A real valued function $\varphi$ of one variable is called a metric transform
if for every metric space $(X,d)$ the composition $d_\varphi = \varphi\circ d$
is also a metric on $X$. We give a complete characterization of the class of
approximately nondecreasing, unbounded metric transforms $\varphi$ such that
the transformed Euclidean half line $([0,\infty),|\cdot|_\varphi)$ is Gromov
hyperbolic. As a consequence, we obtain metric transform rigidity for roughly
geodesic Gromov hyperbolic spaces, that is, if $(X,d)$ is any metric space
containing a rough geodesic ray and $\varphi$ is an approximately
nondecreasing, unbounded metric transform such that the transformed space
$(X,d_\varphi)$ is Gromov hyperbolic and roughly geodesic then $\varphi$ is an
approximate dilation and the original space $(X,d)$ is Gromov hyperbolic and
roughly geodesic.
