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 $\ddot{y}+\alpha f(y)\dot{y}+\beta f(y)\int f(y)dy+\gamma f(y)=0$. 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 $\alpha ,\beta$ and $\gamma$ is presented and for the important case $f=b{y}^n+k$ with $\beta ={\alpha }^2(n+1)/((n+2{)}^2)$ its explicit general solution is found. Moreover, the form invariance is applied to yield exact solutions of same other differential equations.

22 pages, RevTeX; to appear in J. Math. Phys

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