Spacetime, Geometry and Gravitation

1. Front Matter

   Pages i-xiv

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2. Spacetime

   1. Front Matter

      Pages 1-1

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   2. Introduction

      Pages 3-38

   3. What is Curvature?

      Pages 39-53

   4. General Relativity Basics

      Pages 55-63

   5. Spherically Symmetric Gravitational Field

      Pages 65-80

3. Geometry

   1. Front Matter

      Pages 81-81

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   2. Vectors and Tensors

      Pages 83-96

   3. Inner Product

      Pages 97-113

   4. Elementary Differential Geometry

      Pages 115-132

   5. Connection and Curvature

      Pages 133-153

   6. Riemannian Geometry

      Pages 155-179

   7. Additional Topics in Geometry

      Pages 181-207

4. Gravitation

   1. Front Matter

      Pages 209-209

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   2. The Einstein Equation

      Pages 211-235

   3. General Features of Spacetime

      Pages 237-256

   4. Weak Gravitational Fields

      Pages 257-286

   5. Schwarzschild and Kerr Solutions

      Pages 287-315

   6. Cosmology

      Pages 317-327

   7. Special Topics

      Pages 329-350

This is an introductory book on the general theory of relativity based partly on lectures given to students of M.Sc. Physics at my university. The book is divided into three parts. The ?rst part is a preliminary course on general relativity with minimum preparation. The second part builds the ma- ematical background and the third part deals with topics where mathematics developed in the second part is needed. The ?rst chapter gives a general background and introduction. This is f- lowed by an introduction to curvature through Gauss’ Theorema Egregium. This theorem expresses the curvature of a two-dimensional surface in terms of intrinsic quantitiesrelatedtothein?nitesimaldistancefunctiononthesurface.Thestudent isintroducedtothemetrictensor,Christo?elsymbolsandRiemanncurvaturet- sor by elementary methods in the familiar and visualizable case of two dimensions. This early introduction to geometric quantities equips a student to learn simpler topics in general relativity like the Newtonian limit, red shift, the Schwarzschild solution, precession of the perihelion and bending of light in a gravitational ?eld. Part II (chapters 5 to 10) is an introduction to Riemannian geometry as - quired by general relativity. This is done from the beginning, starting with vectors and tensors. I believe that students of physics grasp physical concepts better if they are not shaky about the mathematics involved.

• Christoffel symbol
• Friedman equation
• Gravity
• Kerr solution
• Relativity
• Riemannian geometry
• Schwarzschild solution
• Theorema Egregium
• general relativity
• general theory of relativity
• gravitation
• gravitational wave
• spacetime
• weak gravitational field

From the reviews:

“The book is structured into three parts. The first part … offers preliminary topics on general relativity. … The second part is a comprehensive and accurate incursion in the mathematical background of relativity theory. … The last part, entitled Gravitation, contains six chapters dealing with the Einstein equation … cosmology and special topics. Interesting exercises with complete answers simplify the understanding of the presented material. The textbook is recommended to advanced graduates and researchers.” (Ferenc Szenkovits, Studia Universitatis Babes-Bolyai, Mathematica, Vol. LV (4), December, 2010)

“This book provides the reader with a very gentle introduction to the main ideas and concepts of general relativity. This work is particularly suitable for students who are interested in general relativity or are beginning to study this subject. Moreover, the author presents the reader with a number of problems and provides worked-out solutions, which will surely help the reader to obtain a deeper of interest to students who seek a smooth introduction to general relativity.” (Guglielmo Fucci, Mathematical Reviews, Issue 2011 m)

"This book meets my teaching needs, as it presents a very solid foundation for students who seek a comprehensive introduction into the difficult field of the general theory of relativity. The mathematical background is treated in a rigorous and nevertheless understandable style. Those readers who are novices in the field will welcome the fact that the author provides an abundance of carefully selected exercises together with their solutions. In summary, I will recommend this valuable textbook to my students and use it as the main text for my lectures." - Prof. Dr. P. Wirtz, Faculty of Computer Science and Mathematics, Hochschule Hegensburg, Germany

• Physics Department, Jamia Millia Islamia, New Delhi, India

  Pankaj Sharan

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