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III. Symmetric Tensor in its Own Reference Frame

When talking of a symmetric tensor in general, it is better to characterize it in a reference frame independent of external space considerations. Actually there is but one such reference frame, which can be entirely constructed as based on the tensor only, and this is the reference frame given by its eigenvectors. In this reference frame the tensor is simply reduced to its diagonal form as given by the principal values, and can be represented as a vector. This property has long been used for the representation of the vectors by matrices (Chapman, 1891). While the whole theory is very involved, having close connections with the theory of quaternions, we use it here in a limited sense, only in order to exhibit some physical quantities that may have been historically essential.

     For every plane in Space through a certain point, a tensor has two characteristic scalars: the normal and in-plane (or shear) intensities. In the case of locally isotropic space, V. V. Novozhilov has shown (Novozhilov, 1961) that the averages of these components with respect to the ensemble of planes through a point are invariant quantities that can be written only in terms of the principal components of the tensor. They are

(13)

It so happens that these quantities can be described in vector terms in the reference frame as given by the eigenvectors of our tensor. Indeed, in this reference frame the tensor can be simply represented as a vector having the three eigenvalues as components:

(14)

Now take the so-called octahedral plane of this reference frame. Speaking of the first octant of the reference frame, this is a unique plane in the reference frame cutting all the axes at unit distance from origin, thus having the normal

(15)

Then t_n from equation (13) is the projection along the normal of this plane of the vector

(16)

The other quantity from equation (13) comes around if we consider in-octahedral-plane (shear) components of the vector (14). These components are given by the vector

(17)

Then, a simple calculation gives

(18)

Let us now do the exercise of finding these components for the vectors associated with the tensors from equations (6) and (12). In the first case we have, taking (11) into account,

(19)

On the other hand, in the case of tensor (12) we have

(20)

If the space fabric is locally isotropic, then the two tensors commute and their reference frames coincide. The octahedral plane is then common, so the normal components are collinear. The necessary and sufficient condition for this is that the vector (h_k) is an eigenvector for the fabric tensor from (6). This result has an outstanding connotation. In order to see it, let us extend a little our continuum mechanics considerations. The quantity

(21)

plays the part of energy related to the process of deformation of Space. When written in terms of principal values of the two tensors, it can be decomposed into two parts, one involving the dot product of the corresponding normal vectors with respect to octahedral plane, the other the dot product of the in-plane components. Using equations (19) and (20) these components come out as

(22)

The continuum mechanics wisdom goes here on saying that, of these two quantities, the second one is that responsible for dissipation of energy in local processes. In other words, it is very likely that this energy can also be stored in a piece of Matter. As the angle ψ between the two component vectors characterizing the Space and its fabric in the octahedral plane plays an important part in our considerations of Space-Matter interaction, it is worth calculating it. In order to do this we need to use now the results of another essay from this page. According to those results, from (19), (20) and a simple geometrical consideration we have

(23)

It is important to note that the angle between the two vectors representing the deformation of Space and the fabric of Space in the octahedral plane is not in general zero, and this circumstance depends exclusively on the fabric of Space. The two components in the octahedral plane may be aligned, thus giving us the idea of an optimum storage of energy, in the particular case where u = g, with g denoting the magnitude of the Poynting vector. Now, if θ is the angle between the vectors and , this condition would imply that

(24)

As the quantity in the right hand side here is always greater than or equal to 1, the angle θ between the vectors and cannot be but 90°. Thus, for isotropic Space, in the situations where the fabric of Space and its deformation are aligned in the octahedral plane, the vectors and must be perpendicular to each other. Classically this was always the case of electromagnetic waves in vacuum. Notice that the deformation of Space cannot be perpendicular to the vector characterizing the fabric of Space in the octahedral plane.

 

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