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.