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
y
¨
+αf(y)
y
˙
+βf(y)∫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=b
y
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.