#
Relativity 1

A.Y. 2020/2021

Learning objectives

To provide an introduction to special and general relativity, emphasizing the foundational aspects of both theories, the mathematical rigor in their formulation and the main experimental tests.

Expected learning outcomes

After the course, students will have acquired the fundamentals of special and general relativity theories; furthermore, they will be able to rigorously study the physical phenomena described by them.

**Lesson period:** Second semester
(In case of multiple editions, please check the period, as it may vary)

**Assessment methods:** Esame

**Assessment result:** voto verbalizzato in trentesimi

Course syllabus and organization

### Single session

Responsible

Lesson period

Second semester

As for the pandemic emergency, see the issues on teaching methods ("Metodi didattici") and exams ("Modalità di verifica dell'apprendimento").

**Prerequisites for admission**

The prerequisites for this course are the fundamental notions about linear and multilinear algebra, topology, differential and integral calculus for functions of one or several real variables, Newtonian mechanics of particles and electromagnetism. Such notions are provided by courses of the three-years degree programme in Mathematics at the Milan Uiniversity.

The possession of some basic knowledge about differential manifolds and tensor fields is certainly a benefit for students interested in the present course (even though this is not a mandatory prerequisite); such knowledge is provided, e.g., by the course "Geometry 4" for the three-years degree programme in Mathematics at the Milan University.

For students who do not have some basic knowledge on differential manifolds and tensor calculus it is mandatory to attend the 3-credits part of the present course. This part is as well mandatory for students who do not have a basic knowledge on Riemannian manifolds.

In the teacher's intention, the 3-credits part should be useful even to

students who are familiar with all the above topics (since it describes

them in a language very close to that of the 6-credit part).

The possession of some basic knowledge about differential manifolds and tensor fields is certainly a benefit for students interested in the present course (even though this is not a mandatory prerequisite); such knowledge is provided, e.g., by the course "Geometry 4" for the three-years degree programme in Mathematics at the Milan University.

For students who do not have some basic knowledge on differential manifolds and tensor calculus it is mandatory to attend the 3-credits part of the present course. This part is as well mandatory for students who do not have a basic knowledge on Riemannian manifolds.

In the teacher's intention, the 3-credits part should be useful even to

students who are familiar with all the above topics (since it describes

them in a language very close to that of the 6-credit part).

**Assessment methods and Criteria**

The examination is oral. During the examination the student must expose some of the topics indicated in the program of the course, and reply to related questions from the examiners. The topics to be exposed must be agreed by each student with the teacher before starting the preparation of the oral exam; they must be chosen so as to allow the examiners to check that the student has acquired an overview and specific skills about the whole program of the course. During his oral exam, the student is expected to show a deep understanding of the subjects of his/her exposition, both from the mathematical and from the physical viewpoint.

The examination about the second (optional) part of the course is oral as well and takes place according to the same rules, contextually with the exam on the first part.

Final marks are given using the numerical range 0-30, and communicated immediately after the examination.

For examinations concerning both parts of the course there is a unique final score in the range 0-30, that takes into account the results of both oral exams.

Exams are by appointment; any student who intends to take his exam must contact the teacher by e-mail. According to the general rules holding at the time of the appointment, exams will be in presence, or

online via skype.

The examination about the second (optional) part of the course is oral as well and takes place according to the same rules, contextually with the exam on the first part.

Final marks are given using the numerical range 0-30, and communicated immediately after the examination.

For examinations concerning both parts of the course there is a unique final score in the range 0-30, that takes into account the results of both oral exams.

Exams are by appointment; any student who intends to take his exam must contact the teacher by e-mail. According to the general rules holding at the time of the appointment, exams will be in presence, or

online via skype.

**Relativity 1 (first part)**

**Course syllabus**

1. A CRITICAL ANALYSIS OF THE NEWTONIAN SPACETIME MODEL.

Absolute space and time. Observers. Inertial observers. Incompatibility between the notion of absolute rest and the Galileian

principle of relativity.

2. THE GALILEIAN SPACETIME MODEL.

Spacetime as a bundle on absolute time. Observers. Inertial observers. The four-dimensional affine structure of spacetime.

