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Hydrodynamics of the Vacuum

Hydrodynamics is the appropriate "effective theory" for describing any fluid medium at sufficiently long length scales. This paper treats the vacuum as such a medium and derives the corresponding hydrodynamic equations. Unlike a normal medium the vacuum has no linear sound-wave regime; disturbances always "propagate" nonlinearly. For an "empty vacuum" the hydrodynamic equations are familiar ones (shallow water-wave equations) and they describe an experimentally observed phenomenon -- the spreading of a clump of zero-temperature atoms into empty space. The "Higgs vacuum" case is much stranger; pressure and energy density, and hence time and space, exchange roles. The speed of sound is formally infinite, rather than zero as in the empty vacuum. Higher-derivative corrections to the vacuum hydrodynamic equations are also considered. In the empty-vacuum case the corrections are of quantum origin and the post-hydrodynamic description corresponds to the Gross-Pitaevskii equation. I conjecture the form of the post-hydrodynamic corrections in the Higgs case. In the 1+1-dimensional case the equations possess remarkable `soliton' solutions and appear to constitute a new exactly integrable system.