Introduction to general relativity
GEOPlanet, Warsaw4PHD graduate schools
Winter semester 2023/2024
Course time: Tuesday 9.0011.00, 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
00665 Warszawa
Room 459
 Contact hours: Thursday 14.0016.00
(Contact also possible via Zoom.)
Lectures
None so far.
Problem sheets
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, pseudoRiemannian manifolds)

Prerequisites:
 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

Organization:
 online 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

Textbooks:
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:
korzynski.cft.edu.pl/grcourse.html
Topics
This is a rather ambitious outline, we may not be able to cover all of the material from the last few topics.
 Introduction:
 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, oneforms, tensors, fields, coordinate transformations
 metric, inverse metric
 Einstein's summation convension, index and indexfree 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
 $\Lambda$  cosmological constant
 stressenergy tensor
 Newtonian approximation
 $1/r^2$ 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 $r = 2M$. Peryhelion precession, light bending
 grav. frequency shift
 geometry of $r = 2M$, 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
 RebkaSnyder
 SolarSystem based tests
 Gravity Probe A, B
 Perhaps: 3+1 splitting, ADM formalism