Objective generalized covariant derivative with respect to time in time-varying curvilinear coordinates

Abstract

This paper develops an objective generalized covariant derivative with respect to time from the perspective of space-time duality. It is formulated in arbitrary time-varying curvilinear coordinate systems and serves as a temporal counterpart of the spatial covariant derivative. Reference bases generated by the Eulerian-Lagrangian tensor are used to construct the temporal covariant derivative. The resulting unified component form contains temporal connection terms determined by the Eulerian-Lagrangian tensor, whereas the spatial connection terms involving Christoffel symbols cancel systematically in the corresponding tensor-entity expansion. Hence, this covariant temporal derivative can be expressed without spatial Christoffel symbols even in time-varying curvilinear coordinates. Classical objective rates, including the Oldroyd, Cotter-Rivlin, Green-Naghdi, and Jaumann rates, are unified and interpreted within this framework. In particular, the Green-Naghdi rate shows that such a Christoffel-free component structure is not restricted to Lie-type objective rates, but can also be obtained for a non-Lie-type rate. The formulation clarifies the common covariant structure of Lie-type and non-Lie-type objective rates, and extends the construction from ordinary tensor components to generalized components.