Additional literature:
Lecture notes
"Klassisk feltteori" by Jan Myrheim (kjøpast på
instituttkontoret for kr. 100.).
Additional material can be downloaded from this site.
Alternative textbook:
Introducing Einstein's Relativity av Ray A. D'Inverno. Clarendon Press,
Oxford (2004).
Prerequisites
Fysikk - og matematikkunskapar tilsvarende dei tre første åra
av fysikk-studiet. Det er fullt mogleg å ta kurset i 3.klasse.
Prosjektoppgave:
For dei som skal byrje i 5.klasse til hausten, er det mogleg å ta
prosjektoppgave hjå meg. Eit slikt prosjekt som bygg
på FY3454 kan vere studiar av kvite dvergar eller
nøytronstjerner for ulike tilstandslikningar. Ein skal da
løyse Tolman-Oppenheimer-Volkov likningane (desse likningane blir utleia
frå generell relativitetsteori og
generaliserer vanleg hydrostatisk teori). Det er ei blanding av
analytisk og numerisk arbeid. Sjå og
Prosjekt.
Læringsmål:
The course aims at giving a general introduction to classical field theory,
special and general relativity including applications.
Fagleg innhald:
Gruppeteori. Lagrange og Hamiltonformulering av klassisk
mekanikk og klassisk feltteori. Spesiell relativitetsteori.
Gravitasjon og krumme rom. Generell relativitetsteori.
Svarte hull (Scwarzschild, og Kerr). Kosmologi.
Kvite dvergar og nøytronstjerner.
Forelesningar:
Forelesningar: tuesday 12.15-14.00 (R73) and thursdays 10.15-12.00 (R60).
Forelesning
Første forelesning tirsdag 8. januar kl. 12.15. Siste forelesning
torsdag 8. mai kl. 10.15.
Øvingar:
Exercises are an integral part of the course and will be discussed
during class whenever it falls natural.
Gjennomgått uke:
2: Group theory: Group axioms, abelian, group cyclic groups, subgroups,
center of a group, order of a group and element of a group.
Normal subgroups, Homomorphisms and isomorphisms. Lie groups, in particular
O(n) and U(n). SO(2) and rotations. Generators of Lie groups.
Finite group M.
3: Generators of SO(2)and SU(2). Lie algebra. Exercises 1-5. Classical
mechanics. Lagrangian and Hamiltonian as Legendre transforms.
Lagrange's and Hamilton's equations. Cyclic coordinates and
conserved quantities. Variational formulation of classical mechanics.
Newtonian mechanics and Galilean transformations. Absolute time and
simultaneity.
Lorentz transformations and
Minkowski space (space-time). Metric on space-time. Time dilation.
4: Lorentz contraction. Straight lines maximizes proper time.
Timelike, spacelike and lightlike curves. The relativity of simultaneity.
Exercises 4.2, 4.3, 4.15, and 4.18.
Solutions
Geometric interpretation of boosts as rotations. Lorentz group (boosts and
rotations). Time reversal, parity, and translations. Poincare group.
Four-vectors, scalar products and basis vectors.
5: Four-velocity, four-momentum, four-acceleration, and four-force.
Generalization of Newton I and II. Exercises 5.3, 5.4, 5.6, and 5.7.
Solutions
Lagrangian and Hamiltonian
for a free particle and for a particle in an electromagnetic field.
Minimal substitution.
6: Relativistic particle in electrostatic field and precession.
Notes.
Light rays and affine parameters. Wave four-vector and Doppler effect.
Observers and observations; orthogonal set of basis vectors.
Exercise 5.21. Covariant and contravariant vectors and tensors.
Kronecker delta and raising and lowering of indices.
Field theory and Lagrangian density. Variational principle for fields and
Euler-Lagrange equation.
7: Lagrangian for a scalar field and invariance under
global phase transformations. Maxwells equations in terms of field tensor and
its dual. Gauge invariant Lagrangian for the electromagnetic field.
Local phase transformations, covariant derivative and invariant Lagrangians.
Noether's theorem and conserved current. Energy-momentum tensor. Translational
invariance and conservation of energy and momentum. Phase transformations and
conservation of charge.
