An extra curricular course on "General Relativity, Cosmology and Black Holes" will run throughout the second semester 2011/12. Only basic knowledge of special relativity and Newtonian gravity will be assumed. By default, sessions will take place every friday at 14H00 in the GAP room. Some lecture notes will be handed out and exercises will be set in every session and solved in the following one.

Lecturer: Carlos Herdeiro

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**Session 1** (January 13, Anf. Física) - Review of Newtonian gravity; the inverse square law; successful predictions; one "small" problem: the anomalous perihelion precession of Mercury. (Exercise 1: Orbits of the Kepler problem.)

**Session 2** (January 19) - Conceptual problems of Newtonian gravity and inconsistencies with special relativity: the lack of propagation problem and need for "magnetic"-type effects. The equivalence principle and the features of a relativistic theory of gravity. (Exercise 2: Orbits of the Kepler problem with a sub-leading correction.)

**Session 3** (January 27) - Differential geometry: vectors, co-vectors (1-forms) and tensors: transformation laws under coordinate transformations. (Exercise 3: Transformation of the metric tensor from Cartesian to polar coordinates.)

**Session 4** (February 3) - Covariant derivative: non-tensorial transformation of partial derivatives, linear connection and its transformation rule. (Exercise 4: Covariant derivative of constant, but not covariantly constant vector fields in polar coordinates.)

**Session 5** (February 10) - Covariant derivative of arbitrary tensor fields. Levi-Civita connection and the Christoffel symbols. (Exercise 5: Compute the Christoffel symbols for non-rectangular coordinate systems in Euclidean 3-space.)

**Session 6** (February 17) - Geodesics with an arbitrary parameter. Affinely parameterised geodesics. (Exercise 6: Compute the geodesics on the 2-sphere).

**Session 7 **(February 24) - Lie derivative of tensor fields. (Exercise 7: Compute the vector fields along which the round metric on the 2-sphere has vanishing Lie derivative).

**Session 8** (March 2) - KIlling vectors. (Exercise 8: Compute the Killing vectors on R^2 using Cartesian and polar coordinates and their Lie algebra).

**Session 9** (March 16) - Local flatness theorem. Relation between the metric determinant and the Jacobian determinant. The invariant volume element. The covariant divergence formula in terms of the metric determinant. (Exercise 9: Write down explicitly the covariant divergence in spherical and cylindrical coordinates).

**Session 10** (March 23) - Curvature. Bianchi identities. Einstein equations. (Exercise 10: demonstrate the second Bianchi identities).

**Session 11** (March 30) - Geodesic deviation equation.

**Session 12** (April 13) - Newtonian limit of General Relativity. (Exercise 11: Compute Einstein's tensor for a spherically symmetric ansatz).

**Session 13** (April 20) - Derivation of the Schwarzschild solution. (Exercise 12: Effective Lagrangian for geodesic motion).

**Session 14** (May 4) - Time-like geodesics in the Schwarzschild spacetime.

**Session 15** (May 11) - Computation of the advance of the orbital perihelion

**Session 16** (June 1) - Computation of the deflection of a light ray in Schwarzschild space-time

**Session 17** (June 15) - Gravitational redshift; TEDx presentation "Sussuros do espaço-tempo"; closing of the course