To make the student familiar with the conceptual assumptions of the Theory of General Relativity, both as a geometric theory of space-time and by emphasizing analogies and differences with field theories based on local symmetries that describe the interactions between elementary particles. Illustrate the essential elements of differential geometry necessary to formalize the proposed concepts. Introduce the student to extensions of the theory of interest for current theoretical research.
teacher profile teaching materials
Recap of Special Relativity. Lorentz transformations in Minkowski’s space. Vectors in Minkowski’s space. Basis of the tangent space.
Cotangents space and dual vectors in Minkowski’s space. Basis of cotangent space. Lorentz transformations of vectors and dual vectors. Tensors in Minkowski’s space. Properties of vectors, dual vectors and tensors in Minkowski’s space. Definition of symmetric and antisymmetric tensor. Symmetrization and antisymmetrization of a generic tensor. Metric in Minkowski’s space: definition and properties. Operations related to the metric: scalar products, rising and lowering indices of a tensor, contractions and trace of a tensor. Equivalence between inertial and gravitational mass. Weak Equivalence Principle (WEP), Einstein’s equivalence Principle (EEP).
Basic notions of differential geometry
Introduction to the notion of manifold. Definition and properties of maps. Injective and suriective maps (some examples included). Composition of charts. Invertible charts. Definition of diffeomorphism.
Definition of chart (or coordinate system). Definition of atlas. Definition of manifold. Product of manifolds.
Formal coordinate independent definition of vector. Demonstration that the dimension of the tangent space coincides with the one of the corresponding manifold. Basis (or coordinate system) of the tangent space. Coordinate transformations. Coordinate transformations of the components of a vector. Definition and properties of the tangent field. Definition of one parameter group of diffeomorphisms. Definition of integral curves. Commutator of two vectors. Coordinate independent definition of dual vector (one-form). Cotangent space and corresponding basis. Coordinate transformation of the components of a one-form. Coordinate independent definition of tensor. Demonstration that the partial derivative of a tensor is not a tensor.
Metric: signature and canonical form.
Tensor densities.
Differential forms. Wedge product. Exterior derivative. Closed and exact form. Poincarre Lemma (statement only). Hodge duality. Maxwell equations expressed in term of exterior derivative and hodge duality (only small reference).
Integration over a manifold: volume element in terms of the determinant of the metric.
Maps between manifold: pullback and pushforward. Pullback and pushforward associated to diffeomorphisms. Equivalence between diffeomorphisms and coordinate transformations. Vector field associated to diffeomorphisms. Lie Derivative: definition and general properties. Action of Lie’s derivative on scalars, vectors, one-forms and tensors. General Relativity as diffeomorphism invariant theory. Analogy between gauge transformations and diffeomorphisms.
Symmetries.
Notion of submanifold. Immersed and embedded submanifolds. Notion of hypersurface and boundary of a manifold.
Integration on manifolds again: differential form as generic volume element. Orientation and orientable manifold. Covering of the manifold through partition of unity. Integration of p-forms over submanifold. Demonstrations that the volume element can be expressed in terms of the determinant of the metric. Stokes theorem (no demonstration).
Connection, Covariant Derivative, Curvature
Lie’s Algebra and Lie’s group. Action from the right and from the left. Left- and right-invariant vectors. Structure constants. Examples of Lie groups. Maurer-Cartan forms. Maurer-Cartan’s equations. Action of Lie Groups on manifolds. Definition of free, effective and transitive action. Orbit and stabilizer.
Algebric definition of connection and covariant derivative. General properties of covariant derivatives. Action of coordinate transformations on the connection.
Demonstration that the difference of Christoffel coefficients associated to two different connections transforms as a tensor; torsion tensor, torsion-free and metric connection. Demonstrations that for any given metric exists a connection (metric connections) for which the covariant derivative of the metric is zero.
Formal construction of the covariant derivative from the notion of parallel transport (qualitative introduction).
Fiber bundle. Trivial and locally trivializable bundles. Local trivilizations. Maps between fiber bundles (notions).
Defintion of bundle atlas, G-atlas, G-structure. Fiber Bundle with structure group G. Definition of Principal Bundle. Definition of section of a bundle. Vector bundle and bundle of basis, definition and general properties. Relation between principle bundle, vectorbundle and bundle of frames (definition of associated vector bundle to a principal bundle.
Construction of the covariante derivative on a vectorbundle (only the knowledge of the fundamental logical steps is required for the exam).
Curvature tensor as 2-form on a fiber bundle. Geometrical interpretation of the curvature. Bianchi identity. Fiber metric. Ortogonal basis.
Connections and gauge theories: electromagnetism as simple example.
Soldering form. Choice of the gauge. Ortonormal and metric gauge.
Levi-Civita connection; Riemann’s tensor : definition and properties. Ricci’s tensor and scalar, Weyl’s tensor. Globally and locally inertial coordinates.
Einstein’s theory of gravity
Minimal coupling.
Particle in a gravitational field: affine parameter, self-parallel curves. Geodesic’s equations. Geodesic deviation.
