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
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

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

Compact objects:
-S.L. Shapiro and S.A. Teukolsky, Black Holes, White Dwarfs, and Neutron Stars : The Physics of Compact Objects
-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:
-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

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).