Routes towards Emergent Gravity

We discuss a toy model for an emergent non-relativistic gravitational theory. Within a certain class of Bose-Einstein condensates, it is possible to show that, in a suitable regime, a modified version of non-relativistic Newtonian gravity does effectively describes the low energy dynamics of the coupled system condensate/quasi-particles. Furthermore, we study the role of Lorentz and diffeomorphism invariance in emergent gravity scenarios by developing a toy model showing an emergent Lorentzian signature from a Euclidean setting and simultaneously and emergent gravitational dynamics of the Nordstrom type (scalar gravity). Some lessons about the crucial challenges awaiting in the future the emergent gravity proposal are finally drawn.