Note: as the research develops this post will be updated.
Introduction
Experience shows motion takes place in three dimensions. There are two measures of the extent of motion: length and duration. The length of motion in three dimensions comprises three-dimensional length space. The duration of motion in three dimensions comprises three-dimensional duration space. Length and duration are symmetric concepts, as will be shown below.
Introduction
An independent variable is specified prior to measuring any dependent variable, so an independent variable is the domain of a functionally-related dependent variable. The independent variable is commonly an interval of time. Distance is the independent variable of an inverse square law. In Hooke’s law the independent variable is mass.
A date-time or time-stamp is a combined time-of-day and date on the calendar, which is of interest in history and astronomy. A time interval or elapsed time is the difference between two date-times, which is of interest in science.
A linear reference is of interest in geography and transportation. The stance interval or distance is the difference between two linear references, which is of interest in science.
Distance is an equivalence relation between pairs of (length space) points. Distime is an equivalence relation between pairs of instants.
An elapsed time or distime is the date-time that changes during an event or motion. A travel stance or distance is the change in linear reference during an event or motion.
Variables of time periods and distances are fixed. Variables of elapsed values are increasing from a starting point. Intervals are deltas of elapsed values. E.g., time periods are deltas of time. Distances are deltas of stances, that is, stations or points along a line or curve. Elapsed time and elapsed distance are increasing variables.
Given that there are three dimensions of motion, and that every motion is measured by its length and duration, then motion requires three dimensions of length and three dimensions of duration. Three dimensions of length are called three-dimensional length space. Three dimensions of duration are called three-dimensional duration space.
For example, motion on a two-dimensional surface can be presented as a two-dimensional map scaled in units of length or as a two-dimensional map scaled in units of duration. [The latter time maps are …]
Given a body moving at a specified rate of motion, a map of the motion can be made scaled in units of length or duration.
timescale: an arrangement of events used as a measure of the relative or absolute duration or antiquity of a period of history or geologic or cosmic time.
The length of a standard uniform motion between two points is the distance between the points.
The duration of a standard uniform motion between two instants is the distime between the instants.
Length is the distance traversed in uniform motion between two points or, kinematically, the distance traversed along a curve between two points relative to a standard rate of motion.
Duration is the elapsed time in uniform motion between two instants, or kinematically, the elapsed time along a curve between two instants relative to a standard rate of motion.
Space is the extension of length in three dimensions. Three lengths are required to measure the location of a point relative to another point, which is called displacement.
Time is the extension of duration in three dimensions. Three durations are required to measure the chronation of an instant relative to another instant, which is called dischronment.
Speed is the time rate of change of the curvilinear position of a body, or the distance traversed per time unit. The speed at an instant, or the instantaneous speed, is the time rate of change of the curvilinear position at an instant, which equals dx/dt for length x and time interval t.
Pace is the space rate of change of the curvilinear position of a body, or the elapsed time per length unit. The pace at a point, or the puncstanceous pace, is the space rate of change of the curvilinear position at a point, which equals dt/dx for time interval t and length x.
Velocity is the time rate of change of the position of a body, or the displacement traversed per time unit. The speed at an instant, or the instantaneous velocity, is the time rate of change of the position at an instant, which equals dx/dt for displacement x and time unit t.
Lenticity is the space rate of change of the position of a body, or the elapsed distimement per length unit. The pace at a point, or the puncstanceous lenticity, is the space rate of change of the position at a point, which equals dt/dx for distimement t and length unit x.
The purpose of this paper is to treat duration on a par with length in classical and relativity physics. This is done in three approaches: (1) common experience, (2) frames of reference, (3) mathematical physics.
(1) Consider a timetable listing the duration between stops distributed in two dimensions. Such a timetable depicted on a map with the scale in units of duration shows a representation of multi-dimensional time.
A space rate of motion is the duration of motion per unit of independent length. A time rate of motion is the length of motion per unit of independent duration. The independent duration is called the elapsed time or simply time. The independent length is called here the stance (a stance interval is a distance).
