Friedmanns Equation A description of our Universe

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Friedmanns Equation - A description of our
Universe

• Alexander Merle (Munich University of Technology)
• (AlexanderMerle_at_web.de)
• Katharina Hübner (Leibniz-Gymnasium Altdorf)
• (katharina.huebner_at_onlinehome.de)

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Contents

• 1. Mathematical Basics
• 2. Crash Course on General Relativity
• 3. Friedmanns Equation
• 4. Different Solutions
• 5. The Parameters of our Universe

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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
old coordinates
new indices
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The metric tensor

• Easy and well-known Pythagorean Theorem
• more advanced coefficients depending on the
  coordinates (these are called metric tensor)

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Exemplum gratii The Minkowski-Metric

this gives a flat space (no dependence on the
coordinates) ? special relativity
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2. Crash Course on General RelativityEinsteins
Field Equation

• ... looks complicated but is not!

LHS geometry gs and Rs ( 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
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Energy momentum tensor of an ideal fluid
pressure
density
4-velocity
space components
time component
proper time (time in the rest system )
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3. Friedmanns EquationThe Robertson-Walker-Metri
c

Assumptions mass distribution in the universe is
in average homogeneous and isotropic
curvature (-1, 0 or 1)
scale factor (world radius)
? put this into Einsteins 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
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further assumption homogeneity isotropy ?
density and pressure depend only on time
time-component (i,k0) ?
space-components (i,k1,2,3) ?
? BIG problem only 2 equations for 3 variables
? expedient another equation (equation of state)
needed
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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
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COMBINATION OF ALL THE COMPLICATED FORMULAS
? THEN AFTER A LOT OF WORK WE FINALLY GET
FRIEDMANNS EQUATION
? this is a dynamic (?!?) equation of motion
for the scale factor / world radius R(t)
WHATS LEFT????? ? POSSIBLE SOLUTIONS AND THE
PARAMETERS FOR OUR UNIVERSE PRESENTED BY KATHARINA
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4. Different Solutions (Examples)
Parameters of Friedmanns Equation - ?
Einsteins 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!!!
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A closed universe
? the universe expands to a certain point, stops
and then starts collapsing ? R ? 0 again
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Einsteins static universe
constant radius
? static universe (?!), BUT unstable!
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The de-Sitter-Universe
de-Sitter-Universe
? the universe expands, but the expansion is
decelerated and the radius approaches a constant
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5. The Parameters of our Universe
nothing can stop the expansion
flat space
Revival of ?
t0 NOW
acceleration
Hubble-constant
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What we believe at the moment
? the universe started with the big bang and
expands eternally and even accelerated
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References Thorsten Fließbach, Allgemeine
Relativitätstheorie, Spektrum Verlag 2003 (all
pictures are from this book) Particle Data Group,
Particle Physics Booklet, CERN Library,
2004 Langenscheids Taschenwörterbuch Englisch,
Langenscheid, 1964 BOOKTIP Kip S. Thorne,
Black Holes and Time Warps Einsteins
Outrageous Legacy (German Gekrümmter Raum und
verbogene Zeit Einsteins Vermächtnis) ? best
popular book in this world about general
relativity!