Hi Tom,
I appreciate your willingness to earnestly engage. You write:
"A scalar, by definition, is a magnitude without direction. Time is a scalar in general relativity because it is a simple parameter of reversible direction. When you define scalar as a magnitude increasing in all directions of space and time, this is not what mathematicians generally mean. What your operation requires is an infinity of vectors expressing the same scalar magnitude of spacetime from a fixed point."
You are right about the definition of the scalar, but you are mistaken when you think I define it differently. It is the pseudoscalar magnitude that increases in all directions. In the case of the spatial pseudoscalar, the increase is in all directions of space, as the reciprocal temporal scalar increases. In the case of the temporal pseudoscalar, the increase is in all directions of time, as the reciprocal spatial scalar increases.
The equation for pseudoscalar expansion is the same as the equation for linear distance, except for the dimensions of the quantities involved. The equation for the spatial pseudoscalar expansion is:
s^3/t^0 * nt^0 = ns^3,
where n is the number of discrete units of elapsed time. The inverse of this is the equation of the temporal pseudoscalar expansion:
t^3/s^0 * ns^0 = nt^3,
where n is the number of discrete units of elapsed space.
Thinking in terms of "an infinity of vectors expressing the same scalar magnitude of spacetime from a fixed point," is a spacetime perspective in which the infinite number of vectors are 1D paths of vector motion, or potential vector motion, but the realization of any one vector path must be defined as the motion of something changing locations from the origin to the spherical surface.
However, the strict definition of motion, a change of space per change of time, requires no change of an object's location, only the changing quantities of the equation. Since we assume in the new system that the nature of time is that it is simply one of two, reciprocal, aspects of motion, this requires the existence of a "flow of space," corresponding to the observed "flow of time." Another way to say the same thing is to say that we assume the existence of a space clock as well as a time clock.
You write:
"When you assumed from the outset that a unified space and time are expanding at a constant rate in discrete dimension spheres, you forgot one critical point: after 3 dimensions, we are in hyperspace. Because of this crucial omission, and because you assume that n-dimensional motion is continuous in both space and time (in contradiction to your treatment of discrete dimension spheres), your mathematics is inconsistent."
I don't think so. Here's why: The fourth dimension, time (space), is the reciprocal of the three spatial dimensions, in the equation of motion. Only when we want to define motion in terms of a 1D change of locations does the fourth dimension appear as a coordinate in spacetime, but this is not the concept of pseudoscalar expansion. Neither is it correct to imagine the space/time expansion as occurring in "discrete dimension spheres."
It is only when we seek to measure the expansion, by picking a spacetime point in the expansion, does the "discrete dimension sphere" materialize. The motion is a continuous expansion without contradiction, because the quantization doesn't appear until we introduce scalar changes of "direction" in the expansion.
Once these "direction" reversals are admitted, for a given point in the progression, the unit pseudoscalars follow, as oscillating pseudoscalars. In the case of the spatial pseudoscalar, the physical expansion/contraction is spherical, which means that our dimensions are necessarily expressed in terms of π. The 1D parameter is the circumference of the sphere (there are three orthogonal ones), the 2D parameter is the area of the sphere (three of these may also be defined), and there is one expanded volume, the inverse of the contracted point (the scalar).
Of course, these numbers conform to the mathematics of the binomial expansion in four dimensions, and we can therefore exploit the Clifford algebra, or the geometric algebra, of the Euclidean space, albeit in an unfamiliar manner.
What I mean by that is that we don't regard the four linear spaces of the algebra as n-dimensional vector spaces. We regard them as n-dimensional scalar spaces, components of the infinite dimensional Hilbert space, if you will, but without defining a scalar in terms of the inner product of vectors.
For example, in the unit spherical expansion,
s^3/t^0 * nt^0 = ns^3,
where n = 1, there are three components of the s^3 expanded pseudoscalar. There are three, orthogonal, 1D units, each with two "directions," or six units altogether. There are also three, orthogonal, 2D units, each with two "directions," or twelve altogether, and there is one set of eight 3D units.
Now, as these n-dimensional units are spherical, not cubical, there is a disconnect between the algebra and the geometry of the oscillating pseudoscalar, which the topological approach seeks to overcome, but this fact does not prevent us from exploring other approaches.
