Spacetime, Geometry and Gravitation pp 3-38 | Cite as

# Introduction

Chapter

## Abstract

General Theory of Relativity (or General Relativity) is Einstein’s theory of gravitation given by him in 1915. The name also applies to its later developments. According to the theory spacetime is a Riemannian space whose metric governs the gravitational field. In this equation the quantities

*g*_{μν}determines the gravitational field^{1}. The**Einstein equation**$$
R_{\mu \nu } - \frac{1}
{2}g_{\mu \nu } R = \frac{{8\pi G}}
{2}T_{\mu \nu }
$$

(1.1)

*R*_{μν},*R*are functions of the metric*g*_{μν}and its various derivatives and*T*_{μν}on the right-hand side are determined by distribution of matter.## Keywords

Black Hole Gravitational Potential Proper Time Inertial Frame Lorentz Transformation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Literature

## Original Papers

**Lorentz, Einstein, Minkowski**and**Weyl**,*Principle of Relativity*, Dover books, 1952 A collection of original papers on special and general theory of relativity in English translation with notes by A. Sommerfeld.Google Scholar**C. W. Kilmister**(Ed.),*Special Theory of Relativity*, Pergamon Press, 1970,*General Theory of Relativity*, Pergamon Press, 1973. Collection of almost all the basic papers on relativity. These volumes have introductory chapters and commentary by C. W. Kilmister.Google Scholar

## Historical Matter

**Abraham Pais**,*’subtle is the Lord …’, The Science and the Life of Albert Einstein*, Oxford University Press, 1982 A thorough discussion of development of Einstein’s thinking with an almost day-by-day account.Google Scholar**Edmond T. Whittaker**,*History of theories of Aether and Electricity*, Vol. I and II, Thomas Nelson and Sons, London. Reprinted by Humanities Press, New York, 1973. This is another standard reference for the history of field theories of classical physics.Google Scholar

