Selected Topics of General Relativity

Annotation

This is a slightly advanced  introductory course in General Relativity. It represents the approach to the General Relativity based on the equilibrium combination of mathematics and physics.  The main topics to be considered: variational principle, gravitational waves, experimental tests of GR, black holes and asymptotic symmetries, elements of cosmology.

Course plan

1. Gravity in Newtonian mechanics. Gravitational field of a distributed matter source. Relativity principle of Galileo. Equality of inertial and gravitational masses. Black hole in Newtonian mechanics. Cosmology in Newtonian mechanics.
2. A reminder of the elements of the Riemannian geometry.
3. Derivation of the Einstein equations. The well-defined variational principle in General Relativity. Boundary conditions and the York-Gibbons-Hawking boundary term.
4. Linearized gravity. Gravitational waves. Generation of gravitational waves.  
5. The quadrupole formula. The rate of energy loss formula. Applications for the energy loss in binary systems.
6. Some experimental tests of General Relativity. The deflection of light. The Shapiro time delay effect. The perihelion shift of Mercury.
7. Black holes. 
8. Asymptotically flat spacetimes in four dimensions: Bondi–Metzner–Sachs (BMS) group and gravitational memory effects.
9. Cosmological principle. Standard cosmological model of Friedmann-Robertson-Walker. Dark energy and dark matter.
10.  Simplified analysis of the cosmological perturbations. Formation of galaxies.

Literature

1. Carroll, Sean M.Spacetime and Geometry. Pearson/Addison Wesley, 2004.
2. Wald, Robert M. General Relativity. University of Chicago Press, 1984.
3. Poisson, Eric. A Relativist's Toolkit: The Mathematics of Black-Hole Mechanics. Cambridge University Press, 1973.
4. Will, Clifford. The confrontation between General Relativity and Experiment, 2014, https://arxiv.org/abs/1403.7377.