Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 83 / 3 / i / Solution

Take ${\mathbf{x}}^{\prime }=\mathbf{x}-\mathbf{u}t$, with the same time coordinate in both frames. The normal fluid transforms as ${\mathbf{v}}_n^{\prime }({\mathbf{x}}^{\prime },t)={\mathbf{v}}_n(\mathbf{x},t)-\mathbf{u}$. The field transformation is the unit-modulus Schrödinger boostWriting the phase as $F=-i\mathbf{u}\cdot {\mathbf{x}}^{\prime }-i|\mathbf{u}{|}^{2}t/2$ makes differentiation at fixed primed position transparent:It follows that both $-i{\partial }_t^{\prime }{\psi }^{\prime }$ and ${\nabla }^{\prime 2}{\psi }^{\prime }/2$ gain the same terms $e^F[-i\mathbf{u}\cdot \nabla \psi -u^2\psi /2]$. The density is unchanged, ${\rho }^{\prime }=|{\psi }^{\prime }{|}^2=\rho$, so the nonlinear local term has the same form.

The remaining material derivative transforms asThus the density-damping term acquires only the common phase factor, proving invariance of the complete equation under a Galilean boost. Keeping the normal velocity unchanged in both frames would fail this check; the reference flow must transform along with the condensate.