X-Message-Number: 20786 From: Date: Sat, 4 Jan 2003 17:28:32 EST Subject: Extended indice mechanics --part1_179.13f2d9a3.2b48ba10_boundary Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit 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. --part1_179.13f2d9a3.2b48ba10_boundary Content-Type: text/html; charset="US-ASCII" [ AUTOMATICALLY SKIPPING HTML ENCODING! ] Rate This Message: http://www.cryonet.org/cgi-bin/rate.cgi?msg=20786