General Relativity
FSU Jena, WS 2019/20
Lecture period: 14.10.2019 - 07.02.2020
Holidays: 24.10.2019, 12.11.2019
Break: 23.12.2019 - 03.01.2020
Last updated: 08.09.2019
Topics
- Invitation
- Brief historical summary
- Key concepts
- Course description
- Special relativity
- Principles
- Spacetime and events
- Spacetime invariant and causal structure
- Mikowski metric
- Lorentz transformations
- Time dilation Doppler shift
- 4-velocity, 4-acceleration, tensors
- Action, Lagrangian and equations of motion of a free particle
- Action, Lagrangian density for fields and dynamics equations
- Scalar field and Klein-Gordon equation
- Faraday tensor and Maxwell equations
- Manifolds & tensors
- Manifolds
- Vector, vector field, and dual vectors
- Tensors, Tensor product and contraction
- Metric tensor
- p-forms, wedge product
- Integral of p-forms, Stokes theorem
- Integral of functions
- Abstract index notation
- Curvature & Connection
- Covariant derivative
- Connection associated to the metric
- Parallel transport
- Riemann tensor & Bianchi identities
- Einstein tensor
- Geodesics
- Smooth curve on a manifold
- Geodesic equation
- Exp map and normal coordinates
- Variational approach and Lagrangian
- Geodesic deviation
- Equivalence principle
- Weak equivalence principle (WEP)
- Einstein equivalence principles (EEP, SEP)
- Experimental tests of fundamental principles
- Metric theories
- Geodesic equation as generalization of Newton law
- Einstein equations
- Euristic derivation
- Structure (constraints and dynamical equations)
- Chauchy problem, Harmonic gauge, well-posedness of IVP
- Role of stress-energy tensor
- Energy conditions
- Hilbert action, Lagrangian formulation
- Cosmological constant
- Weak field limit
- Linearization and infinitesiam diffeomosrphism invariace
- Linearized Einstein equations
- Static source and Newtonian limit
- Stationary source, gravito-electric/magnetic forces
- Weak field metric
- Propagation of gravitational waves
- Hilbert and TT gauge
- Effect on test masses, polarizations
- Sources of gravitational waves, quadrupole formula
- Energy of gravitational waves
- Experimental tests of GR
- Metric theories and Equivalence principle
- Perihelion shift of Mercury
- Light deflection
- Shapiro delay
- Gravitational redshift
- Binary systems and binary pulsar
- Gravitational waves from compact binaries
- Symmetries and conservation laws
- Lie derivative
- Symmetries and Killing vectors
- Conserved quantities in geodesic motion
- Conservation of energy-momntum
- Komar currents
- Black holes
- Schwarzschild solution
- Birkhoff theorem
- Geodesic motion
- Hamiltonian equations of motion (*)
- Kruskall extension
- Conformal diagram
- Embedding diagram (*)
- Kerr solution (*)
- Penrose process (*)
- Black-hole thermodynamics (*)
- Compact stars
- White dwarfs
- Newtonian structure equations
- Chandrasekhar limit
- Neutron stars
- TOV equations
- Schwarzschild solution
- Mass-radius diagram
- Gravitational collapse
- Cosmology
- GR at large scales
- Homogeneity and isotropy
- Robertson-Walker geometry
- Friedmann-Robertson-Walker equations
- Big Bang and Hubble law
- Cosmological redshift & horizons
- Cosmological constant
- LambdaCDM cosmology
- Dark matter and dark energy
- Inflation (*)
(*) Optional/Supplementary material.
Books
- R.Wald, "General Relativity" University of Chicago Press, 1984
- C.W.Misner, K.S.Thorne, and J.A.Wheeler, "Gravitation", W. H. Freeman & Company, 1973.
- S.M.Carrol, "Spacetime and Geometry, An Introduction to General Relativity", Addison-Wesley, 2003.
- L.D.Landau, "The Classical Theory of Fields", Course of Theoretical Physics, Vol. 2. Classical Theory, Butterworth-Heinemann, 1980.
- B.F.Schutz, "A First Course in General Relativity", Cambridge University Press, 1985.
- B.F.Schutz, "Geometrical Methods of Mathematical Physics", Cambridge University Press, 1980.
- N.Straumann, "General Relativity: With Applications to Astrophysics", Springer, 2004
- R.A.d'Inverno, "Introducing Einstein's Relativity", Oxford University Press, 1992.
- A.P.Lightman, W.H.Press, S.A.Teukolsky, "Problem Book in Relativity and Gravitation", Princeton University Press, 1975.
There are many MANY good books on GR. To find your favourite you can read the online list here and here and look in the library for those that most attract you.
Lecture notes and reviews
Einstein's papers
- A.Einstein, "Die Grundlage der allgemeinen Relativitätstheorie", AdP 49, 769 (1916)
- A.Einstein, "Prinzipielles zur allgemeinen Relativitätstheorie" AdP 55, 241 (1918) [4 pp.] (EAP, 577; CPE 7, 37)
- Einstein's papers
- Some links to AdP can be found here
Selected papers
Evening readings
- Kip S. Thorne, "Black holes and time warps: Einstein's outrageous legacy." W. W. Norton, 1995.
- Robert M. Wald, "Space, time, and gravity: the theory of the big bang and black holes", University of Chicago Press, 1992.
- Clifford M. Will, "Was Einstein Right?: Putting General Relativity to the Test", 2nd ed. Basic Books, 1993.
Videos
Evaluation
- Exercise sheets will be distirbuted every Monday, collected the next week on Monday and corrected during the exercise class on Thursday.
- Exam admission requires >= 60% exercises.
- A written exam will take place on 17.02.2019 @ 10:00-13:00, HS2 Abbeanum.
- Oral exams will take place in the following days. Time slots are available in the following days and will be booked after the written exam.
Course material
Gravity and institute seminars at TPI
Join the research seminars of the institute, do not be afraid ! You are welcome and can find here all the events.