Matters relating to length and duration in physics and transportation
This post follows on the Gamma factor post here. The form of velocity addition based on the Lorentz transformation is related to a combination of additive and harmonic addition. Galilei velocity addition is Lorentz velocity addition is which equals In this way the Lorentz transformation attempts to combine arithmetic and harmonic addition.
The Michelson-Morley experiment, compared the longitudinal and transverse cases of reflected light, expecting to detect an ether wind (Figure 1). Figure 1. Michelson-Morley apparatus They explain: “Let sa … be a ray of light which is partly reflected in ab, and partly transmitted in ac, being returned by the mirrors b and c, along ba
The rates of motion for translational (linear) and rotational (angular) motion in the time domain and the distance domain are as follows: Time Domain Variable Translational Motion Rotational Motion Location x θ Velocity v = dx/dt ω = dθ/dt Acceleration a = dv/dt α = dω/dt Distance Domain Variable Translational Motion Rotational Motion Chronation z
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). Linear Light Clock A linear 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
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
Dual Euclidean transformations are required to transform six dimensions of length and duration: one Euclidean transformation for length space with time and one Euclidean transformation for duration space with distance. The two Euclidean transformations are: x′ = x − vt and z′ = z − ws where x and x′ are length space vectors, t
