## Symmetry of length and duration

There is a symmetry principle for length and duration: Measures of length and duration are interchangeable and the forms of equations remain the same if all measures of length are interchanged with their corresponding measures of duration and vice versa. The three-dimensionality of length is fully reflected in the three-dimensionality of duration. This is a …

## Transformations with 3-dimensional time

Following the previous post here, we use Jacobian matrices to transform location and chronation vectors between inertial observers. As before, let matrices be written with upper case boldface. Let vectors be written in lowercase boldface and their scalar magnitude without it. Velocity = V, lenticity = W, displacement = x, distimement = z, independent distance = s, independent distime …

## Definitions with 3-dimensional time

In order to combine the three dimensions of length space and three dimensions of duration space in definitions for motion in six dimensions (3+3), it is necessary to use Jacobian matrices. The 3+1 and 1+3 dimensional definitions are simplifications of these. Let matrices be written with upper case boldface. Let vectors be written in lowercase boldface …

## 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 …

## Worlds of motion

Kinecosm is the world of motion, which is the subject of kinematics. Since the extent of motion has two measures: length and duration, the kinecosm has two subworlds: Length space is the three-dimensional world of length, which is commonly called space. Duration space is the three-dimensional world of duration. Chorocosm is length space with time. …

## 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 …

## Rod-clocks and clock-rods

Consider an analogue clock with a dial and a hand, i.e., a pointer, reimagined as two circular bands: The Dial band is at rest relative to the observer and the Hand band turns clockwise at a fixed rate. The pointer on the Hand band points to the marking on the Dial band that indicates the …

## Harmonic motion with inertial reference frames

This post is similar to the post on light clocks here. Simple harmonic motion (SHM) is like the spring below: The equation for describing the period is where T is the period, m is the mass, and k is the spring constant. The displacement for each cycle is zero since it returns to its starting point. …

## Rod-clocks in a frame of reference

Space is the three-dimensional domain in which motion occurs. Time is a one-dimensional domain in which a reference uniform motion occurs. The extent of a motion is measured either as a length or a duration by a rod-clock. A rod-clock is a linear rod combined with a linear clock, like this: The pointer moves in uniform …

## 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 …