General Relativity

Suggested course

Theoretical Physics.

Target skills and knowledge

A basic introduction to the theoretical and phenomenological foundations of general relativity. At the end of the course students should master the basic techniques to find and analyse solutions of Einstein’s field equations.

Learning activities

Lectures. Weekly assignments.

Examination method

Questions on the topics presented during the course and solution of a problem.

Textbooks

• S. Carroll, Spacetime and Geometry: An Introduction to General Relativity, Addison-Wesley, 2003.
• A. Zee, Einstein Gravity in a Nutshell, Princeton University Press, 2013.
• F. de Felice and C. J. S. Clarke, Relativity on Curved Manifolds, Cambridge University Press, 1992.
• S. Weinberg, Gravitation and Cosmology, Wiley, 1972.
• M. Blau, Lecture Notes on General Relativity.

Assignments (2017/18)

First, Second, Third, Fourth, Fifth, Sixth, Seventh (PDF).

Office hour: Fridays, 15:00–16:00, preferably by appointment (also by e-mail or telephone).

1. 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. 2. Rindler spacetime

   2.1 Uniformly accelerated observers. 2.2 Adapted coordinates. 2.3 Rindler spacetime (time dilation, event horizons).

   Blau 1.3

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. 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. 5*. Connection, curvature

   5.1 Connection. 5.2 Parallel transport. 5.3 Curvature.

   Carroll 3.2, 3.6, 3.7

6. 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. 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. 8. Some physics of gravitation

   8.1 Weak-field approximation. 8.2 Gravitational redshift (GPS). 8.3 Gravitational waves. 8.4 Physical effects of gravitational waves (geodesic deviation equation).

   Weinberg 3.4, 3.5; Blau 22.3; Carroll 7.1, 7.2, 7.4; Zee IX.3

9. 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. 10. Schwarzschild’s solution

    10.1 Solving Einstein’s equations using symmetries. 10.2* Finding Schwarzschild.

    Carroll 5.1, 5.3

11. 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. Light emitted from compact stars. Shapiro time delay.

    Carroll 5.4, 5.5; Zee VII.1; Blau 24

12. 12. Schwarzschild black hole

    12.1 Accelerated and freely falling observers. 12.2 Collapse to a 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. 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, 6.2–6.3; gr-qc/9707012

14. 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. 15*. Black hole mechanics

    15.1 Black hole thermodynamics.

16. 16. Cosmology

    16.1 FRW cosmology from Einstein’s equations. 16.2 (Anti-)de Sitter spacetime.

    Carroll 8.2, 8.3; Zee VIII.1

Sections marked * are not part of the examination.

Oral examination: sessions in June, August and September.