Relativistic physics I


prof. RNDr. Jiří Bičák, DrSc., dr.h.c.,    doc. RNDr. Oldřich Semerák, DSc.

fall term 2021, 4/2 exam/credit

Short syllabus:

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.

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).

Time, space, rules:

Thursday 13:10-18:00, with a few breaks, at T1 lecture room (MFF Trója).

In 2021/22, the course is being taught live and in Czech (videos from 2020/21 English run are linked below).

Student seminar talks: one talk per year (per 2 semesters of Relativistic physics) is generally required for credit.
Examinantion in Czech as well as in English can be chosen. Please, contact me by email when you plan to come for exam (

Topics of student seminar talks:

  • (also required at exam, but not all in detail -- see below)
  •   1) Composition of Lorentz transformations, boosts and Thomas precession [Votruba, sections IV.7 and IV.8; also GTR, section 18.2]
          ... Kateřina Mladá (07-10)
          ... you should know "in principle", examined without details
      2) Parallel transport (derived in a different way than in the first semester) [Kuchař, section II.4]
          ... Jan Došek (14-10)
          ... examined, but not in full detail
      3) Angular momentum (spin) and the Fermi-Walker transport [Bičák, Rudenko, sections 1.5, 1.6, 2.2; also GTR, chapter 18 (up to 18.1)]
          ... Jan Kříž, Róbert Jurčo (04-11)
          ... properties of the FW transport in detail, spin-behaviour derivation is not compulsory
      4) Uniqueness of the Riemann tensor [Kuchař, section II.5.8]
          ... Eliška Klimešová (21-10)
          ... understanding required, without details (transformations)
      5) Equivalent criteria for space-time flatness [Kuchař, section II.6]
          ... Alžběta Maleňáková (11-11)
          ... examined almost in detail (tricks in computation of "the integral" are not compulsory)
      6) Pericentre shift, light bending [GTR, section 17.1; or also Dvořák]
          ... Jan Šenk, Jana Švrčková (18-11)
          ... examined almost in detail (final tricks are not compulsory)
      7) Electro-geodesics in the Kerr-Newman space-time, Carter equations [GTR, section 17.3]
          ... Barbora Adamcová (25-11)
          ... examined in semi-detail (need not learn the metric by heart)
      8) Linearized theory of gravitation [Bičák, Rudenko, section 3.1; or GTR, sections 22.1-22.4]
          ... Michael Vávra (02-12)
          ... examined in detail
      9) Plane waves in the linearized gravity [Bičák, Rudenko, sections 3.3 and 3.4; or GTR, sections 22.5 and 22.6]
          ... Monika Dubová (02-12)
          ... examined in detail
    10) Asymptotic form of the field of an isolated source [Bičák, Rudenko, section 3.2]
          ... Tomáš Faikl, Kamil Mudruňka, Milan Vrána (09-12)
          ... should just know what it is about
    11) Wave-fronts in field theories [Bičák, Rudenko, section 4.1]
          ... Jáchym Baláž (16-12)
          ... may be examined in semi-detail
    12) Example of a gravitational wave in an exact theory (sandwich wave) [Bičák, Rudenko, section 4.2]
          ... David Kramár, Vít Beneš (16-12)
          ... may be examined in semi-detail
    13) Thermodynamics, hydrodynamics, electrodynamics, geometrical optics, and kinetic theory [MTW, section 22]
          ... Martin Crhán (06-01)
          ... 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.)

    Links to recordings from 2020/21 English run (mp4):

    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

  • Special message from 25th January (2021):
    I made a mistake in one of the lectures (second part of the semester). I mean quite a serious one, not just a wrong index :-(. Who discovers and explains it will be rewarded somehow (don't know how yet).