The Friedmann-Lemaître-Robertson-Walker metric

The Friedmann-Lemaître-Robertson-Walker (FLRW) metric used to describe the cosmic spacetime is based on the cosmological principle, which assumes homogeneity and isotropy throughout the Universe. It also adopts free-fall conditions via the selection of a constant lapse function, gtt = 1, regardless of whether or not the chosen energy-momentum tensor Tαβ produces an accelerated expansion. This is sometimes justified by arguing that one may shift the gauge, if necessary, transforming the time dt to a new coordinate dt′g ttdt, thereby re-establishing a unitary value for gt′t′. Previously, we have demonstrated that this approach is inconsistent with the Friedmann equations derived using comoving coordinates. In this paper, we advance this discussion significantly by using the Local Flatness Theorem in general relativity to prove that gtt in FLRW is inextricably dependent on the expansion dynamics via the expansion factor a(t), which itself depends on the equation-of-state in Tαβ. One is therefore not free to choose gtt arbitrarily without ensuring its consistency with the energy-momentum tensor. We prove that the use of FLRW in cosmology is valid only for zero active mass, i.e. ρ + 3p = 0, where ρ and p are, respectively, the total energy density and pressure in the cosmic fluid.