442. Differential Geometry and
General Relativity
The course is an introduction to modern Differential Geometry (part
I) and modern General Relativity (part II).
Starting Autumn 2009, the course will
be split into two modules as a result of the Bologna process (more
information here).
MA3429: Differential
Geometry and General Relativity I (5 ECTS)
This module is an introduction to modern Differential Geometry.
Learning
Outcomes:On successful completion of this module, the student
will:
1. be familiar with the basic concepts, methods and results of
modern Differential Geometry
2. be able to recognise differential geometric structures in
Mathematics and Physics, formulate them in the language of Differential
Geometry,
and analyse them using the methods and tools of invariant differential
calculus
3. be able to describe, construct and analyze differential manifolds,
vector bundles, tensor fields and linear connections .
3. be able to apply the techniques of invariant tensor calculus to
basic problems in geometry and physics
4. be able to apply the techniques of vector bundles and connections to
problems in geometry and physics
5. be familiar with the basic concepts of Riemannian and
pseudo-Riemannian geometry.
Prerequisites proposed for MA3429:
MA2023: Metric
spaces I
Familiarity with the notions of abstract topological space, continuous
map between topological spaces, homeomorphism, open and closed
sets,
closure and frontier of a set, compact subset of an abstract
topological space, connectedness and linear (i.e. path-) connectedness
of abstract
topological spaces.
MA2322: Calculus
on manifolds
Familiarity with the notions of differential form, Stokes theorem on
manifolds. Ideally you have been taught something abut the geometry of
curves
and surfaces (Frenet–Serret formulas, Gauss–Codazzi equations,
Gauss's Theorema Egregium) either in this class or in
Mechanics/Advanced Mechanics.
MA2332: Equations
of mathematical physics II
MA2342: Advanced
classical mechanics II
Good understanding of Euler-Lagrange equations, integrals of motion,
variational principle, Newton's theory of gravity (including the
Poisson equation
for the Newtonian gravitational potential).
MA3432: Classical
field theory II
Good understanding of special relativity and classical electrodynamics
in its Minkovsky space formulation.
Note: You
should also be
familiar with the notions of group, actions of
a group on a set, linear representation of a group,
the permutation
group. The course also assumes good understanding of basic linear
algebra (theory of finite and
infinite-dimensional
vector spaces over a field, including the theory of
quadratic forms on real vector spaces) and of multivariate real
analysis.
Course
content (MA3429):
-basic theory of abstract manifolds (chart, atlas,
differentiable structure, tangent space, smooth map,
differential of a
smooth map at a point)
-algebra of smooth real-valued functions on a manifold; vector fields
on a manifold; Lie bracket of vector fields
-basic theory of vector bundles; sections of vector bundles; the
tangent bundle; the differential of a map;
natural operations on vector
bundles (direct sum, tensor product, dual, symmetric and antisymmetric
product)
-reminder on modules over commutative rings and their basic operations
-the module of sections of a vector bundle; its behavior under the
natural operations on bundles
-natural vector bundles on a manifold; tensor fields,
polyvector fields and differential forms on manifolds;
symmetric and
antisymmetric tensor fields
-connections in a vector bundle; curvature of a connection; induced
connections
-pull-back of a vector bundle; pull-back connection
-parallel transport of a connection
-affine connections and their torsion tensor; covariant
derivation of tensor fields; autoparallel curves with respect to an
affine connection
-pseudo-Riemannian manifolds; signature; Riemannian, Minkowskian and
Minkowsky manifolds.
-the Levi-Civita connection, geodesics, variational principle for
geodesics.
-Riemann curvature tensor,
Ricci tensor and
Einstein tensor.
-orientability, volume form determined by a metric
-natural differential operators on a pseudo-Riemannian manifold
Course
materials:
Core textbook:
S. Lang, Fundamentals of differential
geometry
Recommended references:
General:
D. Husemoller, Fibre bundles
An advanced treatment of the theory
of fiber bundles
-G. E. Bredon, Topology and geometry
An advanced treatment which should be
very useful for anyone seriously interested in modern gauge theory,
gravity and string theory
-S. Morita, The geometry of differential forms
A direct treatment of many aspects
with emphasis on differential forms.
-F. W. Warner, Foundations of Differentiable Manifolds and
Lie Groups
A standard reference
-M. Spivak, Calculus on Manifolds: A Modern Approach to
Classical Theorems of Advanced Calculus
For background on analysis and
calculus on manifolds
Pseudo-Riemannian manifolds:
-B. O'Neill, Semi-Riemannian Geometry With Applications to Relativity
-M. Kriele, Spacetime: Foundations
of General Relativity and Differential Geometry
Background:
Category theory:
-S. McLane, Categories for the working mathematician
A standard reference for anyone
interested in post 1950 algebraic, topological and geometric theories
Lnear algebra, rings, modules and
associative algebras:
-Bourbaki, Algebra I (Chapters 1-3)
"the standard reference to all
that mathematics which everyone is already supposed to know"
Differential geometry of curves and
surfaces in R^n:
-V. Toponogov, Differential Geometry of Curves and Surfaces
-W. Kuhnel, Differential
Geometry: Curves - Surfaces - Manifolds
I will not have time to cover
this
topic in class, but this is a classic subject which ought to be part of
the education of any
theoretical physicist and
mathematician. You probably already know something about this from
Mechanics.
Online materials:
An
introduction to general topology
Covering
spaces
The inverse and
implicit function theorems
Orientability
of manifolds
Integration
on manifolds Stokes'
theorem
Lectures on
curves and surfaces in R^3
famous
curves famous
surfaces
special
plane curves
MA4448: Differential Geometry and
General Relativity II (5 ECTS)
This module is an introduction to modern General Relativity. It makes
intensive use of the concepts, methods and techniques taught in MA4047.
