Astronomical Institute, Graduate School of Science, Tohoku University, Sendai 980-8578, Japan
We formulate the Lagrangian perturbation theory to solve the non-linear dynamics of self-gravitating fluid in the framework of the post-Newtonian approximation in general relativity, using the (3+1) formalism. Our formulation coincides with Newtonian Lagrangian perturbation theory developed by Buchert for scales much smaller than the horizon scale and with the gauge invariant linear theory in the longitudinal gauge conditions for the linear regime. These are achieved by using the gauge invariant quantities at the initial time when the linearized theory is valid enough. The post-Newtonian corrections in the solution of the trajectory field of fluid elements are calculated in the explicit forms. Thus our formulation allows us to investigate the evolution of the perturbations involving relativistic corrections from the early regime such as the decoupling time of matter and radiation till the present time. As a result, we are able to show that naive Newtonian cosmology to the structure formation will be a good approximation even for the perturbations with scales not only inside but also beyond the present horizon scale in the longitudinal coordinate. Although the post-Newtonian corrections are small, it is shown that they have a growing transverse mode which is not present in Newtonian theory as well as in the gauge invariant linearized theory. Such post-Newtonian order effects might produce characteristic appearances of the large-scale structure formation, for example, through the observation of anisotropy of the cosmic microwave background radiation (CMB). Furthermore since our approach has an explicit Newtonian limit, it will be also convenient for numerical implementation based on the presently available Newtonian simulation. Our results easily allow us to perform simple order estimation of each term in the solution, which indicates that the post-Newtonian correction may not be neglected in the early evolution of the density fluctuation compared with Newtonian perturbation solutions.