X-Message-Number: 20789
From: 
Date: Sun, 5 Jan 2003 07:52:40 EST
Subject: Extended indice mechanics

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tensor calculus make use of index mechanics, a list of recipes to handle 
multi indice quantities. The basic elements was introduced by Ricci, then 
Einstein added some tools, the most known being this summation rule. There is 
a tool rarely seen, I have found it in a polish book, reduced to a footnote 
place. I think it is interesting for potential applications of non-local 
fields.

The basic idea in indexed quantities is to take some pont, call it zero and 
use it as the start point of all vectors. Any other point V of the space is 
then defined as the vector V starting from the point "0" and ending at point 
V. When projected against a coordinate system, V gives the components: Va 
where the index a is numbering dimensions. The idea of tensor came from the 
fact that a tangent space may be built wwith the point V as point-zero. To be 
sure, a tangent space can be built at each point in the original space.

Ricci was the first to see that a point can contains an infinity ot other 
point, so not only we can get one tangent space at each place of the original 
space, but there can be an infinity of spaces pilled up at the same place, 
each with its own index. So at point V, there is the tangent space Wa, but 
there is too: Wab, Wabc, ... with 2, 3,... indices. That is the basic concept 
of multi-linearity. That is limited to the point V, going to another point, 
there is the same structure, but tensor values will be different. 

A non-local force would need multilinear quantities that are the same at any 
place, a tensor-like object not limited to a point. This is possible if we 
endow the original space with the so-called trivial metric. With it, distance 
is only 0 or 1. There is an extended domain where all points, being at the 
same place or different one in the euclidean metric are seen as being to the 
same place. Outside that domain, all point are at a distance of one from the 
first set.

Given the tensor rank (the number of indice, for example 3) there is only 2 
tensors on that space: W0abc and W1abc. the first being taken in the 
zero-distance domain and the second in the unit distance one. In fact, there 
is only one tensor with stacked indice on 2 levels: 0 and 1.

Tensors indice transform linearly, so there is no much complexity. On the 
other hand, Christofeld symbols have indice with non-linear transform laws. 
When there is more than one derivative operator, there is more levels. For 
example two differential operators have:
Level 0 for the common 0-distance domain.
Level 1 for distance 1 for first differential and distance 0 for the second.
Level 2 for distance 0 for first differential and distance 1 for the second.
Level 3 for distance 1 on both, first and second differential operator.

This may be a bit complex, but there is no way to escape it when dealing with 
force-fields described by covariant derivatives.

In two level systems, level 0 and 1 are clearly defined. On the other side, 
with four levels, from 0 to 3, the order of level 1 and 2 is a bit arbitrary. 
So, the order; 0,2,1,3 seems as good as: 0,1,2,3. I have seen nowhere in 
printed form the logical extension of multi-level index beyond the 0,1 case. 
Neverthless, it seems Nikolas Tesla has used them in one of his nearly 
forgotten machine. He could not have got the idea without knowing such 
mathematical objects, so he may be credited as the implicit inventor of that 
extended system.

Yvan Bozzonetti.

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