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.
Comments welcome during lecture or via email (

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 (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]
          ... Róbert Jurčo, Monika Dubová (09-12)
          ... examined in detail
    10) Asymptotic form of the field of an isolated source [Bičák, Rudenko, section 3.2]
          ... Tomáš Faikl, Kamil Mudruňka, Milan Vrána (16-12)
          ... should just know what it is about
    11) Wave-fronts in field theories [Bičák, Rudenko, section 4.1]
          ... Jáchym Baláž (06-01)
          ... 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 (will see when)
          ... may be examined in semi-detail
    13) Thermodynamics, hydrodynamics, electrodynamics, geometrical optics, and kinetic theory [MTW, section 22]
          ... Martin Crhán (will see when)
          ... 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).