Relativistic physics I

Topics of student seminar talks:

1. Parallel transport (derived in a different way than in the first semester)
   • Resources: [Kuchař, section II.4]
   • Presenting:
   • Exam requirements: examined, but not in full detail
2. Equivalent criteria for space-time flatness
   • Resources: [Kuchař, section II.6]
   • Presenting:
   • Exam requirements: examined almost in detail (tricks in computation of "the integral" are not compulsory)
3. Electro-geodesics in the Kerr-Newman space-time, Carter equations
   • Resources: [GTR, section 17.3]
   • Presenting:
   • Exam requirements: examined in semi-detail (need not learn the metric by heart)
4. Pericentre shift, light bending
   • Resources: [GTR, section 17.1; or also Dvořák]
   • Presenting:
   • Exam requirements: examined almost in detail (final tricks are not compulsory)
5. Uniqueness of the Riemann tensor
   • Resources: [Kuchař, section II.5.8]
   • Presenting:
   • Exam requirements: understanding required, without details (transformations)
6. Angular momentum (spin) and Fermi-Walker transport
   • Resources: [Bičák, Rudenko, sections 1.5, 1.6, 2.2; also GTR, chapter 18 (up to 18.1)]
   • Presenting:
   • Exam requirements: properties of the FW transport in detail, spin-behaviour derivation is not compulsory
7. Composition of Lorentz transformations, boosts and Thomas precession
   • Resources: [GTR, section 18.2]
   • Presenting:
   • Exam requirements: you should know "in principle", examined without details
8. Wave-fronts in field theories
   • Resources: [GTR, section 23.1; also Bičák, Rudenko, section 4.1]
   • Presenting:
   • Exam requirements: may be examined in semi-detail
9. Linearized theory of gravitation
   • Resources: [GTR, sections 22.1-22.4, or Bičák, Rudenko, section 3.1]
   • Presenting:
   • Exam requirements: examined in detail
10. Plane waves in the linearized gravity
    • Resources: [GTR, sections 22.5 and 22.6, or Bičák, Rudenko, sections 3.3 and 3.4]
    • Presenting:
    • Exam requirements: examined in detail
11. Asymptotic form of the field of an isolated source
    • Resources: [Bičák, Rudenko, section 3.2]
    • Presenting:
    • Exam requirements: should just know what it is about
12. Example of a gravitational wave in an exact theory (sandwich wave)
    • Resources: [GTR, sections 23.2 and 23.3; also Bičák, Rudenko, section 4.2]
    • Presenting:
    • Exam requirements: may be examined in semi-detail
13. Thermodynamics, hydrodynamics, electrodynamics, geometrical optics, and kinetic theory
    • Resources: [MTW, section 22]
    • Presenting:
    • Exam requirements: should know basic equations in GR setting

References:

GTR lecture notes
Kuchař: Základy obecné teorie relativity (scanned DJVU)
Bičák, Rudenko: Teorie relativity a gravitační vlny (scanned DJVU)
Misner, Thorne & Wheeler: Gravitation (ask us for a pdf if needed)

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

• 5th October morning: Invariant and coordinate properties of Schwarzschild solution. Notions of radial distance and embeddings.
• 5th October afternoon: Isotropic radial coordinate in Schwarzschild. Nature of the singularities at the horizon and r=0. Question of geodesic maximality. Goals for the analytical extension of Schwarzschild.
• 12th October morning: Analytical extension of Schwarzshild. Kruskal and Penrose-Carter diagrams. "Dynamics" of Schwarzschild spacetime in Kruskal time.
• 12th October afternoon: Tensor densities part 1.
• 12th October afternoon: Tensor densities part 2. Identities for metric determinant derivatives and covariant divergence.
• 19th October morning: Reissner-Nordström solution and its properties. Analytical extension, Kruskal and Penrose-Carter diagrams.
• 19th October afternoon: Maximal extension of Reissner-Nordström in dependence on charge. Cauchy horizons and physicality considerations.
• 26th October: Kerr solution of the Einstein equations. Machian ideas and the Lense-Thirring effect. Kerr metric in Kerr-Schild and Boyer-Lindquist coordinates. Static limit, horizon, curvature singularity and the nonsphericity of important surfaces. Rotational dragging and circular orbits.
• 2th November: Analytical extension of the Kerr metric and its dependence on angular momentum. Kerr-Newman solution and its properties. Gravitational collapse and black holes. Physical formation of black holes. Oppenheimer-Snyder collapse. Hoop and censorship conjectures.
• 9th November: Black hole uniqueness theorems. Price theorem. Black hole thermodynamics and the case of the Kerr-Newman black hole. Carter equations for Kerr-Newman electrogeodesics.
• 16th November: Horizon area in black hole mergers. Temperature of black holes and Hawking radiation. Energy extraction from black holes: Penrose and Blandford-Znajek processes, superradiance. History of opinions on black hole existence. Spherically symmetric stellar equilibria. Evolution of stellar composition and physical mechanism of equilibrium. TOV metric.
• 23th November: Equations of relativistic stellar equilibria (TOV equations). Different notions of stellar mass. Equations of thermal equilibrium. Convection stability.
• 30th November: Radial oscillations of stars. Lagrangian and Eulerian perturbations. Separation of the equations and the resulting Sturm-Liouville problem.
• 7th December: Variational derivation of the TOV equations. Final stages of stellar evolution. Degenerate Fermi gases. White dwarfs and neutron stars and the mass-radius curve.
• 14th December: Variational derivation of the Einstein equations from the Einstein-Hilbert action. Conservation of stress-energy from diffeomorphism invariance. Palatini approach to the variational derivation.
• 4th January: Derivation of the Chandrasekhar and Landau Oppenheimer-Volkoff limits for masses of white dwarfs and neutron stars.

• Boosts & Thomas precession
• Lie derivative & Killing vectors
• Parallel transport
• Spin and Fermi-Walker transport
• Uniqueness of the Riemann tensor
• Equivalent criteria of flatness
• Apsidal precession & light bending
• Motion in Kerr-Newman (Carter equations)
• Linearized theory of gravitation
• Plane waves in linearized theory
• Asymptotic form of an isolated-system field (multipole expansion)
• Wave-fronts in field theories
• Xmas climbing
• Sandwich wave in exact theory
• Non-gravitational physics within GR