Programma

§I. Introduction: inertia and covariance

Galilean Relativity and Special Relativity. The Equivalence Principle. Motivations for general covariance. Local inertial frames.

§II. Dynamical spacetime: basics of General Relativity

Curvilinear coordinates. Vectors and tensors under general coordinate transformations. Parallel transport and Christoffel symbols. Covariant derivatives and metric compatibility. Covariantly conserved vectors and conserved vector densities. Transformation of the Christoffel symbols. Ricci's theorem. Geodesics and their Newtonian limit. Time-like, null and space-like curves. Normal coordinates and local inertial frames. Covariant derivation along curves. Riemann curvature tensor. Algebraic properties. Bianchi identities. Local inertial frames. Characterisation of flat spaces. Geodesic deviation. Einstein-Hilbert action and equations of motion. The Palatini identity. Non-dynamical sources: contracted Bianchi identity, matter energy-momentum tensor and covariant conservation. Minimal coupling to scalar fields and to Maxwell fields.

§III. Linear approximation and gravitational waves

Motivations. Weak fluctuations over flat space-time: linearised Riemann tensor and its abelian gauge invariance. Equations of motion: de Donder gauge and gravitational waves. Comparing to Maxwell's theory.

§IV. Isometries and maximally symmetric spaces

Symmetries of tensors: form-invariance. Killing equations. Integrability condition and maximal number of isometries. The Lie derivative. Killing vectors and conservation laws. The example of Minkowski space. Translational isometries.

§V. Basics of the Schwarschild solution

Schwarzschild metric. Spherical symmetry and Birkhoff's theorem (without proof). Killing vectors of Schwarzschild's metric. Gravitational redshift.

Testi Adottati

-Carroll S Spacetime and Geometry: An Introduction to General Relativity (Addison-Wesley 2014/Cambridge University Press, 2019)
-Dirac P A M General Theory of Relativity (Princeton University Press, 1996)
-Hartle S Gravity: An Introduction to Einstein's General Relativity (Cambridge University Press, 2021)
-Rovelli C, General Relativity - the Essentials, (Cambridge University Press, 2021).
-Weinberg S, Gravitation and Cosmology - principles and applications of the general theory of relativity, (John Wiley \& Sons, 1972).

Modalità Erogazione

Blackboard lectures

Modalità Frequenza

Attendance in person

Modalità Valutazione

Partials: two exemptions with open-ended questions. The exams simulate in written form the performance of an oral exam. The questions are formulated in points, so as to allow students to develop their answer even in the absence of feedback from the teacher. Questions marked as optional are added and are designed to allow students to improve the level of their papers. The evaluation of the tasks includes a scoring grid assigned to each point, given that the teacher keeps the option to evaluate overall aspects of the papers such as the mastery of the subject, the coherence and completeness of the answer and so on. Upon passing both exams students get an exemption from the oral exam. Oral exams: one question pertaining to Part A

Programma

§I. Introduction: inertia and covariance

Galilean Relativity and Special Relativity. The Equivalence Principle. Motivations for general covariance. Local inertial frames.

§II. Dynamical spacetime: basics of General Relativity

Curvilinear coordinates. Vectors and tensors under general coordinate transformations. Parallel transport and Christoffel symbols. Covariant derivatives and metric compatibility. Covariantly conserved vectors and conserved vector densities. Transformation of the Christoffel symbols. Ricci's theorem. Geodesics and their Newtonian limit. Time-like, null and space-like curves. Normal coordinates and local inertial frames. Covariant derivation along curves. Riemann curvature tensor. Algebraic properties. Bianchi identities. Local inertial frames. Characterisation of flat spaces. Geodesic deviation. Einstein-Hilbert action and equations of motion. The Palatini identity. Non-dynamical sources: contracted Bianchi identity, matter energy-momentum tensor and covariant conservation. Minimal coupling to scalar fields and to Maxwell fields.

§III. Linear approximation and gravitational waves

Motivations. Weak fluctuations over flat space-time: linearised Riemann tensor and its abelian gauge invariance. Equations of motion: de Donder gauge and gravitational waves. Comparing to Maxwell's theory.

§IV. Isometries and maximally symmetric spaces

Symmetries of tensors: form-invariance. Killing equations. Integrability condition and maximal number of isometries. The Lie derivative. Killing vectors and conservation laws. The example of Minkowski space. Translational isometries.

§V. Basics of the Schwarschild solution

Schwarzschild metric. Spherical symmetry and Birkhoff's theorem (without proof). Killing vectors of Schwarzschild's metric. Gravitational redshift.

Testi Adottati

-Carroll S Spacetime and Geometry: An Introduction to General Relativity (Addison-Wesley 2014/Cambridge University Press, 2019)
-Dirac P A M General Theory of Relativity (Princeton University Press, 1996)
-Hartle S Gravity: An Introduction to Einstein's General Relativity (Cambridge University Press, 2021)
-Rovelli C, General Relativity - the Essentials, (Cambridge University Press, 2021).
-Weinberg S, Gravitation and Cosmology - principles and applications of the general theory of relativity, (John Wiley \& Sons, 1972).

Modalità Erogazione

Blackboard lectures

Modalità Frequenza

Attendance in person

Modalità Valutazione

Partials: two exemptions with open-ended questions. The exams simulate in written form the performance of an oral exam. The questions are formulated in points, so as to allow students to develop their answer even in the absence of feedback from the teacher. Questions marked as optional are added and are designed to allow students to improve the level of their papers. The evaluation of the tasks includes a scoring grid assigned to each point, given that the teacher keeps the option to evaluate overall aspects of the papers such as the mastery of the subject, the coherence and completeness of the answer and so on. Upon passing both exams students get an exemption from the oral exam. Oral exams: one question pertaining to Part A

Programma

§I. Introduction: inertia and covariance

Galilean Relativity and Special Relativity. The Equivalence Principle. Motivations for general covariance. Local inertial frames.

