The course provides specific advanced knowledge, both, technical and conceptual, in theoretical physics, as detailed below.
D1. The students will become proficient in advanced techniques commonly used in the context of generally covariant theories, and they will learn how to apply them to Einstein's theory of the gravitational field, but, especially, to the generalized gravity theories that are actively discussed in the current literature. Moreover, the students will develop a deep understanding of the fundamental physical properties of the gravitational field, with the aim to apply such an understanding to open, advanced problems, when the role of the gravitational filed cannot be neglected.

D2. The students will gain the ability to apply general covariant theories to concrete open problems in astrophysics, cosmology, and theoretical physics, mastering the technical and conceptual complications that inevitably arise in this field.

D3. The students will acquire the ability to autonomously identify approriate technical tools and methods to solve open problems in the context of general covariant theories. Moreover, they will acquire the independence for a critical and comaprative analysis of the quality of the scientific production i nthis field, by consolidating their understanding of the fundamental principles at the heart of still open problems.

D4. The students will learn the terminology and the appropriate way to discuss, in a mathematically rigorous way, concepts and problems in generally covariant theories, by paying particular attention to the technical and conceptual subtleties that arise in the context of generalized theories of gravity. Moreover, they will be hone the ability of effective communication in other research fields related to the content of the course, both theoretical and applied (mostly, astrophysics and cosmology).

D5. The students will be given skills and tools needed to independently learn new methods and techniques, especially in connection open problems (both, theoretical and applied, especially in astrophysics and cosmology) related to the concepts discussed in the course. Additionally, thanks to the criteria and requirements applied to midterms and final, the course will stimulate the individual initiative and motivation of students to face more challenging topics and problems, which are connected with current research in gravity, and extend consistently the content of the course.

Basics of general relativity, of the related mathematical tools, and of their geometrical interpretation: in particular, Einstein equations, and their structure and physical meaning.

Elements of global analysis; symmetries in generally covariant theories; variational formulation of generally covariant theories (including the Hamiltonian formulation); applications of generally covariant theories to open problems in theoretical physics, astrophysics, and cosmology.

Lecture notes

a review of the stress-energy tensor in generally covariant theories; local and global conservation laws; symmetries and Killing vector fields; coordinate systems adapted to symmetries of spacetime; stationary gravitational field; static gravitational field; review of fundamental concepts about the electromagnetic field; derivation of the Reissner-Nordstroem solution; "foundations of relativistic magneto-hydrodynamics"; maximal extension of the Reissner-Nordstroem solution; general procedure for the maximal extension of timelike two surfaces; rigorous derivation of the most general, spherically symmetric line element; a short introduction to generalized theories of gravity and to motivations for looking for more general Lagrangians for the gravitational field; a first order variational principle for gravity; basics of causal structure: motivations, curves and their characters, chronological and causal past and future, strongly causal spacetimes, achronal sets, edge of an achronal set, domains of dependence, Cauchy surface, partial Cauchy surface, globally hyperbolic spacetimes, asymptotically empty and simple spacetimes, weakly asymptotically empty and simple spacetimes, strongly future asymptotically predictable spacetimes, properties of strongly future asymptotically predictable spacetimes (with examples), black holes and event horizons; ADM formulation: motivations and importance; 3+1 splitting of space-time; 3+1 splitting in terms of spacelike hypersurfaces (lapse and shift functions and decomposition of the four dimensional metric); relation between 4-geometry and 3-geometry covariant derivatives; extrinsic curvature and derivation of the Gauss-Weingarten equations; 3+1 decomposition of the second order, covariant derivatives; anti-symmetric tensor and symbol on pseudo-Riemannian manifolds (proof of the basic relationship with the totally anti-symmetric delta symbol); Einstein tensor as the contracted double dual of the Riemann tensor; 3+1 decomposition of the Riemann tensor; 3+1 decomposition of Einstein equations with one 'temporal' index; momenta conjugated to the 3-metric, super-Hamiltonian, super-Momentum and their properties; Hamiltonian formulation of General Relativity and an introduction to the importance of this formulation in the context of gravity quantization; a class of uniformly accelerated observers in Minkowski spacetime and their properties; Rindler line element; "beyond general relativity: an introduction"; basic ideas about applications to open problems in theoretical physics, astrophysics, and cosmology.

Lectures and assignements.

Additional information can be found starting from the following webpage: https://users.dimi.uniud.it/~stefano.ansoldi

Mid-term and final assignments to be solved individually in written form by the students and to be then discussed at the blackboard. The assignments may be about subjects not fully covered during the course, for the solution of which the students will have to use methods and techniques explained during the lectures. General scoring criteria: 18-22: the student shows a sufficient to fair understanding of the technical aspects, together with the ability to apply them to not previously seen problems, and does not shows major gaps in the conceptual understanding of their physical interpretation; 23-27: the student shows a consistently good to very good understanding of, both, technical and conceptual aspects, and the ability to consistently use them in not previously seen problems; 28-30 with honors: the student shows an optimal to excellent understanding of technical and conceptual aspects, and can apply them with a precise physical interpretation to previously unseen problems.