  The light-cone spreading of entanglement and correlation is a fundamental and
ubiquitous feature of homogeneous extended quantum systems. Here we point out
that a class of inhomogenous Luttinger liquids (those with a uniform Luttinger
parameter $K$) at low energy display the universal phenomenon of curved light
cones: gapless excitations propagate along the geodesics of the metric
$ds^2=dx^2+v(x)^2 d\tau^2$, with $v(x)$ being the calculable spatial dependent
velocity induced by the inhomogeneity. We confirm our findings with explicit
analytic and numerical calculations both in- and out-of-equilibrium for a
Tonks-Girardeau gas in a harmonic potential and in lattice systems with
artificially tuned hamiltonian density.
