Analogue gravity simulation of superpositions of spacetimes

Taking the principles of quantum mechanics as they stand and applying them to gravity, leads to the conclusion that one might be able to generate superpositions of spacetimes, at least formally. We analyze such a possibility from an analogue gravity perspective. We present an analogue toy model consisting of a Bose-Einstein condensate in a double-well potential and identify the states that could potentially be interpreted as superposition of effective spacetimes. These states are unstable and the source of instability from a microscopic point of view can be related to the absence of a well-defined causal structure in the effective geometric description. We explore the consequences of these instabilities and argue that they resonate with Penrose's ideas about the decay that superpositions of states with sufficiently different gravitational fields associated should experience.