General Theory of Relativity (Physics Notes)

General relativity is via it’s formulation a theory heavily based on differential geometry. General Theory of Relativity by Dirac gives a 68 page introduction to the mathematical theory of general relativity. The style of the book is “no nonsense” and “uncluttered”. The book is based on series of lectures given by Dirac at Florida State University in 1973.

The book is aimed at advanced undergraduates, though it will be well suited for beginning graduate students and researchers who want a quick overview of the structure of general relativity.

Chapter 1 to 14 cover topics of differential geometry: parallel displacement, Christoffel symbols, geodesics, covariant derivatives and curvature tensors. All the basics of geometry required for general relativity.

Chapters 15 to 35 deal with the “guts” of general relativity. Topics covered include Einstein’s law of gravity, the Newtonian approximation, gravitational red shift, the Schwarzschild metric & black holes, harmonic coordinates, the field equations with matter, the gravitational action principle, the pesudo energy-tensor for gravity, gravitational waves and the cosmological constant.

Paperback: 68 pages
Publisher: Princeton University Press (January 8, 1996)
Language: English
ISBN-10: 069101146X
ISBN-13: 978-0691011462

Advanced General Relativity

General relativity is one of the cornerstones of modern physics, describing gravitational phenomena in geometric manner. Advanced General Relativity by John Stewart provides an introduction to some of the more advanced mathematical aspects of the theory. The readership is graduate students and researchers who already have some knowledge of general relativity.

Chapter 1 outlines the theory of differential manifolds, the tangent and cotangent spaces, tensor algebra, Lie derivatives, connections, geodesics and curvature tensors. All these geometric ideas are the “bread and butter” of general relativity.

Chapter 2 is where the more advanced topics start. The notion of spinors is introduced here. The Petrov classification and the Newman-Penrose formulation are presented.

Chapter 3 deals with asymptotic properties of space-time. Basically one would expect the space-time far away from an isolated source of gravity to be flat. This chapter deals with the Bondi and ADM mass, asymtopia for Minkowski space-time, asymptotic simplicity and conformal transformations.

Chapter 4 deals with the characteristic initial value problem in general relativity. The idea is to reformulate general relativity as the temporal evolution of 3-spaces.

Two appendices are included. The first deals with Dirac spinors and the second with the Newman-Penrose formalism.

Paperback: 240 pages
Publisher: Cambridge University Press; New Ed edition (26 Nov 1993)
Language English
ISBN-10: 0521449464
ISBN-13: 978-0521449465

Mathematical Physics (Lectures in Physics)

Mathematics is the language of physics. Thus physicists need some background in mathematics. Within mathematics category theory is a collection of unifying ideas that really gets at the heart of mathematical structures. Mathematical Physics by Robert Geroch provides a good grounding in the basic mathematical structures required in physics from a categorical perspective. This book consists of 56 short chapters.

Chapers 1 to 24 discuss algebraic categories: groups, vector spaces, associative algebras, Lie algebras and representations. The main ideas of category theory are laid down in these chapters via motivating examples.

Chapters 25 to 42 take on a topological flavour. Topics about topological spaces include continuous mappings,compactness, connectedness, homotopy, homology, topological groups and topological vector spaces.

Chapters 43 to 56 combine algebra and topology by discussing measure spaces, distributions and Hilbert spaces. Topics here include bounded operators, the spectral theorem, not necessarily bounded operators and self-adjoint operators.

Paperback: 358 pages
Publisher: University Of Chicago Press (September 15, 1985)
Language: English
ISBN-10: 0226288625
ISBN-13: 978-0226288628