Time

[bperet]

I think it is time for a discussion about "time"...

Larson introduced a rather unique concept, that of "coordinate time". The simplest way to understand it is to use a bit of science fiction... the old concept of "anti-matter" and a universe based on anti-matter. In Larson's case, it is more "inverse matter" (the cosmic sector) rather than anti-matter, but the analogy is a good one. Two universes, one with spatial dimensions, one with temporal dimensions, linked by absolute locations as motion.

Coordinate time works exactly the same as coordinate space. It is based on "absolute locations" in the natural reference system, and when perceived from the viewpoint of the cosmic sector, appears identical to the way we view space from the material sector.

When the spatial and temporal coordinate systems become linked, both systems are modified by that linkage, just as mechanical stresses and strains occur when linked components start moving at different speeds and/or in different directions.

Paul deLespinasse, one of the founding members of ISUS, recently sent out a draft of a paper examining the concepts of "rest mass" and "relativistic mass" based on a coordinate time understanding, which offers some interesting ideas.

His paper is attached (with his permission) for your review, where he represents coordinate space and coordinate time as the vertical and horizontal axes of a right triangle, with the hypotenuse being "clock time"--a linkage between them.

When this concept is extended in to the Brehme diagrams, what appears as "frame matching" is basically the normalizing of clock time between reference frames in a relativistic (aka "Metric geometry") system.

[Horace]

In the projective geometry approach to RS, is one 4x4 matrix enough to model time, or do you need a second 4x4 matrix ?

[bperet]

Horace wrote:

In the projective geometry approach to RS, is one 4x4 matrix enough to model time, or do you need a second 4x4 matrix ?

From a material perspective, time would probably better be modeled as a quaternion, using a quaternion transformation matrix.

I would recommend coupling it with a spatial matrix, based in homogeneous coordinates, since they are linked and you would lose information by only modeling half. The homogeneous coordinates handle the rectangular aspects; the quaternions coordinates handle the polar, rotational aspects.

[Horace]

But a quaternion is a subset of the 4x4 projective matrix (which is an octonion I guess), so if you use up most of the elements of the 4x4 matix to define the temporal quaternion, there won't be many elements left over for the spatial aspect, won't it ?

[bperet]

Horace wrote:

But a quaternion is a subset of the 4x4 projective matrix (which is an octonion I guess), so if you use up most of the elements of the 4x4 matix to define the temporal quaternion, there won't be many elements left over for the spatial aspect, won't it ?

OK, now you're making me think... the 4x4 projective matrix would have to include both real and imaginary components, ie: each element would need to be a complex number. The "real" aspect would be spatial, the "imaginary" aspect would be temporal. The transform, itself, would then be able to operate on both space and time simultaneously, as a linked structure.

[Horace]

Bruce,

In complex numbers the real and imaginary parts are completely independent from each other.

If you do not use any operations that involve BOTH the real and imaginary parts (such as an absolute value or a distance), you might as well use 2 separate 4x4 matrices of non-complex numbers.

So the next question is: What operations involving BOTH parts of a complex number, would appear in RSt simulation and what would they signify ?

Regards,

Horace

P.S.

Another issue with complex numbers is the origin of real numbers. Where do the irrationals come from, if RSt postulates natural numbers or rationals at most ?

[bperet]

Horace wrote:

In complex numbers the real and imaginary parts are completely independent from each other.

Well they are two independent axes (Argand diagram), just like X and Y on a graph. But then DO interact in most functions, and that interaction is what I am trying to preserve. The way I view it, the "real" is the spatial, the "imaginary" is the temporal, which are linked together as motion... hence a complex representation seems to be appropriate, accounting for magnitude and geometry in one set.

Horace wrote:

If you do not use any operations that involve BOTH the real and imaginary parts (such as an absolute value or a distance), you might as well use 2 separate 4x4 matrices of non-complex numbers.

Given that the rotations of the atom (imaginary part) affect the fields distributed in space (real part), and spatial interactions can effect atomic structure, I suspect that most operations will involve BOTH the real and imaginary parts (rotation, shear, displacement/distance, etc). I also like the fact that the interaction that occurs between the real and imaginary parts of a complex number, as in multiplication or division, work on a magnitude-only basis, which is what one would expect going across the unit boundaries, where directional information is lost.

If one were to use independent 4x4 matrices, then the net motion of all dimensions of motion combined would be the transmitted effect. I am not sure that is the case, as each scalar dimension, itself, is independent from each other so that there would be three "net" effects transmitted across the unit boundary, not a single magnitude. We can give it a try both ways to see what happens, and which is a better model of atomic behavior.

Horace wrote:

So the next question is: What operations involving BOTH parts of a complex number, would appear in RSt simulation and what would they signify ?

Read Nick Thomas' site on Projective Geometry and counterspace: http://www.nct.anth.org.uk/ which is where I got a lot of the initial ideas from. The various "linkages" he mentions would be the parts in question, such as affine linkage, shear, and the resulting effects. (Looks like he recently updated the site, so I'll have to see what he added/changed).

Horace wrote:

Another issue with complex numbers is the origin of real numbers. Where do the irrationals come from, if RSt postulates natural numbers or rationals at most ?

From what I understand of Larson's RS, the SCALAR dimensions are always rational numbers.

That only leaves the irrationals as a byproduct of the projection into a coordinate reference frame, where assumptions such as perpendicularity and circularity exist, producing irrationals like the sqrt(2) and pi.

Electronics use complex quantities in capacitive and inductive relationships to accurately model the polar effects within the time region, so there is evidence that complex quantities may provide a good, general foundation for the spatio-temporal relationship.

[Horace]

Bruce wrote:

"I suspect that most operations will involve BOTH the real and imaginary parts (rotation, shear, displacement/distance, etc). "

There better be many constraints and linkages between them, so the number of DOFs gets reduced. Otherwise we will be left with 30 DOF for two 4x4 matrces, and this would be too rich for my blood.

[bperet]

Horace wrote:

There better be many constraints and linkages between them, so the number of DOFs gets reduced. Otherwise we will be left with 30 DOF for two 4x4 matrces, and this would be too rich for my blood.

When an origin and infinity are assumed, it reduces the DOF to 18 (affine projection). When the constant scale assumption is made, that drops to 14 (metric projection, same scale on all dimensions) and by setting the scale to unity (Euclidean projection), the DOF drops to 12 (3 translation, 3 turn, 3 rotation, 3 shift).

It can only be reduced beyond that by breaking the linkage between space and time.