Form Invariance of Differential Equations in General Relativity

Abstract

Einstein equations for several matter sources in Robertson-Walker and Bianchi I type metrics, are shown to reduce to a kind of second order nonlinear ordinary differential equation $\ddot{y}+αf(y)\dot{y}+βf(y)\int{f(y) dy}+γf(y)=0$. Also, it appears in the generalized statistical mechanics for the most interesting value q=-1. The invariant form of this equation is imposed and the corresponding nonlocal transformation is obtained. The linearization of that equation for any $α, β$ and $γ$ is presented and for the important case $f=by^n+k$ with $β=α^2 (n+1)/((n+2)^2)$ its explicit general solution is found. Moreover, the form invariance is applied to yield exact solutions of same other differential equations.