The Larson Research Center

I’ve been asked to discuss the new RSM mathematics in the huge discussion forum at the Bad Astronomy and Universe Today website , so I thought I would repost it here due to time constraints.  The first post follows.  I will post the second one later.  My ID over there is Excal.

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 

Well, I’m back, after almost a year of being a away. I had to drop my contributions to this thread due to time constraints. A lot has happened in the past year, but Mike525 recently initiated a thread on the RST perspective of planet formation, which raised the subject of the RST in general, and offered me a chance to enter the dialog again.

However, to avoid hijacking his discussion, I thought it best to return to this thead, where we can discuss the RST in general. In his latest post to Mike525’s thread, antoniseb replied to my post over there with the following request:

The above response was prompted in part, because, in answering an earlier request to “show how simple physics is expressed using Larson’s ideas,” I explained that it is important to recognize that Larson’s theories are developed under a new system of physical theory, not Newton’s system, where everything depends upon the function x(t) and its derivatives.

The difference is that, while the fields of modern physical science, from particle physics to cosmology, are based on the familiar concepts of the vectorial motion of objects, the new system is based on scalar concepts of motion defined without objects, and these ideas are new and unfamiliar.

However, while there is an important connection between these two concepts of motion, generally speaking the laws of physics, involving the motion of objects measured in terms of fixed spatial reference systems, and the familiar transformations of these, are not affected by the advent of the new system.

The domain of the new system is within the entities of radiation, matter, and energy, dealing with their inherent properties of mass, charge, spin, etc., which give rise to the effects of these properties, when scalar and vectorial motion between aggregates of matter exists, which sets the stage for the emerging concepts of force, acceleration, and momentum.

In the new scalar system, therefore, force is understood as a property of motion, not something that can exist autonomously, but this is not so in the vectorial system. The grand goal of Newton’s program of research is to explain nature through the classification of a few elementary particles, by focusing on a few “fundamental” forces of interaction between them.

The mathematical and geometrical concepts used to clothe the concepts of vectorial motion, in a suitable formalism useful for conducting vectorial science, are naturally vectorial concepts, and since vectorial motion is the only motion recognized by the physicists and mathematicians that developed these mathematical and geometrical concepts, we tend to think that these concepts are all that exists, that the existing notions of mathematics and geometry, which pertain to vectorial motion, are concepts regarded as “the best of all possible worlds.”

Well, Larson’s works are changing all that, because the change of context that antoniseb inquires about takes us from a context in which the frame of space and time are paramount, into a new context where the frame of motion is paramount, motion defined in terms of space and time, but where they have no significance apart from their reciprocal relation in the equation of motion.

This means that the distance between the objects in a set of objects, which defines the concept of “space” in one, two, and three dimensions, is really only an emergent concept, the meaning of which is found solely in the history of the motion of the objects that separates them.

It’s hard to overemphasize the impact that the redefinition of the traditional concept of space has on physics, and I’m sorely tempted to begin pointing out some of the more salient implications at this point, but I can’t, if I want to limit the size of this post to any reasonable length.

If you read the previous posts in this thread, you will see that we were discussing Clifford algebras and Hestenes’ Geometric Algebra (GA), and how Hestenes found that there is a surprising geometric interpretation of numbers that can be exploited to form an amazing algebra, not based on the use of imaginary numbers to form the one-dimensional complex numbers, or the two-dimensional quaternions, but rather based on the use of a new definition of vector operations, called the “geometric product,” to form the three-dimensional octonions of the Cl3 Clifford algebra!

According to John Baez, the octonion numbers are the “crazy uncles kept in the attic” by traditional mathematicians (with some intriguing exceptions), because, though they are one of only four known normed division algebras, they are neither ordered, commutative, or associative, making them of little use to physicists, who have stuck with using complex numbers, the essential language of quantum physics.

However, Hestenes has shown that, with a geometric interpretation, these numbers form a set of four n-dimensional numbers, called blades, that can be used to great effect in the equations of theoretical physics, expressed as multivectors that contain one or more n-dimensional blades. The blades correspond to the four dimensions of the octonions (recall the binomial triangle at the fourth (2^3) line, explained in the previous posts of this thread), and constitute four grades of numbers:

• 0D number blades (2^0 = 1 grade, or scalar numbers)

• 1D number blades (2^1 = 2 grade, or vector numbers)

• 2D number blades (2^2 = 4 grade, or bivector numbers)

• 3D number blades (2^3 = 8 grade, or trivector numbers)

Therefore, an equation consisting of a multivector can contain all four of these blades in one! The utility of using this algebra is shown by Pezzaglia Jr, in his paper, Clifford Algebra Derivation of the Characteristic Hypersurfaces of Maxwell’s Equations, where he combines all four of Maxwell’s equations into one multivector, with each of the four blades yielding a fundamental law, or concept, of vectorial physics:

• The scalar blade yields Gauss’s Law

• The vector blade yields Ampere’s Law

• The bivector blade yields Faraday’s Law

• The bivector blade yields the magnetic monopole

However, many mathematicians and physicists have trouble accepting the definition of the geometric product, because it combines the scalar of the inner product of vector multiplication with a special form of the outer product of vectors. How do you combine vectors with scalars?

Hestenes’ GA shows how it can be done and in the process reveals the underlying connection between the numbers of algebra and the magnitudes of geometry, but it also reveals a fascinating drama in the evolution of mathematics as the language of physics.

In order to answer antoniseb’s question, “Is there some way to apply Larson’s concepts in a way that can be expressed mathematically?” I have to first set the context by delving into this mathematical drama, but I can promise you, if you will bear with me, it will be well worth the wait.

Excal