Objective second-rank tensor ($J^{\ast }=QJ{Q}^T$)

An objective spatial second-rank tensor transforms by the displayed rule under every rigid observer change $x^{\ast }=c(t)+Q(t)x$. This permits time-dependent rotations, not merely constant rotations. Cauchy stress tensor is objective because contact traction and surface normal both rotate by Q. The symmetric rate-of-strain tensor is objective because the additional observer spin in the velocity gradient is antisymmetric.