Everyone should understand the distinction between what is moral and what is not moral. I have written briefly about that here. What is legal is not necessarily moral. What is moral is not necessarily legal. What is the relation between the moral law and the civil law? That is something every society must decide for …
Ratio Algebra Let us define an algebra of ratios. A ratio consists of two numeric expressions separated by a colon, and for clarity enclosed in parentheses, i.e., (a : b) with a, b ∈ ℝ. The expression on the left is the antecedent, and the expression on the right is the consequent. (0 : 0) is …
The following is based on Lévy-LeBlond’s Galilei Group and Galilean Invariance, §2 (Nuovo Cimento, Jan. 1973). Let Ω be the complete Newtonian space, the points (events) of which we label by their coordinates in some complete Galilean frame, using the notation y = (x(t), z(s)). (1) The complete proper Galilei group G (or Galilei …
Consider the mean between two quantities: The arithmetic mean is The harmonic mean is Consider a factor γ2 that transforms a harmonic mean into an arithmetic mean: so that which equals the gamma factor of relativity theory. Or consider the mean between these two: The arithmetic mean is The harmonic mean is Consider a factor …
The following is based on the “Derivation of the Wave Equation in Time” here with Faraday’s and Ampere-Maxwell’s laws completed for three dimensions of duration. With electric field e, electric displacement d, magnetic induction b, magnetic intensity h, current density j, length coordinates x, and duration coordinates z, these are as follows: and where the …
Geometrically, motion takes place in a three-dimensional Euclidean space with a one-dimensional parameter. Let σ be a position vector in the space and π be a value of the parameter. Then σ(π) represents the positions of a particle in motion with the parameter π and the position σ. There are two measures of the extent …
This post builds on the post about the Michelson-Morley experiment here. Compare the light clock in the “Derivation of time dilation” (e.g., here). A light clock is a thought experiment in which a light beam reflects back and forth between two parallel mirrors that are a distance D apart (see figure below). When the light …
The common justification for time dilation in the special theory of relativity goes like this: (Sacamol, CC BY-SA 4.0) From Wikipedia: In the frame in which the clock is at rest (see left part of the diagram), the light pulse traces out a path of length 2L and the period of the clock is 2L divided by …
This post is based on Mathematical Aspects of Classical and Celestial Mechanics, Third Edition by Vladimir I. Arnold, Valery V. Kozlov, and Anatoly I. Neishtadt (Springer 2006). Here it is generalized to (3 + 3) dimensions. Motion takes place in two spaces that are three-dimensional and Euclidean with a fixed orientation. Denote them by E3 …
The mean round-trip speed, as in simple harmonic motion, is Galilean invariant. There are two senses in which this is the case: (1) the time is the same in both directions, and (2) the distance covered is the same in both directions. In the first case, the mean round-trip speed equals the arithmetic mean of …
