# Spacetime and Geometry

### An Introduction to General Relativity

"For if each Star is little more a mathematical Point,
located upon the Hemisphere of Heaven by Right Ascension and Declination,
then all the Stars, taken together, tho' innumerable,
must like any other set of points,
in turn represent some single gigantick Equation,
to the mind of God as straightforward as, say, the Equation of a Sphere,---
A lonely, uncompensated, perhaps even impossible Task,---
yet some of us must ever be seeking, I suppose."
-- Thomas Pynchon, Mason & Dixon

In 1996 I taught a one-semester graduate course graduate course in general relativity at MIT. Along the way I typed up a detailed set of lecture notes. The book Spacetime and Geometry is a significantly revised and expanded version of these notes; about half of the finished book is completely new. The lecture notes will continue to be available for free online.

The philosophy of the book is to provide an accessible, useful, and pedagogical introduction to general relativity. In particular, no effort has been made to write a comprehensive reference book. More on the approach taken can be found in the preface.

You can order the book online from Amazon.com or from Addison Wesley.

Contents

• Preface
• Special Relativity and Flat Spacetime
• Prelude
• Space and time, separately and together
• Lorentz transformations
• Vectors
• Dual vectors (one-forms)
• Tensors
• Manipulating tensors
• Maxwell's equations
• Energy and momentum
• Classical field theory
• Exercises
• Manifolds
• Gravity as geometry
• What is a manifold?
• Vectors again
• Tensors again
• The metric
• An expanding universe
• Causality
• Tensor densities
• Differential forms
• Integration
• Exercises
• Curvature
• Overview
• Covariant derivatives
• Parallel transport and geodesics
• Properties of geodesics
• The expanding universe revisited
• The Riemann curvature tensor
• Properties of the Riemann tensor
• Symmetries and Killing vectors
• Maximally symmetric spaces
• Geodesic deviation
• Exercises
• Gravitation
• Physics in curved spacetime
• Einstein's equation
• Lagrangian formulation
• Properties of Einstein's equation
• The cosmological constant
• Energy conditions
• The Equivalence Principle revisited
• Alternative theories
• Exercises
• The Schwarzschild Solution
• The Schwarzschild metric
• Birkhoff's theorem
• Singularities
• Geodesics of Schwarzschild
• Experimental tests
• Schwarzschild black holes
• The maximally extended Schwarzschild solution
• Stars and black holes
• Exercises
• More General Black Holes
• The black hole zoo
• Event Horizons
• Killing Horizons
• Mass, charge, and spin
• Charged (Reissner-Nordström) black holes
• Rotating (Kerr) black holes
• The Penrose process and black-hole thermodynamics
• Exercises
• Perturbation Theory and Gravitational Radiation
• Linearized theory and gauge transformations
• Degrees of freedom
• Newtonian fields and photon trajectories
• Gravitational wave solutions
• Production of gravitational waves
• Energy loss due to gravitational radiation
• Detection of gravitational waves
• Exercises
• Cosmology
• Maximally symmetric universes
• Robertson-Walker metrics
• The Friedmann equation
• Evolution of the scale factor
• Redshifts and distances
• Gravitational lensing
• Our universe
• Inflation
• Exercises
• Quantum Field Theory in Curved Spacetime
• Introduction
• Quantum mechanics
• Quantum field theory in flat spacetime
• Quantum field theory in curved spacetime
• The Unruh effect
• The Hawking effect and black hole evaporation
• Appendices
• Maps between manifolds
• Diffeomorphisms and Lie derivatives
• Submanifolds
• Hypersurfaces
• Stokes's theorem
• Geodesic congruences
• Conformal transformations
• Conformal diagrams
• The parallel propagator
• Non-coordinate bases
Preface
Lecture Notes
Commentary/Reviews
Bibliography