1. The Equivalence Principle.
1.1 Spacetime properties. 1.2 Gravity as Geometry. 1.3 The equivalence principles (WEP, EEP, SEP). 1.4 Experimental tests of the EP. 1.5 Towards General Covariance
[Carroll 2.1, gr-qc/1304.5795, gr-qc/0806.1731, 1501.01644, 1712.01176]
2. Rindler spacetime.
2.1 Uniformly accelerated observers. 2.2 Adapted coordinates. 2.3 Rindler spacetime (Time dilation, event horizons).
[Blau 1.3]
3. Geodesics.
3.1 Accelerated observers. 3.2 Geodesics. 3.3. Massless particles. 3.4 Affine and non-affine parameters. 3.5 Couplings to other fields.
[Blau 1.6-1.8, 2.1-2.2, Zee III.5,IV,V.1, Carroll 3.3-3.4]
4*. Manifolds, Tensors.
4.1 Manifolds. 4.2 Tangent space. 4.3 Cotangent space. 4.4 Tensor fields.
[Notes of the course on group theory, Carroll 2.2-2.5]
5*. Connection, Curvature.
5.1 Connection. 5.2 Parallel transport. 5.3 Curvature.
[Carroll 3.2,3.6,3.7]
6*. Differential Forms.
6.1 Differential forms. (Hodge star, integration of p-forms). 6.2 Vielbeins. 6.3 Connection and Curvature.
[Carroll 2.8-2.10, Zee IX.7, IX.8, Eguchi-Gilkey-Hanson 2.3,2.4,3.1,3.2]
7. Einstein's Equations.
7.1 The Einstein-Hilbert action. 7.2 Einstein's equations. 7.3 The Stress-Energy tensor (stress tensor for a gas of particles).
[Carroll 4.3]
8. Some physics of gravitation.
8.1 Weak field approximation. 8.2 Gravitational redshift (GPS). 8.3 Gravity waves. 8.4 Physical effects of gravity waves (Geodesic deviation equation).
[Weinberg 3.4, 3.5, Blau 22.3, Carroll 7.1, 7.2, 7.4, Zee IX.3]
9. Symmetries.
9.1 Lie Derivatives. 9.2 Killing Vectors. 9.3 Symmetries and conserved charges (Komar integrals*). 9.4 Maximally symmetric spaces.
[Blau 8-9,12.2,12.7, Carroll 3.8,3.9]
10. Schwarzschild's solution.
10.1 Solving Einstein's equations using symmetries. 10.2* Finding Schwarzschild.
[Carroll 5.1,5.3]
11. Physics of the solar system.
11.1 Effective potential for geodesics. 11.2 Geodesics for massive particles and Mercury's precession. 11.3 Geodesics for massless particles. Bending of light rays. Light emitted from compact stars. Shapiro time delay.
[Carroll 5.4,5.5 Zee VII.1, Blau 24]
12. Schwarzschild black hole.
12.1 Accelerated and freely falling observers. 12.2 Collapse to black hole. 12.3 Near-horizon metric. 12.4 Eddington-Finkelstein and Kruskal-Szekeres coordinates. 12.5 Global extensions.
[Carroll 5.6, Blau 25]
13. Black hole physics.
13.1 Time translation in Kruskal coordinates. 13.2 Null hypersurfaces. 13.3 Surface gravity. 13.4* Penrose diagrams.
[Carroll 5.7, Carroll 6.2-6.3, gr-qc/9707012]
14. Other Black Holes.
14.1 Charged black holes. 14.2 Rotating Solutions. 14.3 Frame dragging. 14.4 Penrose process.
[Carroll 6.5-6.7, gr-qc/9707012, Zee VII.5-VII.6, IX.2]
15*. Black hole mechanics.
15.1 Black hole thermodynamics
16. Cosmology.
16.1 FRW cosmology from Einstein equations. 16.2 (Anti) de Sitter spacetime
[Carroll 8.2,8.3, Zee VIII.1]
* Not part of the examination.