  We study the the nonlinear Klein-Gordon (NLKG) equation on a manifold $M$ in
the nonrelativistic limit, namely as the speed of light $c$ tends to infinity.
In particular, we consider an order-$r$ normalized approximation of NLKG (which
corresponds to the NLS at order $r=1$), and prove that when $M=\mathbb{R}^d$,
$d \geq 2$, small radiation solutions of the order-$r$ normalized equation
approximate solutions of the nonlinear NLKG up to times of order
$\mathcal{O}(c^{2(r-1)})$.
