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How does the universe expand?
• What is expansion?• Newtonian derivation of FRW-equations• Problems with FRW: causality• General Relativity derivation of FRW-
equations
What is expansion?
Newtonian Derivation
FRW-equations
Special Relativity:
2 postulates:
• The laws of physics are the same for all observers
• The speed of light is equal to the same constant for all observers
CausalitySpace-time according to O1:The guy who’s watching TV
Space-time according to O2:The guy who is running from O1 to the TV
The OPERA-experiment
GR-derivation of FRW-equations
• Events given by 4-vectors • A tensor, like , is a generalised matrix• Metric tensor defines distances• Indices are raised/lowered by and it’s matrix
inverse: • Energy momentum tensor is
• Einstein’s equation (16 equations!) is:
Homogeneity & Isotropy demand:
Flat space, k = 0 and sum = 180°
Spherical space, k = +1, sum > 180°
Hyperbolical space, k = -1, sum < 180°
The calculation
• Put this into Einstein’s equation (16!):
• Mathematica gives: only 2 independent equations:
k = 0 Flat spacek = +1 Spherical spacek = -1 Hyperbolical space
Conclusion: How Does the universe expand?
Homogeneity & Isotropy:
Gravity:
Equation of state:
Special relativity demands v < c if causality should holdDoes not apply for distant galaxies, since communication is impossible