Characteristic initial data and smoothness of Scri.
II. Asymptotic expansions and construction of conformally smooth data sets^{†}^{†}thanks: Preprint UWThPh20149.
Abstract
We derive necessaryandsufficient conditions on characteristic initial data for Friedrich’s conformal field equations in dimensions to have no logarithmic terms in an asymptotic expansion at null infinity.
PACS: 02.30.Hq, 02.30.Mv, 04.20.Ex, 04.20.Ha
Contents
 1 Introduction
 2 Preliminaries
 3 Asymptotic solutions of Einstein’s characteristic vacuum constraint equations
 4 Metric gauge
 5 Asymptotic Expansions of the unknowns of the CFE on
 A Asymptotic solutions of Fuchsian ODEs
 B Relation between  and gauge
1 Introduction
In this work we continue the work initiated in [7] to analyze the occurrence of logarithmic terms in the asymptotic expansion of the metric tensor and some other fields at null infinity. In part I of this work, where we also described the setting, it has been shown that the harmonic coordinate condition is not compatible with a smooth asymptotic structure at the conformal boundary at infinity, but has to be replaced by a wavemap gauge condition with nonvanishing gaugesource functions.
However, it is expected [5, 12] (compare also [1]) that even for smooth initial data the asymptotic expansion of the spacetime metric at null infinity will generically be polyhomogeneous and involve logarithmic terms which do not have their origin in an inconvenient choice of coordinates. One main object of this note, treated in Section 3, is to study thoroughly the asymptotic behavior of solutions of the Einstein’s vacuum constraint equations and analyze under which conditions a smooth conformal completion of the restriction of the spacetime metric to the characteristic initial surface across null infinity is possible. As announced in [7, Theorem 5.1] we intend to provide necessaryandsufficient conditions on the initial data and the gauge source functions which permit such extensions. In doing so it will become manifest that many, though not all, of the logarithmic terms which arise at infinity are gauge artifacts. The remaining nongauge logarithmic terms can be eliminated by imposing restrictions on the asymptotic behavior of the initial data, captured by what we call nologscondition.
In Section 5, we will show that solutions of the characteristic vacuum constraint equations satisfying the nologscondition lead to smooth initial data for Friedrich’s conformal field equations. The data will be computed in a new gauge scheme developed in Section 4 and will provide the basis to solve the evolution problem and construct spacetimes with a “piece of smooth ”.
In Section 2 we give a summary of [7] where we briefly describe the framework and recall the most important definitions and results of part I. Finally, in Appendix A our proceeding in Section 35 to solve the constraint equations in terms of polyhomogeneous expansions will be rigorously justified, while in Appendix B we compare the peculiarities of different gauge schemes.
2 Preliminaries
We use all the notation, terminology and conventions introduced in part I [7]. For the convenience of the reader, though, let us briefly recall the most essential ingredients and definitions of our framework.
2.1 Notation
Consider a smooth function
If this function permits an asymptotic expansion as a power series in , we denote by , or , the coefficient of in the corresponding expansion. Be aware that sometimes a lower index might denote both the component of a vector, and the th order term in an expansion of the corresponding object. If both indices need to appear simultaneously we use brackets and place the index corresponding to the th order expansion term outside the brackets. We write
if the righthand side is the polyhomogeneous expansion at of the function . Moreover, we write (or , ), , if the function is smooth at .
2.2 Setting
We consider a dimensional manifold . For definiteness we take as initial surface either a globally smooth lightcone or two null hypersurfaces intersecting transversally along a smooth submanifold . Suppose that the closure (in the completed spacetime) of meets transversally in a smooth spherical crosssection.
We introduce adapted null coordinates on (i.e. , where parameterizes the null rays generating , and are local coordinates on ). Then the trace of the metric on becomes
(2.1) 
where
(2.2) 
is a degenerate quadratic form induced by on which induces on each slice an dependent Riemannian metric (coinciding with in the coordinates above). While the components , and depend upon the choice of coordinates off , the quadratic form is intrinsically defined on .