Derivation of the Einstein’s equations from Newton’s limit.
Lagrangian derivations of Einstein’s equations.
General considerations on the structure of Einstein’s equations. Choice of the gauge. Energy conditions.
Symmetries and Killing vectors: version of Noether’s theorem from general relativity. Maximal number of linearly independent Killing vectors on a manifold. Homogenous and isotropic manifold. Spaces at constant curvature. Metric in spaces at constant curvature.
Notable solutions of Einstein’s equations
Static spherically symmetric spacetimes. Determination of Schwarzschild’s metric.
Cosmological solution. Spatially homogeneous and isotropic spacetime. Frieman’s Robertson-Walker metric. Friedman’s equations.
Coordinate singularities. Case of study: Schwarzschild radius. Rindler metric. Kruskal coordinates. Black hole solution.
Perturbation around a background metric. Case of study:perturbation of flat metric. Degrees of freedom. Linearized Einstein’s equations. Choice of the gauge. Linearized Einstein’s equations in vacuum: gravitational waves. Solutions in presence of the source (only few words).
Advanced concepts
Conformal transformations. Cotton’s tensor. Conformally flat metric. Demonstration of the theorem: a metric is conformally flat if and only if Weyl (Cotton) tensor is null. Conformal group. Conformal Killing vectors.
Alternative theories of gravity. Scalar-tensor theories. Jordan and Einstein’s frames.
General Relativity (Addison Wesley, 2004);
2. R. Wald General Relativity (The Chicago Press, 1984);
3. B. Schutz A First Course in General Relativity
(Cambridge Press)
4. B. Schutz Geometrical Methods of Mathematical
Physics (Cambridge Press)
5. S. Weinberg Gravitation and Cosmology-principles and
application of the general theory of relativity (John
Weiley & Sons, 1972);
6. people.sissa.it/~percacci/lectures/general/index.html
Programme
Introductory notionsRecap of Special Relativity. Lorentz transformations in Minkowski’s space. Vectors in Minkowski’s space. Basis of the tangent space.
Cotangents space and dual vectors in Minkowski’s space. Basis of cotangent space. Lorentz transformations of vectors and dual vectors. Tensors in Minkowski’s space. Properties of vectors, dual vectors and tensors in Minkowski’s space. Definition of symmetric and antisymmetric tensor. Symmetrization and antisymmetrization of a generic tensor. Metric in Minkowski’s space: definition and properties. Operations related to the metric: scalar products, rising and lowering indices of a tensor, contractions and trace of a tensor. Equivalence between inertial and gravitational mass. Weak Equivalence Principle (WEP), Einstein’s equivalence Principle (EEP).
Basic notions of differential geometry
Introduction to the notion of manifold. Definition and properties of maps. Injective and suriective maps (some examples included). Composition of charts. Invertible charts. Definition of diffeomorphism.
Definition of chart (or coordinate system). Definition of atlas. Definition of manifold. Product of manifolds.
Formal coordinate independent definition of vector. Demonstration that the dimension of the tangent space coincides with the one of the corresponding manifold. Basis (or coordinate system) of the tangent space. Coordinate transformations. Coordinate transformations of the components of a vector. Definition and properties of the tangent field. Definition of one parameter group of diffeomorphisms. Definition of integral curves. Commutator of two vectors. Coordinate independent definition of dual vector (one-form). Cotangent space and corresponding basis. Coordinate transformation of the components of a one-form. Coordinate independent definition of tensor. Demonstration that the partial derivative of a tensor is not a tensor.
Metric: signature and canonical form.
Tensor densities.
Differential forms. Wedge product. Exterior derivative. Closed and exact form. Poincarre Lemma (statement only). Hodge duality. Maxwell equations expressed in term of exterior derivative and hodge duality (only small reference).
Integration over a manifold: volume element in terms of the determinant of the metric.
Maps between manifold: pullback and pushforward. Pullback and pushforward associated to diffeomorphisms. Equivalence between diffeomorphisms and coordinate transformations. Vector field associated to diffeomorphisms. Lie Derivative: definition and general properties. Action of Lie’s derivative on scalars, vectors, one-forms and tensors. General Relativity as diffeomorphism invariant theory. Analogy between gauge transformations and diffeomorphisms.
Symmetries.
Notion of submanifold. Immersed and embedded submanifolds. Notion of hypersurface and boundary of a manifold.
Integration on manifolds again: differential form as generic volume element. Orientation and orientable manifold. Covering of the manifold through partition of unity. Integration of p-forms over submanifold. Demonstrations that the volume element can be expressed in terms of the determinant of the metric. Stokes theorem (no demonstration).
Connection, Covariant Derivative, Curvature
Lie’s Algebra and Lie’s group. Action from the right and from the left. Left- and right-invariant vectors. Structure constants. Examples of Lie groups. Maurer-Cartan forms. Maurer-Cartan’s equations. Action of Lie Groups on manifolds. Definition of free, effective and transitive action. Orbit and stabilizer.
