Geometry of Friedmann-Lemaître-Robertson-Walker Spacetimes |
Metric
> s.a. FLRW spacetimes; cosmologies
and relativistic cosmological models [geodesics];
spherical symmetry.
* Idea:
A homogeneous and isotropic metric, characterized by one of three type
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
where f 2(χ)
= sin2χ
if k = 1, χ2
if k = 0, and sinh2χ
if k = −1.
* Conformal gauge: The line
element is of the form (using the same definitions for f(χ))
* Singularities:
For k > 0, a point; For k ≤ 0,
an infinite manifold; > s.a. singularities [metric extension].
* Useful quantities:
The Hubble expansion factor, defined as H:= a·/a;
It satisfies \(1\over6\)R = \(\ddot a\)/a + H2
= k/a2.
@ Metric and coordinates: Rindler GRG(81);
Lachièze-Rey A&A(00)ap [embedding in 5M];
Ibison JMP(07) [conformal forms],
a0704 [static form];
Grøn & Johannesen a0911-wd [conformally-flat spacetime coordinates].
@ Singularities: Gruszczak a1011 [differentially singular boundary];
het Lam & Prokopec a1606 [non-singular past-geodesically incomplete spacetimes];
Ling a1810
[k = −1 inflationary FLRW spacetimes, the big bang is a coordinate singularity].
@ Extending the spacetime: Schröter a0906;
Stoica IJTP(15)-a1112;
Belbruno CMDA(13)-a1205;
Stoica IJTP(16),
comment Fernández-Jambrina IJTP(16)-a1603;
Gielen & Turok PRD(17)-a1612 [Feynman propagator].
Connection > s.a. geodesics; holonomy;
Raychaudhuri Equation; relativistic
cosmological models [geodesics].
* In conformal time gauge:
The non-equivalent, non-vanishing components of the metric
connection / Christoffel symbols are
Γ000 = Γ011 = a·/a
Γ122 = −f f '
Γ101 = Γ202 = Γ303 = a·/aΓ022 = f 2 a·/a
Γ133 = −f f ' sin2θ
Γ212 = Γ313 = f '/fΓ033 = f 2 sin2θ a·/a
Γ233 = −sin θ cos θ
Γ323 = cot θ .
Other Geometric Quantities and Topics
> s.a. horizons; types of singularities
[including sudden singularities]; world function.
* Scalar curvature:
Using conformal time,
R = 6 \(\ddot a\)/a − 2 (2ff '' + f '2 − 1)/(af)2 .
@ References:
Ellis & van Elst gq/97-fs [geodesic deviation];
Chen & Van der Veken JMP(07) [non-degenerate surfaces];
Goulart & Novello G&C(08) [Weyl tensor, stability];
Cunningham et al PRD(17)-a1705 [exact geodesic distances].
> Related topics:
see bianchi models.
main page
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send feedback and suggestions to bombelli at olemiss.edu – modified 19 jul 2019