The rate of motion of a body or frame is either speed or pace. Pace is the duration of motion per unit of stance. Pace is the travel time per unit of travel distance (or stance interval). Time is the dependent variable and travel distance is the independent variable. The pace is zero: no travel time per a positive distance. Temporo-spatial rest is a pace of zero.
If the direction is included, the rate is a vector, either velocity or lenticity. Velocity is the displacement per unit of (parametric) time. Lenticity is the dischronment per unit of stance.
Speed is the length of motion per unit of (parametric) time. Speed is the travel distance per unit of time. In racing there is a measure of the time interval per unit of travel distance, which is called the pace. These are inverses with their independent and dependent variables interchanged. Speed is the travel distance per unit of duration (or time interval). Spatio-temporal rest is a speed of zero. A body does not change location (relative to an inertial observer) while time continues.
An independent variable is either bound or free. A bound independent variable is specified, for example, as the length of a race or the time period of a sport. A free independent variable is unspecified and appears to continue at a constant rate indefinitely, such as a clock display. The reading on an odometer connected to a vehicle that travels at a constant rate is an example of an independent distance.
(2) This requires developing a system of reference for six dimensions of length and duration.
Frames of reference are Euclidean. Position in a space frame is called location. The metric between two locations is a spatial distance. The change vector from one location to another is the displacement. Position in a time frame is called chronation. The metric between two chronations is a temporal distance or distime. The change vector from one chronation to another is the dischronment.
The Euclidean metric for space is called length. The Euclidean metric for time is called duration (or time). Because the frames are Euclidean, they are symmetric for translations and rotations, called homogeneous and isotropic, respectively. The secondary frame loses its isotropy because it is fixed in one direction.
A frame of reference (“frame”) is a method to assign every particle a unique position in a coordinate system of points in ℝ3. Such assignment is known continually and universally, without signals, from the universal extent of the frame. The coordinate system is commonly Cartesian.
A length frame is a frame at rest relative to a reference body or observer. A duration frame is a frame in standard uniform motion relative to a space frame. This requires that given the magnitudes r1 and r2 of any two intervals of motion in the length space frame, then the corresponding intervals of the duration frame, t1 and t2, relative to the length space frame satisfy the proportion: r1:r2 :: t1:t2.
The metric of the length space frame is length. Length is the absolute difference between two positions relative to the length space frame. Coordinates relative to the length space frame are in units of length. The metric of the duration frame is time. Duration is the absolute difference between two positions relative to the time frame. Coordinates relative to the duration frame are in units of time. The motion of the secondary frame with respect to the primary frame provides a standard motion for comparison with any other motion.
A system of reference (“reference system”) is a method to assign every event a unique position in a coordinate system of points in ℝ3 × ℝ3. A reference system is composed of a space frame and a time frame, such that the time frame is in standard uniform motion relative to the space frame.
The position of a body in motion is determined from the length and duration frames. An event has a length space position called location and a duration space position called chronation. Length and duration are represented as length and duration space dimensions of a system of reference.
By convention either the length space frame or the duration space frame is primary; the dual frame is secondary. The position and motion of the secondary frame is relative to the primary frame:
The secondary frame moves linearly relative to the primary frame, so the curve of the secondary frame relative to the primary frame is a line, i.e., a single dimension. If the space frame is primary, the system of reference is the time domain, the space frame is called length space, and the duration frame is one dimension of time, called time. If the duration frame is primary, the system of reference is the distance domain, the duration frame is called duration space, and the length frame is one dimension of space, called distance. By convention the length space frame is primary in a time domain and the duration frame is primary in the distance domain.
Representation of the physical universe can be either as three-dimensional length space with independent time or as three-dimensional duration space with independent distance. The former is well-known but the latter is not, and so it is the focus of this paper.
If primary and secondary frames are given, then rates of motion may be defined. If the length space frame is the secondary frame, then distance is the independent variable. If the duration space frame is the secondary frame, then distance is the independent variable.
