By Kenneth Hoffman and Ray Kunze Editorial
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36 Space-Time Structure. Algebra and Geometry 4. Light pyramids The form (2) nullifies in the points that correspond to the three distinguished planes, defined by the equalization: x1 = 0, x2 = 0, x3 = 0. (4) The vectors lying on the plane have the zero meaning of the modulus and in this meaning are isotropic. At the same time, lines, that simultaneously belong to 2 planes (as well as the point of intersection of all the 3) automatically become marked out. As there are only three lines, it is quite natural to try to connect the vectors with the special basis.
At the same time the one-dimension time also obeys the rule, as we can look upon the corresponding line as 2 opposite simply connected domains, divided by a special point, a zero that in a way can be considered to be an extreme singular case of the isotropic cone. Figure 2: Light cones of tree-dimensional time (right) and tree-dimensional pseudoEuclidian space (left) Figure 3: The fragments of unit hyperboloids If we choose 2 facing camera-octants from the 8 of the three-dimensional time and examine their united border we will get a figure depicted on Fig.
Not a scalar poly-product of the four-dimensional time (4) is an invariant to the transformations, but a specific form, that is defined by 2 vectors: (A, B, B, B) (A, A, A, B) + . (8) S(A, B) = 1/2 (A, A, A, A) (B, B, B, B)1/2 Though the form S(A, B) is not an additive quantity of the vectors that belong to the interior of domain of a light pyramid, it complies with other very important D. G. Pavlov Four-dimensional time 49 characteristics of the common scalar product, to be more specific: the symmetry, the rule of multiplication by the vector, the sign distinctness and the triangle rule .