Physics

GENERAL OBJECTIVES: Aim of the course is to introduce the basic notions of the modern theory of gravity, and of its more important conceptual and astrophysical implications. At the end of the course the student should: 1) have acquired the instruments of differential geometry which allow to formulate Einstein's equations and derive its predictions. 2) Have understood what is the role of the equivalence principle between gravitational and inertial mass in the formulation of the theory, and why the gravitational field modifies the spacetime geometry. 3) Have understood how to use the symmetries of a physical problem to simplify Einstein's equations and find solutions. 4) Be able to derive the solution describing the gravitational field external to a non rotating, spherically symmetric body (the Schwarzschild solution), and to show that this solution can also represent a non rotating black hole. 5) Have understood how some of the main predictions of General Relativity can be obtained by solving the geodesic equations, which describe the motion of free particles in a gravitational field. 6) Have understood how to solve Einstein's equations in the weak field limit, to show that spacetime perturbations propagate as gravitational waves. Therefore, at the end of the course the student should: 1) be able to compute how vectors, one-forms and tensors transform under a coordinate transformation; to compute the covariant derivative of these geometrical objects and to solve exercises which involve these operations in tensor equations. 2) Be able to compute how does a vector change when parallely transported along a path in curved spacetime, and to derive the curvature tensor using this operation. 3) Be able to derive Einstein's equations. 4) Be able to derive and interpret some of the most interesting predictions of General Relativity: the gravitational redshift, light deflection near massive bodies, precession of Mercury perihelion, existence of gravitational waves. This course introduces the fundamental concept of a curved spacetime due to the existence of a gravitational field, and discusses the more important aspects of the scientific revolution introduced by Einstein's theory. As such, it is a basic course for the laurea magistrale in Astronomy and Astrophysics, and it is also a matter which should be part of the cultural background of a modern physicist. SPECIFIC OBJECTIVES: A - Knowledge and understanding OF 1) Know the basics of differential geometry OF 2) Know the basics of General Relativity and its most relevant concepts, including that of a black hole and gravitational waves OF 3) Know and interpret the observational applications of the theo B - Application skills OF 4) Be able to perform analytical calculations of differential geometry OF 5) Knowing how to derive Einstein's equations for the gravitational field OF 6) Knowing how to derive and interpret some of the most important effects predicted by General Relativity OF 7) Knowing how to calculate the geodetic motion in the spacetime of a black hole C - Autonomy of judgment OF 8) To fully understand the concept of curved spacetime, change of coordinates, and the consequences of the principles of Equivalence and General Covariance D - Communication skills OF 9) Knowing how to present in written and oral form the main derivations concerning formulas and theorems of differential geometry OF 10) Knowing how to present in written and oral form the main derivations concerning General Relativity: Einstein equations, geodesic motion, metrics of a black hole, gravitational waves E - Ability to learn OF 11) Have the ability to apply the knowledge of the course to understand and derive more advanced topics