3.4 Hyperboloidal initial data

Now the obvious problem is to determine hyperboloidal initial data. That such data exist follows already from Theorem 4 because one can construct hyperboloidal hypersurfaces together with data on them in any of the radiative solutions whose existence is guaranteed by that theorem. However, one can also construct such data sets in a similar way to the construction of Cauchy data on an asymptotically Euclidean hypersurface by solving the constraint equations implied on the Cauchy surface. Let &tidle;Σ be a hyperboloidal hypersurface that extends out to ℐ in an asymptotically flat vacuum space-time, touching ℐ in a two-surface ∂Σ that is topologically a two-sphere. The assumptions on &tidle;Σ are equivalent to the fact that &tidle;Σ := Σ ∪ S is a smooth Riemannian manifold with boundary, which carries a smooth metric hab and a smooth function Ω, obtained by restriction of the unphysical metric and the conformal factor. The conformal factor is a defining function for the boundary ∂ Σ (i.e. it vanishes only on ∂Σ with non-vanishing gradient), and together with the metric it satisfies
− 2 &tidle;hab = Ω hab (30 )
on Σ&tidle;, where &tidle;h ab is the metric induced on Σ&tidle; by the physical metric. Furthermore, let &tidle;χ ab be the extrinsic curvature of &tidle; Σ in the physical space-time. Together with &tidle; hab it satisfies the vacuum constraint equations,
R&tidle; − &tidle;χ &tidle;χab + (&tidle;χa )2 = 0, (31 ) ab a ∂&tidle;a &tidle;χab − ∂&tidle;bχ&tidle;aa = 0, (32 )
where &tidle;∂ a is the Levi–Civita connection of &tidle;h ab and R&tidle; is its scalar curvature.

Several results of increasing generality have been obtained. We discuss only the simplest case here, referring to the literature for the more general results. Assume that the extrinsic curvature is a pure trace term,

&tidle;χ = 1c&tidle;h . ab 3 ab

The momentum constraint (32View Equation) implies that c is constant while the hyperboloidal character of &tidle;Σ implies that c ⁄= 0. With these simplifications and a rescaling of &tidle;hab with a constant factor, the Hamiltonian constraint (31View Equation) takes the form

&tidle; R = − 6. (33 )
A further consequence of the condition (30View Equation) is the vanishing of the magnetic part Bab of the Weyl tensor. For any defining function ω of the boundary, the conformal factor has the form Ω = u− 2ω. Expressing Equation (33View Equation) in terms of the unphysical quantities hab and Ω yields the single second-order equation
8 ω2Δu − 8ω∂a ω∂ u + (ω2R − 4 ωΔ ω + 6 ∂aω ∂ ω) u = − 6u5. (34 ) a a
This equation is a special case of the Lichnérowicz equation and is sometimes also referred to as the Yamabe equation. For a given metric hab and boundary defining function ω, it is a second-order, non-linear equation for the function u. Note that the principal part of the equation degenerates on the boundary. Therefore, on the boundary, the Yamabe equation degenerates to the relation
a 5 u ∂ ω∂a ω = u .

Note also that u = 0 is a solution of Equation (34View Equation) which, however, is not useful for our purposes because it does not give rise to a meaningful conformal factor Ω. Therefore, we require that u be strictly positive and bounded on the boundary. Then the relation above determines the boundary values of u in terms of the function ω. Taking derivatives of Equation (34View Equation), one finds that also the normal derivative of u is fixed on the boundary in terms of the second derivative of ω.

A given metric &tidle;h ab does not fix a unique pair (h ,Ω ) ab. Therefore, Equation (34View Equation) has the property that, for fixed ω, rescaling the metric hab with an arbitrary smooth non-vanishing function πœƒ on Σ ∪ S according to −4 hab ↦→ πœƒ hab results in a rescaling of the solution u of Equation (34View Equation) according to u ↦→ πœƒu and, hence, a change in the conformal factor ٠↦→ πœƒ−2Ω.