## Texts

**Albert Einstein**,*Meaning of Relativity*, Indian Edition by Oxford Book Company, 1965 These are Einstein’s 1921 Princeton lectures, originally published by The Princeton University Press. Every student of relativity should read these 100 odd pocket-book sized pages for the clarity and brevity of the man who discovered the theory. The available editions have appendices containing Einstein’s later unified theories, none of which seem relevant today. But who knows?Google Scholar**Wolfgang Pauli**,*Theory of Relativity*, Pergamon Press 1958 Pauli’s*Mathematical Encyclopedia*review article of 1921 written at the age of 21 by the author. The article gives a complete account of relativity theory till 1921.Google Scholar**Herman Weyl**,*Space-Time-Matter*, Dover, 1952 First published in German in 1918. It was already in its fourth edition in 1920. One of the earliest expositions of relativity by a great mathematician who contributed to the theory. The third edition of 1919 introduces Levi-Civita’s 1917 discovery of infinitesimal parallel displacement.Google Scholar**Lev Landau**and**E. M. Lifshitz**,*Classical Theory of Fields*, Pergamon Press, Reprint 2004 Volume 2 of the famous Course of Theoretical Physics. The first nine chapters are on Electrodynamics, last five on the General Theory of Relativity. These (less than two hundred) pages constitute an introduction which is both deep and brief.Google Scholar**Arther S. Eddington**,*Mathematical Theory of Relativity*, Cambridge University Press, 7th reprint 1963 One of the clearest early expositions, first written as mathematical notes to his delightful and popular book*Space Time and Gravitation*published in 1920.Google Scholar**Peter G. Bergmann**,*Introduction to the Theory of Relativity*, Prentice Hall, 1942 (Now available as a paperback in Dover, New York) This book has a forward by Einstein. It is the complete, perhaps the first, textbook, written with students’ needs in mind. It covers quite ‘advanced’ ideas (for those days) such as the Kaluza-Klein theory, which has made a comeback in theoretical physics recently.Google Scholar**Vladimir A. Fock**,*The Theory of Space, Time and Gravitation*, Macmillan, 1964 The authoritative book by the great Russian physicist who took a critical look at many of the fundamental ideas of relativity.Google Scholar**C. Møller**,*The Theory of Relativity*, Oxford, 1952 Authoritative book on all aspects of the theory.Google Scholar**J. L. Synge**,*Relativity, the General Theory*, North Holland, 1966 This is a book written in a very independent style very unlike other standard books.Google Scholar**C. M. De Witt and B. S. De Witt**(Eds.),*Relativity, Groups and Topology*, Blackie and Sons, 1964 Contains delightful introductory lectures on general theory of relativity given by Synge at Les Houche School in 1963. This collection also includes lectures by Wheeler, Penrose, Sachs, and Misner on various aspects of general relativity.Google Scholar**Charles W. Misner, Kip S. Thorne and John A. Wheeler**,*Gravitation*, Freeman, 1973 The absolute darling of students and researchers in general relativity. Its*twenty fourth*reprint came out in 2002! ‘Merely holding the book in your hand makes you think about gravity!’, we used to say as students in the 1970’s. The presently available paperback is lighter, a little above two kilograms. This large sized (20 cm×25 cm), 1272 page book begins at the beginning and has everything on gravity (up to 1973). There are hundreds of diagrams and special boxes for additional explanations, exercises, historical and biographical asides and bibliographical details. It must have converted a fair number of people into research in general relativity. And conversion is a good word here because S. Chandrasekhar, while reviewing the book, is supposed to have commented on its missionary spirit! What makes it a pleasure to read is that no idea is introduced without its motivation. A student is told why the idea is natural to expect, and, if the natural expectation is wrong, why it is so. It has a cheery delightful style throughout.Google Scholar**Steven Weinberg**,*Gravitation and Cosmology*, John Wiley, 1973 A modern classic which reduces the emphasis on geometry and reinforces the power of the equivalence principle. Extremely readable with a discussion of experimental data (up to 1973).Google Scholar**Paul M. Dirac**,*General theory of Relativity*, Princeton University Press, 1996, Reprinted by Prentice-Hall of India, 2001 Dirac’s 1975 Lectures at Florida State University. As concise, to-the-point as only Dirac could be. This slim 35-section, 70 page booklet is written for a beginner. The book has significantly five sections on the action principle.Google Scholar**S. Chandrasekhar**,*The Mathematical Theory of Black Holes*, Oxford University Press, 1983 The exhaustive treatise on black hole solutions and their properties. If you need anything, anything at all, on the Schwarzschild or the Kerr spacetime it is here.Google Scholar**Robert M. Wald**,*General Relativity*, University of Chicago Press, 1984 The deservedly famous advanced textbook includes extremely readable introductory parts in the first six chapters and advanced topics on researches done in the 1960’s and 1970’s in chapters 7 to 14. A lot of mathematical background is condensed and relegated to Appendices at the end of the book which one cannot do without.Google Scholar**S. W. Hawking and G. F. R. Ellis**,*The large scale structure of space-time*, Cambridge University Press, 1973 A classic on spacetime structure in general relativity, known for its clarity and rigour. All proofs are complete, every concept well defined, most details included. But it requires considerable mathematical maturity to follow the line of thought. The mathematical apparatus used is indispensable for research in the area but the introduction to differential geometry is too brief (forty pages) to be of any actual help to a beginner.Google Scholar**R. Adler, M. Bazin, M. Schiffer**,*Introduction to General Relativity*, Second Edition, McGraw Hill, 1975 A very good textbook although not widely available. Its derivation of the Kerr metric is particularly good.Google Scholar**W. Rindler**,*Essential Relativity*, Springer-Verlag, 1977 This book, written specially for the advanced undegraduate student, is known for conceptual clarity and style. Written in extremely lucid style it is an enjoyable but deep book.Google Scholar**E. F. Taylor**and**J. A. Wheeler**,*Spacetime Physics*, W. H. Freeman, 1963 A delightful undergraduate book on basics.Google Scholar**Bernard F. Schutz**,*A first course in general relativity*, Cambridge University Press, 1985 A good textbook from a beginner’s point of view. It develops the mathematical background of tensor calculus through hundreds of exercises.Google Scholar**Ray d’Inverno**,*Introducing Einstein’s Relativity*, Clarendon Press Oxford, 1992 A textbook with several advanced topics as well.Google Scholar**H. C. O’Hanian**and**R. Ruffini**,*Gravitation and Spacetime*W. W. Norton and Co., 1994 An introductory book which looks at gravity in the most logical and straightforward way. In spirit it is closer to Weinberg’s book. There is a good collection of problems in each chapter. And the book contains*a very detailed*guide to further reading in each chapter.Google Scholar**James B. Hartle**,*Gravity*, Pearson Education, 2002 Written in the spirit of Misner, Thorne, Wheeler, this is the best recent treatment of general relativity available to the advanced undergraduate student. The book is complete with all the exciting experimental data up to the end of the 20th century. This book is a must for every beginner.Google Scholar**S. M. Carroll**,*Spacetime and Geometry: An Introduction to General Relativity*, Addison Wesley, 2004 This is a well-written recent textbook which gives plenty of space to geometry as needed in general relativity. Carroll’s lecture notes on general theory of relativity are also available on the arxiv.org as gr-qc/9712019.Google Scholar**N. Straumann**,*General Relativity — With Applications to Astrophysics*, Springer Verlag, 2004 A thorough recent book. It has a condensed mathematical introduction in Part**III**, used throughout the book.Google Scholar**Eric Poisson**,*A Relativist’s Toolkit*, Cambridge University Press, 2004 A recent advanced book on selected topics in general relativity. Although the topics are advanced, the author has taken pains to provide details and explanation.Google Scholar

