The conformal approach to asymptotic analysis
pp. 571-609
in: Lizhen Ji, Athanase Papadopoulos, Sumio Yamada (eds), From Riemann to differential geometry and relativity, Berlin, Springer, 2017Abstract
Albert Einstein's general theory of relativity is a geometric theory of gravity, using the framework of Lorentzian geometry: an extension of Riemannian geometry in which space and time are united in a real 4-dimensional manifold endowed with an indefinite metric of signature (1, 3) or (3, 1). The metric allows to distinguish between timelike and spacelike directions in an intrinsic manner and, provides a description of gravity via its curvature. The introduction by Minkowski in 1908 of the notion of spacetime was a decisive change of viewpoint which opened the road for Einstein to develop the geometrical framework for the fully covariant theory he was after. Instead of discussing the history of this development and the crucial influence of Riemannian geometry through the help of Marcel Grossmann, this essay explores Roger Penrose's approach to general relativity which bears a remarkable kindred of spirit with Einstein's and perpetuates the geometrical view of the universe initiated by Riemann and Einstein. More specifically, Penrose's approach to asymptotic analysis in general relativity, which is based on conformal geometric techniques, is presented through historical and recent aspects of two specialized topics: conformal scattering and peeling. Other essays in this volume are related to general relativity: Jacques Franchi [15] discusses relativistic analogues of the Brownian motion on various Lorentzian manifolds; Andreas Hermann and Emmanuel Humbert [23] discuss the positive mass theorem, which is closely related to the Yamabe problem in Riemannian geometry; Marc Mars [28] presents some local intrinsic ways of characterizing a spacetime.
