Geometry of Robertson-Walker Spacetimes

Metric > s.a. cosmologies and relativistic cosmological models; spherical symmetry; world function.
* Idea: A homogeneous and isotropic metric, characterized by one of three types of 3D constant curvature spatial geometries (spatially open k = –1, spatially flat k = 0, or spatially closed k = 1), and an arbitrary function a(t) representing the fiducial size of the universe at time t.
* Proper time gauge: The line element is of the form

ds² = –d² + a()² [d² + f ²() (d² + sin² d²)] = –dt² + a(t)² [dr²/(1–kr²) + r² d²] ,

where f ²() = sin² if k = 1, ² if k = 0, and sinh² if k = –1.
* Conformal gauge: The line element is of the form (using the same definitions for f())

ds² = a²(t) [–dt² + d² + f ²() d²] .

* Singularities: For k > 0, a point; For k 0, an infinite manifold.
* Useful quantities: Hubble expansion factor H:= a^·/a; Satisfies (1/6) R = a^··/a + H² = k/a².
@ Metric: Rindler GRG(81); Lachièze-Rey A&A(00)ap [embedding in ⁵M]; Ibison JMP(07) [conformal forms], a0704 [static form].

Connection > s.a. holonomy; relativistic cosmological models [geodesics].
* In conformal time gauge: The non-equivalent, non-vanishing ones are

000 = 011 = a·/a 122 = –f f ' 101 = 202 = 303 = a·/a

022 = f 2 a·/a 133 = –f f ' sin2 212 = 313 = f '/f

033 = f 2 sin2 a·/a 233 = –sin cos 323 = cot .

Other Geometric Quantities and Topics > s.a. types of singularities [including sudden].
* Scalar curvature: Using conformal time,