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