3. THE THEORY OF SPECIAL RELATIVITY.

Light propagation, the Michelson-Morley experiment and the crisis of the Galileian model for spacetime.

The postulates of special relativity. Inertial observers. Aleksandrov's theorem. The transition functions between the spacetime descriptions of different inertial observers ("Lorentz transformations"). The effects of "length contraction" and "time dilatation" predicted by such transformations. Observation of the time dilatation in the decay of elementary particles.

Minkowski spacetime, with its affine and pseudo-Euclidean structures. Spacelike, timelike and null vectors. Proper time.

World line of a particle, or of a light signal. Three-velocity of the particle or of the signal with respect to an inertial observer. Comparison of the three-velocities according to different inertial observers. Light aberration.

Integration of proper time along the world line of a particle. Behaviour of clocks according to special relativity theory. Twin paradox.

Four-dimensional kinematics: four- velocity and four-acceleration of a particle.

Relativistic particle dynamics: invariant formulation and viewpoint of an inertial observer. Solution of the equations of motion in some simple cases. Conservation law of four-monentum in isolated systems. Four-momentum of a photon.

Maxwell's equations for the electromagnetic field: relativistically invariant formulation, by means of the exterior differential calculus and Hodge's duality. Relativistically invariant description of the solutions of Maxwell's equations; advanced and retarded potentials. Doppler's effect.

Some facts on the dynamics of perfect fluids.

The energy-momentum tensor, especially in fluid dynamics and in electromagnetism.

4. THE THEORY OF GENERAL RELATIVITY.

Physical motivations for a geometric theory of gravity. The geometry of spacetime in general relativity. The general notion of observer; the problem of simultaneity with respect to an observer. The Coriolis theorem in general relativity.

Particle dynamics. The case of a freely falling particle: the principle of geodesic motion. The Newtonian law of motion for a freely falling particle as a limit case of the principle of geodesic motion.

The behavior of clocks and the twin paradox in general relativity: experiments of Pound-Rebka and of Hafele-Keating.

Basics on fluid dynamics and electromagnetism in a curved spacetime.

The energy momentum tensor in general relativity. Einstein's equations for the gravitational field. Approximate solution of Einstein's equations for weak fields: the Newtonian theory of gravitation as a limit case of general relativity.

The Schwarzschild solution of Einstein's equations. Motion of a test particle and of a light signal in the Schwarzschild spacetime. Precession of the perihelion of a planet, and deflection of light in the Sun's gravitational field.

REMARK. After attending this course, students will be given the opportunity to look into an advanced topic in the framework of general relativity, following the bibliographic indications provided of the teacher. Students interested in this opportunity will be invited to give a talk on this topic; in case of positive evaluation of the talk by the teacher, this activity will allow the acquisition of 3 extra credits of the F type.

The suggested topics for this activity are the following ones:

1) The GPS system and general relativity.

2) Cosmological models and general relativity.

3) An introduction to black holes.

4) Gravitational waves and their experimental detection.

Absolute space and time. Observers. Inertial observers. Incompatibility between the notion of absolute rest and the Galileian

principle of relativity.

2. THE GALILEIAN SPACETIME MODEL.

Spacetime as a bundle on absolute time. Observers. Inertial observers. The four-dimensional affine structure of spacetime.

3. THE THEORY OF SPECIAL RELATIVITY.

Light propagation, the Michelson-Morley experiment and the crisis of the Galileian model for spacetime.

The postulates of special relativity. Inertial observers. Aleksandrov's theorem. The transition functions between the spacetime descriptions of different inertial observers ("Lorentz transformations"). The effects of "length contraction" and "time dilatation" predicted by such transformations. Observation of the time dilatation in the decay of elementary particles.

Minkowski spacetime, with its affine and pseudo-Euclidean structures. Spacelike, timelike and null vectors. Proper time.

World line of a particle, or of a light signal. Three-velocity of the particle or of the signal with respect to an inertial observer. Comparison of the three-velocities according to different inertial observers. Light aberration.