Learning
Outcomes: On successful completion of this module, the
student will:
1. be familiar with the basic concepts and postulates of General
Relativity, as well as with the observational and experimental evidence
for this theory
2. understand the modern formulation of General Relativity and
its conceptual structure
3. be able to describe and analyze the process of observation and
measurement in General Relativity
4. be familiar with the invariant description of simple matter models
and of electromagnetic fields coupled to gravity
5. be familiar with the simplest exact solutions of General Relativity
6. be able to solve basic kinematics and dynamics problems in General
Relativity.
Prerequisites
proposed for MA4448:
MA4047 :Differential Geometry and General Relativity I
Course
content:
-Minkowsky manifolds, light cones and time orientability; space-times;
timelike, lightlike and spacelike geodesics;
-observers, proper time
-The axioms of general relativity; action principle for gravity coupled
to mater.
-Matter stress-energy tensor, Einstein equations, positivity and
causality conditions
-matter models: free falling particle, dust, real scalar field,
electromagnetic field
-the weak field limit, recovering Newtonian gravity, gravitational waves
-gravitational red shift
-basic special solutions of Einstein's equations (Schwarzschild,
Robertson-Walker)
-motion in a central field; applications to gravitational lensing and
precession of perihelia
Advanced topics (chosen at the
interests of the class and as time allows)
-black hole physics
-Kerr,
Reisner-Nordstrom solutions
-basic cosmology
Course materials:
Core textbook:
R.K. Sachs, General Relativity for
Mathematicians, Graduate Texts in Mathematics vol
48, Springer, 1977
Recommended
references:
-S. Carroll, Spacetime and Geometry: An Introduction to General
Relativity
An excellent and well-balanced
introduction
-Robert Wald, General Relativity
A comprehensive treatement and a
standard reference
-S. Hawking and Ellis, The large scale structure of space-time
Classical reference on global
aspects, causality and and singularities
-Landau and Lifschitz, The Classical Theory of Fields
Standard physics reference though the
mathematic is treated intuitively
-F. de Felice and C. Clarke, Relativity on Curved Manifolds
Modern mathematical treatment
-
J. Stewart, Advanced General Relativity
Advanced topics, goes well beyond
what we cover in this class.
Problems with solutions:
-
A.P. Lightman, W.H. Press, R.H. Price, and S.A. Teukolsky,
Problem Book in Relativity and Gravitation
Special topics
Most of the references below go well
beyond what we can cover in class, but they should be useful for those
who wish to study the subject further (now or in the future).
Cosmology:
-E.W. Kolb and M.S. Turner, The Early Universe
Standard introduction
-S. Dodelson, Modern Cosmology
Systematic graduate-level treatment
Compact objects:
-S.L. Shapiro and S.A. Teukolsky, Black Holes, White Dwarfs, and
Neutron Stars : The Physics of Compact Objects
Standard advanced reference
-S. Chandrasekhar, The
Mathematical Theory of Black Holes
A beautifully written classical
treatment
-C. DeWitt, Les Houches lectures on black holes (1972)
Another classical reference
Fields in GR:
We won't say much about this but it
is a standard graduate level subject
-S. Fulling, Aspects of Quantum Field
Theory in Curved Spacetime
-
N.D. Birrell and P.C. Davis, Quantum Fields in Curved Spacetime
-
R.M. Wald, Quantum Field Theory in Curved Spacetime and Black Hole
Thermodynamics
Geometry of gauge theories:
-M. Nakahara, Geometry, Topology and
Physics
Excellent and well-balanced treatment
for those interested in deeper aspects
Online
materials:
Lecture
notes by Gerard t'Hooft
Sean Carroll's lecture
notes (also see his web page)
Lecture notes
by Matthias Blau
P.K. Townsend's Lectures
on black holes
M. Trodden, S. Carroll,
TASI Lectures: Introduction to Cosmology
Visual
simulations of GR effects near black holes
A
video on galactic supermassive black holes
M Tytgat, Four
Lectures on Cosmology at CERN
For your future reference/graduate students:
Basics of Lie
groups
S. Bradlow, Vector
bundles and introduction to gauge theory
P.
Michor, Gauge theory for fiber bundles
G. Swetlichny,
Preparation for Gauge Theory
For this class:
useful formulas
brief compendium
Problem sets 2010
Problem
set 1
===============
Problem sets 2009:
Set 1: Sove each of last year's exam
problems.
Set 2: Solve all problems in this file.
Exam format:
The exam will consist of two
parts (one for module MA 3429 -- Differential Geometry and one for
module MA4448 -- General Relativity), both of which will be printed on
the same exam sheet.
Each part will have 4 questions (each question worth 25 points), of
which you have to answer 3 questions correctly in order to get a
maximum
score for that part (a score of 75 points will be renormalized to 100%
for each of the two parts). For those
questions which have sub-questions, each sub-question will be numbered
and contain an indication of how many points it is worth. For example:
Part 1: XMA3429: Differential Geometry:
Answer 3 questions out of 4.
Question 1 [25 p] ....
a. [10 p] ...
b. [8 p] ...
c. [7p] ...
Question 2 [25 p] ...
Question 3 [25 p] ...
Question 4 [25 p] ...
Part 2: XMA4448: General
Relativity
Answer 3 questions out of 4.
Question 1 [25 p] ....
a. [10 p] ...
b. [15 p] ...
Question 2 [25 p] ...
Question 3 [25 p] ...
Question 4 [25 p] ...
Students taking only module XMA3429 only need to work on part 1 of the
exam (while ignoring part 2 altogether).