Introduction to general relativity
GEOPlanet, Warsaw4PHD graduate schools
Winter semester 2023/2024
ECTS credits: 3
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.)
Lecture notes and recordings

Lecture 1 (introduction, special relativity)

Lecture 2 (special relativity, Class 1: special relativity)

Lecture 3 (Class 1: special relativity continued, tensor algebra)

Lecture 4 (tensor algebra, introduction to GR: equivalence principle, Mach's principle)
 recording (cloud)
 slides (PDF)
 blackboard (TIFF): [ 2 ] , [ 3 ] , [ 4 ] , [ 5 ] , [ 6 ] , [ 7 ] , [ 8 ] , [ 9 ] , [ 10 ] , [ 11 ] , [ 12 ] (there was no blackboard 1 due to a numeration error)

Lecture 5 (diffetential geometry: manifolds, coordinates and tensors. Blackboard lecture: tensors and coordinate transformations)

Lecture 6 (Blackboard lecture: Problem Sheet 1, coordinate transformations, differentiating tensors. Differential geometry: pseudoRiemannian manifolds, connection, covariant derivative)
 recording
 slides (PDF)
 blackboard (TIFF): [ 1 ] , [ 2 ] , [ 3 ] , [ 4 ] , [ 5 ] , [ 6 ] , [ 7 ] , [ 8 ] , [ 9 ] , [ 10 ] , [ 11 ] , [ 12 ] , [ 13 ] , [ 14 ]

Lecture 7 (Differential geometry: connection, metric/LeviCivita connection, Christoffel symbols, covariant derivative, parallel transport, geodesics. Blackboard lecture: calculating the Christoffels symbols directly and using the variational principle for geodesics)
 recording
 slides (PDF)
 blackboard (TIFF): [ 1 ] , [ 2 ] , [ 3 ] , [ 4 ] , [ 5 ] , [ 6 ] , [ 7 ] , [ 8 ] , [ 9 ] , [ 10 ] , [ 11 ] , [ 12 ]

Lecture 8 (Board lecture: Christoffel symbols. Slides: Curvature tensor. Board lecture  curvature.)
 recording
 slides (PDF)
 blackboard (TIFF): [ 1 ] , [ 2 ] , [ 3 ] , [ 4 ] , [ 5 ] , [ 7 ] , [ 8 ] , [ 9 ] , [ 10 ] , [ 11 ] , [ 12 ] , [ 13 ] (no blackboard 6)

Lecture 9 (Board lecture: Christoffel symbols, curvature). Only one hour of lecture

Lecture 10 (Board lecture: curvature. Slides & blackboard: geodesic deviation equation. Slides: Einstein equations, cosmological constant. Slides & blackboard: linearized gravity)

Lecture 11 (Board lecture: Problem sheet 3, linearized gravity, Newtonian limit)

Lecture 12 (Slides and board lecture: Killing vectors and symmetries; Board lecture + slides: conserved energy, gravitational frequency shift; Board lecture + slides: motions in gravitational field, gravitational deflection of light; Slides: intro to gravitational waves)

Lecture 13 (Slides and blackboard lecture: introduction to gravitational waves, plane gravitational waves, gravitational waves and massive particles; slides: effective energy flux of gravitational waves)
 recording
 slides (PDF)
 blackboard (TIFF): [ 1 ] , [ 2 ] , [ 3 ] , [ 4 ] , [ 5 ] , [ 6 ] , [ 7 ] , [ 8 ] , [ 9 ] , [ 10 ] , [ 11 ] , [ 12 ] , [ 13 ]

Lecture 14 (Slides + blackboard lecture: gravitational wave sources, quadrupole formula, total luminosity; slides: astrophysical sources of GW, detectors, remarks; slides: Schwarzschild solution  intro and derivation)

Lecture 15 (Slides: Schwarzschild solution  derivation, properites. Blackboard lecture: properties of Schwarzschild  asymptotics, gravitational frequency shift, acceleration of static observers; geodesics, perihelion precession)

Lecture 16 (Blackboard + slides: perihelion precession; Blackboard + slides: KruszkalSzekeres coordinates, black holes). Only one hour of lecture
Problem sheets
Please send the solutions via email to korzynski@cft.edu.pl
. I prefer PDF's (for example LaTeXgenerated) or common graphic formats (for example scans of your handwritten notes).

Problem sheet 1
 topic: special relativity
 deadline: Nov 3rd, 2023

Problem sheet 2
 topic: special relativity, tensor algebra
 deadline: Nov 17th, 2023

Problem sheet 3
 topic: coordinate transformations, Christoffel symbols and connection
 deadline: Dec 9th, 2023

Problem sheet 4
 topic: Christoffel symbols and connection, curvature
 deadline: Dec 22nd, 2023

Problem sheet 5
 topic: Gravitational light bending, gravitational redshift
 deadline: extended to Feb 11th, 2024
 solution to problem 1:

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"
 S. Carroll, "Spacetime and Geometry: An Introduction to General Relativity"

Course webpage with notes & problem sheets:
korzynski.cft.edu.pl/grcourse.html
My plan for the incoming lectures

Lecture 12 (Dec 19th, 2023)
 Killing equations, symmetries and geodesics
 conserved energy and Newtonian approximation. Gravitational redshift/blueshift
 gravitational light bending
 gravitational lensing
 gravitational waves  Intro

Lecture 13 (Jan 9th, 2024)
 gravitational waves, sources and detection
 Schwarzschild solution

Jan 15th, 2024  no lecture, winter school holidays

Lecture 14 (Jan 23rd, 2024)
 Schwarzschild solution: horizon, singularity, geodesics

Lecture 15 (Jan 30th, 2024)
 Inertial frame dragging
 Kerr solution

Lecture 16 (Feb 6th, 2024)  1 hour
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