# GENERAL RELATIVITY

**Academic Year 2022/2023**- Teacher:

**ALFIO MAURIZIO BONANNO**

## Expected Learning Outcomes

K*nowledge and understanding*

The student must familiarize with modern research in gravity, and therefore should be able to master basic concepts in BH theory (Event Horizon, Apparent Horizon, Ergoregion, BH entropy, Penrose diagrams).

*Applying knowledge and understanding*

The student should also be able to contextualize the main problems of modern General Relativity in the much broad context of Extragalactic Astrophysics and Theoretical Physics.

*Making judgements*

Ability to decide how to test different gravitational theories with experimental data

*Learning skills*

The student will learn vector calculus in cuved manifolds and Riemannian geometry and familiarize with the astrophysical and cosmological implications of exact solutions of Einstein's field equations.

## Course Structure

The course is divided in two parts. In the first part the student will lear basic concepts of differential geometry. In the second part the mathematical machinery is

applied to develop to general theory of relativity and its applications. Should the circumstances require online or blended teaching, appropriate modifications to what is hereby stated may be introduced, in order to achieve the main objectives of the course.

## Attendance of Lessons

Attending the classes is strongly encouraged## Detailed Course Content

Equivalence principle and its physical implications. Topological Manifolds. Differentiable manifolds. Vector fields and tangent bundle. Connections. Torsion tensor and Riemann tensor. Jacobi geodetic deviation equation. Geodesics. Geometrical formulation of the newtonian gravity. Lorentzian Manifolds. Killing equation and conservation laws in GR. LL hypersurfaces, generators and dynamics. Physical motivation for the Einstein’s field equations. 2+2 formalism and solutions: RN, Schwarzschild, de Sitter. Event Horizon in static spacetimes. Birkhoff theorem. Eddington-Finkelstein coordinates. Kruskal coordinates. Schwarzschild maximal extension. Penrose conformal mapping. RN maximal extension. Geodesic motion in RN. Thorne’s wormhole. Matter matters: TOV equations. Stationary spacetimes. ZAMO observers. Event horizon in stationary spacetimes. Kerr solution in Boyer-Lindquist coordinates. Ergoregion and Event Horizon. Penrose’s extraction mechanism. Kerr maximal extension. No-Hair theorem. Dynamics of LL hypersurfaces in generic spacetimes: Raychaudhuri equation. BH rigidity. The zeroth Law of BH dynamics. Definition of EH in general spacetime. Non stationary spacetimes: accretion and evaporation. Teleological properties of the EH. Hawking’s area theorem and second Law of BH dynamics. First law of BH (thermo) dynamics. QFT in curved spacetime. Quantum creation of particles in an expanding universe. Thermal emission from a BH. Hamiltonian formalism and WdW equation.

## Textbook Information

E. Poisson, A Relativist's Toolkit, The Mathematics of Black-Hole Mechanics, Cambridge UP

R. Wald, General Relativity, Chicago UP

**Introducing** Eintein's **Relativity**: from tensors to gravitational waves. **Ray d'Inverno**. Oxford UP

S. Weinberg, Gravitation and Cosmology, New York: John Wiley and Sons (1972).

## Course Planning

Subjects | Text References | |

1 | Equivalence principle and physical foundations | S. Weinberg, Gravitation and Cosmology, New York: John Wiley and Sons (1972). |

2 | Mathematical tools | Introducing Eintein's Relativity: from tensors to gravitational waves. Ray d'Inverno. Oxford UP |

3 | Exact solutions | R. Wald, General Relativity, Chicago UP |

4 | BH theory | E. Poisson, A Relativist's Toolkit, The Mathematics of Black-Hole Mechanics, Cambridge UP |