Introduction to general relativity
GEOPlanet, Warsaw4PHD graduate schools
Winter semester 2023/2024
Course time: Tuesday 9.00-11.00, via Zoom
Lecturer: Mikołaj Korzyński (Center for Theoretical Physics, PAS)
Warsaw University of Technology (Politechnika Warszawska)
Faculty of Electronics and Information Technology
(Wydział Elektroniki i Technik Informacyjnych)
ul. Nowowiejska 15/19
- Contact hours: Thursday 14.00-16.00
(Contact also possible via Zoom.)
None so far.
None so far.
Course type: This is an introductory course of general relativity with emphasis on astrophysical and cosmological applications
Intended for: graduate students of theoretical physics, astrophysics, astronomy etc.
Why should you take this course?
- Astrophysics, cosmology, relativity PhD students: this a GR course with emphasis of cosmological and astrophysical applications
- Students of quantum information, boson gases etc.: you may expand your general knowledge of theoretical physics and astrophysics, field theory and its mathematical language (differential geometry, pseudo-Riemannian manifolds)
- special relativity
- general theoretical physics knowledge:
- classical mechanics
- Newtionian gravity
- Maxwell's equations and electromagnetic waves would also be welcome
- Python, Jupyter notebooks - general knowledge may be useful
- on-line course (Zoom)
- 30 hours, 2 hours a week. Every second week we will have 1 hour of classes instead of a lecture.
Altogether: 23 hours of lecture, 7 hours of classes
There will be no official course textbook, but I can recommend the following books as additional reading:
- B. Schutz, "A First Course in General Relativity"
- C. W. Misner, K. S. Thorne, J. A. Wheeler, "Gravitation"
- R. Wald, "General relativity"
- E. Poisson, "A Relativist's Toolkit"
- J. B. Hartle, "Gravity: An Intruduction to Einstein's General Relativity"
- J. J. Synge, "Relativity: The General Theory"
Course webpage with notes & problem sheets:
This is a rather ambitious outline, we may not be able to cover all of the material from the last few topics.
- why was GR developed and why is it important
- historical introduction
- Special vs general relativity:
- brief summary of special relativity
- need of a gravity theory consistent with special relativity. Basic idea: gravity is simply geometry in disguise
- equivalence principle, local inertial frames
- Mathematical language = differential geometry:
- manifolds, coordinates, charts, coordinate transformations
- vectors, one-forms, tensors, fields, coordinate transformations
- metric, inverse metric
- Einstein's summation convension, index and index-free notations
- curves, tangent vectors. Light rays and worldlines of particles
- covariant derivatives, parallel transport
- geodesics. Free fall
- curvature tensor, Bianchi identities
- locally flat coordinates. Local inertial frames and equivalence principle
- Einstein equations: how matter curves spacetime
- covariant equations. Coordinate system independence
- - cosmological constant
- stress-energy tensor
- Newtonian approximation
- law, gravitational light bending
- Linearized gravity
- gravitational waves
- Lorentz gauge, 2 polarizations, quadrupole formula
- How gravity affects light propagation in wave picure (WKB approximation), impact on phases and TOA of waves. Gravitational frequency shift, detections of gravitational waves.
- Exact solutions: Schwarzschild metric
- derivation: Lie derivative, flow, pullback, Killing vectors, symmetries
- adapted coordinate system
- solution in standard coordinates
- Schwarzschild as an exterior solution
- geodesics far from . Peryhelion precession, light bending
- grav. frequency shift
- geometry of , horizon and singularity. Kruszkal and other coords
- ISCO, photonsphere
- BH shadow - what does a BH look like in the presence of infalling matter
- spherically symmetric collapse into a BH
- Exact solutions: Kerr metric
- gravitational field of a rotating body
- gravimagnetic effect, geodetic effect. Gravity probe B
- inertial frame dragging
- exact solution: Kerr - just presentation
- Exact solutions: FLRW metric
- FLRW ansatz, scale factor, Friedmann equations
- closed, open, flat solutions
- Hubble diagram, distance measures
- Gravitational lensing
- Lensing by a static Newtonian potential, thin lens approximation
- Convergence, shear
- Strong lensing, caustics, multiple imaging, Einstein arcs
- Tests of GR, classical and new
- Solar-System based tests
- Gravity Probe A, B
- Perhaps: 3+1 splitting, ADM formalism