8: Curves in Euclidean space. Tanget vector, normal and binormal vectors.
Fresnet's equations. Curvature and torsion.
Surfaces in Euclidean space. Vectors, tangent plane and metric. Example:
S^2.
Material.
Exercise 2.5 and 2.7. Problem one spring 2001
Solution.
9:
Covariant derivative of vectors and Christoffel symbols.
Example: plane in polar coordinates.
Parallel vectors
and parallel transport. Example from the plane (cartesian and polar
coordinates). Geodesic curves: tangent vector is parallel transported and
curves extremizing action. Christoffel symbols in terms of metric.
Problem 8.2.
10: Problem 9.2 from Lecture Notes (Parallel transport and geodesics on the
sphere). Light cones and world lines. Example 7.3. Riemann normal coordinates,
local inertial frame and example 7.2
11: Lectures cancelled ("tiltaksveke").
12: Easter holiday.
13: Problem 8.9. Symmetries and Killing vectors. Formulas for length, area...
and problem 7.14. Embedding diagrams and problem 7.20. Spacelike and null
surfaces (example 7.11 and 7.12).
14: Parallell transport and Riemann's curvature tensor. Ricci tensor and
Ricci scalar. Example: sphere. Covariant derivatives of scalars and
tensors of higher rank.Bianchi identity. Metric being covariant constant.
Einstein's field equation and the cosmological constant.
Newtonian gravity and gravitational potential for a point mass distribution.
Field equation in vacuum and Schwarzschild solution. Symmetries: Time
independence and rotational invariance. Coordinate and physical singularity.
Gravitational redshift. Radial motion and effective potential.
15: Radial orbits as functions of proper time and coordinate time.
Singular behavio(u)r of t as r approaches the Scharwschild radius
2m. Escape velocity. Stable and unstable circular orbits.
Angular velocity and Kepler's law. Problem 9.6 and 9.9.
Black holes and the horizon (Schwarzschild radius).
Eddington-Finkelstein coordinates and radial light rays.
Kruskal-Szekeres coordinates. Kruskal diagrams.
Problem 9.7 and 12.14.
16: Problem 12.22 and exam 2006 problem 2. Cosmology as observational science.
Visible matter, radiation, dark matter and dark energy. Expansion of the
Universe and Hubble's law. Isotropic and homogeneous universe models.
Cosmological redshift. Matter, radiation and vacuum energy.
Friedman's equations for scale factor a(t).
Evolution of flat FRW models.
17: Exam 2001 problem 2. Evolution of flat FRW models and deceleration
parameter. Conformal time and particle horizon.
Age and size of the universe. Spatially curved FRW models.
Problems 18.2, 18.5, and 18.11.
18: Problem 3 exam 2006. General FRM models and effective potentials.
Rotating black holes characterized by mass and angular moment.
Properties (singularities, horizons, cyclic
coordinates...)
Lecture cancelled May first (labo(u)r day).
19: Induced two-dimensional metric on surface of constant t and r.
Exercise 15.7.
Stationary observers, rotating observers and the ergosphere.
Eksamen 2007.
Fasit 2007.
Solution to 15.14.
Oppgaver uke:
2: None
3:
Oppgavesett 1.
4: Exercises from chapter four: 4.2, 4.3, 4.15, and 4.18.
5: Exercises 5.3, 5.4, 5.6, and 5.7.
6: Exercise 5.21.
7: Exercise 6.3 from lecture notes.
8: Problem one, spring 2001
Help yourself.
Exercise 2.5 and 2.7.
9: Exercise 8.2.
10:
11: No exercises.
12: Easter holiday. For those of you who are bored: Exercise 7.19 and 8.12.
Solutions.
Solutions.
13: Exercise 8.9. Exercises 7.2, 7.14, and 7.20.
14:
Solutions.
15: Problems 9.6-9.10. Problem 12.14.
16:
Problem 12.22.
Problem 2 Exam spring 2006.
Help yourself.
Exercises chapter 18: 2, 5, 7, 11, 15.
17: Problem 2 Exam spring 2001.
Help yourself.
18:
Problem 3 Exam spring 2006.
Help yourself.
19: Problem 15.7.
