Constant Gaussian curvature, FLRW conditions and a universe without matter

Abstract

The problem of FLRW universes can be seen in the excellent works of A. Friedmann, G. Lemaître, H. P. Robertson, A. G. Walker. The key of obtaining FLRW universes with or without cosmological constant is related to the special form of the contravariant energy–momentum tensor which describes a perfect fluid with components [Formula: see text] because such a “matter” leads to FLRW conditions [Formula: see text] which allow to obtain a texture compatible with Hubble law. The results obtained in the works presented as references show very complete analysis about how the constant curvature 2D metric [Formula: see text] can create possible universes and their evolution. Can we expect to have some other FLRW universes if we use a 2D spatial part as [Formula: see text]? The answer is yes and we prove the special role of the constant ratio [Formula: see text] in the creation of FLRW universes. The solutions obtained lead to FLRW universes which come from surfaces having constant Gaussian curvature related by the previous [Formula: see text] from the rule [Formula: see text]. Therefore, in the case of the 2D metrics [Formula: see text], only the ones having constant Gaussian curvature create FLRW universes. In fact, we find all possible FLWR-type solutions of Einstein field equations coming from constant Gaussian curvature geometries. We complete our study about possible FLRW universes with an example of a spacetime that cannot be “filled” with matter using the above FLRW conditions. This becomes a new example of universe without matter satisfying the Einstein’s field equation with cosmological constant.