Equations of Gravitational Instability are Non-Local

Abstract

Few recent generations of cosmologists have solved non-local newtonian equations of the gravitational instability in an expanding universe. In this approach pancaking is the predominant form of first collapsing objects. Relativistic counterparts of these equations contain the electric and magnetic parts of the Weyl tensor. In the linear theory the magnetic part is associated with gravitational waves. If the magnetic part is ignored, then the newtonian limit of the relativistic equations is reduced to the closed set of the local Lagrangian equations. Recently this fact drew much attention since the gravitational instability in that form would greatly simplify the study of cosmic structure formation. In particular, the filamentary structure of collapsing is predicted. In this paper we resolve the contradiction between the newtonian theory and relativistic version adopted in some recent papers. We show that dropping the magnetic part from the basic relativistic equations is {\it incorrect}. The correct newtonian limit is derived by the $1/c$-expansion of the GR equations and the Bianchi identities for the Weyl tensor. The last ones begin with $\sim 1/c^3$ order, therefore one {\it must} take into account the magnetic part in the post newtonian order $\sim 1/c^3$, which contains non-local terms, related to the non-local gravitational interaction. For the first time we rigorously show that the GR equations with the magnetic part are reduced precisely to the canonic newtonian non-local equations. Thus, the correct treatment of the relativistic version of the gravitational instability resurrects the canonic picture of the structure formation.