Projective geometry and special relativity

Abstract

Some concepts of real and complex projective geometry are applied to the fundamental physical notions that relate to Minkowski space and the Lorentz group. In particular, it is shown that the transition from an infinite speed of propagation for light waves to a finite one entails the replacement of a hyperplane at infinity with a light cone and the replacement of an affine hyperplane -or rest space -with a proper time hyperboloid. The transition from the metric theory of electromagnetism to the pre-metric theory is discussed in the context of complex projective geometry, and ultimately, it is proposed that the geometrical issues are more general than electromagnetism, namely, they pertain to the transition from point mechanics to wave mechanics.

A more direct representation for complex relativity

“…As shown in [6], these are the only two possible types of 3-dimensional subspaces in A 2 . As discussed in [7], this relationship between 1+3 decompositions of R 4 and 3+3 decompositions of A 2 has significance in both projective geometry and special relativity when one regards Π 3 as the rest space for a measurement. Of course, dual statements to all of the foregoing can be made for the vector space A 2 .…”

“…Since such a decomposition is equivalent to a choice of real 3-plane A Re 2 in A 2 , this is physically related, but not equivalent to a choice of 3-plane -i.e., rest space -in R 4 , although we shall not dwell on that fact here. (See Delphenich [7]. )…”

“…Although the space A 2 of bivectors has been given a complex structure, in order to define the conjugation of a bivector one must also assume that A 2 has been given a specific choice of "real + imaginary" decomposition A 2 = Re A in A 2 , this is physically related, but not equivalent to a choice of 3-plane -i.e., rest space -in R 4 , although we shall not dwell on that fact here. (See Delphenich [7].) One can then define the complex conjugate of the bivector F by way of:…”

On the quaternion projective space

“…; A is p × q real matrix be the orthonormal vectors. Hence the sectional curvature of the plane spanned by X, N, is specified by Equation (2).…”

“…In this regard, a fundamental side is the fact that objects at infinity can be represented and handled with projective geometry and this in contrast to the Euclidean geometry. Indeed the projective geometry turns out to be very useful in order to prescribe some complex phenomena in physics [2].…”