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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:

  1. 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?

  2. Were there any other materials that behaves with the similar properties?

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This field is usually referred to as analogue gravity: attempts to model various phenomena of general relativity (black holes, cosmological models) with other physical systems (acoustical, hydrodynamical, optical, etc.).

For a review have a look at:

  • Barcelo, Carlos, Stefano Liberati, and Matt Visser. Analogue gravity. Living reviews in relativity 14.1 (2011): 3, open access doi, pdf.

Could you gave me the exact transformation/relation between BEC and curved space

The effective metric $g_{\mu\nu}$ is written in terms of BEC quantum field condensate in eq. 255 of the above review (and 3 preceding pages for the explanation).

How come the metric for mechanic wave in BEC the same as propagation of space-time in the curved space?

In general, the effective metrics are not the same as the metrics in (vacuum) general relativity. But by carefully choosing the configuration of the BEC (or other system) one can mimic the most interesting features of GR solutions (such as event horizons).

any other materials that behaves with the similar properties

On a classical level curved background could be modeled by propagation of

  • sound waves in moving fluids (acoustics),
  • surface waves in shallow water (gravity waves),
  • light in dielectric media (optics).

on a quantum level (so that one can model quantum phenomena in curved space-time like Hawking radiation) the following systems have been investigated:

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I think the start of this game is W. G. Unruh, “Experimental black hole evaporation”, Phys. Rev. Lett. 46 (1981) 1351. If you do not have access to Phys. Rev. Lett, there is is a Wikipedia article on the "Acoustic metric." There is also a survey of the field by Matt Visser https://arxiv.org/pdf/1206.2397.pdf

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