Relativity on Curved Manifolds (Cambridge Monographs on Mathematical Physics) by F. de Felice (PDF)

Description

General relativity is now essential to the understanding of modern physics, but the power of the theory cannot be exploited fully without a detailed knowledge of its mathematical structure. This book aims to implement this structure, and then to develop those applications that have been central to the growth of the theory.

User’s Reviews

Editorial Reviews: Review “This excellent monograph describes in detail the mathematical structure of Einstein’s general theory of relativity, and the mathematical techniques that are associated with it. It is a completely self-contained exposition, which collects many important general results relating to the classical theory that are not readily available in the literature. It will be of great value to anyone who is already familiar with the theory, and who requires a deeper technical knowledge than is presented in most current textbooks.” Mathematical Reviews”…the authors present a book on general relativity which, on one hand, is mathematically rigorous and, on the other hand, contains good physics by emphasizing what is measurable on curved manifolds. Even those readers who are interested in more applications, such as e.g. gravitational lensing, can find all the tools necessary in this book even though this item cannot be found in the index. Thus the book can strongly be recommended to mathematicians and physicists as well.” Classical and Quantum Gravity”This is an excellent book and admirably lives up to the promise implied in its title in giving a thorough treatment of the mathematical structure underlying the theory of general relativity.” The Observatory”…a valuable reference work, particularly for its mathematical introduction and its treatment of the other topics mentioned above.” Robert M. Wald, Foundations of Physics”…provides more in details in various formalisms than do many comparable relativity books. It should prove to be a very good supplementary book for a graduate course on general relativity, or it might serve as a reference book for researchers…” K.K. Lee, Physics in Canada Book Description This is a self-contained exposition of general relativity with emphasis given to tetrad and spinor structures and physical measurement on curved manifolds.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Sophisticated monograph detailing mathematical structures pertinent to Relativity. The exposition, both mathematical and physical, is crisp and somewhat innovative. Not intended as a first (elementary) textbook on general relativity (say, Hartle or Schutz). It as not as extensive as the inevitable classic: Gravitation by Misner, Thorne, Wheeler. The text does provide a unique outlook on the differential structures central and relevant to relativistic physics. Preliminary chapter provides summary of the primary trilogy: local, global, infinitesimal. A tour:(1) Chapter One: insightful excursion through the terrain of topology and tensors, culminating with exterior forms. Second: differentiation (tensor fields, Lie derivatives, exterior derivatives), nicely accomplished. Third: Curvature, that means Riemann, Ricci, Weyl and Bianchi. Exceptional exposition.(2) These preliminaries aside, the fun begins: tetrads and spinors (two chapters) are absolutely essential and delightful. Read: “…since quantum field theories should be described on them (space-times), we demand that they admit a spinor structure” and “…property of admitting a spinor structure is closely related to the assumed paracompactness together with the possibility of defining…concepts of observer and measurement.” (page 129).(3) The full richness of this text now unfolds: field equations, geometry, conservation laws and congruences. We Read: “…these (congruences) describe in most cases the space-time evolution of physical systems.” (page 250). I highlight chapter eight, Geometry of Congruences, being one of the finest expositions to touch upon tetrads, null-rays, Raychaudhuri equation and singularities, about which read “…still unclear whether singularities are always associated with unbounded physical quantities ” and “…a central role is played by the concept of trapped surface.” Physical Measurements (chapter nine), Read: “we shall apply…to…mechanics, electrodynamics, and fluid dynamics…and shall interpret as genuine relativistic effects (curvature effects) those quantities which have no part in a special relativistic treatment and contain the curvature either explicitly or implicitly.” (page 274).(4) Final two chapter are lucid: spherically (Schwarzschild) and axially (Kerr) symmetric solutions. Read: “…being itself endowed with energy and momentum, the gravitational interaction generates more curvature itself, these solutions (of the vacuum einstein equations) therefore describe self-sustaining gravitational fields.” (page 194) and “A necessary prerequisite to a discussion on the equations of motion in general relativity, is a realization of the uncertainties which underlie any global definition of energy, momentum and angular momentum of a gravitating system.” (page 206) and ” A distinctive feature of relativity is that velocities do not sum up linearly, but according to a law which prevents them from appearing larger than the velocity of light, whatever observer we refer to.” (see pages 295-297). Read: “…it is generally believed that Black-Holes are physically realistic…” but “on the contrary, White-Holes would turn into black holes due to the gravitational action of the radiation which appears infinitely blue-shifted.”(5) This text is a nice complement to existing treatments. My brief summary does not begin to do justice to the enrichment offered between the covers of this treatise. Clarke’s earlier book, Elementary General Relativity (1979), provides preliminary to this technical monograph. Highly recommended for those interest in mathematical treatment of General Relativity.

⭐この本はあまり有名ではないみたいですが、隠れた良書です。一言で述べるとかゆい所に手の届く本です。 多くの本において、式の計算で抜けている部分などが記述されています（Kerr時空など）。個人的に参考になったのは、数学に部分のFrobeniusの定理の証明の部分、Killing vectorsの解説、定曲率空間でのcurvatureの導出、角運動量等に関連しているKomar formula、10章から11章にかけて球対称性と軸対称性の解についての説明（およそ110ページ費やしてます）です。 数学の解説部分では記述に独特の記法を用いているので、初学者がこの本で数学を身につける場合、他書と対比したとき混乱を招くかもしれません。

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