this we can, and will, imposê (2.17) The conditionĝ µν = η µν is a matter of choice. The conditions R = 0 and H σ = 0 are classical, and can be realized by solving wave equations. The condition κ = 0 is a choice of parameterization of the generators of C i − . The fact that s can be made a negative constant on C i − is justified in Appendix A, see Remark A.3. As already pointed out, the value s = −2 is a matter of convenience, and can be achieved by a constant rescaling of Θ and of the field d αβγ δ . Consider the set of fields Ψ = (g µν , L µν , C µνσ ρ , ξ µνσ , Θ, s) . (2.18) We will denote byΨ := (g µν ,L µν ,C µνσ ρ ,ξ µνσ ,Θ,s) and define λ AB to be the solution of the equation Here, and elsewhere, the symbol D A denotes the covariant derivative ofg AB dx A dx B . Let s AB denote the unit round metric on S 2 . We will need the following result [13, Theorem 6.5]: Theorem 2.1 Consider a set of smooth fields Ψ defined in a neighborhood U of i − and satisfying (2.8)-(2.13) in I + (i − ). Define the data (2.19) by restriction of Ψ to C i − , suppose thatΘ = 0 ands = −2. Then the fields and where λ AB is the solution of (2.21) satisfying λ AB = O(r 3 ), or differs from that solution by O(r ∞ ) terms. Our first main result is the following: Theorem 2.4 Letg µν be a smooth metric defined near i − such that for small r we haveg LetL µν be the Schouten tensor ofg µν , letξ αβγ be the Cotton tensor ofg µν and letC αβγβ be its Weyl tensor. Assume that (L µν ,ξ µνρ ) solves the approximate constraint equations. Then there exist smooth fields (g µν , L µν , C µνσ ρ , Θ, s) defined in a neighbourhood of i − such that the fields solve the conformal field equations (2.1)-(2.6) in I + (i − ), satisfy the gauge conditions (2.17), with

43)
with the conformal factor Θ positive on I + (i − ) sufficiently close to i − , and with Proof: We will apply Theorem 2.1 to a suitable evolution of the initial data. For this we need to correct Ψ by smooth fields so that the restriction to the light-cone of the new Ψ satisfies the constraint equations as needed for that theorem. Subsequently, we define new fieldš g µν =g µν + δg µν ,Ľ µν =L µν + δL µν ,ξ µνσ =ξ µνσ + δξ µνσ , as follows: We let δg µν be any smooth tensor field defined in a neighborhood of i − which is O(|y| ∞ ) and which satisfies δg µν = η µν −g µν .
Indeed, it follows from e.g. [4, Equations (C4)-(C5)] that the y-coordinates components δg µν of g are O(r ∞ ), and existence of their smooth extensions follows from [5, Lemma A.1]. This extension procedure will be used extensively from now on without further reference.
To continue, we let δξ αβγ be any smooth tensor defined in a neighborhood of i − , with y-coordinate-components which are O(|y| ∞ ), such that 4. δξ 01A = −ξ 01A +g BCξ BAC ; 5. δξ 00A is the solution vanishing at r = 0 of the system of ODEs whereρ is the unique bounded solution of A1B . Finally, we let δL µν be any smooth tensor field defined in a neighborhood of i − , the y-components of which are O(|y| ∞ ), such that: we emphasise that the η AB -trace-free part of L AB coincides thus with the η AB -trace-free part ofL AB ); 4. δL 00 is the solution vanishing at r = 0 of the system of ODEs 4(∂ 1 + r −1 )(δL 00 +L 00 ) = λ AB ω AB − 2D AL 0A − 4rρ (2.46) (it follows from [4, Appendix B] that δL 00 is O(r ∞ )).
A solution exists by [7,Théorème 2]. It follows by construction that the hypotheses of Theorem 2.1 hold, and the theorem is proved. ✷ An alternative way of obtaining solutions of our problem proceeds via the following system of conformal wave equations:  A smooth metricg will be called an approximate solution of the constraint equations (2.53)-(2.60) ifC α βγδ =Θd α βγδ for some smooth functionΘ vanishing on C i − and for some smooth tensord α βγδ , whereC α βγδ is the Weyl tensor ofg, and if (2.53)-(2.60) hold on the light-cone of i − up to terms which are O(r ∞ ), whereL µν is the Schouten tensor ofg, and where ω AB and λ AB are, possibly up to O(r ∞ ) terms, given by (2.20)-(2.21).
Our second main result is the following:

61)
with the conformal factor Θ positive on I + (i − ) sufficiently close to i − , and with Proof: We will apply Theorem 2.5 to a suitable evolution of the initial data. For this we need to correct (g µν ,L µν ,d µνσρ ) by smooth fields so that the new initial data on the light-cone satisfy the constraint equations as needed for that theorem. The construction of the new fieldš g µν =g µν + δg µν ,Ľ µν =L µν + δL µν ,ď µνσρ =d µνσρ + δd µνσρ , (2.62) is essentially identical to that of the new fields of the proof of Theorem 2.4, the reader should have no difficulties filling-in the details. We emphasise that the trace-free part of δL AB is chosen to be zero, hence the trace-free part ofĽ AB coincides with the trace-free part ofL AB on the light-cone. Once the fields (2.62) have been constructed, we let (g µν , L µν , d µνσρ , Θ, s) be a solution of (2.47)-(2.51) with initial data (g µν ,L µν ,d µνσρ ,Θ,s) := (η µν ,Ľ µν ,ď µνσρ , 0, −2) .
A solution exists by [7,Théorème 2]. It follows by construction that the hypotheses of Theorem 2.5 hold, and the theorem is proved. ✷ 3 Proof of Theorem 1.1 We are ready now to prove Theorem 1.1: Let ς be the cone-smooth tensor field of the statement of the theorem. Thus, there exists a smooth tensor fieldd αβγδ with the algebraic symmetries of the Weyl tensor so that ς is the pull-back of Letψ MN P Q be a totally-symmetric two-index spinor associated tod αβγδ in the usual way [12, Section 3] (compare [14]). Set θ 0 = dt, θ 1 = dr. Let γ be a generator of C 0 , and let θ 2 , θ 3 be a pair of covector fields so that {θ µ } forms an orthonormal basis of T * M over γ and which are η-parallel propagated along γ. Then ς can be written as ς = ς ab θ a θ b , with a, b running over {2, 3}. By construction, the coordinate components ς AB of ς coincide with the coordinate componentsd 1A1B of the restriction to the light-cone ofd 1A1B , and thus define a unique field ω AB by integrating (2.54) with the boundary condition ω AB = O(r 2 ). Let the basis {e µ } be dual to {θ µ }, set m = e 2 + √ −1e 3 . Then the radiation field ψ 0 of [12, Equation (5.3)], defined usingψ MN P Q , equals (Under a rotation of {e 3 , e 4 } the field ψ 0 changes by a phase, and defines thus a section of a spin-weighted bundle over C O \ {O}.) Conversely, any radiation field ψ 0 of [12] arises from a unique cone-smooth ς ab as above.
It has been shown in [12,Propositions 8.1 & 9.1] that the radiation field ψ 0 , hence ς, defines a smooth Lorentzian metricg such that the resulting collection of fields (g µν ,L µν ,d µνσρ ,Θ,s) satisfies (2.1)-(2.6) up to error terms which are O(|y| ∞ ), withg µν | CO = η µν . The construction in [12] is such that the field ς calculated from the fieldd αβγδ of (3.1) coincides with the field ς calculated from the fieldd αβγδ associated with the metricg. Hence the fields ω AB associated withd αβγδ andd αβγδ are identical. The conclusion follows now from Theorem 2.8. ✷ 4 Alternative data at I − Recall that there are many alternative ways to specify initial data for the Cauchy problem for the vacuum Einstein equations on a (usual) light-cone, cf. e.g. [6].
Similarly there are many ways to provide initial data on a light-cone emanating from past timelike infinity. In Theorem 1.1 some components of the rescaled Weyl tensor d µνσρ have been prescribed as free data. As made clear in the proof of that theorem, this is equivalent to providing some components of the rescaled Weyl spinor ψ MN P Q , providing thus an alternative equivalent prescription. Our Theorems 2.4 and 2.8 use instead the components (2.33) of the rescaled Schouten tensorL µν . These components are related directly to the free data of Theorem 1.1 via the constraint equation (2.54). It is clear that further possibilities exist. Which of these descriptions of the degrees of freedom of the gravitational field at large retarded times is most useful for physical applications remains to be seen.

A The s = −2 gauge
We start with some terminology. We say that a function f defined on a space- where |y| := n µ=0 (y µ ) 2 . A similar definition will be used for functions defined in a neighbourhood of O on the future light-cone For this, we parameterize C O by coordinates y = (y i ) ∈ R n , and we say that a function f defined on a neighbourhood of O within C O is o m (r k ) if f is a C m function of the coordinates y i and if for 0 ≤ ℓ ≤ m we have lim r→0 r ℓ−k ∂ µ1 . . . ∂ µ ℓ f = 0, where We further set A function ϕ defined on C O will be said to be C k -cone-smooth if there exists a function f on space-time of differentiability class C k such that ϕ is the restriction of f to C O . We will simply say cone-smooth if k = ∞.
The following lemma will be used repeatedly: with where f i1...ip and f ′ i1...ip−1 are numbers. The claim remains true with k = ∞ if (A.1) holds for all k ∈ N. ✷ Coefficients f p of the form (A.2) will be said to be admissible.
One of the elements needed for the construction in [12] is provided by the following result: Proof: In dimension n let g ′ = φ 4/(n−2) g, then So, in dimension n = 4, and with φ = θ we obtain We overline restrictions of space-time functions to the light-cone. The equation ℓ µ ℓ ν S µν [θ 2 g] = 0 takes thus the form In coordinates adapted to the light-cone as in [1, Appendix A], so that ℓ µ ∂ µ = ∂ r , with r an affine parameter, this reads Setting ϕ := ∂ r θ θ , this can be rewritten as It is useful to introduce some notation. As in [1], we underline the components of a tensor in the coordinates y µ , thus: etc, where (x µ ) := (y 0 − | y|, | y|, x A ), with x A being any local coordinates on S 2 . We write interchangeably x 1 and r.
The initial data for ϕ are To show that θ is cone-smooth, it suffices to prove that ψ := rϕ is C k -cone-smooth for all k, as follows immediately from the expansions of Lemma A.1, together with integration term-by-term in the formula ln θ a = We shall proceed by induction. So suppose that ψ is C k -cone-smooth. The result is true for k = 0 since every solution of (A.5) is continuous in all variables, and rϕ tends to zero as r tends to zero, uniformly in Θ i ∈ S 2 .
The right-hand side is C k -cone-smooth. We conclude that ψ is C k+1 -conesmooth. This finishes the induction, and proves that θ is cone-smooth. The existence and uniqueness of a solution θ of (A.4) which equals θ on C O follows now from [7, Théorème 2]. ✷