Invariance transformations of the class y″ = F(x) y ^N of differential equations arising in general relativity

Abstract

We have found all transformations {x, y, F} → {$\tilde x,\tilde y,\tilde F$} which leave the class y″ = F(x) y ^N of differential equations form-invariant (and are elements of a Lie group) by constructing their generators (Theorem 1). The corresponding finite transformations are written down.

For N = 1 (i.e. for the linear case y″ = F(x) y), these transformations are not of much interest and value. Besides scalings and the linear superposition of solutions, they consist of transformations which transform any two functions F(x) and $\tilde F(\tilde x)$into each other - but to map, say y″ = F(x) y into $\tilde y''$= 0, one has to solve exactly the original differential equation.

For N ≠ 0, 1, 2 the general transformation can be explicitely given (Theorem 3).

The most interesting case is N = 2. Here the transformation law includes a kind of superposition principle; again, the finite transformations could be explicitely given. It turns out that in the vast literature on this case N = 2, the transformation method has in fact been used in a disguised form, without reference to Lie groups.