Long wavelength iteration of Einstein’s equations near a spacetime singularity

We analyze the behavior of a very inhomogeneous spacetime near the singularity by using the recently developed long wavelength iteration scheme of Einstein’s equations. Near the singularity, the local anisotropy cannot be neglected and we give the first order and third order solutions for any perfect fluid adiabatic index. We also clarify the links between a recently developed long wavelength iteration scheme of Einstein’s equations, the Belinski-Khalatnikov-Lifschitz (BKL) general solution near a singularity and the anti-Newtonian scheme of Tomita’s. We determine the regimes when the long wavelength or anti-Newtonian scheme is directly applicable and show how it can otherwise be implemented to yield the BKL oscillatory approach to a spacetime singularity.