# relativity

## 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 […]

## 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

## 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

## Lorentz transformation derivation fails

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

## 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

## Galilean invariance of the wave equation

This post follows James Rohlf’s Modern Physics from α to Z0 (p.104-105). See also the slides here. The Galilean transformations are applied here to 3D space and 3D time in this case because both space and time are independent arguments. Start with the standard configuration for relativity in which motion is parallel to the x-t axis. The

## Wave equation with space and time duality

The following is based on section 3.3.2 of Electricity and Magnetism for Mathematicians by Thomas A. Garrity (Cambridge UP, 2015). See also blog post Relative Motion and Waves by Conrad Schiff. The classical wave equation is consistent with the Galilean transformation in the context of space and time duality. Reflected electromagnetic waves are also consistent with

## Michelson-Morley experiment

This post relates to a previous post here. The Michelson-Morley experiment is a famous “null” result that has been understood as leading to the Lorentz transformation. However, an elementary error has persisted so that the null result is fully consistent with classical physics. The Michelson-Morley paper of 1887 [Amer. Jour. Sci.-Third Series, Vol. XXXIV, No.

## Michelson-Morley re-examined

Revised 2022-08-23. There are many expositions of the famous Michelson-Morley experiment (for example here) but they all assume the variable in common is time, which is not the case. In fact, distance is the variable in common, and so the experiment is temporo-spatial (1+3). Let us examine the original experiment as it should have been

## Principle of relativity

The relativity of uniform motion was stated by Galileo in the 17th century, though it was known to Buridan in the 14th century. Galileo’s statement of the principle of relativity is in terms of ships in uniform motion: … so long as the motion is uniform and not fluctuating this way and that. You will