space & time

Matters relating to length and duration in physics and transportation

Derivation of the wave equation

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 …

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Motion from geometry to algebra

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 …

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Light clock in motion

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 …

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Dilation of time or distance

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 …

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Newtonian mechanics generalized

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 …

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Invariance of round-trip speed

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 …

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Dual Euclidean transformations

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 …

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Squares of opposition

The traditional Aristotelian square of opposition is like that of first-order logic apart from existential import: Or in words: Outer negation is the contradictory, i.e., affirm/deny, and inner negation is the contrary, i.e., all/none. For quantifiers (or other operators) there is a duality square: Outer negation is negation of the whole quantifier; inner negation is …

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