Consider the Minkowski space and the isotropic cone. Consider two points M and M ‘on the surface of the isotropic cone. Try to determine: Is there a translation of the uniqueness of the point M to point M ‘, that is, only if known Lorentz transform M into M ‘. Conversion must be orthogonal to the transformation of inputs into the orthogonal group for which there is an invariant of two points, that is, the interval that gives us the right to ask metric form. Consider how to obtain the orthogonality condition: it begins with the degeneracy of the canonical quadratic form.
The form should be not degenerate, then use the familiar formula. So as we consider the surface of the isotropic cone, the shape of our identity zero, and hence degenerate. This means that we must have a form on one coordinate less than the dimension of space. (All this is well-known facts, see references.) If the point M define the coordinates x, y, z, t, and the point M ‘define the coordinates x’, y ‘, z’, t ‘, then the Lorentz transformation (let’s not paint all the known factors) appear to be (1) t = At ‘+ Bx’, x = Dt ‘+ Ex’, y = y ‘, z = z’, to form not identically zero, and that it was not four coordinates (since the dimension of four), we need to fix, for example, the coordinate z = z , z ‘= z ‘. Section of the form for x, y, z, t at z , and the form for x ‘, y’, z ‘, t’ in the z ‘, and then replace all the coordinates: (2) T = t / z , X = x / z , Y = y / z and T ‘= t’ / z ‘, X’ = x ‘/ z ‘, Y ‘= y’ / z ‘, it is clear that we have quadratic forms in the canonical form different from zero (we will not paint them).
In (2) (1), then: (3) T = AT ‘+ BX’, X = DT ‘+ EX’, Y = Y ‘, (3) coincide exactly with the known Lorentz transformations, and hence orthogonal. QED But we see that the introduction of an arbitrary coefficient N for all at the same time coordinate changes in (3) does not happen, Indeed, if (4) t = N (At ‘+ Bx’), x = N (Dt ‘+ Ex’), y = Ny ‘, z = Nz’, then equation (3) do not change, while maintaining their orthogonality, but equation (1) will not be the only ones. Interval recorded in the (4) does not change, because it – the identity zero, the study of the orthogonality of the known formulas is not carried out because the form is singular, but once we arrive at is not a degenerate form, coordinate transformations are orthogonal. It should be noted this is only possible on the surface of the isotropic cone. References: 1) NV Efimov, “Higher geometry”. 2) GE Shilov, “Mathematical analysis. Finite linear spaces.