Simultaneity, causality, and spectral representations - Abstract

Recently Zangari and Censor discussed the non-uniqueness of the spatiotempo ral world-view, and proposed a representative alternative based on the Four ier transform as a mathematical model. It was argued that this so called sp ectral representation, by virtue of the invertibility of the Fourier transf orm, is fully equivalent to our conventional spatiotemporal world-view, alt hough in the two systems the information is ordered in a radically differen t manner. Criticism of the new conception can be traced back to the fundamental, prin ciples of simultaneity and causality, whose role in the spectral domain has not been sufficiently demonstrated. These questions are carefully investig ated in the present study. Simple but concise examples are used to verbally and graphically clarify th e mathematics involved in integral transforms, like the Fourier transform u nder consideration. The transition from the spatiotemporal domain to the spectral domain entail s not only a different patterning of data points. What is involved here is that every point in one domain is affecting all points in the other domain, and to follow what happens to simultaneity and causality under such circum stances is not a trivial feat. Even for the general reader, the discussion based on the simple examples should suffice to critically follow the argume nts as they unfold. For completeness, the general mathematical formulations are given too. In order to follow the footprints of the spatiotemporal simultaneity and ca usality concepts into the spectral domain, a special strategy is implemente d here: Certain spatiotemporal situations are stated, and then their outcom e in the spectral domain is examined. For example, it is shown that if a ca usal sequence of events is flipped over in time, thus reversing the order o f cause and effect, in the spectral domain the associated spectrum will bec ome a mirror image of the original one. The claim that the spectral transfo rms are invertible, consequently no information is lost in the spectral wor ld-view, is thus substantiated. These ideas are extended to situations involving both space and time. Of pa rticular interest are cases where relatively moving observers are involved, each at rest with respect to an appropriate spatial frame of reference, me asuring proper time in this frame. In such cases, time and space are intert wined, hence simultaneity and causality must be appropriately redefined. Bo th the Galilean, and the Special Relativistic Lorentzian transformations in the spatiotemporal domain, and their corresponding spectral domain Doppler transformations, fit into our argument. Special situations are assumed in the spatiotemporal domain, and their consequent footprints in the spectral domain are investigated. Although a great effort is made to keep the presen tation and notation as simple as possible, in some places more sophisticate d mathematical concepts, such as the Jacobian associated with the change of integration variables, must be incorporated. Here the general reader will have to accept the (mathematical) facts without proof.