so that, for example, g − 1 / 2 ≃ 1 − ∇ · u . In view of these relations we will, in the rest of this paper, replace ˆ H$_{u}$ [ g ] by ˆ H$_{u}$ [ u ], with the understanding that u completely de- termines the metrics. We also notice that, by virtue of Eq. (26), we have Finally, it is possible to prove (see Appendix B) that the first order force density is exactly given by the ex- pression where δ ˆ H$_{u}$ [ u ] δu$_{µ}$ ( r ) is its functional derivative calculated with respect to a virtual (time-independent) variation of the displacement field. The vertical bar ∣ 1 mandates that we keep only the first-order in u part of the bracketed expression. With the help of this identity we see that the equation of motion for the displacement field takes the form The main reason for introducing the co-moving refer- ence frame is that in this frame we can make a simple approximation, which enormously simplifies the task of linearizing the stress tensor. This will be discussed in the next section. For the time being we proceed in a formally exact manner. To begin with, we observe that the gen- eral relation between the stress tensor in the lab frame and that in the co-moving frame is where After expanding the stress tensor in the co-moving frame to first order in the displacement field, The same result could have been derived almost imme- diately by using the more sophisticated machinery of the generally covariant Lagrangian formalism introduced in Ref. 20. In fact, Eq. (39) is simply a linerized version of the equation of motion for an infinitesimal fluid element, Eq. (39) of Ref. 20. As a reality check, let us ask ourselves whether the sys- tem can support excitations in which the displacement field is uniform in space: u ( r , t ) = u ( t ). Clearly in this case the strain vanishes and there is no change in met- rics, so δ ˆ H$_{u}$ [ u ] /δ u is null. As a result, after setting the external field V$_{1}$ = 0 we get the equation it is easy to see that in the lab frame we have From this we get and, after some algebra, which has the solution u$_{µ}$ ( t ) ∝ cos( ωt + φ ) if and only if the potential is of the harmonic form V$_{0}$ ( r ) = 1 $_{2}$mω$^{2}$r$^{2}$. This is just a statement of the harmonic potential theo- rem 27 according to which a many-body system in a har- monic potential performs a rigid simple harmonic motion with frequency ω imposed by the curvature of the har- monic potential. We have now shown that the harmonic potential is the only potential with this property. where we have made use of the equilibrium condition (8) and the definition (27) of the strain tensor. It is convenient at this point to introduce the first-order stress force density so that the equation of motion (10) takes the form Equation (39) is formally exact, but it still contains the time-dependent state | ˜ ψ ( t ) 〉 , which of course is not known. In spite of the simple behavior of the density (constant) and the current density (null), the evolution of the many-body wave function in the co-moving frame is far from trivial. Nevertheless, a simple and physically appealing approximation suggests itself. Namely, we as- sume that the wave function ˜ ψ is time-independent (just as the density), and coincides with the ground-state wave B. The elastic approximation 5 δ ˆ H$_{u}$ [ g ] δg $^{µν}$( r ) = − 1 2 δ ˆ H$_{u}$ [ u ] δu$_{µν}$ ( r ) . (29) F µ, $_{1}$( r , t ) = −〈 ˜ ψ ( t ) | δ ˆ H$_{u}$ [ u ] δu$_{µ}$ ( r ) | ˜ ψ ( t ) 〉 ∣ 1 , (38) m∂ 2 t u$_{µ}$ ( r , t ) + u · ∇ ∂$_{µ}$V$_{0}$ = − 1 n$_{0}$ ( r ) 〈 ˜ ψ ( t ) | δ ˆ H$_{u}$ [ u ] δu$_{µ}$ ( r ) | ˜ ψ ( t ) 〉 ∣ 1 − ∂$_{µ}$V$_{1}$ ( r , t ) . (39) P$_{µν}$ ( r , t ) = ( ∂$_{µ}$ξ$_{α}$ )( ∂$_{ν}$ξ$_{β}$ ) ˜ P$_{αβ}$ ( ξ ( r , t ) , t ) , (30) ˜ P$_{µν}$ ( ξ , t ) = P$_{µν,}$$_{0}$ ( ξ ) + ˜ P$_{µν,}$$_{1}$ ( ξ , t ) , (32) P$_{µν,}$$_{1}$ = ˜ P$_{µν,}$$_{1}$ − u · ∇ P$_{µν,}$$_{0}$ − ( ∂$_{µ}$u$_{α}$ ) P$_{αν,}$$_{0}$ − ( ∂$_{ν}$u$_{α}$ ) P$_{µα,}$$_{0}$ . (33) ∂$_{ν}$P$_{µν,}$$_{1}$ = ∂$_{ν}$ [ ˜ P$_{µν,}$$_{1}$ − ( ∂$_{µ}$u$_{α}$ ) P$_{αν,}$$_{0}$ − ( ∂$_{ν}$u$_{α}$ ) P$_{µα,}$$_{0}$ ] − ∂$_{ν}$ ( u · ∇ P$_{µν,}$$_{0}$ ) , (34) n$_{1}$∂$_{µ}$V$_{0}$ + ∂$_{ν}$P$_{µν,}$$_{1}$ = n$_{0}$ u · ∇ ∂$_{µ}$V$_{0}$ + ∂$_{ν}$ [ ˜ P$_{µν,}$$_{1}$ + ( ∇ · u ) P$_{µν,}$$_{0}$ − ( ∂$_{µ}$u$_{α}$ ) P$_{αν,}$$_{0}$ − 2 u$_{αν}$P$_{µα,}$$_{0}$ ] (35) F µ, 1 ≡ − ∂$_{ν}$ [ ˜ P$_{µν,}$$_{1}$ +( ∇ · u ) P$_{µν,}$$_{0}$ − ( ∂$_{µ}$u$_{α}$ ) P$_{αν,}$$_{0}$ − 2 u$_{αν}$P$_{µα,}$$_{0}$ (36) m∂ 2 t u$_{µ}$ ( r , t ) + u · ∇ ∂$_{µ}$V$_{0}$ = F µ, $_{1}$( r , t ) n$_{0}$ ( r ) − ∂$_{µ}$V$_{1}$ ( r , t ) . (37) m∂ 2 t u$_{µ}$ ( t ) + u ( t ) · ∇ ∂$_{µ}$V$_{0}$ = 0 (40) ˜ P$_{µν}$ ( r , t ) = − 1 √ g ( r , t $_{)}$〈 ˜ ψ ( t ) | δ ˆ H$_{u}$ [ u ] δu$_{µν}$ ( r ) | ˜ ψ ( t ) 〉 , (31) where | ˜ ψ ( t ) 〉 is the quantum state in the co-moving frame. 44