Partition events by those with the same secondary coordinate. This forms an equivalence relation. Because the secondary coordinate is linear, blocks of equivalent events form a total order.
First Law of Dynamics: There exists an elementary reference system such that a body continues in its state of motion unless compelled or constrained otherwise.
Second Law of Dynamics: The rate of change of momentum of a body over time is directly proportional to the force applied and occurs in the same direction as the applied force. The rate of change of levamentum of a body over stance is directly proportional to the release applied and occurs in the same direction as the applied release.
Third Law of Dynamics: All forces or releases between two bodies exist in equal magnitude and opposite direction.
(3) Abstraction
The space where the motion takes place is three-dimensional and Euclidean with a fixed orientation. We shall denote it by E3. We fix some point o ∈ E3 called the “origin of reference”. Then the position of every point s in E3 is uniquely determined by its position vector os = r (whose initial point is o and end point is s). The set of all position vectors forms the three-dimensional vector space ℝ3, which is equipped with the scalar product ‹ , ›. [Mathematical Aspects of Classical and Celestial Mechanics, Third Edition, Arnold, Kozlov, & Neishtadt, p.1]
The duration space in which motion takes place has the same three-dimensional structure as the abstract space above. The combined vector space is ℝ3 × ℝ3. The abstractions for length and duration space are unconnected unless there is defined a fixed relationship between them. Examples of such a fixed relationship include a default or extremum rate of motion. Let us begin without such a relationship.
The point event E has six coordinates (x1, x2, x3; t1, t2, t3) = (x; t), where first three coordinates are length space, the second three coordinates are duration space, x is the vector of (spatial) location, and t is the vector of chronation. This may be reduced to either (x; t) or (x; t) depending on whether the space frame or time frame is primary.
The distance between events E1 (x11, x12, x13; t11, t12, t13) and E2 (x21, x22, x23; t21, t22, t23) equals (√((x21 – x11)2 + (x22 – x12)2 + (x23 – x13)2); √((t21 – t11)2 + (t22 – t12)2 + (t23 – t13)2)) = (p; q), where p and q are scalars and (p; q) is a two-dimensional scalar. Such a 2D scalar may be seen by expressing the coordinates of E as ((xi1, ti1); (xi2, ti2); (xi3, ti3)).
For convenience, consider linear motion along the x-t axis. Let frame K with axes x = x1, x2, and x3 be a length space frame of observer P. Let frame L with axes t = t1, t2, and t3 be a duration space frame of observer P, with standard uniform motion û parallel to the x and t axes, where û is the standard velocity or lenticity.
The transformations for observer K’s length space frame to observer L’s duration space frame, with observer L’s length space frame moving with velocity v relative to K’s time frame are:
x′ = x + vt and t = t′,
where x and x′ are the x-axis coordinates, and t and t′ are the t-axis coordinates of length space frames K and L, respectively. The transformations for observer K’s length space frame to observer L’s length space frame is
t′ = t + wx and x = x′.
The time domain equations of motion with constant acceleration can easily be derived from the definitions for time, location, velocity, and acceleration. Similarly, the distance domain equations of motion with constant relentation can easily be derived from the definitions for distance, chronation, lenticity, and relentation.
The time domain weighted equations of motion with mass, m, as the weighting factor and constant acceleration can be easily derived from the definitions for time, weighted location, weighted velocity or momentum, and weighted acceleration or force.
In order to develop the distance domain weighted equations of motion we must determine the weighting factor. Because of the inverse relation between the time domaiin and the distance domain, the inverse of mass should be the appropriate weighting factor. I have called this the vass. With the vass, n, as the weighting factor and constant relentation, the distance domain weighted equations of motion are easily determined from the definitions for distance, weighted chronation, weighted lenticity or levamentum, and weighted relentation or release.
Realisation
“Ever while time flows on and on and on, / That narrow noiseless river” ‒ Christina Rossetti, A Life’s Parallels