ations in what follows. For a three dimensional super- solid there are conservation laws for mass, three compo- nents of momentum, and energy; however, by ignoring thermal fluctuations we can omit the energy conserva- tion law. In addition the conservation laws, there are three broken translational symmetries (due to the crys- tallinity) and one broken gauge symmetry (due to the Bose-Einstein condensation). Thus, a three dimensional supersolid without thermal fluctuations has eight con- servation laws and broken symmetries (nine, if conser- vation of energy is included). Correspondingly, there are eight hydrodynamic modes: two pairs of the ordi- nary transverse propagating modes, a pair of longitudi- nal first sound modes, and a pair of longitudinal second sound modes (note that a solution of the hydrodynamic equations with dispersion ω = ± ck would count as two modes–a pair of modes, with one propagating and a sec- ond counter-propagating). The appearance of the longi- tudinal second sound modes is one of the key signatures of a supersolid. We start by establishing the notation for the small fluc- tuations away from equilibrium. The equilibrium value of the densities will be denoted with a subscript of ‘0’, and the density fluctuations will be denoted by δρ , so that ρ = ρ$_{0}$ + δρ and ρ$_{s}$$_{ij}$ = ρ$_{s}$$_{0}$$_{ij}$ + δρ$_{s}$$_{ij}$ . For the lattice fluc- tuations, let u denote the (small) deformation field away from the unstrained solid (i.e. the difference between the Lagrangian and the Eulerian coordinates): Then the deformation tensor becomes where w$_{ij}$ ≡ ∂$_{i}$u$_{j}$ . Finally, the velocity of the normal component is obtained by linearizing the inverted Lin’s constraint, Eq. (23), so that to lowest order in the strain field the normal velocity is the time derivative of the dis- placement field, Expanding the Lagrangian density, Eq. (21), up to second order in the small quantities δρ , δρ$_{s}$$_{ij}$ , w$_{ij}$ , ∂$_{i}$φ and ∂$_{t}$φ , we obtain where ρ$_{n}$$_{0}$$_{ij}$ ≡ ρ$_{0}$$_{ij}$ − ρ$_{s}$$_{0}$$_{ij}$ and the thermodynamic re- lation for f , Eq. (22), is used. In the above expansion we have dropped constants which do not contribute to and from Eq. (28) we obtain The minus sign is necessary so that the total defect den- sity is conserved–in the bulk of the crystal vacancies and interstitials are created and destroyed in pairs (ignoring surface effects). Then we have From Eq. (28) it is clear that the density fluctuation is an independent hydrodynamic variable–it is not slaved to the lattice deformation, as would be the case for a commensurate solid, where δρ = − ρ$_{0}$ ∇ · u . Indeed, in a real (incommensurate) crystal a density fluctuation can be produced by lattice deformations or by point defects (vacancies and interstitials). To highlight the role of de- fects we will introduce the defect density fluctuation δρ$_{Δ}$ as our independent hydrodynamic variable, instead of δρ . The local defect density is defined as the difference be- tween the density of vacancies, ρ$_{V}$ , and the density of interstitials, ρ$_{I}$ : where we have used the linearized Josephson equation ( ∂$_{t}$φ = − µ$_{0}$ − δµ ) and the identity Now we are in a position to study the hydrodynamic modes of a supersolid. The quadratic Lagrangian den- sity, Eq. (27), produces three linearized equations of mo- tions which are equivalent to the continuity equation, the Josephson equation and the momentum conserva- tion equation. Before proceeding further, it is useful to rewrite the quadratic Lagrangian density in terms of the defect density fluctuation by using one of the equations of motion. By varying the action with respect to δρ we obtain the equations of motion, and have neglected terms pro- portional to ∂f /∂ρ$_{s}$$_{ij}$ because they are of higher order [see Eq. (12)]; consequently, the quadratic Lagrangian turns out to be independent of fluctuations of the su- perfluid density. However, we have kept the first two terms in Eq. (27); although they are total derivatives and would seem to be irrelevant to the equations of mo- tion, they are non-trivial for topological defects such as vortices or dislocations. In fact, one can show 34 that the first term produces the Magnus force on a vortex 39 and the second term generates the Peach-Koehler force on a dislocation. 40,41 We will defer the discussion of these effects to a subsequent publication. 34 4 x = R + u . (24) R$_{ij}$ = δ$_{ij}$ − w$_{ij}$ , (25) v$_{n}$$_{i}$ = ∂$_{t}$u$_{i}$. (26) L quad SS = − ρ$_{0}$∂$_{t}$φ − λ$_{0}$$_{ij}$ w$_{ij}$ − µ$_{0}$δρ − δρ∂$_{t}$φ − 1 2 ρ$_{0}$ ( ∂$_{i}$φ ) 2 − ∂µ ∂w$_{ij}$ ∣ ρ δρ w$_{ij}$ − 1 2 ∂µ ∂ρ ∣ w$_{ij}$ ( δρ ) 2 + 1 2 ρ$_{n}$$_{0}$$_{ij}$ ( ∂$_{t}$u$_{i}$ − ∂$_{i}$φ ) ( ∂$_{t}$u$_{j}$ − ∂$_{j}$ φ ) − 1 2 ∂λ$_{ij}$ ∂w$_{lk}$ ∣ ρ w$_{ij}$ w$_{lk}$ , (27) δµ = ∂µ ∂w$_{ij}$ ∣ ρ Δ w$_{ij}$ + ∂µ ∂ρ$_{Δ}$ ∣ w$_{ij}$ δρ$_{Δ}$, (31) δρ = ∂ρ ∂w$_{ij}$ ∣ ρ Δ w$_{ij}$ + ∂ρ ∂ρ$_{Δ}$ ∣ w$_{ij}$ δρ$_{Δ}$, (32) ρ$_{Δ}$ = ρ$_{I}$ − ρ$_{V}$. (30) ∂x ∂y ∣ z = − ∂z ∂y ∣ x ∂x ∂z ∣ y . (29) δρ = ∂ρ ∂w$_{ij}$ ∣ µ w$_{ij}$ + ∂ρ ∂µ ∣ w$_{ij}$ δµ, (28)