In fact, this disconnect is the familiar discrete versus continuous mystery that has plagued mankind from the beginning of science, and it is why I think Lynds' essay is so important to this contest.
Regardless, however, you ask:
"What happens when you move up to dimension 4? That's the inconsistency in your technique. You've forgotten that of the 16 components of the Riemann tensor metric (remember, you have assumed a space-time combination, or Minkowski space, Euclideanized in d= 1,2,3 from the beginning), 6 components are redundant. With the 3-sphere, i.e., a 3-dimension sphere in 4 dimensions (skipping a lot of technical details here), the sphere packing & kissing number is 24. These packings, in dimensions 1--4 (also in d=8 & 24) are known and proved. The "3D expansion" compatible with your Euclidean model is then 8 2 (=10, or 16-6); i.e., 8 components of space and 2 of time. This works for continuous functions in general relativity, but not for your discrete treatment of dimensional spheres, and not for infinite-dimension Hilbert space."
The direct answer to your question is simple in structure, but complicated in detail, like a maze. Let me try to explain, but I don't know how successful I will be. The pseudoscalar/scalar expansion is four-dimensional, from 0D to 3D. No higher dimensional magnitudes, with two "directions," exist, as observation shows, even though mathematically, we can easily increase the powers of two ad infinitum.
However, as the oscillating pseudoscalar is a composite, n-dimensional, entity, up to the fourth dimension, its four component dimensional spaces (0, 1, 2, and 3, corresponding to the spaces of points, lines, areas, and volumes), can be repeated, or compounded, infinitely, in principle.
In this way, the next dimension (the fifth dimension in the binomial expansion of Clifford algebras), becomes the new 0D space, the sixth dimension becomes the new 1D space, the seventh dimension becomes the new 2D space, and the eighth dimension becomes the new 3D space. At this point, we have effectively doubled the tetraktys, to what we call the second tetraktys, which simply describes denser points, lines, areas and volumes, but still maintains the dimensions of the first tetraktys (that's why I have always been fascinated with Carl Brannen's density operators).
What's important to recognize is that the new system is not based on understanding oscillation in terms of rotation, but in terms of pseudoscalar expansion/contraction. The difference is stark, not just in a switch from quadrantal functions to binary functions, but in terms of eliminating the need for imaginary numbers and thus the whole idea of what constitutes the reals, the complexes, the quaternions, and the octonions, let alone the septillions. However, the good news is that it explains the repeating function of 8 and the mystery of 1, 2, 4 and 8, together with 0, 1, 3 and 7, in the Russian doll like fashion of Bott periodicity that so mystifies the topologists.
While this may be too iconoclastic even for the brave hearts in here, it's all an inescapable consequence of developing the consequences of the fundamental postulates of the reciprocal system, which has to be reassuring for the thoughtful.
You write:
"At last, I understand why you took Peter Lynds' essay--which is completely empty of physics--so seriously. Your universe is massless, and assumes mass created by a fundamental motion of space and time, while Lynds assumes motion without space and time. Both of these views obviate the exchange of mass -energy between bodies, which defines motion and makes the analysis of mechanics possible."
Your conclusion is premature, Tom, because the new system subsumes the legacy system. Once mass is manifest, all the known relations of mass and energy in mechanics emerge in the new system. What may cause some confusion, though, is the definition of energy in mechanics, from a 0D scalar, to a 1D "scalar" via the concepts of force and work.
Finally, you state:
"You write that "...mass and energy should be quantifiable, in terms of ...changing space and time quantities." Well, of course they are quantifiable, and quantified, in special relativity (E=mc^2), which treats only uniform motion. General relativity incorporates the change properties by dealing with acceleration. That's what acceleration is, a change in the direction of motion."
Again, one has to take into account the new system's definition of scalar motion, where change in the "direction" of an increasing scalar quantity, to a decreasing scalar quantity, is not the same as a change in the direction of the vector of a moving mass. The difference is crucial to understand. Whereas acceleration, as a component of rest mass, is hard to come by (see Hestenes essay), acceleration of a moving mass is common place.
Merry Christmas,
Doug
