Observer-based invariants for cosmological models

Abstract

We consider the equivalence problem for cosmological models in fourdimensional gravity theories. A cosmological model is considered as a triple (M, g, u) consisting of a spacetime (M, g) and a preferred normalized time-like vector field u tangent to a congruence of fundamental observers. We introduce a modification of the Cartan-Karlhede algorithm by restricting to frames adapted to u and including the covariant derivatives of u along with the Riemann tensor and its covariant derivatives. To fix the frame we make use of quantities relative to the fundamental observers, such as the anisotropic pressure tensor, energy flux vector, electric and magnetic parts of the Weyl tensor and the kinematical quantities of u. This provides a simpler way to construct a list of invariants relative to the fundamental observers that completely characterizes the model, independent of coordinates. As an illustration of the algorithm, we consider several well-known cosmological models from General Relativity.

EBWeyl: a code to invariantly characterize numerical spacetimes

“…More general sets of invariants were then considered in [35,38]. Recently, the specific equivalence problem for cosmological models has been addressed in [40]; in [41] a more refined classification for Petrov type I spacetimes has been proposed. For a classical and rather detailed account of invariants and the characterization of spacetimes we refer the reader to [39].…”

“…In conclusion, E µν and B µν and the scalars built from them can be useful for the physical interpretation of simulations and for the comparison of different codes, as well as for comparison of the fully nonlinear results of a simulation with the result of perturbation theory. Although our codes can compute E µν and B µν in any spacetime and in any frame, for the case of cosmology and especially for comparison with perturbation theory results E µν{u} and B µν{u} , computed in the matter rest frame, seem particularly useful [40].…”

“…Physical interpretations and simulation comparisons then need to be based on invariants [9,19,32], i.e. scalars independent from the coordinate choice: these are quantities characterizing the spacetime [28,[33][34][35][36][37][38][39][40][41] and should be, at least in principle, observable, i.e. measurable quantities [42].…”

On the equivalence of spacetimes, the Cartan-Karlhede algorithm

“…In spite of this dramatic example, in several families of spacetimes one can distinguish its members by looking at the curvature invariants, examples of which are given in Refs. [3,4]. In addition, it is interesting that more recently it has been proposed that the event horizons of black holes could be identified by the vanishing of certain curvature invariants [5].…”