Duration: 19 weeks.

Number of lectures per week: 3

Assessment:
End-of-year Examination: One 3-hour examination

Description: The course is an introduction to general relativity.

Course content:

Differential Geometry

• basic theory of abstract manifolds (chart, atlas, differentiable structure, tangent space, smooth map, differential of a smooth map at a point)
• algebra of smooth real-valued functions on a manifold; vector fields on a manifold; Lie bracket of vector fields
• basic theory of vector bundles; sections of vector bundles; the tangent bundle; the differential of a map; natural operations on vector bundles (direct sum, tensor product, dual, symmetric and antisymmetric product)
• reminder on modules over commutative rings and their basic operations
• the module of sections of a vector bundle; its behavior under the natural operations on bundles
• natural vector bundles on a manifold; tensor fields, polyvector fields and differential forms on manifolds; symmetric and antisymmetric tensor fields
• connections in a vector bundle; curvature of a connection; induced connections
• pull-back of a vector bundle; pull-back connection
• parallel transport of a connection
• affine connections and their torsion tensor; covariant derivation of tensor fields; autoparallel curves with respect to an affine connection
• pseudo-Riemannian manifolds; signature; Riemannian, Minkowskian and Minkowsky manifolds.
• the Levi-Civita connection, geodesics, variational principle for geodesics.
• Riemann curvature tensor, Ricci tensor and Einstein tensor.
• orientability, volume form determined by a metric
• natural differential operators on a pseudo-Riemannian manifold

General Relativity

• Minkowsky manifolds, light cones and time orientability; space-times; timelike, lightlike and spacelike geodesics;
• observers, proper time
• The axioms of general relativity; action principle for gravity coupled to mater.
• Matter stress-energy tensor, Einstein equations, positivity and causality conditions
• matter models: free falling particle, dust, real scalar field, electromagnetic field
• the weak field limit, recovering Newtonian gravity, gravitational waves
• gravitational red shift
• basic special solutions of Einstein's equations (Schwarzschild, Robertson-Walker)
• motion in a central field; applications to gravitational lensing and precession of perihelia

Advanced topics (chosen at the interests of the class and as time allows): black hole physics; Kerr, Reisner-Nordstrom solutions; basic cosmology.

Core textbook:

R.K. Sachs, General Relativity for Mathematicians, Graduate Texts in Mathematics vol 48, Springer, 1977

Mar 13, 2009