The Post-Newtonian Approximation

This chapter is devoted to the study of the motion of multiple isolated bodies under their mutual gravitational interaction and of the accompanying emission of gravitational radiation, a subject of great astrophysical interest. Only approximation methods,

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The Post-Newtonian Approximation

Il y a seulement des problèmes plus ou moins résolus. —H. Poincaré (1908)

6.1 Motion and Gravitational Radiation (Generalities) The study of the motion of multiple isolated bodies under their mutual gravitational interaction and of the accompanying emission of gravitational radiation is obviously of great astrophysical interest. The first investigations on this subject by Einstein, Droste, de Sitter and others were made very soon after the completion of GR. Since then a vast amount of work has been invested to tackle the problems of motion and gravitational radiation of isolated systems (for an instructive review see [163]). For testing GR, and applying it to interesting astrophysical systems, such as binary systems of neutron stars and black holes, we need analytical approximations and reliable numerical methods to construct global, asymptotically flat solutions of the coupled field and matter equations, satisfying some intuitively obvious properties: The matter distributions should describe a system of celestial bodies (planets, white dwarfs, neutron stars, . . .), and there should be no relevant incoming gravitational radiation. It goes without saying that this is a formidable task. In this chapter we shall only discuss approximation methods that make use of some small parameters for sufficiently separated bodies. But before starting with this, we want to address some conceptual issues, in particular the notion of asymptotic flatness. We shall afterward treat in detail only the first post-Newtonian approximation, but an outline of the general strategies of approximation methods will also be given. The detailed implementations by various groups become—with increasing order—rapidly very complicated. For interested readers some important recent papers and reviews will be cited. In the final section we shall apply the developed tools to analyze very precise binary pulsar data. N. Straumann, General Relativity, Graduate Texts in Physics, DOI 10.1007/978-94-007-5410-2_6, © Springer Science+Business Media Dordrecht 2013

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6 The Post-Newtonian Approximation

6.1.1 Asymptotic Flatness For the description of an isolated system, like an oscillating star, it is very sensible to ignore the rest of the universe. Then spacetime should resemble far away from the matter source more and more Minkowski spacetime. What that exactly means was finally formulated by R. Penrose in an elegant geometrical manner (see [164]). His definition of asymptotic flatness emphasizes the conformal structure of spacetime. Guided by the examples discussed in Sect. 4.8, the following concepts emerged: Definition A spacetime (M, g) is asymptotically simple, provided all null geodesics are complete and (M, g) can conformally be embedded into a Lorentz manifold ˜ g) (M, ˜ such that the following holds: (i) M is an open submanifold of M˜ with smooth boundary ∂M =: I . (ii) The function Ω in the conformal relation g˜ = Ω 2 g on M can be extended smoothly to M˜ such that Ω = 0, dΩ = 0 on I . (iii) The boundary I