More on geometry: tensor densities, covariant divergence, Fermi-Walker transport. Invariant and coordinate features of the Schwarzschild solution, analytic extension of the metric. Pericentre precession and light bending in the Schwarzschild field. Reissner-Nordström solution of the Einstein equations. Kerr and Kerr-Newman solutions of the Einstein equations, Carter equations for electro-geodesic test motion. Gravitational collapse and black holes: black-hole uniqueness theorems, formation of black holes, laws of black-hole (thermo)dynamics, extraction of energy from black holes. Relativistic theory of stellar equilibria: description of a static and spherically symmetric star, equations of stellar equilibria, radial oscillations and stability. Final stages of stellar evolution: degenerate fermion gas, white dwarfs and neutron stars; Chandrasekhar limit. Linearized theory of gravitation, plane gravitational waves. Basics of Lagrangian approach: variational derivation of Einstein equations, energy-momentum tensor and its conservation; the case with generic (torsion-free) connection.
A follow-up of the basic general-relativity course (NTMF111), mainly suitable for theoretical physicists and astrophysicists at the turn of their bachelor and master studies.
The course continues by NTMF038 – Relativistic physics II (in spring term).
Monday 14:00-16:15 and Thursday 14:00-16:15, at T1 lecture room (MFF Trója). No lessons on 17th Nov. (state holiday) and on 27th Nov. (Faculty open-door day)
In 2025/26, the course is being taught in English. Videos from 2020/21 "covid" English run are linked below.
Student seminar talks: one talk per year (per 2 semesters of Relativistic physics) is generally required for credit. A few homeworks will be assigned, they are voluntary yet their (correct) solution will be taken as a certain "plus" at the exam.
Examinantion in Czech as well as in English may be chosen.
Emails: oldrich.semerak@mff.cuni.cz, vojtech.witzany@matfyz.cuni.cz.
1) Parallel transport (derived in a different way than in the first semester)
[Kuchař, section II.4]
... Paulína Dujavová, Filip Brutovský (October 16)
... examined, but not in full detail
2) Equivalent criteria for space-time flatness
[Kuchař, section II.6]
... Vojtěch Dienstbier, Jiří Kohl (October 23)
... examined almost in detail (tricks in computation of "the integral" are not compulsory)
3) Electro-geodesics in the Kerr-Newman space-time, Carter equations
[GTR, section 17.3]
... Jan Kropáček (October 30)
... examined in semi-detail (need not learn the metric by heart)
4) Pericentre shift, light bending
[GTR, section 17.1; or also Dvořák]
... Nikita Ustinov, Vojtěch Votruba (November 6)
... examined almost in detail (final tricks are not compulsory)
5) Uniqueness of the Riemann tensor
[Kuchař, section II.5.8]
... Jakub Koňárek (November 13)
... understanding required, without details (transformations)
6) Angular momentum (spin) and Fermi-Walker transport
[Bičák, Rudenko, sections 1.5, 1.6, 2.2; also GTR, chapter 18 (up to 18.1)]
... Matěj Ruml, Vilém Horák (November 20)
... properties of the FW transport in detail, spin-behaviour derivation is not compulsory
7) Composition of Lorentz transformations, boosts and Thomas precession
[Votruba, sections IV.7 and IV.8; also GTR, section 18.2]
... Michal Šrank, Marek Miškeřík (???)
... you should know "in principle", examined without details
8) Wave-fronts in field theories
[GTR, section 23.1; also Bičák, Rudenko, section 4.1]
... Lukáš Pudil, Matěj Ptáček (December 1 and 8)
... may be examined in semi-detail
9) Linearized theory of gravitation
[GTR, sections 22.1-22.4, or Bičák, Rudenko, section 3.1]
... Šimon Húdek (December 1)
... examined in detail
10) Plane waves in the linearized gravity
[GTR, sections 22.5 and 22.6, or Bičák, Rudenko, sections 3.3 and 3.4]
... Lutz Jonas Leimenstoll, Valentina Rosa (December 4)
... examined in detail
11) Asymptotic form of the field of an isolated source
[Bičák, Rudenko, section 3.2]
... Jan Slovák (???)
... should just know what it is about
12) Example of a gravitational wave in an exact theory (sandwich wave)
[GTR, sections 23.2 and 23.3; also Bičák, Rudenko, section 4.2]
... Jonas Klimbacher (???)
... may be examined in semi-detail
13) (possibly) Thermodynamics, hydrodynamics, electrodynamics, geometrical optics, and kinetic theory
[MTW, section 22]
... (???)
... should know basic equations in GR setting
GTR: Relativistic Physics
Votruba: Základy speciální teorie relativity
Kuchař: Základy obecné teorie relativity
Bičák, Rudenko: Teorie relativity a gravitační vlny (skripta)
Dvořák: Obecná teorie relativity a moderní fyzikální obraz vesmíru (skripta)
Misner, Thorne & Wheeler: Gravitation
(Kuchař, Bičák & Rudenko, and MTW we have as files, can send you.)
5th October morning, 5th October afternoon
12th October morning, 12th October afternoon, 12th October afternoon, Boosts & Thomas precession
19th October morning, 19th October afternoon, Lie derivative & Killing vectors, and, if you did not recognize what I had on T-shirt
26th October lecture, Parallel transport
2th November lecture, Spin and Fermi-Walker transport
9th November lecture, Uniqueness of the Riemann tensor
16th November lecture, Equivalent criteria of flatness, Apsidal precession & light bending
23th November lecture, Motion in Kerr-Newman (Carter equations), and dark side of the T-shirt
30th November lecture, Linearized theory of gravitation
7th December lecture, Plane waves in linearized theory
14th December lecture, Asymptotic form of an isolated-system field (multipole expansion)
Wave-fronts in field theories,
Xmas climbing (contains upsetting scenes!),
and
"Nesem vám noviny" (Silvester-Eve edit)
singing of carols via Zoom was not recorded, but we made it, even with a decent piano accompaniment
4th January -- Chandrasekhar limit, Sandwich wave in exact theory, Non-gravitational physics within GR
