# Conformal factors and light rays

Suppose $(\mathcal{M},g)$ is a $3$-dimensional Riemannian manifold and let $\gamma \in \mathcal{M}$ denote an arbitrary curve in $\mathcal{M}$. Does there exist a conformal factor $c>0$ such that $cg = (dx^1)^2+ (dx^2)^2+(dx^3)^2+O_{ab}(|x'|^{\infty})dx^adx^b$ where $(x^1,x^2,x^3):=(x^1,x')$ denotes the Fermi coordinates about $\gamma$? If not, are there classes of curves for which this property holds? thanks,