The Geometry of Spacetime

General Relativity permits many possible Universes each with a unique geometry. The overall geometry of spacetime depends upon the total density of matter in the Universe. There is some critical density, ρ_c, for which the Universe has Euclidean geometry. A Euclidean Universe would resemble a flat, infinite (although 4-dimensional) sheet.

If the ratio of the Universe's total matter density to the critical density, denoted Ω₀ , is greater than 1, the Universe is closed and has positive curvature like a sphere. On such a surface the angles of a triangle sum up to be greater than 180° and parallel geodesics eventually converge. Alternatively, if Ω₀ is less than 1, the Universe has negative curvature like a saddle. In a hyperbolic Universe the angles of a triangle sum to less than 180° and parallel geodesics diverge.