basic notions about k-linear alternating maps over vector
spaces; basis and dimension of the vector space of k-linear
alternating forms; some canonical isomorphisms; exterior product of
forms; exterior algebra over vector spaces;
basic notions about the tensor algebra over vector spaces; dimension
and basis of the space of tensors of the (r,s) type;
symmetric and anti-symmetric tensors; symmetrization ed anti-symmetrization
of tensors; forms as totally anti-symmetric tensors;
orientation and choice of an orientation;
scalar product.
Basic notions of differential geometry.
Basic notions about topological spaces, open sets, topology.
Coverings, open coverings, locally finite open coverings, refinements;
compact and paracompact topological spaces.
Differentiable structure and differentiable manifold; (differentiable)
functions and maps on/between manifolds; (differentiable) curves on manifolds.
Differentiable partition of unity; existence of differentiable partitions
of unity (statement without proof).
Tangent vectors and tangent space; differential of a map at a point,
cotangent vectors and cotangent space; coordinate basis in tangent and
cotangent space; components of tangent and cotangent vectors; basis change
in tangent and cotangent spaces and transformation law for the components;
coordinate change and change in associated coordinate basis in tangent
and cotangent spaces; transformation law for the components.
Space of k-forms at a point; basis change and transformation
law for components.
Space of tensors at a point; basis change and transformation law for components.
Structures on manifolds:
vector bundles, tangent bundle, cotangent bundle, exterior algebra,
tensor bundle; bundles as differentiable manifold and their dimension;
meaning of the local triviality of bundle; parallelizable manifold;
sections of bundles;
vector fields on an open set and along a curve; differentiable vector
fields; coordinate expression of a vector field in a given coordinate
system; characterization of differentiable vector fields; space of
differentiable vector fields on a manifold; integral curve of a vector
field, existence and uniqueness of integral curves; basic concepts
concerning the flow associated with a vector field (definition without
proof);
form/tensor fields over an open set and along a curve; coordinate
expression of a form/tensor field; differentiable form/tensor fields,
characterization of differentiable form/tensor fields; space of differentiable
form/tensor fields on a manifold; the differential of a map as a 1-form
field on the manifold.
Exterior derivative: existence and uniqueness (statement without proof),
properties.
Maps between manifolds and associate maps between corresponding structures
over manifolds: pull-back and push-forward;
diffeomorphisms as a special case; pull-back and exterior
derivative, properties.
Lie derivative of a vector field; coordinate basis expression of the
Lie derivative of a vector field; generalization of the Lie derivative
to forms and tensors; coordinate expression of the Lie derivative; Lie
derivative of a vector field and Lie brackets.
Orientation of a differentiable manifold, meaning and characterization
of orientation (statement without proof).
Integration on manifolds; local definition of the integral; local expression
of the integral and its independence from the choice of coordinates; global
definition if the integral by means of the partition of unity; independence
of the integral from the choice of partition of unity; Stokes theorem;
corollary to Stokes theorem.
Riemannian and Lorentzian metric; Riemannian and Lorentzian manifold;
Existence of Riemannian metrics on manifolds; isometries between manifolds.
Natural volume element on a Riemannian/Lorentzian manifold and its coordinate
expression by means of the Levi-Civita tensor.
Connection at a point and on a manifold; connection properties and local
expression by means of connection symbols; covariant derivative of a vector
field and covariant derivative of a vector field along a curve; parallel
vector fields along a curve; compatibility condition for the connection
on a Riemannian/Lorentzian manifold; necessary and sufficient condition
for the compatibility of a connection on a Riemannian/Lorentzian manifold;
symmetric connection; characterization of a symmetric connection in a
coordinate basis and properties of the associated connection symbols;
existence and uniqueness of the symmetric connection compatible with the
Riemannian/Lorentzian metric.
Covariant derivative of form/tensor fields.
Relations between covariant derivative and exterior derivative and between
covariant derivative and Lie derivative on a manifold.
Symmetries on manifolds, Killing vector fields and Killing equation.
Self-parallel curves and geodesics; affine reparametrization; local
expression of the geodesic equation with an affine parametrization; existence
and uniqueness of geodesics (statement without proof) and definition of
the exponential map; differential of the exponential map; exponential
map as a local diffeomorphism; basic notions of normal coordinate systems.
Curvature: Riemann and Ricci tensors and their properties; coordinate
expression of the Riemann tensor; Riemann and Ricci tensor in the case
of the unique symmetric compatible connection: additional properties,
curvature scalar and Einstein tensor.
Spacetime, special relativity, general relativity.
