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Friedmann’s Equation - A description of our Universe. Alexander Merle (Munich University of Technology) (AlexanderMerle@web.de) & Katharina Hübner (Leibniz-Gymnasium Altdorf) (katharina.huebner@onlinehome.de). Contents:. 1. Mathematical Basics 2. Crash Course on General Relativity
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Friedmann’s Equation - A description of our Universe Alexander Merle (Munich University of Technology) (AlexanderMerle@web.de) & Katharina Hübner (Leibniz-Gymnasium Altdorf) (katharina.huebner@onlinehome.de)
Contents: 1. Mathematical Basics 2. Crash Course on General Relativity 3. Friedmann’s Equation 4. Different Solutions 5. The Parameters of our Universe
1. Mathematical BasicsWhat the hell is a tensor??? An ordered set of numbers, which is characterized by its behavior in coordinate transformations: new coordinates Sum convention!!! sum over ‘i’ and ‘k’ old indices new indices old coordinates
The metric tensor Easy and well-known: Pythagorean Theorem more advanced: coefficients depending on the coordinates (these are called ‘metric tensor’)
Exemplum gratii: The Minkowski-Metric this gives a flat space (no dependence on the coordinates) special relativity
2. Crash Course on General RelativityEinstein’s Field Equation ... looks complicated but is not! LHS: geometry g’s and R’s (= functions of the metric g) gravitation in form of curvature RHS: PHYSICS!!! T: ‘energy momentum tensor’ (contains the complete energy of the system) every energy contributes to the gravitation
Energy momentum tensor of an ideal fluid pressure density 4-velocity space components time component proper time (time in the rest system )
3. Friedmann’s EquationThe Robertson-Walker-Metric Assumptions: mass distribution in the universe is in average homogeneous and isotropic curvature (-1, 0 or +1) scale factor (‘world radius’) put this into Einstein’s equation and combine it with the energy-momentum-tensor of an ideal fluid with time-independent pressure and density ‘equation of motion’ for the whole universe
further assumption: homogeneity & isotropy density and pressure depend only on time time-component (i,k=0) space-components (i,k=1,2,3) ‘BIG’ problem: only 2 equations for 3 variables expedient: another equation (equation of state) needed
Separate pressure into two parts: pressure by matter and by radiation two equations of state two equations of state and combine that with the two previous equations two ‘conservation laws’ constant quantities combine that with the physical constant to get rid of the ‘disturbing’ physical units:
COMBINATION OF ALL THE COMPLICATED FORMULAS THEN AFTER A LOT OF WORK WE FINALLY GET FRIEDMANN’S EQUATION this is a dynamic (?!?) “equation of motion” for the scale factor / world radius R(t) WHAT’S LEFT????? POSSIBLE SOLUTIONS AND THE PARAMETERS FOR OUR UNIVERSE PRESENTED BY KATHARINA
4. Different Solutions (Examples) Parameters of Friedmann’s Equation: - : Einstein’s Cosmological Constant has an influence on the development of the universe for high values of R(t) - k: curvature (+1, 0 or -1) describes the geometrical properties of the whole space HERE: some examples for all solutions we get: dR/dt as R 0 BIG BANG!!!
A closed universe the universe expands to a certain point, stops and then starts collapsing R 0 again
Einstein’s static universe constant radius static universe (Λ!), BUT: unstable!
The de-Sitter-Universe de-Sitter-Universe the universe expands, but the expansion is decelerated and the radius approaches a constant
5. The Parameters of our Universe nothing can stop the expansion flat space ‘Revival’ of Λ t0: NOW acceleration Hubble-constant
What we believe at the moment the universe started with the big bang and expands eternally and even accelerated
References: Thorsten Fließbach, Allgemeine Relativitätstheorie, Spektrum Verlag 2003 (all pictures are from this book) Particle Data Group, Particle Physics Booklet, CERN Library, 2004 Langenscheid’s Taschenwörterbuch Englisch, Langenscheid, 1964 BOOKTIP: Kip S. Thorne, ‘Black Holes and Time Warps: Einstein’s Outrageous Legacy’ (German: ‘Gekrümmter Raum und verbogene Zeit: Einsteins Vermächtnis’) best popular book in this world about general relativity!