Tetrad-based perturbative approach to inhomogeneous universes: A general relativistic version of the Zel’dovich approximation

A new approximation scheme in general relativity is developed to describe nonlinear inhomogeneous universes containing irrotational dust. The parallel-transported basis of the orthonormal tetrad frame is employed and a second-order differential equation is obtained for the perturbations of the spatial basis vectors, with nonlinear corrections as a source term. The equation can be solved iteratively in a way very similar to that in the Lagrangian perturbation theory in Newtonian cosmology. The first-order solution is presented, which contains Szekeres’ exact solution as a special case. A general relativistic version of the ‘‘Zel’dovich approximation’’ is proposed with emphasis on the formal similarity to the Newtonian treatment.