1The Schwarzschild radius• Things get funny at:• To find the path of light: set ds = 0• To make it easy, set dθ= dφ= 0Î light travelling radiallyThe Schwarzschild metric:Coordinate speed of light (dr/dt)/cr =• R-W metric: most general solution for universe obeying Cosmological Principle.• Smooth distribution of matter.• Same everywhere.• Same everywhere at any given time.• Curvature• Can be found from local measurements• By bug on sphere Î Smooth curvature, same everywherePositive Curvature(K > 0)Negative Curvature(K < 0)Flat(K = 0)Curved Spaces & the Robertson-Walker Metric21RK =2Geometry of a 2D Spherical Surface ÎÎ21RK =Geometry of a 2D Spherical Surface To get R-W metric:• Add time• Add another dimensionÎÎwhere r(t) = R(t)ϖand21RK =3Dynamics of a 3D Surface (our Universe)in an Expanding 4D Space To get R-W metric:• Add time• Add another dimensionr(t) = R(t)ϖPositive Curvature(K > 0)Negative Curvature(K < 0)Flat(K = 0)222ρπ381kcRGdtdRR−=−Friedmann EquationÎDynamics of a 3D Surface (our Universe)in an Expanding 4D Space r(t) = R(t)ϖ222ρπ381kcRGdtdRR−=−• Dynamics and curvature both due to mass-energy density.• For Friedmann Eqn withoutCosmological Constant:closedflatopenoodtdRHdtdRRH==1HDynamicsCurvatureDensityCollapses backPositiveρo> ρc,oOozes to stop at t = ∞Flatρo= ρc,oExpands foreverNegativeρo< ρc,oempty4Why is the empty universe not flat??Cosmological principle Î same age since Big Bang everywhere in the
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