General Relativity

(a.y. 2014/15-2017/18)

1º anno, 2º semestre (LM Fisica)

General Information

Suggested course: Theoretical Physics

Target skills and knowledge: This course will cover a basic introduction to the theoretical and phenomenological foundations of the General Theory of Relativity. At the end of the course students should be able to master basic techniques to find and analyze solutions to Einstein field equations.

Planned learning activities and teaching methods: 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, C.J.S. Clarke, Relativity on curved manifolds. --: Cambridge University Press, 1992.
  • S. Weinberg, Gravitation and Cosmology. --: Wiley, 1972.
  • M. Blau, Lecture Notes in General Relativity.

Assignments

First 2017/18, Second 2017/18, Third 2017/18, Fourth 2017/18, Fifth 2017/18, Sixth 2017/18, Seventh

Class schedule

Lectures: Lectures will be delivered at the Physics and Astronomy department:

Day Time Room
THU 10.30 - 12.15 Aula LuF
FRI 10.30 - 12.15 Aula LuF

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

Syllabus

    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]

    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, hep-th/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.

Exams

Oral examination

  • Winter Session
  • Summer Session
    • To be defined
  • Fall Session
    • To be defined