Cylindrically symmetric dust spacetime

We present an explicit exact solution of Einstein’s equations for an inhomogeneous dust universe with cylindrical symmetry. The spacetime is extremely simple but nonetheless it has new surprising features. The universe is “closed” in the sense that the dust expands from a big-bang singularity but recollapses to a big-crunch singularity. In fact, both singularities are connected so that the whole spacetime is “enclosed” within a single singularity of general character. The big-bang is not simultaneous for the dust, and in fact the age of the universe as measured by the dust particles depends on the spatial position, an effect due to the inhomogeneity, and their total lifetime has no non-zero lower limit. Part of the big-crunch singularity is naked. The metric depends on a parameter and contains flat spacetime as a non-singular particular case. For appropriate values of the parameter the spacetime is a small perturbation of Minkowski spacetime. This seems to indicate that flat spacetime may be unstable against some global non-vacuum perturbations. Exact solutions of Einstein’s equations for an inhomogeneous dust with an Abelian G2 on S2 group of motions seem to be difficult to find, and very few of them are known, see [1] and references therein. Among them, there are some Petrov type-D known cases belonging to the general class found by Szekeres [2], some examples appear in [3, 4]. However, it seems that no exact solution for an inhomogeneous dust with cylindrical symmetry ([5] and references therein) has been singled out. In this short letter we prove that, actually, there is a very simple cylindrically symmetric dust universe included in the Szekeres family, which surprisingly has not been studied or considered hitherto. Some plausible reasons for this overlooking will also be analized: apparently, some ∗Also at Laboratori de F́ısica Matemàtica, Societat Catalana de F́ısica, IEC, Barcelona.