This chapter discusses the properties of the conformal Einstein field equations and the behaviour of their solutions in a suitable neighbourhood of spatial infinity. This analysis is key in any attempt to extend the semiglobal existence results for Minkowski-like spacetimes of Chapter 16 to a truly global problem where initial data is prescribed on a Cauchy hypersurface. An interesting feature of the semiglobal existence Theorem 16.1 is that the location of the intersection of the initial hyperboloid with null infinity does not play any role in the formulation of the result. This observation suggests that the essential difficulty in formulating a Cauchy problem is concentrated in an arbitrary (spacetime) neighbourhood of spatial infinity. The subject of this chapter can be regarded, in some sense, as a natural extension of the study of static spacetimes in Chapter 19 to dynamic spacetimes – a considerable amount of the discussion of the present chapter is devoted to understanding why this is indeed the case. A further objective of this chapter is to understand the close relation between the behaviour of the gravitational field at spatial infinity and the so-called peeling behaviour discussed in Chapter 10. The main technical tool in this chapter is the construction of the so-called cylinder at spatial infinity – a conformal representation of spatial infinity allowing the formulation of a regular initial value problem by means of which it is possible to relate properties of the initial data on a Cauchy hypersurface with the behaviour of the gravitational field at null infinity.

Despite recent developments in the understanding of the behaviour of solutions to the Einstein field equations in a neighbourhood of spatial infinity, several key issues still remain open.

Cauchy data for the conformal field equations near spatial infinity

To begin to understand the difficulties behind the formulation of a standard initial value problem for a Minkowski-like spacetime, it is convenient to look at the behaviour of Cauchy data for the conformal equations near spatial infinity.

In what follows, initial data sets which are asymptotically Euclidean and regular in the sense of Definition 11.2 will be considered. As the discussion in this chapter will be mainly concerned with the behaviour of the data in a neighbourhood of spatial infinity, it will be assumed, without any loss of generality, that the manifold has only one asymptotic end.

This chapter discusses the global existence and stability of de Sitter-like spacetimes, that is, vacuum spacetimes with a de Sitter-like value of the cosmological constant. This class of spacetimes admits a conformal extension with a spacelike conformal boundary; see Theorem 10.1. The construction of de Sitter-like spacetimes provides, arguably, the simplest application of the conformal field equations to the analysis of global properties of spacetimes. The original discussion of the analysis presented in this chapter was given in Friedrich (1986b). The results of this seminal analysis were subsequently generalised to the case of Einstein equations coupled to the Yang-Mills field in Friedrich (1991). The methods used in the proof of the stability of the de Sitter spacetime can be adapted to analyse the future non-linear stability of Friedman-Robertson-Walker cosmologies with a perfect fluid satisfying the equation of state of radiation; see Lübbe and Valiente Kroon (2013b).

The global existence and stability theorem proven in this chapter can be formulated as follows:

Theorem (global existence and stability of de Sitter-like spacetimes). Small enough perturbations of initial data for the de Sitter spacetime give rise to solutions of the vacuum Einstein field equations which exist globally towards the past and the future. The solutions have the same global structure as the de Sitter spacetime. Thus, perturbations of the de Sitter spacetime are asymptotically simple.

Intuitively, the last statement in the theorem can be read as saying that the resulting spacetimes have a Penrose diagram similar to the one of the de Sitter spacetime; see Figure 15.1. Accordingly, these spacetimes provide non-trivial (i.e. dynamic) examples of asymptotically simple spacetimes. A detailed formulation of the above result is given in the main text of the chapter; see Theorem 15.1.

To illustrate the comparative advantages of the hyperbolic reduction procedures discussed in Chapter 13, two versions of the proof are provided. The first one makes use of gauge source functions and follows the original proofs in Friedrich (1986b, 1991). The second proof makes use of conformal Gaussian systems and is based on the analysis given in Lübbe and Valiente Kroon (2009).

This chapter studies the existence and stability of Minkowski-like spacetimes, that is, solutions to the vacuum Einstein field equations with vanishing cosmological constant. The main result of this chapter is very similar in spirit to the main result concerning the global existence and stability of de Sitter-like spacetimes of Chapter 15. There is, however, a key difference: while the results in Chapter 15 are global in nature, the ones in the present chapter are semiglobal. More precisely, the spacetimes to be discussed arise as the development of suitable initial data on hyperboloidal hypersurfaces – an examination of the Penrose diagram of the Minkowski spacetime in Figure 16.1 reveals that these hypersurfaces are not Cauchy hypersurfaces of the spacetime. Accordingly, only a portion of the whole spacetime can be recovered from this type of initial value problem. The main result of this chapter can be formulated as follows:

Theorem (semiglobal existence and stability of Minkowski-like spacetimes). Small enough perturbations of hyperboloidal initial data for the Minkowski spacetime give rise to solutions to the vacuum Einstein field equations which exist globally towards the future and have an asymptotic structure similar to that of the Minkowski spacetime.

