I went to a lecture where the speaker explained that people could study space and cosmology and curved space by using BEC states.
She tole me that the sound wave (mechanic wave) in BEC was equivalent to light (energy wave) in space. And an analogy could be sued to study black hole (Observation of quantum Hawking radiation and its entanglement in ananalogue black hole by Jeff Steinhauer https://www.nature.com/news/artificial-black-hole-creates-its-own-version-of-hawking-radiation-1.20430 )
My question was that:
Could you gave me the exact transformation/relation between BEC and curved space and the invariance relation between them? How come the metric for mechanic wave in BEC the same as propagation of space-time in the curved space?
Were there any other materials that behaves with the similar properties?