  The spin stiffness $\rho_{\rm s}$ of the repulsive Hubbard model that occurs
in the hydrodynamic theory of antiferromagnetic spin waves is shown to be the
same as the thermodynamically defined stiffness involved in twisting the order
parameter. New expressions for $\rho_{\rm s}$ are derived, which enable easier
interpretation, and connections with superconducting weight and gauge
invariance are discussed.
