Canonical frame-indifferent transport operators with the four-dimensional formalism of differential geometry

Differential geometry, within its four-dimensional formalism, has proven its ability to describe physical fields and their variations in space and time while ensuring the covariance of any physical law. This description is here applied to the motion of a material continuum within the classical hypotheses of Newtonian physics. In this context, we show that the rate of a tensor as seen by a point of space-time is uniquely defined by the covariant rate; this quantity is not invariant with respect to superposed rigid body motions. The rate of a tensor as seen by a moving particle of matter is uniquely defined by the Lie derivative of the tensor. This operator is invariant with respect to superposed rigid body motions. Both, the covariant rate and the Lie derivative are independent of the observer and could thus be used in a constitutive model within a four-dimensional formalism. We show next that the projection of the Lie derivative of the Cauchy stress tensor within an inertial 3D Cartesian frame corresponds to Truesdell's transport and that the other 3D objective stress transports, if they have the dimension of a rate, do not correspond to a time derivative of this tensor. The Truesdell transport is thus the only objective transport that represents a frame-indifferent time derivative of the Cauchy stress tensor.