The Friedmann-Equation - A description of our Universe

Friedmann’s Equation
- A description of our Universe
Alexander Merle (Munich University
of Technology)
([email protected])
&
Katharina Hübner (LeibnizGymnasium Altdorf)
([email protected])
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 Basics
What the hell is a tensor???
An ordered set of numbers, which is
characterized by its behavior in coordinate
new coordinates
transformations:
Sum
convention!!!
→ sum over ‘i’
and ‘k’
new indices
old coordinates
old indices
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 Relativity
Einstein’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 Equation
The 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 timeindependent 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:
- Λ : 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
acceleration
‘Revival’ of Λ
t : NOW
flat space
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!