About the principle of special relativity; inertial reference systems,
principle of relativity, principle of the constancy of the speed of light
and their experimental foundations.
Physical meaning of geometrical propositions, operational definition of
physical concepts e analysis of the concept of simultaneity from the point
of view of its operative definition.
Reconciliation of the apparent contradiction between the relativity principle
in its particular form and the principle of the constancy of the speed
of light: basics about Lorentz transformation equations.
Spacetime structure in special relativity: light cones, signals, inertial
observers.
About the principle of general covariance; problems and ambiguities
in the definition of inertial reference systems; general coordinate systems.
The equivalence principle in its weak form and Einstein's lift Gedanken
experiment; (local)equivalence between uniform gravitational
fields and inertial fields; gaussian coordinate systems, description
of physical properties by means of geometrical properties of the spacetime
continuum; relation between the principle of general covariance,
Einstein's equivalence principle and theory of the gravitational field.
Relation between the operational definition of the concepts of space and
time, the exchange of signals between observers and the causal structure
of spacetime.
Basics about experimental tests of the principles of general relativity
and about its practical consequences.
Construction of a system of uniformly accelerated observers in Minkowski
spacetime and derivation of its properties: red-shift, event horizon and
causal spacetime structure for an accelerated observer in Minkowski spacetime;
Rindler coordinates, line element in Rindler coordinates and its relation
with the Minkowskian line element; interpretation of Rindler spacetime
in light of Einstein's lift Gedanken experiment.
The stress-energy tensor.
Lagrangian formulation for the non-relativistic dynamics of a point
particle; energy conservation principle.
Basics of relativistic kinematics: action integral for the dynamics
of a relativistic particle; variational principle with fixed boundaries
and derivation of the equation of motion; variational principle with variable
boundary and definition of momentum; relativistic angular momentum tensor
and interpretation of its components in relation with the group of Lorentz
transformations (interpreted as rotations in spacetime); relativistic
center of mass.
Basics about the Lagrangian formulation of a field theory (emphasis
about Lorentz invariant theories) from a variational principle: Euler-Lagrange
equations; definition of the stress-energy tensor associated with the
fields and its law of local conservation; symmetry of the stress-energy
tensor and interpretation of its components; conservation laws in integral
form and relativistic constraint between flow of the energy density and
momentum density.
Definition of the stress-energy tensor for a field theory in presence
of general covariance; local form of the conservation law; interpretation
of the local conservation law and invariance under diffeomorphisms; conserved
quantities in the presence of symmetries (Killing vectors) and analysis
about integral (global) conservation laws in presence of general covariance;
interpretation of the Lorentz invariant case as a particular case in view
of the above considerations.
Einstein Equations.
Heuristic derivation of Einstein equations in a static spacetime
by the analysis of the weak field limit.
Einstein equations (in two equivalent forms); non-linearity of the equations
and peculiarities of their relation with sources.
Basics of the structural analysis of Einstein equations and their characterization
with respect to the associated Cauchy problem.
The metric tensor and the relations of its components with the properties
of the gravitational field and of the particular choice of the reference
system; relation between components of the metric tensor and possibility
of synchronization of clocks.
Classical limit of Einstein equations and classical limit of the geodesic
equation. Physical meaning of the connection coefficients and of the metric
tensor as it results from the classical limit of Einstein equations.
Basics about the large scale structure of spacetime.
characterization of the type of vectors on a Lorentzian manifold; characterization
of curves at a point; general characterization of curves as timelike,
null, spacelike and causal.
Classification of geodesics and invariance of their characterization.
mathematical formulation of spacetime as a Lorentzian manifold; physical
interpretation of geometrical entities associated with the concept of
manifold: exponential map, geodesics, normal coordinate systems and their
relations with the physical concepts of free motion, causal structure
and equivalence principle.
Basics about the definition of black hole from the point of view of
causal structure (introduction and definition of the following fundamental
concepts and their physical meaning, without proofs):
future/past directed timelike, spacelike null and causal curves;
past/future endpoint of a causal curve; past/future inextendible causal
curves;
chronological and causal past/future of an event and of a set of
events;
strongly causal spacetime;
achronal set and edge of an achronal set; past/future domain of
dependence of a closed achronal set; domain of dependence of a closed
achronal set; Cauchy surface and partial Cauchy surface; globally
hyperbolic spacetime;
asymptotically empty and simple spacetime and associate unphysical
space; properties of an asymptotically empty and simple spacetime:
asymptotically null past/future infinity and characterization in terms
of geodesics; weakly asymptotically empty and simple spacetime;
strongly future asymptotically predictable spacetime from a partial
Cauchy surface and its properties;