This result was first proved in Friedrich (1986b) and subsequently extended to the Einstein-Yang-Mills equations in Friedrich (1991). The original proof of the result made use of the standard conformal Einstein field equations and is similar to the argument given for the de Sitter spacetime in Section 15.3. In this chapter a proof of the theorem is given which makes use of the extended conformal field equations and conformal Gaussian systems following the ideas in Lübbe and Valiente Kroon (2009). This approach allows for a more detailed and explicit discussion of the structure of the conformal boundary.

The restriction of the analysis of the present chapter to the hyperboloidal initial value problem may seem mysterious at first sight. As will be discussed in some detail in Chapter 20, the initial data for the conformal Einstein field equations on an asymptotically Euclidean (Cauchy) hypersurface is generically singular at spatial infinity – the various issues associated to this singular behaviour are usually known as the problem of spatial infinity.

This chapter discusses the basic theory of characteristic problems for the conformal field equations. Characteristic problems have been of great conceptual value in the development of the modern theory of gravitational radiation. Indeed, the seminal works by Bondi et al. (1962) and Sachs (1962b), in which the modern understanding of gravitational waves was established, were carried out in a setting based on a characteristic initial value problem; see also Sachs (1962c) and Newman and Penrose (1962). The connection between characteristic problems and the notion of asymptotic flatness, already present in the seminal work by Penrose (1963), was further elaborated in Penrose (1965, 1980). From a mathematical point of view, the realisation that the characteristic initial value problem for the Einstein field equations leads to a symmetric hyperbolic evolution system for which the machinery of the theory of partial differential equations (PDEs) is available was first established in Friedrich (1981b). In Friedrich (1981a, 1982) these ideas were subsequently extended to a situation in which part of the data is prescribed at null infinity –a so-called asymptotic characteristic initial value problem, the subject of this chapter. These results established the local existence of analytic solutions and were later extended to the smooth case by (1996b) using the method of reduction to a standard Cauchy problem by Rendall (1990); see Section 12.5.3.

There are two basic types of asymptotic characteristic problem for the conformal Einstein field equations. The first type is the so-called standard asymptotic characteristic problem – introduced in Friedrich (1981b) – where initial data are prescribed on null infinity and a null hypersurface intersecting null infinity in a two-dimensional surface with the topology of a 2-sphere; see Figure 18.1, left. In the second type – the so-called characteristic problem on a cone, first discussed in Friedrich (1986c) – one prescribes information on a null cone down to its vertex; see Figure 18.1, right. For reasons discussed in Section 12.5, characteristic problems on a cone are more technically involved. Existence results have been obtained in and Paetz (2013).

The standard and characteristic initial value problems have several structural properties in common. Moreover, the characteristic problem on a cone can be regarded as a limiting case of the standard characteristic problem.

Conformal notions provide valuable tools for the analysis of global properties of spacetimes. In Part IV of this book it has been shown how a conformal point of view leads to proofs of the global existence and non-linear stability of de Sitterlike spacetimes, of the semiglobal existence and non-linear stability of Minkowskilike spacetimes, and how they provide a systematic procedure for the construction of anti-de Sitter-like spacetimes. Moreover, conformal methods provide a robust framework for the analysis of the gravitational field of isolated systems in a neighbourhood of both null and spatial infinity.

The application of conformal methods in general relativity is a mature area of research with a considerable number of open problems. Several of these have been discussed in various places of this book. Unavoidably, there are other problems and aspects of the subject which, for reasons of space, could not be covered in the main text. This last chapter presents a list of ideas and problems which, in the opinion of the author, may play a role in the future development of the subject.

Stability of cosmological models

The global non-linear stability of the de Sitter spacetime was discussed in Chapter 15. This exact solution can be regarded as a basic cosmological model. The analysis of Chapter 15 can be extended to include a non-vacuum matter content with good conformal properties: for example, the Maxwell, Yang-Mills and conformally coupled scalar field; see Friedrich (1991) and Lübbe and Valiente Kroon (2012). More recently, the ideas behind these proofs have been adapted in Lübbe and Valiente Kroon (2013b) to provide an analysis of the future stability of Friedman-Robertson-Walker cosmological models with a perfect fluid with the equation of state of radiation; see Section 9.4. A natural question is whether conformal methods could be adapted to more general matter models, that is, matter models with a non-vanishing trace.

This book discusses an approach to the study of global properties of solutions to the equations of general relativity, the Einstein field equations, in which the notion of conformal transformation plays a central role. The use of conformal transformations in differential geometry dates back, at least, to the work of Hermann Weyl in the 1920s. Their application to global questions in general relativity, as presented in this book, stems from the seminal work of Roger Penrose in the 1960s in which the close connection between the global causal structure of the solutions to the equations of general relativity and conformal geometry was established. Penrose's key insights are that the close relation between the propagation of the gravitational field and the structure of light cones which holds locally in a spacetime is also preserved in the case of large scales and that the asymptotic behaviour of the gravitational field can be conveniently analysed in terms of conformal extensions of the spacetime. In the following decade Penrose's ideas were polished, extended and absorbed into the mainstream research of general relativity by a considerable number of researchers – finally leading to the influential notion of asymptotic simplicity. The subject reached its maturity when this formal theory was combined with the methods of the theory of partial differential equations (PDEs). This breakthrough is mainly due to the work of Helmut Friedrich in the early 1980s, who – through the conformal Einstein field equations – showed that ideas of conformal geometry can be used to establish the existence of large classes of solutions to the Einstein field equations satisfying Penrose's notion of asymptotic simplicity. As a result of this work it is now clear that Penrose's original insights hold for large classes of spacetimes and not only for special explicitly known solutions.