Integration of proper time along the world line of a particle. Behaviour of clocks according to special relativity theory. Twin paradox.

Four-dimensional kinematics: four- velocity and four-acceleration of a particle.

Relativistic particle dynamics: invariant formulation and viewpoint of an inertial observer. Solution of the equations of motion in some simple cases. Conservation law of four-monentum in isolated systems. Four-momentum of a photon.

Maxwell's equations for the electromagnetic field: relativistically invariant formulation, by means of the exterior differential calculus and Hodge's duality. Relativistically invariant description of the solutions of Maxwell's equations; advanced and retarded potentials. Doppler's effect.

Some facts on the dynamics of perfect fluids.

The energy-momentum tensor, especially in fluid dynamics and in electromagnetism.

4. THE THEORY OF GENERAL RELATIVITY.

Physical motivations for a geometric theory of gravity. The geometry of spacetime in general relativity. The general notion of observer; the problem of simultaneity with respect to an observer. The Coriolis theorem in general relativity.

Particle dynamics. The case of a freely falling particle: the principle of geodesic motion. The Newtonian law of motion for a freely falling particle as a limit case of the principle of geodesic motion.

The behavior of clocks and the twin paradox in general relativity: experiments of Pound-Rebka and of Hafele-Keating.

Basics on fluid dynamics and electromagnetism in a curved spacetime.

The energy momentum tensor in general relativity. Einstein's equations for the gravitational field. Approximate solution of Einstein's equations for weak fields: the Newtonian theory of gravitation as a limit case of general relativity.

The Schwarzschild solution of Einstein's equations. Motion of a test particle and of a light signal in the Schwarzschild spacetime. Precession of the perihelion of a planet, and deflection of light in the Sun's gravitational field.

REMARK. After attending this course, students will be given the opportunity to look into an advanced topic in the framework of general relativity, following the bibliographic indications provided of the teacher. Students interested in this opportunity will be invited to give a talk on this topic; in case of positive evaluation of the talk by the teacher, this activity will allow the acquisition of 3 extra credits of the F type.

The suggested topics for this activity are the following ones:

1) The GPS system and general relativity.

2) Cosmological models and general relativity.

3) An introduction to black holes.

4) Gravitational waves and their experimental detection.

**Teaching methods**

The course is usually based on classroom lectures. During these lectures the teacher shows slides with his notes on the course, that are accurately commented; these notes (written in Italian) are also available on the Ariel web site of the course.

The global pandemy makes likely the adoption of online teaching metods: these could flank, or replace completely, the classroom lectures. Detailed information on online teaching methods will be

made available after the assumption of general operative decisions

about the second semester 2020/21.

The global pandemy makes likely the adoption of online teaching metods: these could flank, or replace completely, the classroom lectures. Detailed information on online teaching methods will be

made available after the assumption of general operative decisions

about the second semester 2020/21.

**Teaching Resources**

The program is exhaustively treated by the notes of the teacher (in Italian), available on the Ariel web site of the course.

For completeness, some classical references on the same

topics are listed hereafter.

* S. Benenti, "Modelli matematici della meccanica I, II" , Celid.

* R. d' Inverno, "Introduzione alla relatività di Einstein", CLUEB.

* B.A. Dubrovin, A.T. Fomenko, S.P. Novikov, "Geometria contemporanea", Editori Riuniti.

* G. Ferrarese, "Lezioni di Relatività generale", Pitagora Editrice.

* T. Frankel, "The geometry of physics", Cambridge University Press.

* J.B. Hartle. "Gravity. An introduction to Einstein's general relativity", Addison Wesley.

* S.W. Hawking, G.F.R. Ellis, "The large scale structure of space- time", Cambridge University Press.

* L.D. Landau, E.M. Lifsits, "The classical theory of fields", Pergamon Press.

* C.W. Misner, K.S. Thorne, J.A. Wheeler, "Gravitation", Freeman and Company.

* C. Moller, The Theory of Relativity, Oxford University Press.

* R.M. Wald, "General relativity", University of Chicago Press.

* S. Weinberg, " Gravitation and cosmology", Wiley and Sons.

