Revisiting geodesic observers in cosmology

Abstract

Geodesic observers in cosmology are revisited. The coordinates based on freely falling observers introduced by Gautreau in de Sitter and Einstein-de Sitter spaces (and, previously, by Gautreau and Hoffmann in Schwarzschild space) are extended to general FLRW universes. We identify situations in which the relation between geodesic and comoving coordinates can be expressed explicitly in terms of elementary functions. In general, geodesic coordinates in cosmology turn out to be rather cumbersome and limited to the region below the apparent horizon.

Friedmann-Lemaître-Robertson-Walker cosmology through the lens of gravitoelectromagnetism

“…Freely-falling observers are determined up to a Lorentz boost (e.g., [30]). In FLRW universes, it is natural to consider freely falling radial observers, to which are associated special coordinates in cosmology [18,28,29,[48][49][50]. Since the FLRW universe is approximated locally with an osculating de Sitter space, which is locally static, one can introduce Martel-Poisson observers and their special subclass, the Painlevé-Gullstrand observers [18].…”

“…where f ≡ 1−H 2 0 R 2 and p is a parameter labelling different charts (it is straightforward to check that the differential d T is exact). The physical meaning of p is obtained by writing the equation of outgoing ( Ṙ > 0) radial timelike geodesics [18,28]…”

“…The special parameter value p = 1 gives Painlevé-Gullstrand coordinates (see [28] for a discussion of different radial geodesic observers in FLRW cosmology) and it is now clear that it corresponds to vanishing initial velocity v 0 = 0 of the freely-falling observer at the origin. With p = 1, the de Sitter line element (4.15) assumes the Painlevé-Gullstrand form [18]…”

Regular Frames for Spherically Symmetric Black Holes Revisited

“…κ is the surface gravity, r * is defined in (19). Then, one can check that in variables t , τ the metric takes the form:…”

“…from which we can extract the Hubble law for the velocity of the flow V = Hr. It is known that this form is valid not only for de Sitter solution, but for an arbitrary Friedmann cosmology [19]. Note that the fact that the resulting diagonal metric (79) appears to be a homogeneous one explicitly is connected with a particular form of the function f in Equation (3) and particular value e 0 = 1 which leads to factorizable expression for r. Meanwhile, in the next section we will see that there exists another family of homogeneous metrics existing for an arbitrary function f .…”