Text 4. Objective Tensors
The physical properties of materials are not dependent on the coordinate frame selected. It is intuitively clear that whether the observer is at rest or in motion, the material properties he observes should be the same. If this viewpoint is accepted, then the measurements made in one frame of reference are sufficient to determine the material properties in all other frames that are in rigid motion with respect to one another.
In the formulation of physical laws, it is desirable to employ, as far as possible, quantities that are independent of the motion of the observer. Such quantities are called objective or material frame-indifferent. For example, the location of a point will appear different to observers located at different places. Similarly the velocity of a point is dependent on the velocity of the observer. Therefore, these quantities are not objective. On the other hand, the distance between two points and the angles between two directions are independent of the rigid motion of the frame of reference (the observer). Newton's laws of motion have long been known to be valid only in a special frame of reference called the galilean frame. A galilean frame differs from a fixed reference frame by a constant translatory velocity. Attempts to free the principles of mechanics from the motion of the observer were resolved by Einstein's theory of general relativity.
We wish to stay in the domain of classical mechanics with regard to basic axioms. However, we would like to employ the principle of objectivity in the description of material properties.
Let a rectangular frame F be in relative rigid motion with respect to another one, F´. A point with rectangular coordinates at time t in F will have the rectangular coordinates at time t' in F´. Since the frames are in rigid motion with respect to each other, we have
(2.10.1) t' = t – a
where a is a constant allowing us to select the origin of time different in x' than in x, and Q(t) and b(t) are functions of time alone, of which Q(t) is subject to
(2.10.2) QklQml = QlkQ,m =
These conditions are the usual conditions satisfied by the cosine directors of x' with respect to x. From (2.10.2) it follows that
(2.10.3) det Q = ±1
The rigid motions exclude the minus sign on the right-hand side, that is
(2.10.4) det Q =1