Relativity posts

Gamma factor between means

Consider the mean between two quantities: The arithmetic mean is The harmonic mean is where which equals the gamma factor of the Lorentz transformation. The geometric mean is Then the factor γ2 transforms a harmonic mean into an arithmetic mean: The inverse γ factor transforms an arithmetic mean into a geometric mean: so that The […]

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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). 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

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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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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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General Galilean invariance

The following is generalized from the explanation of Galilean invariance here. Chorocosm (inertial frames) Among the axioms from Newton’s theory are: (1) There exists an original inertial frame in which Newton’s laws are true. An inertial frame is a reference frame in uniform motion relative to the original inertial frame. (2) All inertial frames share

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Time transformation

The length part of the Galilean transformation is: with the relative velocity v. The time part of the Galilean transformation is: so that time is the same for all observers. Einstein made time relative and symmetric with length (at least in one dimension) by assuming an absolute speed of light, c. With β = v/c

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Lorentz with round-trip light

This builds on the post Lorentz transformation derivations but given the round-trip light postulate (RTLP) here which states: The mean round-trip speed of light in vacant space is a constant, c, which is independent of the motion of the emitting body. From this empirical principle the round-trip Lorentz transformations may be derived, which are of the

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