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II. Mathematics of the Problem

We found it very convenient (for historical as well as for logical reasons) to represent such a mathematical entity related to the fabric of Space in the form of a 3×3 matrices involving a vector and a scalar as follows:

(1)

The first one of these matrices accounts for the stresses in the case where some uncontrollable strains are involved, as represented by the vector ; the second matrix accounts for the strains in the case where some uncontrollable stresses are involved, as represented by the vector . Let us discuss this representation a little bit closer. Start with the first of these matrices. The description of the other one is the same with an obvious replacement of the corresponding letters. The principal values of the matrix are, as well known, the roots of the third degree equation

(2)

where I_k are invariants with respect to the orthogonal group, in case the matrix is an orthogonal tensor like here (I stands for “Invariant”). Boldface lower case letters denote the matrices from equation (1), with e the 3×3 identity matrix. In general the expressions of these coefficients are

(3)

Here TR(•) is the operation of taking the trace of a matrix. For the case in point we then have

(4)

so that the principal values (eigenvalues) of the matrix are

(5)

The vector is the eigenvector corresponding to ξ₁ (not normalized). The other two eigenvectors are identical and arbitrarily oriented in a plane perpendicular to the first one. The very same can be said of the second matrix from equation (1), namely σ. The characteristic vector of that matrix is , corresponding to the eigenvalue bs².

     Now, in reality our fabric filling the Space is part of the first kind, part of the second kind and we have to find a way to describe this situation. The most obvious choice, in line with the classical formalism of electromagnetic field, is a linear combination of the matrices from equation (1). This combination will absorb the quantities a and b in its coefficients, so that we can characterize what we think is the real fabric of Space by the matrix:

(6)

with λ, μ – two real coefficients representing the contributions of the two component matrices to the real situation. We are interested in the principal values of this matrix, which we calculate according to the previously outlined method. First of all we need the powers of the matrix t, which are:

(7)

As we only use the traces of these powers, we actually do not have to calculate so many matrices for the trace is a linear operation and is invariant to permutation of matrices in a monomial. Thus we can write

(8)

With the help of these we calculate the three orthogonal invariants of the matrix (6) by using the equations (3) for the matrix t. The detailed calculations will be omitted here. The final result is

(9)

with the following notations

(10)

In short, the principal values of the matrix t are

(11)

It can be readily checked that the quantity under the square root is always positive. Thus, we have two negative and one positive eigenvalues. The principal vector corresponding to t₁ is which, for obvious historical reasons, will be called it the Poynting vector of the matrix t.

     Now, it will be really profitable if we can relate this intrinsic image of the fabric to the behavior of the Space itself. The Space undergoes deformation, which we can recognize and measure through the limit between Space and the Matter and we need to relate its deformation to the previous construction of the material properties. In order to do this we have first to characterize the deformation of Space. Traditionally the deformation is described by the variation of the metric tensor, as given by a deformation tensor, but we can simplify this image by associating this process with the extension of our influence in Space. Namely, one can assume that locally the space is Euclidean, and the extension of the Space measure of local region is a bigger region having a constant curvature metric. Then a recent result of Coll and collaborators (Coll, Llosa, Soler, 2002) shows that any deformation of space leads to a metric tensor of the form

(12)

Here ξ and η are two functions and (h_k) is a vector. The principal values of the metric tensor (12) are , the last one corresponding to the eigenvector Thus, in completing the image of a ‘dynamical’ Space, we just have to accept that the deformation (12) of the Space is somehow related to the matrix t representing its fabric.

     Here we enter the realm of conjectures that cannot take us too far along the way, for we do not have a third term of the equation: the Matter partially filling the Space. All we can say at this point is that detecting the qualities of the fabric of Space needs close characterization of the Matter. Practically we need something material, a certain object, appropriate for the task. We think that at least the geometrical characteristics of this object can be described, and historically there have been civilizations on Earth that realized it from technological point of view.

 

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