Motion as a Matrix

bperet's picture
Submitted by bperet on Wed, 02/25/2009 - 15:12

I have been experimenting with various ways to model the Reciprocal System, and finally created this "equation of unity":

\begin{vmatrix}<br /> S_x &amp; S_y &amp; S_z &amp; C_t<br /> \end{vmatrix}\times \begin{vmatrix}<br /> a &amp; b &amp; c &amp; d\\ <br /> e &amp; f &amp; g &amp; h\\ <br /> i &amp; j &amp; k &amp; l\\ <br /> m &amp; n &amp; o &amp; 1<br /> \end{vmatrix}\times \begin{vmatrix}<br /> T_x\\ <br /> T_y\\ <br /> T_z\\<br /> C_s <br /> \end{vmatrix} = 1

Where absolute locations in space (row) and time (column) are linked to the compound, scalar speeds (4x4 matrix). ALL numbers are complex quantities, not integers.

Location in space is represented by a homogeneous coordinate, with the 4th coordinate being clock time (scale factor for spatial location), and contain only the "real" aspect of the complex quantity: |X + i0, Y + j0, Z + j0, 1|

Location in time is represented by a quaternion column, containing the imaginary component: |1, 0 + iZ, 0 + jY, 0 + kX| (needs to be read bottom-up in the column).

The 4x4 matrix contains the various speeds of the three, scalar dimensions along the diagonal, with transforms for rotation (turn) and shift (translation) multiplied in, as complex quantities. One interesting result is that spatial location is altered by temporal location, and vice versa, and that no single scalar dimension is directly represented in the system, as Larson claimed, but it is the net motion of all three dimensions that is represented. (Not a problem most of the time, since two of the dimensions are usually at unity--identity--and have no effect).

The projection of the real aspect of the motion is what we call space, and can observe and measure.

The projection of the imaginary aspect of the motion is what we call time, and measured indirectly as force.

Forums:

StevenO's picture

StevenO

Fri, 02/27/2009 - 15:41

Very interesting Bruce. These kind of matrices can be easily fed into programs like MathLab and such for further experimentation.

The Universe unlocks its secrets in geometry

bperet's picture

bperet

Sat, 02/28/2009 - 12:42

Most computers model spatial points as a column vector, so I've updated my matrix equation of motion to reflect this. I have also added labels for the various components:

\begin{vmatrix} 0+iT_x &amp; 0+jT_y &amp; 0+kT_z &amp; 0+wC_s \end{vmatrix}<br /> \cdot<br /> \begin{vmatrix}<br /> A &amp; r_{z} &amp; r_y &amp; S_{tx}\\ <br /> -r_{z} &amp; B &amp; r_{x} &amp; S_{ty}\\ <br /> -r_y &amp; -r_{x} &amp; C &amp; S_{tz}\\ <br /> T_{tx} &amp; T_{ty} &amp; T_{tz} &amp; 1+0w<br /> \end{vmatrix}<br /> \cdot<br /> \begin{vmatrix} S_x+i0 \\ S_y+j0 \\ S_z+k0 \\ C_t+w0 \end{vmatrix} = 1

Row vector is a temporal "absolute location", formed by the intersection of three planes.

Column vector is a spatial "absolute location", formed by the coordinates of a point.

A,B,C are the three, scalar dimensions of motion.

The "r" values are the rotations/shifts for the respective dimensions (sine component). The cosine component is multipled across the two adjacent, scalar dimensions. For example, rotation about Z includes a cosine component to A and B.

tx,ty,tz are translations. The "S" translations are the linear, spatial translation we are familiar with. The "T" translations are the temporal "translation", which we would view as a phase angle of the turn (unbounded angle).

Ct is Clock time. Notice that it is REAL, not imaginary (temporal), thus making clock time appear as though it was a "4th dimension" to space. It is actually a scale factor to space.

Cs is Clock space, an imaginary component that scales temporal vectors, and acts as a 4th dimension to time.

Horace's picture

Horace

Wed, 03/04/2009 - 03:14

Bruce,

Could you elaborate a little about the invariants in this equation of motion?

Horace's picture

Horace

Tue, 03/03/2009 - 04:14

...and while you are at it, it would be nice if you could post some inequalities (or other constraints between the variables in your equation), that would define the:

  • space-time region
  • time-space region
  • space region
  • time region

I'm sure it would make jhize and Steven0 happy too

bperet's picture

bperet

Wed, 03/04/2009 - 11:05

Horace wrote:

Could you elaborate a little about the invariants in this equation of motion?

"Invariants" are just geometric properties that remain constant under a set of conditions. In the RS, Larson postulates a single invariant, "magnitudes are absolute". From this, the invariant magnitude of the progression is unity (1). The magnitudes of any motion are also invariant, so a proton with speeds of 1/2-1/2-2/1, whenever or whereever measured, are still 1/2-1/2-2/1.

With the application of a polar coordinate system, other invariants arise based on "magnitudes are absolute", which are radial length, phase angle and frequency. The progression can also be considered as a "unit sphere", with a phase angle of zero and unit frequency (in natural units of angle, not length). This natural datum is represented by a complex quantity in the bottom right of the transformation matrix as ( 1 + 0i ).

