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 null geodesics of the metric ds(2) = d x(2) - v (x)(2)d t(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.