Algebric definition of connection and covariant derivative. General properties of covariant derivatives. Action of coordinate transformations on the connection.
Demonstration that the difference of Christoffel coefficients associated to two different connections transforms as a tensor; torsion tensor, torsion-free and metric connection. Demonstrations that for any given metric exists a connection (metric connections) for which the covariant derivative of the metric is zero.
Formal construction of the covariant derivative from the notion of parallel transport (qualitative introduction).
Fiber bundle. Trivial and locally trivializable bundles. Local trivilizations. Maps between fiber bundles (notions).
Defintion of bundle atlas, G-atlas, G-structure. Fiber Bundle with structure group G. Definition of Principal Bundle. Definition of section of a bundle. Vector bundle and bundle of basis, definition and general properties. Relation between principle bundle, vectorbundle and bundle of frames (definition of associated vector bundle to a principal bundle.
Construction of the covariante derivative on a vectorbundle (only the knowledge of the fundamental logical steps is required for the exam).
Curvature tensor as 2-form on a fiber bundle. Geometrical interpretation of the curvature. Bianchi identity. Fiber metric. Ortogonal basis.
Connections and gauge theories: electromagnetism as simple example.
Soldering form. Choice of the gauge. Ortonormal and metric gauge.
Levi-Civita connection; Riemann’s tensor : definition and properties. Ricci’s tensor and scalar, Weyl’s tensor. Globally and locally inertial coordinates.
Einstein’s theory of gravity
Minimal coupling.
Particle in a gravitational field: affine parameter, self-parallel curves. Geodesic’s equations. Geodesic deviation.
Derivation of the Einstein’s equations from Newton’s limit.
Lagrangian derivations of Einstein’s equations.
General considerations on the structure of Einstein’s equations. Choice of the gauge. Energy conditions.
Symmetries and Killing vectors: version of Noether’s theorem from general relativity. Maximal number of linearly independent Killing vectors on a manifold. Homogenous and isotropic manifold. Spaces at constant curvature. Metric in spaces at constant curvature.
Notable solutions of Einstein’s equations
Static spherically symmetric spacetimes. Determination of Schwarzschild’s metric.
Cosmological solution. Spatially homogeneous and isotropic spacetime. Frieman’s Robertson-Walker metric. Friedman’s equations.
Coordinate singularities. Case of study: Schwarzschild radius. Rindler metric. Kruskal coordinates. Black hole solution.
Perturbation around a background metric. Case of study:perturbation of flat metric. Degrees of freedom. Linearized Einstein’s equations. Choice of the gauge. Linearized Einstein’s equations in vacuum: gravitational waves. Solutions in presence of the source (only few words).
Advanced concepts
Conformal transformations. Cotton’s tensor. Conformally flat metric. Demonstration of the theorem: a metric is conformally flat if and only if Weyl (Cotton) tensor is null. Conformal group. Conformal Killing vectors.
Alternative theories of gravity. Scalar-tensor theories. Jordan and Einstein’s frames.
Core Documentation
1. S. Carrol Space time and Geometry: An Introduction toGeneral Relativity (Addison Wesley, 2004);
2. R. Wald General Relativity (The Chicago Press, 1984);
3. B. Schutz A First Course in General Relativity
(Cambridge Press)
4. B. Schutz Geometrical Methods of Mathematical
Physics (Cambridge Press)
5. S. Weinberg Gravitation and Cosmology-principles and
application of the general theory of relativity (John
Weiley & Sons, 1972);
6. people.sissa.it/~percacci/lectures/general/index.html
Type of delivery of the course
The lectures are through the exposition of the concepts, intended both as definition and mathematical equations, which are necessary to full fil the aims of the course and might be possibly object of the exam. The students are invited to interrupt at any time the teacher to ask the repetition of some explanations as well as to pose more general questions. The students are often invited to do, during their studying time, small exercises, mostly consisting in doing or completing the demonstration of mathematical relations illustrated during the lectures. The aim is to make the students more confident with the topics of the program of the course. The teacher will solve these exercise during the lecture time whether he receives explicit requests from one or more students. The students have as well the opportunity of asking, at the beginning of each lectures, further clarifications on the previous lectures. The teacher makes periodically available to the students, through the web page of the course, a sort of diary of lectures: the teacher provides, for each lecture, the detailed list of the discussed topics as well as the indications of which among the adopted texts have been used for that lecture. Unless explicitly stated, this list of topics will be object of the exam and will compose the definitive program of the course which will be provided at the end of the lectures.Type of evaluation
The exam is oral. The exam is passed if the student demostrates sufficient competence on three of the topics listed in the definitive programm of the exams. The competence in the exam topics is determined by asking questions, typically 1-2 per topic, consisting in the exposition of concepts and/or definitions as well as reproducing/deriving equations or, more in general, mathematical relations. For this last kind of questions the student is expected to use a written support, typically paper and pen, provided by the teacher or blackboard, if the latter is available and the student express this preference. The grade is determined, after agreement with the members of the exam commission, at the end of the exam itself and proposed to the student.