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

Many different forms of the de Sitter metric in different coordinate systems are used in the general relativity literature. Two of them are the most common, the static form and the cosmological (exponentially expanding) form. The staticity and non-stationarity of these two different forms are traced back to the noncomoving and comoving nature of the corresponding coordinate systems. In this paper using the quasi-Maxwell form of the Einstein field equations and a definition of static spacetimes based upon them, we look at these two different forms of the same solution from a new perspective which classifies them as a special case in a general one-parameter family of solutions. Specifically it is proved that, {\it irrespective of the spacetime symmetry, a one-element perfect fluid in any frame noncomoving with the fluid could be the source of a static spacetime, only if its equation of state is that of a dark fluid namely $p = -\rho = const.$}. These static solutions, which include the well-known de Sitter spacetime, are called de Sitter-type spacetimes. To exemplify we consider static axially and cylindrically symmetric de Sitter-type spacetimes and their dynamic (cosmological) versions. It is shown how despite the seemingly natural expectation based on the presence of $\Lambda$ as their only parameter, the nonspherical expansions of these genuinely different solutions should be expected indeed. To the best of our knowledge the dynamic version of the cylindrically symmetric de Sitter-type spacetime is introduced here for the first 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 in de Sitter-type spacetimes.