This book develops the theory of the conformal Einstein field equations from the ground up and discusses their applications to the study of asymptotically simple spacetimes. Special attention is paid to results concerning the existence and stability of de Sitter-like spacetimes, the semiglobal existence and stability of Minkowski-like spacetimes using hyperboloidal Cauchy problems and the construction of anti-de Sitter-like spacetimes from initial boundary value problems. These results belong to the canon of modern mathematical relativity.

The notion of spinors arises naturally in the construction of a relativistic firstorder equation for a quantum wave function – the so-called Dirac equation. Spinors are the most basic objects to which one can apply a Lorentz transformation. The seminal work in Penrose (1960) has shown that spinors constitute a powerful tool to analyse the structure of the Einstein field equations and their solutions. Most applications of spinors in general relativity make use not of the Dirac spinors but of the so-called 2-spinors. The latter are more elementary objects, and indeed, the whole theory of the Dirac equation can be reformulated in terms of 2-spinors. In the sequel, 2-spinors will be very often simply called spinors.

The purpose of this chapter is to develop the basic formalism of spinors in a spacetime. Accordingly, one speaks of spacetime spinors, sometimes also called SL (2, C) spinors; see, for example, Ashtekar (1991). A discussion of spinors in the presence of a singled-out timelike direction, the so-called space spinor formalism, is given in Chapter 4. One of the motivations for the use of spinors in general relativity is that they provide a simple representation of null vectors and of several tensorial operations. Although spinors will be used systematically in this book, they are not essential for the analysis. All the key arguments could be carried out in a tensorial way at the expense of lengthier and less transparent computations.

The presentation in this chapter differs sligthly in focus and content from that given in other texts; see, for example, Penrose and Rindler (1984); Stewart (1991); O'Donnell (2003). For reasons to be discussed in the main text, a systematic use of the so-called Newman-Penrose formalism will be avoided – although the basic notational conventions of Penrose and Rindler (1984), the authoritative work on the subject, are retained.

Algebra of 2-spinors

In what follows let (M, g) be a spacetime. The present discussion begins by analysing spinorial structures at a given point p of the spacetime manifold M. The concept of a spinor is closely related to the representation theory of the group SL (2,C). This group has two inequivalent representations in terms of two-dimensional complex vector spaces which are complex conjugates of each other; for a discussion of this aspect of the theory, see, for example, Carmeli (1977); Sexl and Urbantke (2000).

This chapter discusses the construction of anti-de Sitter-like spacetimes, that is, solutions to the vacuum Einstein field equations with an anti-de Sitter-like value of the cosmological constant λ. Following the general discussion in Chapter 10, an anti-de Sitter-like value of the cosmological constant implies a timelike conformal boundary. This feature of anti-de Sitter-like spacetimes marks the essential difference between the analysis contained in this chapter and the ones given in Chapters 15 and 16 for de Sitter-like and Minkowski-like spacetimes, respectively.

While the de Sitter and Minkowski spacetimes are both globally hyperbolic, and, accordingly, perturbations thereof can be constructed by means of suitable initial value problems, the anti-de Sitter spacetime is not-globally hyperbolic; see the discussion in Section 14.5. Consequently, anti-de Sitter-like spacetimes cannot be solely reconstructed from initial data. One needs to prescribe some boundary data on the conformal boundary. Thus, the proper setting for the construction of anti-de Sitter-like spacetimes is that of an initial boundary value problem. In this spirit, one of the key objectives of this chapter is to identify suitable boundary data for the conformal Einstein field equations.

For both the de Sitter and Minkowski spacetimes it is possible to obtain conformal representations which are compact in time so that global existence of perturbations can be analysed in terms of problems which are local in time. However, the conformal representations of the anti-de Sitter spacetime discussed in Chapter 6 involve an infinite range of time. As a consequence, the main result of this chapter is local in time and makes no assertions about the stability of the anti-de Sitter spacetime. The main result of this chapter can be formulated as follows:

Theorem (local existence of anti-de Sitter-like spacetimes). Given smooth anti-de Sitter-like initial data for the Einstein field equations on a three-dimensional manifold S with boundary and a smooth three-dimensional Lorentzian metric ℓ on a cylinder [0, τ•) ×∂S for some τ• > 0, and assuming that these data satisfy certain corner conditions, there exists a local-in-time solution to the Einstein field equations with an anti-de Sitter-like cosmological constant such that on {0}×S it implies the given anti-de Sitter-like initial data. Moreover, this solution to the Einstein field equations admits a conformal completion such that the intrinsic metric of the resulting (timelike) conformal boundary belongs to the conformal class[ℓ].