General Relativity

summer term 2023
Lecturer: Maturi
Link to LSF
125 participants


The black hole Gargantua from the film Interstellar

Priv.-Doz. Dr. Matteo Maturi (ITA/ZAH, ITP)
Head tutor: Nadine Nußbaumer (

Space and time:

  • Summer semester, 2023
  • From April 17th to July 19th
  • Monday 09:15-11:00 (INF308/HS 2)
  • Wednesday 09:15-11:00 (INF308/HS 2)

Gravity is the weakest of all forces in nature and yet it shapes our universe on all scales, from humans bind to Hearth up to defining the dynamics of the entire universe. The lectures will open the path in the understanding on gravity as described by the theory of General Relativity. The students will learn about the properties of flat and curved space-times, the behaviour of massive and massless particles in presence of gravity, black holes a different kinds, gravitational waves, the dynamic of the universe, etc...

PTP2, PTP3 is helpful but not mandatory.

The lectures will be held in person and will be recorded. Lecture notes about what will be present at the blackboard and additional material will be provided. The notes will be complementary to other material/books and present full derivations. I will start slow to build a solid background. The lectures and exercise classes are held in English and will be recorded.

To get credit points for the lectures it is necessary to enroll through this website.
If you are still missing the credentials (immatriculation, ID number, ...) and can not log-in, ask the head tutor to be enroll manually.

Material and exercises:
You find everything in this page.


Question session:

Wednesday, July 26th, 2023, from 3pm to 5pm; room R 068, Philosophenweg 12

Written. Thursday, July 27th, 2023 from 2pm to 5pm; Lecture halls INF 227 HS1 and HS2


Exam review: 

Tuesday, August 1st, 2023, from 2pm to 4pm; Lecture hall nHS, Philosophenweg 12

Retake exam:
Written. Wednesday, October 11th, 2023 from 2pm to 5pm; Lecture halls gHS, Philosophenweg 12


Retake exam review:

Friday, October 13th, from 2pm to 4pm; Seminar room (SR) on the 3rd floor, Philosophenweg 12

Admission to the exam:
Attend at least 50% of the tutorials AND gain 3 points by: Presenting an exercise (1 point) or actively participating in the discussion during the tutorials (max 1 point per tutorial). If < 3 points are acquired, it is required to hand in 3 full exercise sheets that will be graded.

Lehre, Studium und Forschung:
Lecture Token MKTP3.1 (8CP): LSF

First Ever ray tracing simulation of a black hole (1979 © Jean-Pierre Luminet/CNRS Phototheque). This is the computer used to realize it: IBM7040. At the time programs was loaded on the computer with a punch card.


First ever 'picture' of a shadow of a black hole (EHT collaboration)


First ever direct detectin of a gravitational wave (Ligo and Virgo collaboration)
GW150914 GWsignalsPlots

Index of the lectures

In blue the parts already covered during the lectures

PART 1: Intro

Newtonian gravity:
    1. Newtonian gravity: idea and problems
The equivalence principle:
    1. The equivalence principle,  gravity ↔ non inertial frames
    2. Predictions: gravitational redshift and lensing

More then Newtonian gravity
    2. The most general classical non-relativistic gravitational field
    3. The link between Φ α r
-1 and the Euclidean space

PART 2: flat space-time

Special relativity: Minkowski space-time
    1. Special relativity, the need, the idea and the the Lorentz transforms
    2. The Lorentz geometry and the Lorentz group
    3. Groups, Lie-groups, Lie algebra applied to the Lorentz transformation
    4. Relativistic mechanics

Attempting a relativistic linear theory of gravity
    1. Dynamic of a particle in the field: perihelion shift problem
    2. Relativistic linear theory: dynamic of the field

Approaching general relativity: gravity ↔ non inertial frames
    1. Recalling the equivalence principle
    2. Non-inertial frames and the equivalence principle: example, a rotating frame
    3. Connection between gravity and the metric of space-time

PART 3: curved space-time

Curved space-time
    1. Getting formal: scalars, vectors, one-forms and tensors
    2. Manifolds, geometry, Riemanian geometry
    3. The tangent space
    4. Connection and covariant derivatives
    5. Torsion

    6. Link between the connection and the metric tensor
    7. Parallel transport and the geodesic equations
    8. Curvature Riemann tensor and Einstein tensor
    9. Geodesic deviation equation

    10. Conserved quantities, killing vectors and Lie derivatives
    11. Strong-equivalence principle; electrodynamics in curved space-time

Field equations
    1. The source of gravity: energy momentum tensor
    2. Einstein field equations, Einstein's approach

    3. Einstein field equations, Hilbert's approach
    4. Is there one single theory of gravity?
    5. Linearized field equations
    6. Nearly Newtonian regime and gravitomagnetic field

PART4: applications

Gravitational waves
    1. Gravitational waves
    2. Generation of gravitational waves

Spherically symmetric systems
    1. Schwartzshild metric
    2. Schwartzshild black-holes
    4. Kruskal coordinates

    5. Reissner-Nordström (electrically charged black-holes)

Axially symmetric systems
    1. Kerr metric (rotating spherical objects)

Cosmology: isotropic and homogeneous universe
    1. Friedmann(-Lamaitre)-Robertson-Walker metric (FLRW)
    2. The cosmological constant and dark energy


Practice groups

  • Group 01 (Adrian Schlosser)
    12 participants
    INF 227 / SR 2.404, Tue 09:15 - 11:00
  • Group 02 (Shayan Hemmatyar)
    12 participants
    INF 227 / SR 2.403, Tue 11:15 - 13:00
  • Group 03 (Yannic Pietschke)
    13 participants
    Philos.-weg 12 / R 068, Wed 14:15 - 16:00
  • Group 04 (Selin √úst√ľndag)
    16 participants
    Philos.-weg 12 / kHS, Thu 09:15 - 11:00
  • Group 05 (Carlos Pastor Marcos)
    11 participants
    Philos.-weg 12 / kHS, Thu 11:15 - 13:00
  • Group 06 (Jiahan Shi)
    10 participants
    Philos.-weg 12 / R 060, Fri 09:15 - 11:00
  • Group 07 (Shaun Fell)
    10 participants
    Philos.-weg 12 / R 060, Fri 14:15 - 16:00
  • Group 08 (Youssef Aly)
    11 participants
    Philos.-weg 12 / R 070, Tue 9:15 - 11:00
  • Group 09 (Felix Heinze)
    11 participants
    Philos.-weg 12 / R 056, Fri 14:15 - 16:00
  • Group 10 (Quentin Coppee)
    10 participants
    Philos.-weg 12 / R 068, Wed 16:15 - 18:00
  • Group 11 (Vito Aberham)
    9 participants
    INF 501 / FP (R. 102), Fri 9:15 - 11:00
General Relativity
summer term 2023
Link zum LSF
125 participants