Note: as the research develops this post will be updated.

Here is a formulation of Newtonian physics in six dimensions (3+3), three dimensions of space and three dimensions of time, that is effectively either 3+1 or 1+3 dimensions of space and time.

A *frame of reference *(“frame”) is a six-dimensional physical system relative to which the location of physical bodies can be determined. Frames are composed of two idealized constructions. These frames do not come with clocks.

Start with two idealized physical structures, one static, the other kinetic, with the kinetic structure in constant motion relative to the static structure. Each body or observer has one of each structure associated with it. The structures are *dual* to one another: (a) a *static structure*, which is at rest relative to its associated body or observer; and (b) a *kinetic structure*, which is in uniform motion relative to its associated body or observer at a fixed rate and direction, which are established by convention. The position of a body on a structure is determined by contiguity with the structure and is known universally, without signals, from the universal extent of each structure.

The position of a particle relative to each structure is compared, either the kinetic to the static or the static to the kinetic. Length is a result of comparing a point on the kinetic structure to two points on the static structure:

Duration is a result of comparing a point on the static structure to two points on the kinetic structure:

Then represent length and duration as space and time dimensions of a frame of reference:

Convention determines whether the static frame or the kinetic structure is *primary*; the dual structure is *secondary*. The secondary structure moves relative to the primary structure, with its origin tracing the path of a line within the primary structure. The position of the secondary structure origin along its path is a parameter on that path. If the primary structure is the static structure, the parameter is called *time*. If the primary structure is the kinetic frame, the parameter is called *stance*.

The spatial position of an event observed is its location in the observer’s static structure. The temporal position of an event observed is its chronation in the observer’s kinetic frame.

Distances follow the Euclidean (*L*_{2}) norm. Two positions (locations) of the static frame relative to the same point on the kinetic frame mark out a spatial distance, or *length*. Two positions (chronations) of the kinetic frame relative to the same point on the static frame mark out a temporal distance, or *duration*.

The dual frame rate of motion is either a speed unit or a pace unit. Three motion units can be defined from two of these units. Given the duration unit, Δ*t*_{u}, and the speed unit, *v*_{u}, the length unit is their product: Δ*t*_{u}*v*_{u} = *s*_{u}. Given the length unit, Δ*s*_{u}, and the *pace unit*, *w*_{u}, the duration unit is their product: Δ*s*_{u}*w*_{u} = *t*_{u}.

For convenience, we shall often consider linear motion along the *x-t* axis. The transformations for observer K’s static frame to observer L’s static frame, with observer L’s static frame moving with velocity *v* relative to K’s static frame, and observer L’s kinetic frame moving with legerity *w* relative to K’s kinetic frame are:

*x*′ = *x* + *vt* and *t*′ = *t* + *wx*,

where *x* and *x*′ are the *x*-axis coordinates, and *t* and *t*′ are the *t*-axis coordinates of static frames K and L, respectively.

Let frame K with axes *x*, *y*, and *z* be a static frame of observer P. Let frame L with axes *t*, *s*, and *r* be a kinetic frame of observer P, with standard uniform motion **û** parallel to the *x* and *t* axes, where **û** is the standard velocity or legerity.

The point event E has six coordinates (*x*_{i}, *y*_{i}, *z*_{i}; *t*_{i}, *s*_{i}, *r*_{i}) = (**x**; **t**), where first three coordinates are rectilinear in space, the second three coordinates are rectilinear in time, **x** is the vector of location in space, and **t** is the vector of chronation in time. This may be reduced to either (**x**; *t*) or (*x*; **t**) depending on whether the static frame or kinetic frame is primary.

Other notes:

Speed is the travel distance per unit of travel time. In racing there is a measure of the travel time per unit of travel distance, which is called the *pace*. These are not exactly inverses since the denominated in different units. Note that a faster motion is indicated by a lower pace since it takes a shorter time to travel the same distance.

Velocity is a vector quantity whose magnitude is a body’s speed and whose direction is the body’s direction of motion. What is the opposite concept, a vector quantity whose magnitude is a body’s *pace* and whose direction is the body’s direction of motion? The dictionary lacks a word for this concept; I propose calling it *legerity* from French *légèreté*, agility.

Motion is a form of change, and change is characterized by difference. A body at rest does not change. A body in motion changes. But a body is at rest only with respect to another body at rest; they change the same way. A body is in motion only with respect to another body in motion; they change in different ways.

What is a body but something physical with some consistency; some attribute must not change. If something changes completely, it is not a body, or at least not a single body. What is the length of a body? It is the difference between one end and the other end. This difference is a change, a motion, with respect to the body or with respect to an observer at rest with respect to the body.

Since a body at rest is not moving, only one representation is required. But a body in motion requires two or more representations. Uniform motion requires a minimum of two representations. Accelerated motion requires a minimum of three representations. The order of representations indicates the temporal direction of motion.

An observer of an analogue clock is at rest relative to the clock dial but not the clock hands, yet the hands indicate the time for the observer. That is, there is the frame of reference for the clock dial and another frame of reference for the clock hands. This requirement for two frames of reference to measure time motives the following definitions.