The book aims to give a mathematical presentation of the theory of general relativity (that is, spacetime-geometry-based gravitation theory) to advanced undergraduate mathematics students. Mathematicians will find spacetime physics presented in the definition-theorem-proof format familiar to them. The given precise mathematical definitions of physical notions help avoiding pitfalls, especially in the context of spacetime physics describing phenomena that are counter-intuitive to everyday experiences.In the first part, the differential geometry of smooth manifolds, which is needed to present the spacetime-based gravitation theory, is developed from scratch. Here, many of the illustrating examples are the Lorentzian manifolds which later serve as spacetime models. This has the twofold purpose of making the physics forthcoming in the second part relatable, and the mathematics learnt in the first part less dry. The book uses the modern coordinate-free language of semi-Riemannian geometry. Nevertheless, to familiarise the reader with the useful tool of coordinates for computations, and to bridge the gap with the physics literature, the link to coordinates is made through exercises, and via frequent remarks on how the two languages are related.In the second part, the focus is on physics, covering essential material of the 20th century spacetime-based view of gravity: energy-momentum tensor field of matter, field equation, spacetime examples, Newtonian approximation, geodesics, tests of the theory, black holes, and cosmological models of the universe.Prior knowledge of differential geometry or physics is not assumed. The book is intended for self-study, and the solutions to the (over 200) exercises are included.
Contents:
- Smooth Manifolds
- Co-/Tangent Space
- Co-/Tangent Bundles
- Tensor Fields
- Lorentzian Manifolds
- Levi-Civita Connection
- Parallel Transport
- Geodesics
- Curvature
- Form Fields
- Integration
- Minkowski Spacetime Physics
- Matter
- Field Equation
- Black Holes
- Cosmology
Readership: Advanced undergraduate and beginning graduate students in Mathematics and Physics.
Key Features:
- No previous familiarity with differential geometry is assumed
- No prior deep knowledge of physics is needed (a general introductory college course covering Newton's laws of motion and of gravitation should suffice)
- All the physics is presented using mathematical concepts, and the theory of general relativity is presented in definition-theorem-proof style
- Over 200 exercises with complete solutions are included, so that the book can be used for self-study
- 83 pictures are included, making the book user-friendly. Complete derivations of calculations are included, so that the book is self-contained and easy to follow
- The semi-Riemannian geometry needed to present the theory of relativity is developed in the modern coordinate-free language. Concrete computations using tensor components are not ignored however, and are taught in tandem via the exercises
- The differential geometric concepts are illustrated with spacetime examples right from the beginning, so that all the mathematics being learnt is put in context right from the outset. It also makes the subject more appealing and more focused towards the goal
- Notational differences with the physics literature are highlighted, building a dictionary towards gaining familiarity with the notation used by the physicists
- The essential material from the 20th century spacetime-geometry-based view of gravity is covered
