TABLE OF CONTENTS

Differential geometry

The concept of a vector
 Vector operations
 Projections and basis vectors
 Towards tangent space and all that
 Index notation

Manifolds and tensors
 Tangent space and vectors fields
 Tensors
 Manifolds and metric
 Examples of metrics
 Geodesics
 Covariant derivative
 Parallel transport and geodesics

Curvature
 Infinitesimal parallelogram
 Riemann, Ricci and Weyl tensors
 Geodesic deviation equation
 EulerLagrange equations
 Further reading
 Exercises

The concept of a vector

Einstein field equations

Some physics background
 Newton's theory of gravity
 Special relativity
 Maxwell equations
 Matter tensors

Geometry and gravity
 Geodesics and Newton's law
 Curvature and the Poisson equation
 Field equations of General Relativity
 The principle of minimal gravitational coupling

Weak gravity
 Linearised Riemann and Ricci tensors
 Gauge transformations
 Linearised Einstein field equations
 Gravitational waves
 Variational approach to general relativity
 Further reading
 Exercises

Some physics background

Schwarzschild solutions
 Spherical symmetry and Birkhoff's theorem
 The Schwarzschild solution
 The Schwarzschild interior solution
 Geodesics in Schwarzschild spacetime

Testing general relativity – the classical tests
 Perihelion precession of Mercury
 Light deflection by the Sun
 Gravitational redshift of light
 Radar echo or gravitational time delay

The Schwarzschild radius
 Radial null geodesics
 EddingtonFinkelstein coordinates
 KruskalSzekeres coordinates
 Black holes
 Further reading
 Exercises

Cosmology

Classical and Modern Cosmology
 Cosmological principle
 Geometry of constant time hypersurfaces
 FriedmannLemaitreRobertsonWalker metric
 Particle horizons
 Field equations

Cosmological solutions
 Matter dominated universe
 Radiation dominated universe
 The Einstein static universe
 De Sitter universe

Physical cosmology
 Cosmological parameters
 Redshift
 Distances in cosmology
 Distance redshift relationships
 The universe today
 Shortcomings in cosmology

Inflation
 Accelerated expansion
 Scalar fields in cosmology
 Slowroll inflation
 Further reading
 Exercises

Classical and Modern Cosmology
 Solutions to exercises