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## How does the universe expand?

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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**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**Causality**Space-time according to O2: The guy who is running from O1 to the TV Space-time according to O1: The guy who’s watching TV**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!): • Mathematicagives: only 2 independent equations: k = 0 Flat space k = +1 Spherical space k = -1 Hyperbolical space**Conclusion: How Does the universe expand?**Homogeneity & Isotropy: Gravity: Equation of state: Special relativity demands v < c if causality should hold Does not apply for distant galaxies, since communication is impossible