**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.