The n-Body Problem in General Relativity

Description

Guérard des Lauriers, Michel Louis, O.P., studied under him in Rome.

MR44932 This excellent monograph on the n-body problem in the general theory ofrelativity was prepared about ten years ago, but its appearance now is nonethe less timely for those who have worried themselves with one or anotheraspect of the problem. Its major achievements are two: a derivation of theequations of motion of n point masses, free from the subtle errorsbesetting most of the standard treatments; and a careful discussion of thepossible contributions, in the Einsteinian approximation, of the finitesize and internal constitution of the bodies involved. While the treatmentof this latter topic does not profess to the attainment of complete rigor,it is shown on the basis of plausible hypotheses that the effects of thefinite extension of the bodies may be absorbed into the Newtonian terms bya first-order redefinition of their masses.
A short historical preface is followed by a lucid although concise chapteron the fundamental concepts and equations of the general relativity theoryof gravitation. Chapter II examines the structure of the differentialequations of thegravitational and motional fields for a system of incoherent masses, andshows that they can in fact be written in normal form, to which theCauchy-Kowalewski existence theorem applies. This is followed by a carefuldevelopment of the standard successive approximation method, in steps ofrelative order (v/c)2, to the stage required to bring out theEinsteinian effects in the motion. After a hurried side-glance at theinterior problem, Levi-Civita marches on to the nub of the matter inChapter V, "Reduction to Ordinary Differential Equations''. Here theauthor avoids the conceptual and computational pitfalls into which DeSitter, Chazy and others—among them, upon occasion, Levi-Civitahimself—have plunged. Here too is a reasoned justification for MarcelBrillouin's "Principle of Effacement'', whereby the contribution of a bodyto the gravitational field may be ignored in computing its own equations ofmotion. The monograph ends with a detailed analysis of the 2-body problem(perihelion advance and motion of the center of mass), into which isincorporated a specific account of the absorption into the Newtonian massof contributions to the field due to the finite extension of thegravitating bodies. Reviewed by H. P. Robertson