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Foundations of mechanics for time-space

The first edition of New Foundations for Classical Mechanics (1986) by David Hestenes included “Foundations of Mechanics” as Chapter 9. This was removed for lack of space in the second edition, but is available online as a pdf here. This space-time foundation may serve as a guide for the foundation of mechanics for time-space. To do so requires terms and correspondences in addition to switching space and time:

space → time space, time → space length, particle → eventicle, body → time body, instant → spot, clock → odologe, reference frame → reference timeframe.

Let’s focus on section 2 “The Zeroth Law of Physics” and start with the second paragraph on page 8, revising it for time-space:

To begin with, we recognize two kinds of entities, eventicles and time bodies which are composed of eventicles. Given a time body, R, called a reference timeframe, each eventicle has a geometrical property called its time position with respect to R. We characterize this property indirectly by introducing the concept of Time Space. For each reference timeframe R, a time space T is defined by the following postulates:

  1. T is a 3-dimensional Euclidean space.
  2. The time position (with respect to R) of any eventicle can be represented as a point in T.

The first postulate specifies the mathematical structure of a time space while the second postulate supplies it with a physical interpretation. Thus, the postulates define a physical law, for the mathematical structure implies geometrical relations among the time positions of distinct eventicles. Let us call it the Law of Temporal Order.

Notice that this law asserts that every eventicle has a property called time position and it specifies properties of this property. But it does not tell us how to measure time position. Measurement is a separate matter, since it entails correspondence rules as well as laws. In actual practice the reference timeframe is often fictitious, though it is related indirectly to a physical time body. Our discussion is simplified by feigning that the reference timeframe is always a real time body.

We turn now to the problem of formulating the scientific concept of space length. We begin with the idea that space length is a measure of motion, and motion is a change of time position with respect to a given reference timeframe. The concept of space length embraces two distinct relations: spatial order and temporally remote coincidence. To keep this clear we introduce each relation with a separate postulate.

First we formulate the Law of Spatial Order:

The motion of any eventicle with respect to a given reference timeframe can be represented as an orbit in time space.

This postulate has a semantic component as well as a mathematical one. It presumes that each eventicle has a property called motion and attributes a mathematical structure to that property by associating it with an orbit in time space. Recall that an orbit is a continuous, oriented curve. Thus, an eventicle’s orbit in time space represents an ordered sequence of time positions. We call this order a spatial order, so we have attributed a distinct spatial order to the motion of each eventicle.

To define a physical, space length scale as a measure of motion, we select a moving eventicle which we call an eventicle odologe. We refer to each successive time position of this eventicle as a spot. We define the space length interval Δs between two spots by

Δs = cΔt,

where c is a positive numerical constant and Δt is the arc time of the odologe’s orbit between the two spots. Our measure of space length is thus related to the measure of duration in time space.

To use this space length scale as a measure for the motions of other eventicles, we need to relate the motions of eventicles at different time positions. The necessary relation can be introduced by postulating the

Law of Coincidence:

At every spot, each eventicle has a unique time position.

This postulate determines a correspondence between the points on the orbit of any eventicle and points on the orbit of an eventicle odologe. Therefore, every eventicle orbit can be parametrized by a space length parameter defined on the orbit of an eventicle odologe.

Note that this postulate does not tell us how to determine the time position of a given eventicle at any spot. That is a problem for the theory of measurement.

So far our laws permit orbits which are nondifferentiable at isolated points or even at every point. These possibilities will be eliminated by Newton’s laws which require differentiable orbits. We include in the class of allowable orbits, orbits which consist of a single time point through some space length interval. An eventicle with such an orbit is said to be fixed with respect to the given reference timeframe through that space length interval. Of course, we require that the eventicles composing the reference timeframe itself be fixed with respect to each other, so the reference timeframe can be regarded as a rigid time body.

Note that the pace of an eventicle is just a comparison of the eventicle’s distimement to the distimement of an eventicle odologe. The pace of a eventicle odologe has the constant value 1/c = Δt/Δs, so the odologe moves uniformly by definition. In principle, we can use any moving eventicle as an odologe, but the dynamical laws we introduce later suggest a preferred choice. Any moving eventicle defines a periodic process, because it moves successively over time intervals of equal length. It should be evident that any real odologe can be accurately modeled as an eventicle odologe. By regarding the eventicle odologe as the fundamental kind of odologe, we make clear in the foundations of physics that the scientific concept of space length is based on an objective comparison of motions.

We now have definite formulations of time space and space length, so we can define a reference system as a representation x for the possible time position of any eventicle at each space length r in some space length interval. Each reference system presumes the selection of a particular origin for space and time space and particular choices for the units of distance and duration, so each time position and space length is assigned a definite numerical value. The term “reference system” is sometimes construed as a system of procedures for constructing a numerical representation of time space and space length.

After we have formulated our dynamical laws, it will be clear that certain reference systems called alacrital systems have a special status. Then it will be necessary to supplement our Law of Coincidence with a postulate that relates coincident points in different alacrital systems. That is the critical postulate that distinguishes classical mechanics from special relativity, but we defer discussion of it until we are prepared to handle it completely. It is mentioned now, because our formulation of time space and space length will not be complete until such a postulate is made.

It is convenient to summarize and generalize our postulates with a single law statement, the Zeroth (or Temporospatial) Law of Physics:

Every real time body has a continuous history in time space and space length.

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