February 23, 2021
Special Relativity, Proper Velocity, k-calculus, Transformation Equations
There are three key parts of the paper. The first is equations 7-25. Using the convention that the speed of light is 1, the special relativity transformation equations are solved without introducing velocity. With constants a and b in the transformation, the equations derived are
x′ = ax – bt, t′ = at – bx, x = ax′ + bt′, t = at′ + bx′ , a² − b² = 1. The second key part is equations 57-60. It is shown that b in the transformation equations corresponds to Proper Velocity. Einstein’s definition of velocity, v, is b/a. As a = √(b² + 1), the value of a is always more than b. Thus b/a is always less than 1. This means v is always less than the speed of light. The value b has no such restriction. The third key part follows where it is shown that the constants a and b in the transformation equations are related to Herman Bondi's k-calculus where k = a + b; 1/k = a - b. This abstract dated 22 February 2021 replaces the original abstract published in 2004, which is in an appendix to the paper. The paper is otherwise unchanged apart from some formatting and numbering corrections.