## Part III - Larson’s New System of Physical Theory

As we have seen, according to Hestenes, the grand goal of Newton’s program of research is to “describe and explain all properties of all physical objects.” Hestenes identifies two fundamental assumptions that are necessary to accomplish this ambitious goal. First, it is assumed that all physical objects are composites of particles, and, second, it is assumed that the behavior of a particle is governed by its interactions with other particles. Thus, the program seeks “to explain the diverse properties of objects in terms of a few kinds of interactions among a few kinds of particles.” 

Hestenes explains that the power of this approach lies in the fact that particles, and their interactions, can be precisely defined in terms of the motion of the particle; that is, the description of its existence at a given location, at a given moment of time, consists of a mathematical function, x(t). Using this function and the principles of calculus, physicists can define the velocity and momentum of any particle at a given location, x, and at a given instant of time, t, and quantify interactions between them in terms of the acceleration and force aspects of these motions, because the derivatives and integrals of the calculus enable them to precisely model both the existence of the particles and the interactions between them, and consequently predict their behavior.

However, there is an additional assumption in this program that Hestenes fails to clearly elucidate. It is that the description and explanation of the behavior of the particles, in terms of their interactions, assumes a knowledge of the inertial, magnetic, and electrical properties of the particles. In other words, the Newtonian program that seeks to describe nature in terms of the existence of particles and their interactions from moment to moment, must assume first that particles exist with a given value of mass, magnetic moment, and charge; that is, these values must be put into the system, before the power of the system, to investigate the properties of particles of matter and to classify them and their interactions accordingly, can be applied!

Thus, given two particles, particle A and B, the properties of one of which, particle A, are unknown, we can calculate its properties, if we know its interaction with B, the properties of which are known. Or, alternately, if we know the properties of particle A, we can predict its interaction with B. However, because the foundation of the system is based on the function x(t), a function of space and time, and the dimensions of inertia, magnetic, and electrical properties are not known in terms of space and time, the function x(t), used to describe the behavior of particles and their interactions is an observed relationship; that is, f, the force, or quantity of acceleration, is simply a unit of acceleration, expressed in terms of space and time, multiplied by mass, a measured quantity with an unknown relationship to space and time.

In other words, we don’t know why the total quantity of acceleration is determined by the number of mass units of a particle. We just know that it is, and we use this knowledge in our system. Without the knowledge of the mass (or magnetic moment, or charge, if applicable) of a particle, we must know the total acceleration, the force acting on it. Without a knowledge of the force, we must know the mass. For instance, given the mass, we can find the momentum of a particle, located at point x, at time t, specified for all t in a given interval of time, because it is a function of x(t), the velocity. Conversely, given the momentum, we can find the mass of a particle located at point x, at time t, for the same reason.

What this means is that Hestenes’ characterization of the grand goal of Newton’s program, as an effort to “describe and explain all properties of all physical objects … in terms of a few kinds of interactions among a few kinds of particles,” is entirely limited to the description and explanation of how the given properties of particles are related to one another.

Moreover, because the function x(t), which is the fundamental relationship in the system, upon which everything is based, is a relation of space and time, the definition of space and time is crucial to its operation. Therefore, the definition of space and time that the system employs, Newton’s definition initially, is crucial to the operation of the system. Consequently, whenever the system has failed to produce the correct results, it is the definition, or interpretation of the definition, of the nature of space and time, that has been the target that physicists naturally zero in on.

Reference Frame Transformations

Fundamentally, this problem is seen as a challenge of coping with the frame of reference that the definition of space, as a set of points that satisfies the postulates of geometry, creates. Early on, they had to contend with relatively moving frames of reference of space and time, wherein corrections were needed to ensure that comparisons were made in proper inertial frames to preserve the integrity of the function x(t). Thus, we see the Galilean transformations, the Lorentzian transformations, and, then, the gauge transformations devised to cope with increasingly sophisticated issues of space and time.

However, we don’t need to know the details of these issues and their resolutions to know that they are issues stemming from the definition of space and time, used in the Newtonian system of physical theory. This is clear to all, as we’ve seen in the discussion of David Gross’s interview on PBS, in part II of this article. Hence, the question, how can we define space and time in such a way as to avoid these problems, is clearly coming more and more into focus. As we saw in the beginning of this series, the major recalcitrant problems that face modern physicists are once again being attributed to the background reference of space and time, required by the function x(t). Indeed, it introduces into the latest and most modern of research projects in Newton’s program, namely GR, QFT, and string theory, irreconcilable contradictions. So, this time physicists don’t just seek to change it, as was done in the past, but they seek to eliminate it altogether.

