Simultaneity, Causality, and Spectral Representations
Dan Censor
Recently Zangari and Censor discussed the non-uniqueness of the spatiotemporal world-view, and proposed a representative alternative based on the Fourier transform as a mathematical model. It was argued that this so called spectral representation, by virtue of the invertibility of the Fourier transform, is fully equivalent to our conventional spatiotemporal world-view, although in the two systems the information is ordered in a radically different manner. Criticism of the new conception can be traced back to the fundamental principles of simultaneity and causality, whose role in the spectral domain has not been sufficiently demonstrated. These questions are carefully investigated in the present study. Simple but concise examples are used to verbally and graphically clarify the mathematics involved in integral transforms, like the Fourier transform under consideration. The transition from the spatiotemporal domain to the spectral domain entails 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 circumstances is not a trivial feat. Even for the general reader, the discussion based on the simple examples should suffice to critically follow the arguments as they unfold. For completeness, the general mathematical formulations are given too. In order to follow the footprints of the spatiotemporal simultaneity and causality concepts into the spectral domain, a special strategy is implemented here: Certain spatiotemporal situations are stated, and then their outcome in the spectral domain is examined. For example, it is shown that if a causal sequence of events is flipped over in time, thus reversing the order of cause and effect, in the spectral domain the associated spectrum will become a mirror image of the original one. The claim that the spectral transforms are invertible, consequently no information is lost in the spectral world-view, is thus substantiated. These ideas are extended to situations involving both space and time. Of particular interest are cases where relatively moving observers are involved, each at rest with respect to an appropriate spatial frame of reference, measuring proper time in this frame. In such cases, time and space are intertwined, hence simultaneity and causality must be appropriately redefined. Both 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 presentation and notation as simple as possible, in some places more sophisticated 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.