In fact, we will be interested merely in the asymptotic behavior of the restriction of the spacetime metric to . A regular lightcone or two transversally intersecting null hypersurfaces with appropriately specified initial data, though, guarantee that the vacuum constraint equations have unique solutions.
Throughout this work we use an overline to denote a spacetime object restricted to . The symbol “” will be used to denote objects associated with the Riemannian metric .
Definition (cf. [7])
We say that a smooth metric tensor defined on a null hypersurface given in adapted null coordinates has a smooth conformal completion at infinity if the unphysical metric tensor obtained via the coordinate transformation and the conformal rescaling is, as a Lorentzian metric, smoothly extendable at .
The components of a smooth tensor field on will be said to be smooth at infinity whenever they admit a smooth expansion in the conformally rescaled spacetime at and expressed in the coordinates.
As remarked in [7], Definition 2.2 concerns only fields on and is not tied to the existence of an associated spacetime. Moreover, it concerns both conditions on the metric and on the coordinate system.
The Einstein equations split into a set of evolution equations and a set of constraint equations which need to be satisfied on the initial surface. In the characteristic case the data for the evolution equations are provided by the trace of the metric on the initial surface. Due to the constraints not all of its components can be prescribed freely. There are various ways of choosing free data [6, 7]. Here we focus on Rendall’s scheme [10], where the free data are formed by the conformal class of the tensor (and the function introduced below).^{1}^{1}1 In the case of two transversally intersecting null hypersurfaces these data need to be supplemented by corresponding data on and certain data on the intersection manifold . By choosing a representative they can be viewed as a oneparameter family, parameterized by , of Riemannian metrics on . A major advantage of this scheme in particular in view of Section 4 is that it permits a separation of physical and gauge degrees of freedom. Some comments on how things change for other approaches to prescribe characteristic initial data are given in [7], cf. Remark 3.3.
Einstein’s vacuum constraint equations in a generalized wavemap gauge are obtained from Einstein’s vacuum equations assuming that the wavegauge vector
(2.3) 
vanishes. We use the hatsymbol “” to indicate objects associated with some target metric , which we assume for convenience to be of the form
(2.4) 
on , where is the unit round metric on the sphere . By we denote the components of a vector field, the gauge source functions, which can be arbitrarily prescribed. They reflect the freedom to choose coordinates off the initial surface, and thus allow us to analyze smoothness of the metric tensor at infinity in arbitrary coordinates.
For given initial data the wavemap gauge constraints form a hierarchical system of ODEs along the null generators of the cone (cf. [2]):
(2.5)  
(2.6)  
(2.7)  
(2.8)  
(2.9)  
(2.10) 
where and denote the expansion and the shear of , respectively,
(2.11) 
Apart from the function turns out to be another gauge degree of freedom, reflecting the freedom to choose , which can be prescribed conveniently.
Integrating these equations successively one determines all the components of (note that and ). The relevant boundary conditions follow either from regularity conditions at the vertex [2] when is a cone, or from the remaining data specified on and (cf. e.g. [6, 10]) in the case of two characteristic surfaces intersecting transversally.
Our ultimate goal is to find necessaryandsufficient conditions on the initial data given on , such that the resulting spacetime has a smooth conformal completion at infinity à la Penrose. This requires the following ingredients:

To exclude the appearance of conjugate points or coordinate singularities on , the constraint equations need to admit a nondegenerated global solution on . This will be the case if and only if the functions and are of constant sign,
(2.12) 
The metric needs to be smoothly extendable as a Lorentzian metric, which means that the functions and need to have a sign
(2.13) This assumption excludes conjugate points at the intersection of with .

The components of need to be smooth at . For this one has to make sure that their asymptotic expansions contain no logarithmic terms and have the correct order in terms of powers of .