Nevertheless, it’s vital to point out, that, if the problems and challenges that persist in the Newtonian program, are so obviously connected with the nature of space and time, then maybe that’s a clue as to why our efforts to describe and explain how the inertial, magnetic and electrical properties of the constituents of radiation, matter and energy are related to one another, are in trouble. Maybe, the most important lesson we have learned in the course of pursuing the Newtonian research program is that the nature of these properties is intimately related to the nature of space and time. The fact that needed corrections to the frame of reference used are constantly arising, indicates that the properties themselves, not just the relations between them, have something to do with space and time intrinsically. Certainly, if we don’t understand the true nature of these properties, we will find it quite difficult to completely understand the laws underlying their existence.

Therefore, perhaps the time has come to consider a change in the existing research program, to change it from a program that seeks to describe and explain how, given the properties of particles, they relate to one another in a reference frame of space and time that describes their motion by reference to that frame only, to a program that seeks to describe and explain their properties in terms of an intrinsic space/time relation comprising a discrete instance of motion, employing the only known relation of space and time, reciprocity.

At least we should hope, given the trouble that the traditional concept of space and time has brought to mankind’s efforts to understand the physical universe, there ought to be a better way. Indeed, it would really be helpful, if we had an absolute reference, against which we could measure magnitudes of motion, thus avoiding the need for corrections altogether. However, as we know, there can be no special rest frame of reference relative to which all x(t) motion can be defined. The closest to such a thing, and the solution Einstein turned to, was Mach’s idea of the array of fixed stars forming an absolute reference, but this concept too, is, in the final analysis, not satisfactory. The only relevant knowledge we have is that the assumption of the constant velocity of light works; that is, as Frank Wilczek puts it, we know that there appears to exist a “constant-velocity symmetry” in the physical laws of the universe. Consequently, we probably would be well advised to go back to the drawing board and see if we can discover if this symmetry means something more than we have realized, if there is something to it that we might have overlooked.

Exploring Space/Time Reciprocity

Again, the only known relationship of space and time is motion. In this relationship, time is the reciprocal of space. Larson’s positing of this relationship as the sole constituent of the physical universe, existing in discrete units, and in three dimensions, leads to a concept of a progression of space, which corresponds to the familiar progression of time. That we can observe evidence of such a space progression in the motion of the distant galaxies, receding away from us, and each other, at extreme velocities, lends credence to the assumption. Larson simply adds one other assumption to this motion: that it exists in discrete units. This means that we can quantify it as the ratio of two magnitudes that either are constantly increasing, or constantly decreasing.

In the case in which both space and time are increasing at the same rate, we can express the progression ratio as the unit ratio, ds/dt = 1/1; that is, for every increase in the number of space units, there exists a corresponding increase in the number of time units. This then is the initial condition of the motion. Notice the perfect symmetry inherent in this relationship of space and time. Also notice that the symmetry can be broken exactly two ways: the progression of one aspect or the other can be larger than the unit ratio. For instance, the progression of time can be greater than unity, in which case ds/dt = 1/n, or the progression of space can be greater than unity, in which case ds/dt = n/1, where n > 1.

Of course, the magnitude of this universal motion depends upon the size of the discrete space and time units that we select. Since the constant speed of light plays such a central role in physical phenomena, it is reasonable to assume then that the magnitude of the unit ratio, ds/dt = 1/1, is equal to c. This means that, if we can find the correct size of either the space unit, or the time unit, we can calculate the size of the unit of the reciprocal aspect. Larson selected the Rydberg constant for this purpose, since it also seems to play a central role in the phenomena of radiation of the hydrogen atom. The Rydberg frequency is 3.2899 x 10^15 Hertz, so the reciprocal of this frequency is a time unit equal to 3.03961 x 10^-16 seconds. The accepted value of the Rydberg constant has changed slightly since Larson’s day, so this figure differs slightly from his.

For reasons which will be explained below, the actual unit of time, which Larson called the natural unit of time, that enters into the unit progression ratio, is half this value, which he calculated as 1.520655 x 10^-16 seconds, as seen in his publications. Hence, the natural unit of space is then calculated as c divided by the quantity of time, or 4.558816 x 10^-6 cm. So, the physical situation, at this point, is a constant increase of space/time, the magnitude of which is 2.997930 x 10^10 cm/sec, the speed of light (here, using the figures in Larson’s publications, which have been slightly modified since.)

However, it’s important to note, that this is nothing but a ratio of the discrete units of a scalar progression. There is no information in the equation, ds/dt = 1/1, indicating dimensions, and there is only one progression ratio. Nevertheless, the assumptions in the first postulate, given in part I of this article, are that the motion of the universe exists in discrete units and in three dimensions. So, given that this universal progression is the initial state of our theoretical universe, the question is, how do we proceed and end up with discrete units of motion existing in three dimensions?