* H. Weyl, "Space, time, matter", Dover.

For completeness, some classical references on the same

topics are listed hereafter.

* S. Benenti, "Modelli matematici della meccanica I, II" , Celid.

* R. d' Inverno, "Introduzione alla relatività di Einstein", CLUEB.

* B.A. Dubrovin, A.T. Fomenko, S.P. Novikov, "Geometria contemporanea", Editori Riuniti.

* G. Ferrarese, "Lezioni di Relatività generale", Pitagora Editrice.

* T. Frankel, "The geometry of physics", Cambridge University Press.

* J.B. Hartle. "Gravity. An introduction to Einstein's general relativity", Addison Wesley.

* S.W. Hawking, G.F.R. Ellis, "The large scale structure of space- time", Cambridge University Press.

* L.D. Landau, E.M. Lifsits, "The classical theory of fields", Pergamon Press.

* C.W. Misner, K.S. Thorne, J.A. Wheeler, "Gravitation", Freeman and Company.

* C. Moller, The Theory of Relativity, Oxford University Press.

* R.M. Wald, "General relativity", University of Chicago Press.

* S. Weinberg, " Gravitation and cosmology", Wiley and Sons.

* H. Weyl, "Space, time, matter", Dover.

**Relativity 1 (mod/02)**

**Course syllabus**

This part of the course concerns the basic differential-geometric notions involved in a rigorous formulation of relativity theory. The subjects illustrated include: multilinear algebra and tensors, differentiable manifolds, tensor fields, Lie derivative, exterior differential, distributions and Frobenius theorem, vector bundles and connections, Riemannian and pseudo-Riemannian manifolds.

**Teaching methods**

The course is usually based on classroom lectures. During these lectures the teacher shows slides with his notes on the course, that are accurately commented; these notes (written in Italian) are also available on the Ariel web site of the course.

The global pandemy makes likely the adoption of online teaching metods: these could flank, or replace completely, the classroom lectures. Detailed information on online teaching methods will be

made available after the assumption of general operative decisions

about the second semester 2020/21.

The global pandemy makes likely the adoption of online teaching metods: these could flank, or replace completely, the classroom lectures. Detailed information on online teaching methods will be

made available after the assumption of general operative decisions

about the second semester 2020/21.

**Teaching Resources**

The program is exhaustively treated by the notes of the teacher (in Italian), available on the Ariel web site of the course

For completeness, some classical textbooks on the same

topics are listed hereafter.

* R. Abraham, J.E. Marsden, ''Foundations of Mechanics'', Addison Wesley.

* F Brickell, R. S. Clark, ``Differentiable manifolds. An introduction'', Van Nostrand Reinhold

* B.A. Dubrovin, A.T. Fomenko, S.P. Novikov, "Geometria contemporanea", Editori Riuniti

* T. Frankel, "The geometry of physics", Cambridge University Press.

* S.W. Hawking, G.F.R. Ellis, "The large scale structure of

space- time, Cambridge University Press.

, Cambrifspace- time", Cambridge University Press

For completeness, some classical textbooks on the same

topics are listed hereafter.

* R. Abraham, J.E. Marsden, ''Foundations of Mechanics'', Addison Wesley.

* F Brickell, R. S. Clark, ``Differentiable manifolds. An introduction'', Van Nostrand Reinhold

* B.A. Dubrovin, A.T. Fomenko, S.P. Novikov, "Geometria contemporanea", Editori Riuniti

* T. Frankel, "The geometry of physics", Cambridge University Press.

* S.W. Hawking, G.F.R. Ellis, "The large scale structure of

space- time, Cambridge University Press.

, Cambrifspace- time", Cambridge University Press

Relativity 1 (first part)

MAT/07 - MATHEMATICAL PHYSICS - University credits: 6

Practicals: 24 hours

Lessons: 28 hours

Lessons: 28 hours

Relativity 1 (mod/02)

MAT/07 - MATHEMATICAL PHYSICS - University credits: 3

Practicals: 12 hours

Lessons: 14 hours

Lessons: 14 hours

Professor(s)