Happy New Year Tom,
It's great to start the New Year off talking theoretical physics. Hopefully, I can convince you that the foundational premise of an RST-based physical theory does not contradict known physics and certainly does not redefine motion with a preferred inertial frame outside the universe.
True, to accept the premise of the RST, one has to think outside the box, but this does not mean outside the physical universe, only outside the established cultural and scientific universe. If we go back and revisit the foundations of our modern thinking and consider things like the Pythagorean theorem and squaring the circle, we can see that there is actually a lot of room for thinking outside the box, and, if we do it at that fundamental point, the ramifications it might have downstream, for currently accepted scientific thought, is potentially very serious, even iconoclastic.
For example, the resolution of the crisis that the ancient Greeks faced, vis-à-vis the square root of two, is central to our modern way of thinking, for both quantum physics (QP) and covariant physics (CP). In the case of QP, this takes the form of rotation in the complex plane, and, in the case of CP, it takes the form of the fourth coordinate in the spacetime metric, which is equivalent to the radius of the unit circle in the complex plane, x4 = ict..
However, the basis of all this thinking is the relation of the right triangle's hypotenuse to its sides, when the sides are length 1. While it's true that the length of this hypotenuse is the square root of 2, it's also true that the ratio of the perpendicular sides is 1:1. What I'm saying is that the RST is a system of physical theory that has, as its foundation, the latter truth, rather than the former truth.
I'm thinking that, if the representations of the symmetry groups of the two systems can be equated, this might go a long way in getting your attention, as far as the new system's potential application to theoretical physics goes, so let me explore that a bit.
Based on the square root of 2 (SR2), QP quantizes the infinite positions on the circumference of the unit circle into a rotation group (U(1)), with its identity element of 1 implied by the complex number of size one, which then is used to work out the physics of the wave equation. Similarly, for the higher-dimensional group SU(2), and, with some difficulty, SU(3).
On the other hand, based on the unit ratio of the sides (URS), the RST quantizes the infinite sizes of the unit expansion into an equivalent, but unnamed, set of groups, with the identity element of 1, derived from the URS. The important thing to understand in both cases is that SR2 squared is equal to 2/1 and the inverse of this is equal to 1/2, which provides the fundamental symmetry of both systems, since 1/2 * 2/1 = 2/2 = 1/1 = 1, satisfying one of the most important requirements of a group.
However, the devil is in the details, as they say. Referring to figure 5 in my essay, it's easy to see that this squaring operation, in real numbers, expands the SR2 radius, r' = SR2, to the larger radius, r'' = 2, and the inverse of this is the smaller radius, r = 1, requiring us to deal with three circles, not one.
Nevertheless, thanks to the ad hoc invention of imaginary numbers, we can combine them with real numbers, to form the almost magical complex numbers, which we can then use in our operations and still remain on the circumference of the unit circle, which suits our purposes in QP quite well.
Notice, though, that the (x, y) rectangular coordinates of r, r' and r'' are equal to the square root of (.5, .5), (1, 1) and (2, 2) respectively. These coordinate values are not indicated in the figure, but you can quickly calculate the ratios of the radii, r : r' = 1/SR2 = SR.5, r': r' = SR2/SR2 = 1 and r'': r' = 2/SR2 = SR2, which are equivalent to the coordinate values.
Consequently, while we know that we can't square the circle, we see from this that we can square the coordinates of the three circle radii, and on this basis get the ratio of the three squares, which corresponds to the ratio of the radii of the three circles, 1/2, 1/1, 2/2.
But now the question, "So what?" arises. Well, the crux of the answer is that, while the mathematics of vector rotation in the complex plane can clearly be used to form representations of the three groups, U(1), SU(2) and SU(3), it should also be possible to use the mathematics of scalar expansion/contraction in the pseudoscalar sphere to form them.
If this is a valid conclusion, then all we need to get from the geometry and algebra of the new system, to its physics, is something like the wave equation of QP, or the covariant equation of CP; that is, the concept of energy. My assertion is that energy, with dimensions t/s, is the inverse of motion, with dimensions s/t, and since the inverse of the spatial pseudoscalar is the temporal pseudoscalar, I figure that we are in good shape, both mathematically and physically, from a fundamental point of view.
However, you protest my assertion that you have set up a straw man, in which the importance of the fundamentals just described are not fully appreciated, by characterizing the new concept as a geometric model that, while "static," nevertheless attempts to derive inertia from this geometry and thus impart energy to the model, something that can't be done from geometry alone.
But, while I need to correct the misconception that we are only dealing here with geometry, the science of space, and not with time, the science of algebra, and not with space/time, the science of physics, I first have to point out that your protest illustrates exactly our dissatisfaction with QP and CP, doesn't it?
The fact that we have to introduce free parameters, such as mass and charge, into our current theories is what drives Hawking to characterize the standard model as "ugly and ad hoc," from the point of view of a unified theory of physics. And, on this basis, the same judgment that indicts QP theory also indicts CP theory, in my mind.
The fact is that we want to move beyond QP and CP, and, as Peter Woit maintains, this may require more than a unification of the theories of physics; it may require unification of physics with mathematics.
Can you see a little of what I mean, Tom? If the observed expansion of space and the observed flow of time constitute motion, by definition, then all the energy of the universe is there as well, and, given the mathematics of right lines and circles, we need only seek to understand how it is quantized and manifest as mass aggregates, magnetic fields, point charges, and radiation that have the relationships and interactions that we observe, when and where we observe them.
LOL. Sounds so easy!
Doug
