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Foundations of mechanics for 3D time with linear references

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:

position space (spatiocosm) → temporocosm, time order → length order, particle → transicle, (space) position → time position, body (object) → body (subject), instant → waypoint, clock → odometer, 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, transicles and subject bodies which are composed of transicles. Given a subject body, R, called a reference timeframe, each transicle has a geometrical property called its time position with respect to R. We characterize this property indirectly by introducing the concept of a Temporocosm. For each reference timeframe R, a temporocosm 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 transicle can be represented as a point in T.

The first postulate specifies the mathematical structure of a temporocosm 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 transicles. Let us call it the Law of Temporal Order.

Notice that this law asserts that every transicle 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 subject. Our discussion is simplified by feigning that the reference timeframe is always a real subject body.

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

First we formulate the Law of Spatial Order:

The motion of any transicle with respect to a given reference timeframe can be represented as an orbit in a temporocosm.

This postulate has a semantic component as well as a mathematical one. It presumes that each transicle has a property called motion and attributes a mathematical structure to that property by associating it with an orbit in a temporocosm. Recall that an orbit is a continuous, oriented curve. Thus, a transicle’s orbit in a temporocosm 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 transicle.

To define a physical, linear reference scale as a measure of motion, we select a moving transicle which we call a transicle odometer We refer to each successive time position of this transicle as a waypoint. We define the linear reference interval Δs between two waypoints by

Δs = cΔt,

where c is a positive numerical constant and Δt is the time reference interval of the odometer’s orbit between the two waypoints. Our measure of linear reference interval is thus related to the measure of duration in a temporocosm.

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

Law of Coincidence:

At every waypoint, each transicle has a unique time position.

This postulate determines a correspondence between the points on the orbit of any transicle and points on the orbit of a transicle odometer. Therefore, every transicle orbit can be parametrized by a linear reference parameter defined on the orbit of a transicle odometer.

Note that this postulate does not tell us how to determine the time position of a given transicle at any waypoint. 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 across some linear reference interval. A transicle with such an orbit is said to be fixed with respect to the given reference timeframe through that linear reference interval. Of course, we require that the transicles composing the reference timeframe itself be fixed with respect to each other, so the reference timeframe can be regarded as a rigid subject body.

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

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

After we have formulated our dynamical laws, it will be clear that certain reference systems called facilial 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 facilial 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 a temporocosm 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 movement has a continuous history in a temporocosm.

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