Now we define the trace-free part sab = Ο•ab − 1habΟ•cc 3 of the projection Ο•ab of the trace-free part of the unphysical Ricci tensor onto Σ, and consider the equations

( ) −1 1 sab = − Ω ∂a∂bΩ − 3habΔ Ω , (35 ) ( ) −1 1- Eab = − Ω Rab − 3habR − sab , (36 )
which follow from the Equations (20View Equation) and (16View Equation), respectively. Together with the fields hab, Ω, s = 1ΔΩ 3, they provide initial data for all the quantities appearing in the evolution equations under the given assumptions. As they stand, these expressions are formally singular at the boundary and one needs to worry about the possibility of a smooth extension of the field to ∂ Σ. This question was answered in [5], where the following theorem was proved:

Theorem 7: Suppose (Σ, h) is a three-dimensional, orientable, compact, smooth Riemannian manifold with boundary ∂Σ. Then there exists a unique solution u of Equation (34View Equation), and the following conditions are equivalent:


The function u as well as the tensor fields sab and Eab determined on the interior Σ&tidle; from h and Ω = u−2ω extend smoothly to all of Σ.


The conformal Weyl tensor computed from the data vanishes on ∂Σ.


The conformal class of h is such that the extrinsic curvature of ∂Σ with respect to its embedding in (Σ,h) is pure trace.

Condition (3) in Theorem 7 is a weak restriction on the conformal class of the metric h on Σ, since it is only effective on the boundary. It is equivalent to the fact that in the space-time that evolves from the hyperboloidal data, null-infinity ℐ is shear-free. Interestingly, the theorem only requires Σ to be orientable and does not restrict the topology of Σ any further.

This theorem gives the answer in a highly simplified case because the freedom in the extrinsic curvature has been suppressed. But there are also several other, less restrictive, treatments in the literature. In [3Jump To The Next Citation Point4] the assumption (30View Equation) is dropped, allowing for an extrinsic curvature that is almost general apart from the fact that the mean curvature is required to be constant. In [101] also this requirement is dropped (but, in contrast to the other works, there is no discussion of smoothness of the implied conformal initial data), and in [103] the existence of hyperboloidal initial data is discussed for situations with a non-vanishing cosmological constant.

The theorem states that one can construct the essential initial data for the evolution once Equation (34View Equation) has been solved. The data are given by expressions that are formally singular at the boundary because of the division by the conformal factor Ω. This is of no consequences for the analytical considerations if Condition (3) in Theorem 7 is satisfied. However, even then it is a problem for the numerical treatments because one has to perform a limit process to get to the values of the fields on the boundary. This is numerically difficult. Therefore, it would be desirable to solve the conformal constraints directly. It is clear from Equations (82View Equation, 83View Equation, 84View Equation, 85View Equation, 86View Equation, 87View Equation, 88View Equation, 89View Equation, 90View Equation, 91View Equation, 92View Equation) that the conformal constraints are regular as well. Some of the equations are rather simple but the overall dependencies and interrelations among the equations are very complicated. At the moment there exists no clear analytical method (or even strategy) for solving this system. An interesting feature appears in connection with Condition (3) in Theorem 7 and analogous conditions in the more general cases. The necessity of having to impose this condition seems to indicate that the development of hyperboloidal data is not smooth but in general at most 2 π’ž. If the condition were not imposed then logarithms appear in an expansion of the solution of the Yamabe equation near the boundary, and it is rather likely that these logarithmic terms will be carried along with the time evolution, so that the developing null-infinity loses differentiability. Thus, the conformal boundary is not smooth enough and, consequently, the Weyl tensor need not vanish on ℐ which, in addition, is not necessarily shear-free. The Sachs peeling property is not completely realized in these situations. One can show [3] that generically hyperboloidal data fall into the class of “poly-homogeneous” functions, which are (roughly) characterized by the fact that they allow for asymptotic expansions including logarithmic terms. This behaviour is in accordance with other work [165] on the smoothness of ℐ, in particular with the Bondi–Sachs type expansions, which were restricted by the condition of analyticity (i.e. no appearance of logarithmic terms). It is also consistent with the work of Christodoulou and Klainerman.

