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
- Euler-Lagrange equations
- Further reading
- Exercises
-
The concept of a vector
-
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
- 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
- Eddington-Finkelstein coordinates
- Kruskal-Szekeres coordinates
- Black holes
- Further reading
- Exercises
-
Classical and Modern Cosmology
- Cosmological principle
- Geometry of constant time hypersurfaces
- Friedmann-Lemaitre-Robertson-Walker 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
- Slow-roll inflation
- Further reading
- Exercises