Vacuum Hydrodynamics

The Higgs vacuum – with its constant background field – is not ‘empty’ but is a kind of medium. Any ordinary medium, viewed at sufficiently long length scales, has a hydrodynamic description and can propagate sound waves. The vacuum medium is unusual; the speed of sound is formally infinite and there is no linear sound-wave regime. Instead, long-wavelength disturbances are described by some intrinsically nonlinear hydrodynamic equations. The purpose of this Letter is more to raise questions than to answer them. The premise is that the vacuum of the Standard Model, with its non-zero background field 〈φ〉 6 = 0, is a kind of medium [1]. The only other assumption is that a “mean free path” scale λmfp exists and is finite. (While λmfp must be long compared to particle-physics length scales, whether it is a millimeter or an astronomical or cosmological scale is another question.) My point is that at sufficiently long length scales L ≫ λmfp any fluid medium can be described by hydrodynamics (see, for example, [2, 3, 4]). That is, it can be treated as an ideal fluid up to corrections of order λmfp/L [5]. In the hydrodynamic regime the only relevant modes are those directly linked to conservation laws. In particular, energy-momentum conservation equations lead directly to longitudinal pressure waves — sound waves. Let us briefly review the derivation. The energy-momentum tensor for a perfect fluid has the form [c = 1, gμν = diag(1,−1,−1,−1)] T μν = (E + P )uu − Pg , (1) where E is the energy density, P is the pressure, and uμ is the flow 4-velocity u = (u0, ~u), with u 2 0 − ~u = 1. (2) The energy-momentum conservation equation ∂μT μν = 0 yields ∂ ∂t [ (E + P )u0 ] + ∂ ∂xj [ (E + P )uu0 ] − ∂P ∂t = 0, (3) ∂ ∂t [ (E + P )u0u ] + ∂ ∂xj [ (E + P )uu ] + ∂P ∂xi = 0. (4) Consider a small perturbation about a static, homogeneous equilibrium state E0, P0, ~u = 0. If the pressure variation δP and the energy variation δE are sufficiently small they must be proportional, with their ratio fixed by the thermodynamic derivative ∂P ∂E ∣