Solutions of the hyperboloidal initial value problem provide pieces of space-times that are semi-global in the sense that their future or past developments are determined depending on whether the hyperboloidal hypersurface intersects + ℐ or − ℐ. However, the domain of dependence of a hyperboloidal initial surface does not include space-like infinity and one may wonder whether this fact is the reason for the apparent generic non-smoothness of null-infinity. Is it not conceivable that the possibility of making a connection between + ℐ and − ℐ across 0 i to build up a global space-time automatically excludes the non-smooth data? If we let the hyperboloidal initial surface approach space-like infinity, it might well be that Condition (3) in Theorem 7 imposes additional conditions on asymptotically flat Cauchy data at spatial infinity. These conditions would make sure that the development of such Cauchy data is an asymptotically flat space-time, in particular that it has a smooth conformal extension at null-infinity.

These questions give some indications about the importance of gaining a detailed understanding of the structure of gravitational fields near space-like infinity. One of the difficulties in obtaining more information about the structure at space-like infinity is the lack of examples that are general enough. There exist exact radiative solutions with boost-rotation symmetry [22Jump To The Next Citation Point]. They possess a part of a smooth null-infinity, which, however, is incomplete. This is a general problem because the existence of a complete null-infinity with non-vanishing radiation restricts the possible isometry group of a space-time to be at most one-dimensional with space-like orbits [14]. Some of the boost-rotation symmetric space-times even have a regular i0; thus they have a vanishing ADM mass. Other examples exist of space-times that are solutions of the Einstein–Maxwell [39] or Einstein–Yang–Mills [15] equations. They have smooth and complete null-infinities. However, they were constructed in a way that enforces the field to coincide with the Schwarzschild or the Reissner–Nordström solutions near i0. So they are not general enough to draw any conclusions about the generic behaviour of asymptotically flat space-times near 0 i.

Recently, the still outstanding answer to the question as to whether there exist at all global asymptotically simple vacuum space-times with smooth null-infinity could be answered in the affirmative. Corvino [35Jump To The Next Citation Point] has shown that there exist asymptotically flat metrics on 3 ℝ with vanishing scalar curvature that are spherically symmetric outside a compact set. His method of construction consists of a gluing procedure by which a scalar flat and asymptotically flat metric can be deformed in an annular region in such a way that it can be glued with a predetermined degree of smoothness to Schwarzschild data while remaining scalar flat. Very recently a generalisation of this construction to non-time-symmetric initial data has been presented by Corvino and Schoen [36Jump To The Next Citation Point].

Such a metric satisfies the constraint equations implied by the vacuum Einstein equations on a space-like hypersurface of time symmetry. It evolves into a space-time that is identical to the Schwarzschild solution near space-like infinity. Within this space-time, which exists for some finite time interval, one can now find hyperboloidal hypersurfaces on which hyperboloidal initial data are induced. This hypersurface can be chosen close enough to the initial Cauchy surface so that it intersects the domain of dependence of the asymptotic region of the initial hypersurface where the data are Schwarzschild. This means that the data implied on the hyperboloidal hypersurface are also Schwarzschild close to ℐ so that the conditions of Theorem 7 are clearly satisfied.

If one could now apply Friedrich’s stability result (Theorem 6), then the existence of global asymptotically flat space-times with smooth extension to null-infinity would have been established. However, one assumption in this theorem is that the data be “sufficiently small”. Therefore, one needs to check that one can in fact perform Corvino’s construction in the limit of vanishing ADM mass while maintaining a sufficiently “large” Schwarzschild region around i0 so that one can find enough hyperboloidal hypersurfaces intersecting the Schwarzschild region in the time development of the data. ChruΕ›ciel and Delay [33] (see also the erratum in [32]) have adapted Corvino’s construction to that situation, and show that it is possible to construct time-symmetric initial data that are Schwarzschild outside a fixed compact region and with a fixed degree of differentiability in the limit mADM → 0. From these data one can construct hyperboloidal hypersurfaces of the type required in Theorem 6 on which hyperboloidal data are induced which satisfy the smallness criterion of that theorem. Hence, there exists a complete space-time in the future of the initial Cauchy surface that admits a conformal extension to null-infinity as smooth as one wishes. Since the argument can also be applied towards the past, one has shown the existence of global space-times with smooth ℐ.

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