General Theory of Relativity

Enrollment into the program.
Positive result from colloquiums or written exam is necessary to enter the oral exam.

Newton law of motion and its symmetries, Galilean transformations and Galilean group.
Equations of EM field and their symmetries, Lorentz transformation and Poincare group.
Classical mechanics with Poincare symmetry - special relativity.
Equations of source free EM field and its symmetries - Poincare invariance.
Motion of charged particles in EM field, gradient invariance and conservation of charge.
Lagrange function for particle motion in weak gravitational field, gauge invariance.
Equations of (weak) source free gravitational field.
Gauge transformations and non-inertial systems (uniformly accelerating system, uniformly rotating system, meaning of inertial forces).
Stress-energy tensor as the source of gravitational field, gravitational constant, gauge invariance and continuity equation.
Stress-energy tensor for ideal gas, meaning of its components. Equation of state in rest frame. Continuity equations as equations of motion for continuum. Generalization to viscous fluids and Navier-Stokes equation.
Motion of charged particles and stress-energy tensor of EM field.
(Pseudo) stress-energy tensor of gravitational field, why the theory becomes non-linear. Gravitational potential energy and energy of gravitational waves.
Classical tests of general theory of relativity and solutions of equations of motion for particles in the field of "point" mass.
Non-linear Einstein equations, stationary spherically symmetric solution and black holes.

A. Čadež: Teorija gravitacije, DMFA Založništvo, 2011,
R. D’Inverno: Introducing Einstein’s Relativity, Clarendon Press, 1992,
Gron, Oyvind, Hervik, Sigbjorn: Einstein’s General Theory of relativity With Modern Applications in Cosmology, Springer, 2007.

Objectives:
The objective of the course is discussion of two classical field theories: electromagnetism and gravitation. Discussion of symmetry consequences and non-linear theory of gravity.

Competences:
Knowledge and understanding of uniform description of classical fields of electromagnetism and gravity. Knowledge and understanding of fundamental principles of general theory of relativity.

Knowledge and understanding:
Unification of electrodynamics and classical mechanics in the Minkowski space. Equations of weak gravitational field and uniform description of classical fields of electromagnetism and gravity. Basic meaning of stress-energy tensor and non-linearity of gravity.

Application:
Mutual framework for unified description of both fundamental classical fields.

Reflection:
Properties of classical fields and their coupling with sources and stress-energy tensor illustrate fundamental symmetries of laws of nature.

Transferable skills:
Understanding and capability to solve equations of classical fields in various limits. Understanding the meaning of symmetries in field theories and use of differential geometry.

Lectures, exercises, and consultations.

2 colloquiums in exercises or final written exam
2 tests in theory or final theory exam
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

prof. J. Novak: [1] M. Marques, M. Oertel, M. Hempel, J. Novak : New temperature dependent hyperonic equation of state: Application to rotating neutron star models and I − Q relations, Physical Review C 96, 045806 (2017). preprint: arXiv:1706.02913, [2] D. Chatterjee, A.F. Fantina, N. Chamel, J. Novak, M. Oertel : On the maximum mass of magnetized white dwarfs, Monthly Notices of the Royal Astronomical Society 469, 95-109 (2017). preprint: arXiv:1610.03987. [3] A. Sourie, N. Chamel, J. Novak, M. Oertel : Global numerical simulations of the rise of vortex-mediated pulsar glitches in full general relativity, Monthly Notices of the Royal Astronomical Society 464, 4641-4657 (2017). preprint: arXiv:1607.08213. [4] A. Sourie, M. Oertel, J. Novak : Numerical models for stationary superfluid neutron stars in general relativity with realistic equations of state, Physical Review D 93, 083004 (2016). preprint: arXiv:1602.06228. [5] A. Le Tiec, J. Novak : Theory of gravitational waves in An Overview of Gravitational Waves: Theory, Sources and Detection editors G. Auger and E. Plagnol, World Scientific Publishing (2017) preprint: arXiv:1607.04202.