All the fields which appear in Friedrich’s conformal field equations (which provide an evolution system which, in contrast to Einstein’s field equations, is wellbehaved at ) need to be smooth at .

Finally, an appropriate wellposedness result for the conformal field equations is needed.
Point 1 and 2 have been addressed in [7], cf. Proposition 2.2 below. Point 3 will be the subject of Section 3, while point 4 will be investigated in Section 5. Point 5 will be addressed elsewhere.
From now on we shall consider exclusively conformal data and gauge functions and for which (2.12) and (2.13) hold. Let us summarize some of the results established in [7] (adapted to the smooth setting on which we focus here) which provide sufficient conditions such that (2.12) and (2.13) hold in the case where represents a regular lightcone :^{2}^{2}2 In the conventions of [7] the functions , , and need to be positive.
2.3 A priori restrictions
Before we analyze thoroughly the asymptotic behavior of the vacuum Einstein constraint equations and derive necessaryandsufficient conditions concerning smoothness of the solutions at infinity it is convenient to have some a priori knowledge regarding the lowest admissible orders of the gauge functions, and to exclude the appearance of log terms in the expansion of “auxiliary” fields such as . In [7] we have shown that the following equations need to be necessarily fulfilled in some adapted null coordinate system to end up with a trace of a metric on which admits a smooth conformal completion and infinity, and connection coefficients which are smooth at :
(2.14) 
Moreover, we may assume the initial data to be of the asymptotic form
(2.15) 
for some smooth tensor fields on . If is not of the form (2.15), it can either be brought to it via a conformal rescaling and an suitable choice of , or it leads to a metric which does not have a smooth conformal completion at .
At this stage we do not know whether a spacetime which admits a conformal completion at infinity is compatible with polyhomogeneous rather than smooth expansions of the functions and . It will turn out that this is not the case. However, we note that it follows from the constraint equations and the above a priori restrictions that
if then .  (2.16) 
3 Asymptotic solutions of Einstein’s characteristic vacuum constraint equations
It is useful to introduce some notation first: Let be a rank2tensor on the initial surface . We denote by , or , its tracefree part w.r.t. the metric . Consider now the expansion coefficients at infinity, which are tensors on . We denote by , or , the tracefree part w.r.t. the unit round metric . Finally, we set
Let us make the convention to raise indices of the expansion coefficients with the standard metric. Moreover, we set
A ring on a covariant derivative operator or a connection coefficient indicates that the corresponding object is associated to .
3.1 Asymptotic solutions of the constraint equations
The object of this section is twofold: First of all we will show that the Einstein wavemap gauge constraints (2.5)(2.10) can be solved asymptotically in terms of polyhomogeneous expansions at infinity of the solution. This is done by rewriting the equations in a form to which Appendix A applies. The second aim is to make some general considerations concerning the appearance of logarithmic terms in the asymptotic solutions of (2.5)(2.10) for initial data of the form (2.15). We want to extract necessaryandsufficient conditions leading to the trace of a metric on which admits a smooth conformal completion at infinity in the sense of Definition 2.2.
Our starting point are initial data with an asymptotic behavior of the form (2.15) and gauge functions
(3.1) 
for which , , and have a sign on and , respectively. We further require that the asymptotic expansion of contains no logarithmic terms, i.e. . A violation of one of these assumptions would not be compatible with a spacetime which admits a smooth conformal completion at infinity as follows from the a priori restriction and the following fact: The considerations below reveal that (2.15), and imply , and that (2.16) applies. So we are imposing no restrictions when assuming .
Consider the shear tensor,
(3.2) 
whose asymptotic expansion we express in terms of the expansion coefficients of the initial data
(3.3)  
(3.4) 
A global solution, and thereby also the value of the “asymptotic integration functions”, to each of the constraint ODEs is determined by regularity conditions at the vertex of a lightcone, or by the data on the intersection manifold for two intersecting characteristic surfaces. However, the integration functions, which depend on the initial data , the gauge functions and the boundary conditions at the vertex or the intersection manifold, appear difficult to control.