Nothing is Perfect

Notice that, in this initial state of unit motion, there is no reference frame, no structure against which the magnitude of the unit motion can be referenced. The state of the unit progression is all that exists at this point. It is the equivalent of nothing. In order for something to exist, in order for discrete (separate) units of motion to exist, there must be a deviation from this initial state of uniform motion. Earlier, it was pointed out that there are two possibilities, or “directions,” in which the perfect symmetry of unit motion could be “broken.” One possibility exists when the space/time progression ratio is greater than unity, (1+n)/1 and the other when it is less than unity, 1/(1+n).

Of course, this is a result of our interpreting the definition of magnitude in an “operational” sense, as opposed to a “quantitative” sense. The difference is enlightening. Hestenes traces the history of mathematical development with respect to the Clifford algebras and the geometric product, which Clifford developed by combining the algebraic ideas of Grassmann and Hamilton, and he explains the meaning of the two interpretations that was uncovered by Clifford:

Clifford may have been the first person to find significance in the fact that two different interpretations of number can be distinguished, the quantitative and the operational. On the first interpretation, number is a measure of ‘how much’ or ‘how many’ of something. On the second, number describes a relation between different quantities.

It is the operational relation between the units of the space/time progression that constitutes the magnitude of scalar motion, and it is only because of this that the possibility exists that it can have two “directions.” The two “directions” may be designated positive and negative, up and down, left and right, or whatever. The important thing to understand, however, is that the special meaning of “direction,” with respect to this magnitude, has a scalar meaning of “direction” that is not geometrical. For this reason, the distinction will be made between the two by placing quotation marks around the word to indicate its scalar meaning, as opposed to its geometric meaning.

There is a point of deep significance here. The story is told of a young school girl in a poor Welsh village who, when she was introduced to negative numbers, “got into a crying jag:” 

…one day in a grammar school maths lesson I got into a crying jag over the notion of minus numbers. Minus one threw out my universe, it couldn’t exist, I couldn’t understand it. This, I realised tearfully, under coaxing from an amused (and mildly amazed) teacher, was because I thought numbers were things. In fact, cabbages. We’d been taught in Miss Myra’s class to do addition and subtraction by imagining more cabbages and fewer cabbages. Every time I did mental arithmetic I was juggling ghostly vegetables in my head. And when I tried to think of minus one I was trying to imagine an anti-cabbage, an anti-matter cabbage, which was as hard as conceiving of an alternative universe.

This anecdote ought to be included in every textbook published for elementary math teachers. People have reported similar experiences when they were introduced to imaginary numbers by teachers who couldn’t, or didn’t bother, to adequately explain the idea behind the imaginary “number” ‘i’. These stories demonstrate vividly that just because we can grasp an idea abstractly, by divorcing it from physical concepts, doesn’t necessarily mean that we understand the meaning underlying the concept, only that we can understand how to use it as a method.

Certainly, the idea of negative quantities is absurd, but, nevertheless, we have learned that we can use it to great advantage, although it took mankind centuries to make this giant mental leap for the first time, and it was not done without a lot of hand wringing and pain. Fortunately, there is a way that we can easily teach children the meaning of negative numbers, as well as how to use them abstractly, but only if we are willing to ferret out its meaning by thinking on our own, it definitely cannot be accomplished by reading textbooks. The key is to understand that space doesn’t exist as stuff that has properties, like cabbages, or fabric, and, for this reason, the early Greeks were wise in keeping the ideas of magnitude and numbers separate from one another.

It turns out that the ideas of operationally defined magnitudes as opposed to quantitatively defined magnitudes, and the proper use of real numbers in these respective definitions, can help us avoid a lot of grief, not only for naïve school children, trying to learn mathematics, but also for the sophisticated adults they later turn into, who then try to formulate physical theories.

Under the current definition of space, as a set of points satisfying the postulates of geometry, negative space doesn’t exist. But under the new definition of space, as the reciprocal aspect of time, in the equation of motion, a negative magnitude of motion does exist, and just as positive space can be generated by positive motion, so inverse space, or time, can be generated by negative motion. But this is getting ahead of ourselves. The important thing to understand now is that two “directions” of scalar motion exist in the universe of motion, under the new definition.

Not only is it important to understand the two possible “directions” of scalar motion, but it’s also very important to understand that the datum for these possible magnitudes, the “zero” reference from which they are measured, is 1/1, not zero. The easiest way to keep this in mind is to think of the motion as an old fashioned pan balance, where equal weights on either side balance out, so that 1:1 is actually zero, and 1:2 or 2:1 is a magnitude of one, in two different “directions.”

Larson calls this unit motion, where the space/time progression ratio is balanced, the natural reference system of the RST. It is an absolute reference system from which magnitudes of scalar motion may be reckoned, providing, at long last, the widely sought basis for a background free definition of motion, and much, much more.