TABLE OF CONTENTS

  1. Differential geometry
    1. The concept of a vector
      1. Vector operations
      2. Projections and basis vectors
      3. Towards tangent space and all that
      4. Index notation
    2. Manifolds and tensors
      1. Tangent space and vectors fields
      2. Tensors
      3. Manifolds and metric
      4. Examples of metrics
      5. Geodesics
      6. Covariant derivative
      7. Parallel transport and geodesics
    3. Curvature
      1. Infinitesimal parallelogram
      2. Riemann, Ricci and Weyl tensors
      3. Geodesic deviation equation
    4. Euler-Lagrange equations
    5. Further reading
    6. Exercises
  2. Einstein field equations
    1. Some physics background
      1. Newton's theory of gravity
      2. Special relativity
      3. Maxwell equations
      4. Matter tensors
    2. Geometry and gravity
      1. Geodesics and Newton's law
      2. Curvature and the Poisson equation
      3. Field equations of General Relativity
      4. The principle of minimal gravitational coupling
    3. Weak gravity
      1. Linearised Riemann and Ricci tensors
      2. Gauge transformations
      3. Linearised Einstein field equations
      4. Gravitational waves
    4. Variational approach to general relativity
    5. Further reading
    6. Exercises
  3. Schwarzschild solutions
    1. Spherical symmetry and Birkhoff's theorem
    2. The Schwarzschild solution
    3. The Schwarzschild interior solution
    4. Geodesics in Schwarzschild spacetime
    5. Testing general relativity – the classical tests
      1. Perihelion precession of Mercury
      2. Light deflection by the Sun
      3. Gravitational redshift of light
      4. Radar echo or gravitational time delay
    6. The Schwarzschild radius
      1. Radial null geodesics
      2. Eddington-Finkelstein coordinates
      3. Kruskal-Szekeres coordinates
      4. Black holes
    7. Further reading
    8. Exercises
  4. Cosmology
    1. Classical and Modern Cosmology
      1. Cosmological principle
      2. Geometry of constant time hypersurfaces
      3. Friedmann-Lemaitre-Robertson-Walker metric
      4. Particle horizons
      5. Field equations
    2. Cosmological solutions
      1. Matter dominated universe
      2. Radiation dominated universe
      3. The Einstein static universe
      4. De Sitter universe
    3. Physical cosmology
      1. Cosmological parameters
      2. Redshift
      3. Distances in cosmology
      4. Distance redshift relationships
      5. The universe today
      6. Shortcomings in cosmology
    4. Inflation
      1. Accelerated expansion
      2. Scalar fields in cosmology
      3. Slow-roll inflation
    5. Further reading
    6. Exercises
  5. Solutions to exercises