While driven interacting quantum matter is generically subject to heating and scrambling, certain classes of systems evade this paradigm. I will discuss such an exceptional class in periodically driven critical (1 + 1)-dimensional systems with a spatially modulated, but disorder-free time evolution operator. Instead of complete scrambling, the excitations of the system remain well-defined. Their propagation is analogous to the evolution along light cones in a curved space-time defined by Schwarzschild black holes. The Hawking temperature serves as an order parameter which distinguishes between heating and non-heating phases. Beyond a time scale determined by the inverse Hawking temperature, excitations are absorbed by the black holes resulting in a singular concentration of energy at their center. I will discuss how these results can be obtained analytically within conformal field theory and complementary by means of numerical calculations for an interacting XXZ spin-1/2 chain. Finally, I will show that much of this phenomenology survives when the driving is quasiperiodic instead of periodic, thus proving to be surprisingly robust.