§II. Dynamical spacetime: basics of General Relativity

Curvilinear coordinates. Vectors and tensors under general coordinate transformations. Parallel transport and Christoffel symbols. Covariant derivatives and metric compatibility. Covariantly conserved vectors and conserved vector densities. Transformation of the Christoffel symbols. Ricci's theorem. Geodesics and their Newtonian limit. Time-like, null and space-like curves. Normal coordinates and local inertial frames. Covariant derivation along curves. Riemann curvature tensor. Algebraic properties. Bianchi identities. Local inertial frames. Characterisation of flat spaces. Geodesic deviation. Einstein-Hilbert action and equations of motion. The Palatini identity. Non-dynamical sources: contracted Bianchi identity, matter energy-momentum tensor and covariant conservation. Minimal coupling to scalar fields and to Maxwell fields.

§III. Linear approximation and gravitational waves

Motivations. Weak fluctuations over flat space-time: linearised Riemann tensor and its abelian gauge invariance. Equations of motion: de Donder gauge and gravitational waves. Comparing to Maxwell's theory.

§IV. Isometries and maximally symmetric spaces

Symmetries of tensors: form-invariance. Killing equations. Integrability condition and maximal number of isometries. The Lie derivative. Killing vectors and conservation laws. The example of Minkowski space. Translational isometries.

§V. Basics of the Schwarschild solution

Schwarzschild metric. Spherical symmetry and Birkhoff's theorem (without proof). Killing vectors of Schwarzschild's metric. Gravitational redshift.

Testi Adottati

-Carroll S Spacetime and Geometry: An Introduction to General Relativity (Addison-Wesley 2014/Cambridge University Press, 2019)
-Dirac P A M General Theory of Relativity (Princeton University Press, 1996)
-Hartle S Gravity: An Introduction to Einstein's General Relativity (Cambridge University Press, 2021)
-Rovelli C, General Relativity - the Essentials, (Cambridge University Press, 2021).
-Weinberg S, Gravitation and Cosmology - principles and applications of the general theory of relativity, (John Wiley \& Sons, 1972).

Modalità Erogazione

Blackboard lectures

Modalità Frequenza

Attendance in person

Modalità Valutazione

Partials: two exemptions with open-ended questions. The exams simulate in written form the performance of an oral exam. The questions are formulated in points, so as to allow students to develop their answer even in the absence of feedback from the teacher. Questions marked as optional are added and are designed to allow students to improve the level of their papers. The evaluation of the tasks includes a scoring grid assigned to each point, given that the teacher keeps the option to evaluate overall aspects of the papers such as the mastery of the subject, the coherence and completeness of the answer and so on. Upon passing both exams students get an exemption from the oral exam. Oral exams: one question pertaining to Part A

Programma

§I. Introduction: inertia and covariance

Galilean Relativity and Special Relativity. The Equivalence Principle. Motivations for general covariance. Local inertial frames.

§II. Dynamical spacetime: basics of General Relativity

Curvilinear coordinates. Vectors and tensors under general coordinate transformations. Parallel transport and Christoffel symbols. Covariant derivatives and metric compatibility. Covariantly conserved vectors and conserved vector densities. Transformation of the Christoffel symbols. Ricci's theorem. Geodesics and their Newtonian limit. Time-like, null and space-like curves. Normal coordinates and local inertial frames. Covariant derivation along curves. Riemann curvature tensor. Algebraic properties. Bianchi identities. Local inertial frames. Characterisation of flat spaces. Geodesic deviation. Einstein-Hilbert action and equations of motion. The Palatini identity. Non-dynamical sources: contracted Bianchi identity, matter energy-momentum tensor and covariant conservation. Minimal coupling to scalar fields and to Maxwell fields.

§III. Linear approximation and gravitational waves

Motivations. Weak fluctuations over flat space-time: linearised Riemann tensor and its abelian gauge invariance. Equations of motion: de Donder gauge and gravitational waves. Comparing to Maxwell's theory.

§IV. Isometries and maximally symmetric spaces

Symmetries of tensors: form-invariance. Killing equations. Integrability condition and maximal number of isometries. The Lie derivative. Killing vectors and conservation laws. The example of Minkowski space. Translational isometries.

§V. Basics of the Schwarschild solution

Schwarzschild metric. Spherical symmetry and Birkhoff's theorem (without proof). Killing vectors of Schwarzschild's metric. Gravitational redshift.

Testi Adottati

-Carroll S Spacetime and Geometry: An Introduction to General Relativity (Addison-Wesley 2014/Cambridge University Press, 2019)
-Dirac P A M General Theory of Relativity (Princeton University Press, 1996)
-Hartle S Gravity: An Introduction to Einstein's General Relativity (Cambridge University Press, 2021)
-Rovelli C, General Relativity - the Essentials, (Cambridge University Press, 2021).
-Weinberg S, Gravitation and Cosmology - principles and applications of the general theory of relativity, (John Wiley \& Sons, 1972).

Modalità Erogazione

Blackboard lectures

Modalità Frequenza

Attendance in person

Modalità Valutazione

Partials: two exemptions with open-ended questions. The exams simulate in written form the performance of an oral exam. The questions are formulated in points, so as to allow students to develop their answer even in the absence of feedback from the teacher. Questions marked as optional are added and are designed to allow students to improve the level of their papers. The evaluation of the tasks includes a scoring grid assigned to each point, given that the teacher keeps the option to evaluate overall aspects of the papers such as the mastery of the subject, the coherence and completeness of the answer and so on. Upon passing both exams students get an exemption from the oral exam. Oral exams: one question pertaining to Part A