## Important Reviews and Internet Sources

**S. W. Hawking**and**W. Israel**,*Einstein Centenary Survey*, Cambridge University Press, 1979Google Scholar**S. W. Hawking**and**W. Israel**,*300 Years of Gravitation*, Cambridge University Press, 1987Google Scholar**Living Reviews on Relativity**: (http://relativity.livingreviews.org/) An internet source of reviews which are periodically updated. In addition there are research articles and reviews available on (http://arxiv.org/) in the “gr-qc” (general relativity and quantum cosmology) section.Google Scholar

## Books on Mathematics

- For the convenience of a physics student all texts on the general theory of relativity do try to give an introduction to Riemannian geometry with varying degrees of pedagogical attention or success.
**B. F. Schutz**,*Geometrical Methods in Mathematical Physics*, Cambridge University Press, 1980 An introduction to geometry and topology used in gravity and gauge theories.Google Scholar **Y. Choquet-Bruhat, C. De Witt-Morette**and**M. Dillard-Bleick**,*Analysis, Manifolds and Physics*, North Holland., 1989 A thorough introduction to modern differential geometry as needed by physicists. It has rigourous approach illustrated by examples from physics.Google Scholar**C. Isham**,*Modern Differential Geometry for Physicists*, World Scientific, 1989 Another thorough introduction to differential geometry as used in gravity and gauge theories.Google Scholar**H. K. Nickerson, D. C. Spencer**and**N. E. Steenrod**,*Advanced Calculus*, Von Nostrand, 1959 An undergraduate textbook for introduction to vectors, tensors, forms and differentiable manifolds based on lectures at Princeton University.Google Scholar**I. M. Singer**and**J. A. Thorpe**,*Lecture Notes on Elementary Topology and Geometry*, Undegraduate Texts in Mathematics, Springer-Verlag, 1976 Another classic with undergraduate students in mind.Google Scholar**F. W. Warner**,*Foundations of Differentiable Manifolds and Lie Groups*, Springer-Verlag, 1983 A more advanced introduction.Google Scholar**B. A. Dubrovin, A. T. Fomenko**and**S. P. Novikov**,*Modern Geometry — Methods and Applications*, Springer-Verlag, 1992 A good relaxed introduction to geometry in three volumes.Google Scholar**S. S. Chern, W. H. Chen**and**K. S. Lam**,*Lectures on Differential Geometry*, World Scientific, 1999 The Peking University lectures in 1980 by the great mathematician S. S. Chern.Google Scholar**N. J. Hicks**,*Notes on Differential Geometry*, Von Nostrand, 1965 This is an all-time favourite. A slim, 175 page book which introduces Riemannian geometry through hypersurfaces and theorema egregium.Google Scholar

## Books on Astrophysics and Cosmology

- Astrophysics and Cosmology are two important branches of physics where general theory of relativity is applied. The subject of cosmology has been in very rapid growth in the last ten years.
**T. Padmanabhan**,*Theoretical Astrophysics*, Cambridge University Press, Vol. I, 2000, vol. II, 2001, Vol. III, 2002 This is a thorough introduction in three volumes.Google Scholar **J. V. Narlikar**,*An Introduction to Cosmology*, Cambridge University Press, 2002 For cosmology there are many recent texts but as introduction, this is perhaps the best. It gives a clear, detailed, account of all concepts with their historical background.Google Scholar**V. Mukhanov**,*Physical Foundations of Cosmology*, Cambridge University Press, 2005 These are two of the many recent textbooks on cosmology.Google Scholar

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