Introduction to general relativity

GEOPlanet, Warsaw4PHD graduate schools

Winter semester 2023/2024

ECTS credits: 3

Course time: Tuesday 9.00-11.00, via Zoom

Lecturer: Mikołaj Korzyński (Center for Theoretical Physics, PAS)

• email: korzynski@cft.edu.pl

• office:

Warsaw University of Technology (Politechnika Warszawska)  
Faculty of Electronics and Information Technology  
(Wydział Elektroniki i Technik Informacyjnych)  
ul. Nowowiejska 15/19  
00-665 Warszawa  
Room 459

• Contact hours: Thursday 14.00-16.00
  (Contact also possible via Zoom.)

Lecture notes and recordings

Lecture recordings: YouTube channel

• Lecture 1 (introduction, special relativity)

  • recording (cloud)
  • slides (PDF)
  • blackboard (TIFF)

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

  • recording (cloud)
  • slides (PDF)
  • blackboard (TIFF): [ 1 ] , [ 2 ] , [ 3 ] , [ 4 ] , [ 5 ] , [ 6 ] , [ 7 ] , [ 8 ]

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

  • recording (cloud)
  • slides (PDF)
  • blackboard (TIFF): [ 1 ] , [ 2 ] , [ 3 ] , [ 4 ] , [ 5 ] , [ 6 ] , [ 7 ] , [ 8 ] , [ 9 ]
  • Jupyter notebook .ipynb: ultrarelativistic luminous object passing by with constant velocity.

• 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)

  • recording (cloud)
  • slides (PDF)
  • blackboard (TIFF): [ 1 ] , [ 2 ] , [ 3 ] , [ 4 ] , [ 5 ] , [ 6 ] , [ 7 ] , [ 8 ] , [ 9 ] , [ 10 ]

• Lecture 6 (Blackboard lecture: Problem Sheet 1, coordinate transformations, differentiating tensors. Differential geometry: pseudo-Riemannian 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/Levi-Civita 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

  • recording
  • blackboard (TIFF): [ 1 ] , [ 2 ] , [ 3 ] , [ 4 ] , [ 5 ] , [ 6 ] , [ 7 ]

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

  • recording
  • slides (PDF)
  • blackboard (TIFF): [ 1 ] , [ 2 ] , [ 3 ] , [ 4 ] , [ 5 ] , [ 6 ] , [ 7 ]

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

  • recording
  • blackboard (TIFF): [ 1 ] , [ 2 ] , [ 3 ] , [ 4 ] , [ 5 ] , [ 6 ] , [ 7 ] , [ 8 ] , [ 9 ] , [ 10 ]

• 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)

  • recording
  • slides (PDF)
  • blackboard (TIFF): [ 1 ] , [ 2 ] , [ 3 ] , [ 4 ] , [ 5 ] , [ 6 ] , [ 7 ] , [ 8 ] , [ 9 ]

• 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)

  • recording
  • slides (PDF)
  • blackboard (TIFF): [ 1 ] , [ 2 ] , [ 3 ] , [ 4 ] , [ 5 ]
  • emission.ipynb: Jupyter notebook calculating the strain and total GW luminosity of binary systems on circular orbits. You may check the GW emission from your favourite system.

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

  • recording
  • slides (PDF)
  • blackboard (TIFF): [ 1 ] , [ 2 ] , [ 3 ] , [ 4 ] , [ 5 ] , [ 6 ] , [ 7 ] , [ 8 ]
  • schwarzschild.ipynb: Jupyter notebook calculating the effective potential for geodesics in Schwarzschild and the tortoise coordinate

• Lecture 16 (Blackboard + slides: perihelion precession; Blackboard + slides: Kruszkal-Szekeres coordinates, black holes). Only one hour of lecture

  • recording
  • slides (PDF)
  • blackboard (TIFF): [ 1 ] , [ 2 ] , [ 3 ] , [ 4 ] , [ 5 ]

Problem sheets

Please send the solutions via email to korzynski@cft.edu.pl. I prefer PDF's (for example LaTeX-generated) or common graphic formats (for example scans of your hand-written notes).

1. Problem sheet 1

   • topic: special relativity
   • deadline: Nov 3rd, 2023

2. Problem sheet 2

   • topic: special relativity, tensor algebra
   • deadline: Nov 17th, 2023

3. Problem sheet 3

   • topic: coordinate transformations, Christoffel symbols and connection
   • deadline: Dec 9th, 2023

4. Problem sheet 4

   • topic: Christoffel symbols and connection, curvature
   • deadline: Dec 22nd, 2023

5. Problem sheet 5

   • topic: Gravitational light bending, gravitational redshift
   • deadline: extended to Feb 11th, 2024
   • solution to problem 1:
     • solar-deflection.ipynb
     • deflection.png

General information

1. Course type: This is an introductory course of general relativity with emphasis on astrophysical and cosmological applications

2. 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)
     
     

3. 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
     
     

4. Organization:

   • 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
     
     

5. 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"
     
     

6. Course webpage with notes & problem sheets:
   korzynski.cft.edu.pl/gr-course.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

  • Kerr solution continued

Topics

This is a rather ambitious outline, we may not be able to cover all of the material from the last few topics.

1. Introduction:
   • why was GR developed and why is it important
   • historical introduction
     
     
2. 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
     
     
3. 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
     
     
4. Einstein equations: how matter curves spacetime
   • covariant equations. Coordinate system independence
   • $\Lambda$ - cosmological constant
   • stress-energy tensor
     
     
5. Newtonian approximation
   • $1/r^2$ law, gravitational light bending
     
     
6. 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.
     
     
7. 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
     
     
8. 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
     
     
9. Exact solutions: FLRW metric
   • FLRW ansatz, scale factor, Friedmann equations
   • closed, open, flat solutions
   • Hubble diagram, distance measures
     
     
10. Gravitational lensing
    • Lensing by a static Newtonian potential, thin lens approximation
    • Convergence, shear
    • Strong lensing, caustics, multiple imaging, Einstein arcs
      
      
11. Tests of GR, classical and new
    • Rebka-Snyder
    • Solar-System based tests
    • Gravity Probe A, B
      
      
12. Perhaps: 3+1 splitting, ADM formalism