Relativistic Zel’dovich approximation in a spherically symmetric model

We compare relativistic approximation methods, which describe gravitational instability in the expanding universe, in a spherically symmetric model. Linear perturbation theory, second-order perturbation theory, the relativistic Zel'dovich approximation, and the relativistic post-Zel'dovich approximation are considered and compared with the Lema\^{}tre-Tolman-Bondi solution in order to examine the accuracy of these approximations. We consider some cases of inhomogeneous matter distribution while the homogeneous top-hat model has been usually taken in the previous Newtonian works. It is found that the Zel'dovich-type approximations are generally more accurate than the conventional perturbation theories in the weakly nonlinear regime. The applicable range of the Zel'dovich-type approximations is also discussed.