IDEAL characterization of isometry classes of FLRW and inflationary spacetimes

Abstract

In general relativity, an IDEAL (Intrinsic, Deductive, Explicit, ALgorithmic) characterization of a reference spacetime metric g0 consists of a set of tensorial equations T [g] = 0, constructed covariantly out of the metric g, its Riemann curvature and their derivatives, that are satisfied if and only if g is locally isometric to the reference spacetime metric g0. The same notion can be extended to also include scalar or tensor fields, where the equations T [g, φ] = 0 are allowed to also depend on the extra fields φ. We give the first IDEAL characterization of cosmological FLRW spacetimes, with and without a dynamical scalar (inflaton) field. We restrict our attention to what we call regular geometries, which uniformly satisfy certain identities or inequalities. They roughly split into the following natural special cases: constant curvature spacetime, Einstein static universe, and flat or curved spatial slices. We also briefly comment on how the solution of this problem has implications, in general relativity and inflation theory, for the construction of local gauge invariant observables for linear cosmological perturbations and for stability analysis.

Approaches to linear local gauge-invariant observables in inflationary cosmologies

“…For constant curvature spacetimes (like de Sitter or anti-de Sitter), the components of the linearised Riemann tensor also do the job, provided its tensor indices are appropriately raised or lowered before linearisation [36,37]. Unfortunately, no simple generalisation of these results exists for an arbitrary spacetime, which explains the necessity of the works [4,6] and their present conclusion to produce the answer for spatially flat inflationary cosmologies. Any other spacetime, or family of spacetimes, like in the Bianchi homogeneous anisotropic cosmology classification, needs to be analysed separately.…”

“…the first two of which rely on u µ being proportional to ∇ µ φ. The logic behind the precise choice of the CDK tensors is explained in [6]. For our purposes, it is more convenient to replace these tensors by the slightly different set {W µνρσ ,Ẑ µν ,D µν , S, T}, where the first three tensors are defined as…”

“…Thus, in one direction we propagate the proof of completeness, and in the other direction we propagate a geometric interpretation. In Sections 2 and 3, we respectively review the relevant results from [6] and [4], reproducing the explicit formulas that we will need. In Section 4, the two sets of observables are related by explicit transformations.…”

Intrinsic, deductive, explicit, and algorithmic characterization of the Szekeres-Szafron solutions

“…He partially performed this type of analysis for the spherically symmetric spacetimes [16,17], a result we attained in two recent papers [18,19]. This kind of IDEAL characterization has also been achieved for other geometrically significant families of metrics and for physically relevant solutions of the Einstein equations [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35]. The use of the appellation IDEAL (as an acronym) seems to be adequate because the conditions obtained are Intrinsic (depending only of the metric tensor), Deductive (not involving inductive or inferential methods or arguments), Explicit (expressing the solution non implicitly) and ALgorithmic (giving the solution as a flow chart with a finite number of steps).…”

IDEAL characterization of higher dimensional spherically symmetric black holes

“…Replacing the sphere by hyperbolic space, we obtain so-called pseudo-Schwarzschild wormhole spacetimes [19]. Other IDEAL characterizations for geometries of interest in General Relativity include (4-dimensional) Schwarzschild [6], Reissner-Nordström [5], Kerr [7], Lemaître-Tolman-Bondi [9], Stephani universes [10] (see references for complete lists and details), and most recently FLRW and inflationary spacetimes (in any dimension) [2]. Of course, for completeness, we have to mention the classic cases of constant curvature spaces (cf.…”