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Space – Continuous or Discrete?

“Although there have been suggestions that spacetime may have a discrete structure, I see no reason to abandon the continuum theories that have been so successful.” S.W.Hawking.1

The quotation from Hawking exemplifies the position of many theoretical physicists on the question of how space and time fit into the overall picture of matter and the real world. Perhaps, not all would agree that continuity has been as successful as it might have been. Discreteness might be even more successful.
The imperative for this project springs from the implied questions: Does spacetime have a discrete structure and what might the microscopic world be like if it did? How can discrete spacetime be understood? Such questions are unlikely to be satisfied by single answers because there is no reason to suppose that discrete spacetime can be treated theoretically in only one way. The present project explores just one discrete interpretation of atoms in spacetime—a theory of microscopic realism.

Nineteenth-century physical theory, as it applied to space, time and matter, displayed a certain aesthetic appeal, which did not quite survive the early twentieth-century transition to spacetime brought about by Einstein and Minkowski. Three dimensional matter floated in three dimensional space, in a comparatively harmonious manner. It seemed natural that space inside matter was the same as that surrounding it—their metrical properties were the same. That type of spatial harmony allowed the internal space relations of objects to be continuous with their external space relations. The implementation of the objective coupling of space with time ended that simple appeal.

One of the problems encountered in focusing attention on novel aspects of space and time is one’s preconceived notions and the prejudices they engender. So as to better follow the argument and entertain what are strange concepts it is as well to set aside both mathematical models of space, time and spacetime together with the stubborn fact of a fixed spacetime background. Just as objective space and time, as they are currently understood, have never been detected, direct observation of the behaviour of space and time within a novel framework should not be expected. As Bruce Kellett once said, we take for granted the concept of an absolute spacetime background, on which all our theories are painted.

Space Concepts Evolve

Concepts of space and time have evolved since the time of Leucippus. He and Democritus considered space to be simply void—nothingness, in which material objects moved. To Plato and Aristotle, the very concept of empty space was ridiculous—there was simply no evidence for it. Everything was composed of just four elements—air, earth, fire and water—where did void fit into the scheme?

Space is fixed, uniform and motionless, according to Newton. It does not move aside to permit the motion of matter. It follows therefore that the properties of space within matter and outside it are the same. Three dimensional matter occupies three dimensional space!

Space is nothing but relations among objects, according to Leibniz, which is entirely consistent with the view expressed by Einstein when he said that the existence of space depends upon the presence of matter. Which follows from the identity of space and the gravitational field; the latter depends entirely upon the presence of matter.
If space does not merely depend for its existence on the presence of matter, as though determined by some scheme of preestablished harmony, but also physically arises from matter then the possibility exists that the arrival of matter coincides with the arrival of space.

If Leibniz’s space concept is to be adopted, as it is in the theory of the inheritance of order, in which space has no objective reality but is nothing more than geometrical relations among objects, then the arrival of objects implies the arrival of space. Leibniz and Einstein are at one, but with one small stumbling block. What does the arrival of matter mean and how is it to be understood? Always presuming it is not simply some variant of the motion of objects. If the arrival of matter means what it seems to mean, then some de novo element is implied. This opens up questions as to the prior occupancy of the newly arrived object’s location and whether the object formerly occupied some other location.

It is not sufficient to develop a theory of the arrival of matter and then baldly state that the arrival of space is thereby explained—on the grounds that the existence of space depends upon the presence of matter—they must each be independently explained, especially if space is nothing more than geometrical relations among objects. It becomes a causality principle. Objects cannot be permitted by theory to arrive just anywhere; their arrival must obey certain well-established physical laws.

A corollary of the arrival of space is that its arrival was preceded by its absence. This in turn opens the way for the logical possibility that space is not always everywhere! Which is the key to the background independence of the theory. Objects of the standard model, when rendered discrete in the new scheme, encounter a phase of their own existence during which they are independent of the spacetime background—they lose their geometric relations with the rest of the world. Diffeomorphism invariance is the symmetry of loop quantum gravity (LQG) which leads to background independence. The spacetime framework of the present theory (which embraces both particles and gravitation) has a close resemblance to that of loop quantum gravity.

Twnety-First Century Space

In the present scheme, the importance of loop quantum gravity (LQG) is its spacetime framework. It is unique among theories of matter developed in the last century. The picture of LQG comes from mathematics, not from observed facts of the real world. Despite the deep aesthetic appeal of LQG its mathematical origin conceals a major problem, which Russell expressed perfectly when he said: "Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true." He might well have added: despite its inherent beauty. A major problem about mathematics is that it tells us that physical reality begins at the Planck scale, when as far as facts are concerned it doesn't.

The starting point for a scheme which seeks answers to these questions and a natural accommodation of this seemingly counter-intuitive arrangement and behaviour of space and material objects is an understanding of time in the system, but not as a continuous parameter. The approach follows the spacetime synthesis of Einstein and Minkowski. But instead of beginning with a geometrical framework of space and time and fitting objects and points into it, space and time are considered as adjectives of atoms and their components. In other words we do not employ the concept of objective space or time. Instead, spacetime reduces to distance, duration and direction as elements of relations among objects.

Atoms and their component parts—electrons, protons and neutrons—are spatially small. They are seemingly as small as they might be, consistent with their properties. They have the smallest width, height and depth consistent with being the component parts of atoms. We do not believe that the real world consists of individual electrons which each extend spatially across the universe—they are tiny. From this analysis it is concluded that the spatial extension of atomic constituents of material objects, as they are currently understood, is subject to a brevity constraint. This is the basis for the twentieth-century claim that matter, at the level of its components, is spatially discrete.

By contrast with such a brevity constraint on spatial extension, for twentieth-century science, the temporal extension of the same objects is free of such a constraint. The durations of the constituents of atoms extend indefinitely and time is made to be a continuous parameter; it is not an internal property of matter.
If time is to be coupled with space as spacetime in the Minkowski sense, but retain the characters given it by Leibniz, it must not become an object. If it is to function in an adjectival manner then it must become duration. And it can only be the duration of something. Like “energy” according to Feynman, which must always be the energy of something, duration must always be the duration of something. Time, like space and energy have never been detected free of the objects to which they belong or relate. Thus, coupled time and space, as spacetime, must remain the property of objects, not a basis for the retention of the objective notion of space, time or spacetime, in the Newtonian sense.

This analysis places space and time in a subordinate relation to material objects; exactly as Einstein did for the gravitational field in his theory of general relativity. And yet their intimate and entwined interdependence prevents their theoretical treatment in isolation from each other. Accordingly, the new theory’s space and time concepts are developed as outlined here. They are derivative of matter whose fundamental elements are subject to a brevity imperative which equally applies to its spatial and temporal extension. Time and space are subordinate to material objects and yet they are internal elements of the constraints on the behaviour of matter. The crucial role of this interrelationship for a discrete model of microscopic reality restores the Newtonian aesthetic in which four dimensional matter occupies four dimensional spacetime.

1. Stephen Hawking and Roger Penrose. 1996. The Nature of Space and Time Princeton: Princeton University Press. p4.

Comments and questions are welcome to pjf@it.net.au

© Peter Fimmel 2002-2007

Last page update 11/05/07