epistemology, science, kinds of knowledge, methodology
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
The purpose of science is to discover laws, which are then applied to predict and explain phenomena, develop technology, and make things. This occurs through a cycle of material induction and formal deduction. Induction consists of making observations, defining terms, and proposing postulates. Deduction consists of taking the terms and definitions from induction, possibly with
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
The Euclidean geometry is a category with point positions as the objects and Euclidean transformations as the morphisms. In kinematics there are two Euclidean geometries: that of length and that of duration. They are in turn categories. Length space is a category with point locations as the objects and Euclidean transformations as the morphisms. Duration
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
Attempted derivations of the Lorentz transformation in the previous post here, which is similar to the light wavefronts approach here, do not work. The reason is that independent and dependent variables are treated alike, but they are not. I suspect this applies to all derivations of the Lorentz transformation. Let us look at the first