3.1.1 Expansion of
We start with the constraint equation (2.5) for the function ,
(3.5) 
In order to enable an easier comparison to Appendix A we make the transformation , with , and treated as scalars, and rewrite the ODE as a firstorder system. The equation then reads with and ,
or, when the leading order term is diagonalized,
with
The results in Appendix A (we have, in the notation used there, , , and, since there is no source, ) shows that this ODE can be solved asymptotically via a polyhomogeneous expansion with . It also reveals that the coefficients , , i.e. and , can be regarded as integrations functions, and that logarithmic terms do not appear if and only if (cf. condition (A.19))
(3.6) 
Recall that denotes the term (term) in the asymptotic expansion of the corresponding field. Using and we observe that (3.6) holds automatically. In particular we have . Furthermore, since
the coefficients and can be identified as the integration functions. As indicated above, though being left undetermined by the equation itself, they have global character.
In the following we shall set for convenience
(3.7) 
Inserting the expansion into (3.5) and equating coefficients gives the expansion coefficients by a hierarchy of equations,
(3.8)  
(3.9) 
while
(3.10)  
Consider the conformal factor relating the dependent Riemannian metric and , . We find
(3.11)  
We conclude that
(3.13)  
To sum it up, (2.15) and imply that no logarithmic terms appear in the conformal factor relating and , the latter one thus being smoothly extendable at as a Riemannian metric on whenever a global solution of the Raychaudhuri equation exists on with .
3.1.2 Expansion of
We consider the constraint equation (2.6) which determines ,
(3.14) 
where and, using (2.4),
(3.15)  
Again, we express the ODE by . Its asymptotic form reads
(3.16)  
We want to make sure that the asymptotic solution of can be written as power series. In the notation of Appendix A we have and , and condition (A.10) which characterizes the absence of logarithmic terms reads
(3.17) 
Assuming (3.17) we insert the expansion into (3.14) to obtain the expansion coefficients in terms of , , and ,
(3.18) 
where denotes the globally defined integration function.
As mentioned above, for to have a smooth conformal completion at infinity the function needs to be of constant sign which is the case if and only if the gauge source function is chosen so that (compare Proposition 2.2)
(3.19) 
For the inverse of we then find
(3.20) 
Note that in the special case where we have and the positivity of follows from that of .
Since can be prescribed arbitrarily and the value of does not depend on that choice, (3.19) is not a geometric restriction. Similarly, (3.17) can be fulfilled by an appropriate choice of . The gauge freedom associated with the choice of can be used to control the behavior of and to get rid of the log terms in its asymptotic expansion (only the two leading order terms in the expansion of are affected, compare the discussion in Section 3.3).
3.1.3 Expansion of
The connection coefficients are determined by (2.7),
(3.21) 
To compute the covariant derivative of associated to , we first determine the asymptotic form of the Christoffel symbols,
with
(3.22)  
(3.23)  
Invoking (3.3) that yields
(3.24) 
where
(3.25)  
(3.26) 
Substituting now the coefficients by their asymptotic expansions we observe that (3.21) has the asymptotic structure,
Nicely enough, the ODEs for , , are decoupled. For comparison with the formulae in Appendix A we rewrite them in terms of ,
Appendix A tells us (with and ) that there are no logarithmic terms in the expansion of if and only if (A.10) holds,
(3.27) 
The asymptotic expansion (3.10) of implies
(3.28) 
such that (3.27) can be written as
(3.29) 
Note that , on which we have not imposed conditions yet, drops out, so there is no gauge freedom left which could be appropriately adjusted to fulfill this equation. The impact of (3.29) will be analyzed in Section 3.2, where it becomes manifest that it does impose geometric restrictions on the initial data. We refer to (3.29) as nologscondition.
Whenever the nologscondition holds, which we assume henceforth, the covector field can be expanded as a power series,
(3.30) 
The coefficients