Why the Rh = ct cosmology is unphysical and in fact a vacuum in disguise like the milne cosmology

Abstract

Recently, Melia and his coworkers have proposed the so-called R_h = ct cosmology where the scale factor of the universe is a(t) ∝ t and the spatial part is flat. This model also implies a fixed equation of state (EOS) ρ = −(1/3)ρc^2 of the universe. Here, we look at this proposal from a fundamental angle. First, we note that Melia cosmology looks strikingly similar to the old Milne cosmology where a(t) ∝ t and the spatial part is negatively curved. It is known that though Milne cosmology is a valid Friedmann solution, it actually corresponds to ρ = 0 and can be described by a globally static Minkowski metric. Secondly, we note that for the Melia model, Ricci & Kretschmann scalars assume their perfect static form hinting that it too may tacitly correspond to vacuum. This is also necessitated by the fact that even at the moment of ‘big bang’, Melia cosmology EOS is ρ = −(1/3)ρc^2 rather than the radiation-dominated ρ = +(1/3)ρc^2. To compare Melia universe with the Milne universe, we express Melia metric too in curvature/Schwarzschild coordinates. Finally, by using the fact for such coordinate transformations dx′^4 = Jdx^4, where J is the appropriate Jacobian, we explicitly show that Melia metric is static, which for k = 0 case implies vacuum. This shows that even apparently meaningful general relativistic solutions could be illusory as far as physical reality is concerned.