Lorentz transformations with arbitrary line of motion

Abstract

Sometimes it becomes a matter of natural choice for an observer (A) that he prefers a coordinate system of two-dimensional spatial x-y coordinates from which he observes another observer (B) who is moving at a uniform speed along a line of motion, which is not collinear with A's chosen x or y axis. It becomes necessary in such cases to develop Lorentz transformations where the line of motion is not aligned with either the x or the yaxis. In this paper we develop these transformations and show that under such transformations, two orthogonal systems (in their respective frames) appear nonorthogonal to each other. We also illustrate the usefulness of the transformation by applying it to three problems including the rod-slot problem. The derivation has been done before using vector algebra. Such derivations assume that the axes of K and K' are parallel. Our method uses matrix algebra and shows that the axes of K and K' do not remain parallel, and in fact K and K' which are properly orthogonal are observed to be non-orthogonal by K' and K respectively.

The development of physics learning tools to improve critical thinking skills

“…Physics is one of the fields of science that is still considered difficult for students because it relates to abstracting calculations [5]- [7]. Physics learning materials that are still considered difficult by students include Newton's Laws [8], [9], Elasticity [10], [11], Light [12], Free body diagrams [13], [14], Electricity and Magnetism [15], Static fluid [16], Circular motion [17], Projectile motion [18], Force and motion [19], Temperature and heat [20], Vector [21], [22], Lorentz force [23] and many another physics material. Students' motivation to learn physics is still shallow because teachers do not instill appropriate scientific  ISSN: 2252-8822 concepts and strategies in teaching physics [24].…”

Some studies on Lorentz transformation matrix in non-cartesian co-ordinate system

“…Our knowledge of special theory of relativity tells us that Lorentz transformation matrix is a mathematical tool for transforming co-ordinates, velocities and many other characteristics of matter in Cartesian co-ordinate system when there is a relative velocity between two frames of references. In literature it is found that several workers have put forward the matrix for Lorentz transformation of coordinates in cases of 3D and 4D spaces [1][2][3]. The velocity of matter was, also, dealt with by some other workers [4].…”

Comment on ‘Analysis of scissors cutting paper at super luminal speeds’

“…In a closely related paper [3], examining the rod-slot paradox, we have shown that the leading edge of the rod enters the slot first as observed by the rod, and the trailing edge of the rod enters the slot first as observed by the slot. Here also if the rod can be visualized as a blade and the slot as a sheet of paper we can see that the paper getting cut at super luminal speed but in different directions as observed by the rod (blade) and the slot (paper) Both the 'collision of inclined rods' and the 'rod and slot' problems have also been analyzed and solved by transformational techniques of spatial and space-time rotations [4] yielding the same results as in [2] and [3] Conclusion: While it is interesting to note that paper can be cut at super luminal speeds, it is additionally intriguing that the direction of the progression of the cutting is reversed as observed by observers stationary with respect to the paper compared to the direction of the progression of the cutting as observed by observers stationary with respect to the blade,…”