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Many different forms of the de Sitter metric in different coordinate systemsare used in the general relativity literature. Two of them are the most common,the static form and the cosmological (exponentially expanding) form. Thestaticity and non-stationarity of these two different forms are traced back tothe noncomoving and comoving nature of the corresponding coordinate systems. Inthis paper using the quasi-Maxwell form of the Einstein field equations and adefinition of static spacetimes based upon them, we look at these two differentforms of the same solution from a new perspective which classifies them as aspecial case in a general one-parameter family of solutions. Specifically it isproved that, {\it irrespective of the spacetime symmetry, a one-element perfectfluid in any frame noncomoving with the fluid could be the source of a staticspacetime, only if its equation of state is that of a dark fluid namely $p =-\rho = const.$}. These static solutions, which include the well-known deSitter spacetime, are called de Sitter-type spacetimes. To exemplify weconsider static axially and cylindrically symmetric de Sitter-type spacetimesand their dynamic (cosmological) versions. It is shown how despite theseemingly natural expectation based on the presence of $\Lambda$ as their onlyparameter, the nonspherical expansions of these genuinely different solutionsshould be expected indeed. To the best of our knowledge the dynamic version ofthe cylindrically symmetric de Sitter-type spacetime is introduced here for thefirst time. Finally it is noted that the identification of the geometric term$\Lambda g_{ij}$ with a perfect fluid with equation of state $p = -\rho =const.$, although mathematically consistent, obscures the crucial role of the(dark) fluid's velocity in defining a preferred (comoving) coordinate system inde Sitter-type spacetimes.

Dark fluid or cosmological constant : Why there are different de Sitter-type spacetimes