Introduction to Cosmological Models
by bob jantzen [January, March, May 1988]
University of Rome "La Sapienza", Dipartimento di Fisica, Corso di Fisica
Teorico (Ruffini)
Five part notes from lectures given in 3 separate visits.
- Part I: orthogonal coordinates on flat or constant curvature
manifolds with metric, FRW geometry, simplest spacetime splittings,
Gaussian normal coords, intrinsic and extrinsic curvature (January
1988)
(icm1.pdf: 37 pages, 1.163 MEG)
contents:
- What is Cosmology? Overview
- Flat Euclidean/Lorentz geometry for N = 2
- N > 2: (pseudo-)spherical, (pesudo-)cylindrical coordinates
- Orthogonal coordinates on M4 and constant
curvature subspaces
- Maximally symmetric spaces
- Friedmann-Robertson-Walker spacetimes
- Higher dimensions?
- Special coordinates on de Sitter spacetime
- Special coordinates on anti de Sitter spacetime
- Gaussian normal coordinates and constant curvature, Einstein
curvature
- Interpretation of extrinsic curvature
- Arbitrary signature flat spaces and pseudo-orthogonal groups
Part 1 is designed to generalize the student's knowledge of the
geometry of special relativity to the case of cosmological
spacetimes, from the point of view of metric and coordinates,
with some mention of groups of motion. Only geometries of
constant curvature are considered.
The geometry of symmetry groups and its application to less
symmetric spacetime geometries relevant in cosmological models
in 4 or more dimensions will be developed in part 2.
- Part II: Symmetries and Lie groups (March 1988)
(icm2.pdf: 29 pages, 731K)
contents:
- Generating vector fields, Killing vector fields, Lie
algebras, matrix groups
- Lie group actions (transformation groups), orbits, quotient
spaces, homogeneous space
- Lie derivatives, invariance
Part 2 is designed to cover the topic of symmetry groups, in
particular, isometry groups (groups of motions of metrics).
Metrics with transitive isometry grups (all points equivalent
under the actions of the group) are studied as homogeneous
spaces useful as spatial sections of a spacetime or as fiber
spaces in higher dimensional spacetimes or simply in ordinary
gauge theories, both of which are important in current
unification schemes. These applications will be covered in part
3.
- Part III: Differential geometry, classical mechanics, matrix
groups, rigid body dynamics
(icm3.pdf: 20 pages, 883K)
contents:
- Another glance at differential geometry: classical mechanics
and phase spaces
- Additional useful facts about matrix groups: canonical
parametrizations, invariant field computation, adjoint group,
SO(3,R) and SU(2)
- Rigid body dynamics
Part 3 studies in more detail the geometry of Lie groups and
group invariant metrics and applications to the rigid body
problem to bridge the new ideas with an elementary physics style
perspective.
- Part IV: Fiber bundles, gauge groups (May 1988)
(icm4.pdf: 20 pages, 663K)
contents:
- Loose ends from part III: active Euler angles, frame
Lagrange derivatives
- What is a fiber bundle? Why bother?
- Kinds of fiber bundles, familiar examples: tensor bundles
- Local trivializations, horizontal and vertical subspaces and
connections
- Structural group, sections
- Bundle of frames
- U(1) bundle and electromagnetism
- Nonabelian generalization
- Bundle metric, higher-dimensional Einstein equations, Weyl
transformations
- End
Part 4 gives an impression of how fiber bundles work using the
tangent and cotangent bundle examples of part 3 and a circle
bundle of electromagnetism and the bundle of frames. These are
then placed in the context of higher dimensional spacetimes and
unified field theories.
This part was written "on the fly" without notes relying on
memory so they could easily stand being rewritten someday.
Maybe.
(Someday never came...)
- Part V: Anisotropic cosmological models, gravitational dynamics
(May 1988)
(icm5.pdf: 7 pages, 222K)
contents:
- Tying things together
- Anisotropic cosmological models
- Gravitational dynamics, de Witt metric
- Generalization of Kepler and rigid body dynamics
- Bye
Part 5 tries to connect up these ideas with more general ones in
general relativity and cosmology.