Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.mrao.cam.ac.uk/~cjld1/pages/geometryandcosmology.pdf Дата изменения: Mon Mar 24 21:27:12 2003 Дата индексирования: Tue Oct 2 07:55:41 2012 Кодировка: Поисковые слова: п п п п п п п п

Geometry and Cosmology
Chris Doran Anthony Lasenby www.mrao.cam.ac.uk/~clifford


Non-Euclidean Geometry
· Axioms of geometry date back to Euclid's Elements · Among these is the parallel postulate:
­ Given a line l and a point P not on l, there exists a unique line m in the same plane as P and l which passes through P and does not meet l

· Non-Euclidean geometry arises by removing the uniqueness requirement
Geometry and Cosmology


Non-Euclidean Geometry
· Developed by Lobachevskii (17921856) and Bolyai (1802-1860) · In modern terminology this defines hyperbolic geometry · A homogeneous, isotropic, unbounded space of constant negative curvature · An elegant view of this geometry was constructed by PoincarИ (1854-1912)
Geometry and Cosmology


PoincarИ Disc
· Points contained in a disc of unit radius · Boundary of the disk represents set of points at infinity · Lines (geodesics) are represented by circles which intersect the unit circle at right angles · All geodesics through the origin are straight lines (in Euclidean sense)
Geometry and Cosmology


PoincarИ Disc
Set of lines through A which miss l Disc

l

Right-angle intersection

Plot constructed in Cinderella
Geometry and Cosmology


Distance
· The metric in the disc representation is

· This is a conformal representation ­ only differs from flat by a single factor · Distortions get greater as you move away from the centre · Can define tesselations
Geometry and Cosmology


Geometry and Cosmology

Circle Limit 3 M. Escher


de Sitter Space
· Now suppose that the underlying signature is Lorentzian · Construct a homogeneous, isotropic space of constant negative curvature · This is de Sitter space · 2D version from embedding picture

Geometry and Cosmology


Embedding View

time

null geodesic ­ straight line in embedding space
Geometry and Cosmology

space


Lorentzian View

Circle mapped onto a line via a stereographic projection. Extend out assuming null trajectories are at 45o
Geometry and Cosmology


Lorentzian View
`Perpendicular' intersection Timelike geodesic Boundary Hyperbolae Spacelike geodesic Null cone Always at 45
Geometry and Cosmology

o


The Cosmological Constant
· Start with FRW equations

· Introduce the dimensionless ratios

Geometry and Cosmology


The Cosmological Constant
· Write · Evolution equations now

· Define trajectories via

Geometry and Cosmology


Cosmic Trajectories
Dust de Sitter phase Radiation

Big Bang
Geometry and Cosmology


The de Sitter Phase
· End of the universe enters a de Sitter phase · Should really be closed for pure de Sitter · Only get a natural symmetric embedding onto entire de Sitter topology if

· Says that a photon gets ј of the way across the universe
Geometry and Cosmology


A Preferred Model
Current Observations Dust Arrive at a model quite close to observation For dust (=0) predict a universe closed at about the 10% level
Geometry and Cosmology

Critical


Summary
· de Sitter geometry is a natural extension of non-Euclidean geometry · Has a straightforward construction in a Lorentzian space · Can form a background space for a gauge theory of gravity · Appears to pick out a preferred cosmological model · But is this causal?
Geometry and Cosmology