Università Roma Tre

Introductory notions
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.

1. S. Carrol Space time and Geometry: An Introduction to
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