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Conformal mappings
Published in Martin Vermeer, Antti Rasila, Map of the World, 2019
Unfortunately, there is no direct equivalent to complex numbers for three-dimensional space. For this purpose, one has tried to use the quaternions. Quaternions are numbers Q = a + ix + jy + kz, with calculation rules: ij=k,jk=i,ki=j,ji=−k,kj=−i,ik=−j,i2=j2=k2=−1. Quaternions are like complex numbers, but not nearly as useful. Their inventor was Sir William Rowan Hamilton (1805 – 1865) of Dublin6 . He was a famous mathematican and physicist who invented a new way to formalize Newton's classical mechanics, in which the path of an object is the same “shortest path” as in optics the path of a light ray. He was about a century ahead of his time, and not until the advent of quantum theory did it become clear, that a wave phenomenon guides the motion of also other particles than photons. Similarly also the general theory of relativity showed that the world line describing the motion of an object is a geodesic in space-time.
Quantum Dynamics of Tribosystems
Published in Dmitry N. Lyubimov, Kirill N. Dolgopolov, L.S. Pinchuk, Quantum Effects in Tribology, 2017
Dmitry N. Lyubimov, Kirill N. Dolgopolov, L.S. Pinchuk
World line is a curve in space-time that shows the motion of a classical point particle, as well as the propagation of light beams; it is a continuous sequence of events corresponding to the particle position in space at each moment of time [49].
The Sagnac effect and the role of simultaneity in relativity theory
Published in Journal of Modern Optics, 2021
Gianfranco Spavieri, George T. Gillies, Espen Gaarder Haug
In the literature, the term ‘synchronization’ has not always been used in accordance with its actual, original meaning. When in an inertial reference frame S we use the world line to denote the position s of a photon or an event in space-time, we implicitly assume that, throughout frame S, we have a set of synchronized clocks that mark simultaneously the same common time t. If superluminal, quasi-infinite speed signals were available, there would be no problem in synchronizing spatially separated clocks. However, in the absence of such signals, if space is homogeneous and isotropic in S and the one-way speed of light is c, clocks can be synchronized by means of the Einstein procedure equally well as if synchronized by means of an infinite speed signal. In this case, two synchronized clocks, spatially separated by the distance L, will mark simultaneously the same time t when either a light signal, sent to both clocks from the middle position L/2 reaches them, or an infinite speed signal is sent from one clock to the other. From an ideal perspective, although an infinite speed signal does not exist (except possibly within the context of quantum entanglement), the concept is still useful in principle for clarifying and understanding the concept of simultaneity.