This course will provide knowledge, both, conceptual and technical, that is a prerequisite to reach an advanced preparation in theoretical physics, as detailed below.
D1. The students will become proficient in diverse differential geometry methods and techniques of common use in physics. They will also learn how to apply them to Einstein's theory of the gravitational field. The students will also acquire a deep understanding of the fundamental properties of the gravitational field as described by general relativity.

D2. The students will acquire the ability to apply abstract mathematical concepts to physical theories and models, in particular, they will understand the physical meaning of abstract models in connection with the theory of the gravitational field and general relativity.

D3. The students will develop the ability to independently identify appropriate technical/mathematical tools to rigorously and clearly model physical problems (mostly, in connection with general relativity and other gravity theories, but, possibly, also in other areas of theoretical physics).

D4. The students will develop the terminology and the correct approach needed to express physical ideas and concepts in a rigorous mathematical way, mostly in the general context of gravity theory, and, more specifically, in general relativity.

D5. The students will be given tools and skills that are needed to independently face the study of diverse gravitational theories, which are more general than the one discussed during the course. They will also be able to apply the same, or similar, skills, in other areas of theoretical physics.

Advanced calculus and linear algebra; basic ideas of differential geometry; special relativity; analytical mechanics and classical (non gravitational) field theory.

Elementary concepts in differential geometry; connections on manifolds; Riemmanian and pseudo-Riemannian manifolds. Foundations of general relativity and their mathematical formulation. Stress-energy tensor. Einstein equations.

Lecture notes.

differentiable manifolds, differentiable functions/maps over manifolds; tangent vectors at a point as derivations; the tangent space; holonomic basis; cotangent space; tensors' spaces; tangent, cotangent and tensor bundles as differentiable manifolds; bundle sections; connections on
manifolds (covariant derivative); components of the connection in a given basis of the tangent space; Christoffel symbols; differentiable curves on manifolds; integral curves of vector fields; the flux associated to a vector field; Lie derivative; Lie brackets and their properties; symmetric
connections; characterization of symmetric connections (theorem); covariant derivative along a curve and parallel transport; curvature tensors (Riemann and Ricci tensors) and their geometric interpretation; component expression of the covariant derivative and of the "second" covariant derivative; extension of the covariant derivative to generic tensor fields; components of the covariant derivative of a 1-form; components of the
covariant derivative of a generic tensor; components of the Riemann tensor; symmetry properties of the Riemann tensor; auto-parallel curves on manifolds; the exponential map; exponential map as a local diffeomorphism; a review about scalar products on vector spaces (especially
pseudo-Euclidean ones); pseudo-Euclidean scalar product on manifolds (metric); compatibility of connection and metric: definition and geometric interpretation; characterization of compatibility for symmetric connections; metric connection, uniqueness and connection coefficients in terms of the metric components; the concepts of space and time from Galilean to special relativistic physics; basic principles of general relativity: Einstein elevator "Gedankenexperiment", the equivalence principle (strong and weak form), general covariance; representation of the basic principle of general relativity in terms of manifold concepts; motion of free test particles and autoparallel curves/geodesics; the equation for autoparallel curves/geodesics written in terms of a generic parameter; non-relativistic limit of the geodesics equation, physical interpretation of the connection coefficients and geometrization of the concept of force; higher order variational principles, constrained systems and some examples of higher order Lagrangians that do not result in higher order equations of motion; general ideas about gauge invariance, the Gauss constraints, and the order of a differential equation; variational principles: from finite to infinite degrees of freedom; the field equations for a (collection of) scalar field(s) from a variational principle in Minkowski space; from energy in a classical Lagrangian system, to the stress-energy tensor in special relativity; interpretation of the components of the stress-energy tensor; energy density, and energy flow; momentum and momentum flow (pressures); differential and integral conservation laws in special relativity; conservation of the stress-energy tensor; identification of the simplest Lagrangian for the metric field; derivation of the field equations for a pure metric theory (without other fields/sources): Einstein equations in vacuo; the structure of Einstein equations in vacuo: identification of dynamical degrees of freedom and gauge degrees of freedom; the non-relativistic limit of Einstein equations in vacuo: interpretation of the metric field as a generalization of the Newtonian gravitational potential; coupling matter sources to the gravitational field in the Lagrangian formulation; minimal coupling; covariant stress-energy tensor and Einstein equations in presence of sources; universality of the gravitational interaction; infinitesimal coordinate transformations and covariant conservation of the stress-energy tensor.

Lectures and assignments.

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

Written final examination consisting i) in a short dissertation about fundamental concepts and ideas discussed in the course, and ii) one/two problems and/or one/two proofs of technical results related to the material presented during the course. General scoring criteria. 18-22: the student shows a sufficient to fair understanding of the technical aspects, 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 their physical interpretation; 28-30 with honors: the student shows an optimal to excellent understanding of technical and conceptual aspects, and their physical interpretation.