Foundations of mechanics for length space or duration 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 room 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 either space-time or time-space. To do so requires introducing abstract terminology, notably:

position space → position geometry; time → event order; particle → point body; instant → point event; clock → event order indicator; simultaneity → correspondence; reference frame → frame.

The application of this abstract theory is to interpret the 3D position geometry with event order as either 3D length space with temporal event order (space-time) or 3D duration space with stantial event order (duration-base). It could also be applied to derivatives or integrals of these, e.g., a velocity space.

Let’s focus on section 2 “The Zeroth Law of Physics” and start with the second paragraph on page 8, revising it for 3D length space or 3D duration space:

To begin with, we recognize two kinds of bodies, point bodies and bodies which are composed of point bodies. Given a body R called a frame, each point body has a geometrical property called its position with respect to R. We characterize this property indirectly by introducing the concept of 3D Position Geometry, or Relative Geometry, if you prefer. For each frame R, a position geometry P is defined by the following postulates:

  1. P is a 3-dimensional Euclidean geometry.
  2. The position (with respect to R) of any point body can be represented as a point in P.

The first postulate specifies the mathematical structure of a 3D position geometry 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 positions of distinct point bodies. Let us call it the Law of Geometric Order.

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

We turn now to the problem of formulating the scientific concept of event order. We begin with the idea that event order is a measure of motion, and motion is change of position with respect to a given frame. The concept of event order embraces two distinct relations: event order and distant correspondence. To keep this clear we introduce each relation with a separate postulate.

First we formulate the Law of Event Order:

The motion of any point body with respect to a given frame can be represented as an orbit in 3D position geometry.

This postulate has a semantic component as well as a mathematical one. It presumes that each point body has a property called motion and attributes a mathematical structure to that property by associating it with an orbit in 3D position geometry. Recall that an orbit is a continuous, oriented curve. Thus, a point body’s orbit in 3D position geometry represents an ordered sequence of positions. We call this order an event order, so we have attributed a distinct order to the motion of each point body.

To define a physical event order scale as a measure of motion, we select a moving point body which we call a point body event order indicator. We refer to each successive position of this point body as a point event. We define the event order interval Δz between two point events by

Δy = cΔz,

where c is a positive numerical constant and Δy is the arclength of the event order’s orbit between the two point events. Our measure of event order is thus related to the metric measure in 3D position geometry.

To use this event order scale as a measure for the motions of other point bodies, we need to relate the motions of point bodies at different points. The necessary relation can be introduced by postulating the

Law of Correspondence:

At every point event, each point body has a unique position.

This postulate determines a correspondence between the points on the orbit of any point body and points on the orbit of an event order. Therefore, every point body orbit can be parametrized by an event order parameter defined on the orbit of a point body event order indicator.

Note that this postulate does not tell us how to determine the position of a given point body at any point event. 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 point during some event order interval. A point body with such an orbit is said to be not moving with respect to the given frame for that event order interval. Of course, we require that the point bodies composing the frame itself be not moving with respect to each other, so the frame can be regarded as a persistent body.

Note that a point body’s rate of motion is just a comparison of the point body’s relative position to the relative position of a point body event order indicator. The rate of motion of the point body event order indicator has the constant value c = Δy/Δz, so the event order indicator moves uniformly by definition. In principle, we can use any moving point body as an event order indicator, but the dynamical laws we introduce later suggest a preferred choice.

It is sometimes asserted that a periodic process is needed to define an event order indicator. But any moving point body automatically defines a periodic process, because it moves successively over position intervals of equal extent. It should be evident that any real event order indicator can be accurately modeled as a point body event order indicator. By regarding the point body event order indicator as the fundamental kind of event order indicator, we make clear in the foundations of physics that the scientific concept of event order is based on a consistent comparison of motions.

We now have definite formulations of 3D position geometry and event order, so we can define a reference system as a representation y for the possible position of any point body at each event order z in some event order interval. Each reference system presumes the selection of a particular frame and point body event order indicator, so y is to be interpreted as a point in the 3D position geometry of that frame. Also, a reference system presumes the selection of a particular reference point for event order and 3D position and particular choices for the units of distance and duration, so each position and event order is assigned a definite numerical value. The term “reference system” is sometimes construed as a system of procedures for constructing a numerical representation of 3D position geometry and event order.

After we have formulated our dynamical laws, it will be clear that certain reference systems called inertial systems have a special status. Then it will be necessary to supplement our Law of Correspondence with a postulate that relates corresponding events in different inertial systems. That is the critical postulate that distinguishes Newtonian theory 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 3D position geometry and event order 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 Law of Physics:

Every real body has a continuous history in 3D position geometry and event order.