In projective geometry, the initial invariant is the cross-ratio. What that means is that the cross-ratio never changes, regardless of how you scale something, rotate it, or slide it around in space. In other words, it is a magnitude where direction is meaningless--a scalar magnitude, just like Larson postulates in the RS.

That is the only invariant in the equation--everything else is variable.

But once you begin the projection process from "scalar" to "coordinate" motion, assumptions are compounded and additional invariants are created, that only hold true for the projective stratum and those below. At the top, we have magnitude as invariant, so that carries all the way down to the Euclidean stratum. (But note that "Unity" is POSTULATED, it may not be true in actual Nature, as what we view as unity may just be a projection of something else... just makes a convenient starting point.)

The next invariant is the creation (assumption) of a plane at infinity. Infinity needs to stay in one place, otherwise the whole idea of "parallel" goes out the window. Two "reciprocal" concepts come into play here: the inverse relation between space and time, and the geometric "duality" where points and planes are reciprocals in 3D.

If we were to create a plane at infinity as a spatial, column vector, the X,Y,Z coordinates would have to be infinite (undefined). Any number, no matter how large, would just be a plane at that distance, and parallel lines meeting at that plane would always have a slight, convergent angle. Infinity is not a number, but "everything", which is why it is undefined in computer systems. But... reciprocal relationship to the rescue! Rather than expressing the plane at infinity with everything, what if we were to express its reciprocal--nothing--which we have a "value" for, zero. All that needs to be done is to flip everything around... rather than a plane at infinity in space, we create a point at zero in time, and switch from a column to row vector: | 0 0 0 1 |. This is the bottom row of the transformation matrix--the "real" aspect of the complex quantities being the Euclidean plane at infinity--a temporal "dual" point at zero. And since it is technically in time, no matter what we do to in in space, scale, rotate, translate, it remains untouched and invariant.

The inverse also holds true for temporal coordinates, which are planar, requiring a "point at infinity" for parallel planes to converge. Not as difficult a concept to understand as you may think, since you're looking at it right now... your eyes use a "vanishing point" perspective, where the center of your focus is a "point at infinity", where parallel lines appear to converge. Look down a long, straight railroad track and you'll see what I mean... the rails appear to converge to a point, even though we know they are parallel and do not converge. In the transform, the temporal point at infinity is the column on the right of the transformation matrix, as a quaternion | 1 0i 0j 0k|.

With the point/plane at infinity, a new invariant property is created: parallelism, and with that, and the ratio of "lengths" in a specified direction (basically still a cross-ratio, but one of the points/planes is now at "infinity"). Just remember the system of measure... length in space, angle in time!

I'm sure you can see the advantage of having the zero/infinity relationship defined in both forms, rectangular and polar, within the same transformation, as it allows the plotting of coordinate space + coordinate time in the same motion.

bperet's picture

bperet

Wed, 03/04/2009 - 11:35

Horace wrote:

...and while you are at it, it would be nice if you could post some inequalities (or other constraints between the variables in your equation), that would define the:

  • space-time region

  • time-space region

  • space region

  • time region

This equation only works with a MATERIAL SECTOR OBSERVER, which parallels the sensory mechanisms we use to observe and measure space (sensation) and time (intuition). You have to place the camera somewhere, in order to get a picture. Because of the camera position, and the geometric duality, it must be understood that "what you see isn't what is actually there." From a Material perspective, the Cosmic sector is inside-out. Yet, if you move your camera into the Cosmic sector, you will see the Material sector as inside-out.

The term "region" is somewhat misleading, as it isn't actually a place (in space nor in time), but a condition. The condition of "net speed less than unity" is Larson's "time-space region", the Material Sector, more commonly known as space/time.

The space-time region, the Cosmic sector, is the condition of "net speed greater than unity", from a Material observer perspective. If you were a Cosmic sector observer, all the ratios would be inverted, and it would appear that you were the condition of "less than unity"!

So, for your inequalities...

time-space region:

\begin{vmatrix}<br /> A &amp; r_{z} &amp; r_y &amp; S_{tx}\\ <br /> -r_{z} &amp; B &amp; r_{x} &amp; S_{ty}\\ <br /> -r_y &amp; -r_{x} &amp; C &amp; S_{tz}\\ <br /> T_{tx} &amp; T_{ty} &amp; T_{tz} &amp; 1+0w<br /> \end{vmatrix} &lt; 1+0w

space-time region:

\begin{vmatrix}<br /> A &amp; r_{z} &amp; r_y &amp; S_{tx}\\ <br /> -r_{z} &amp; B &amp; r_{x} &amp; S_{ty}\\ <br /> -r_y &amp; -r_{x} &amp; C &amp; S_{tz}\\ <br /> T_{tx} &amp; T_{ty} &amp; T_{tz} &amp; 1+0w<br /> \end{vmatrix} &gt; 1+0w

The condition: (equation) = 1 only occurs if EVERY motion in the Universe were compounded into the transformation matrix, which would then resolve to the identity matrix, 1.

The space and time regions are again, just conditions. (I don't know how to represent an inequality in one aspect of a complex quantity, so I'm just substituting variables):

The time region (where one or more of a,b,c must not be equal to zero):

\begin{vmatrix}<br /> 1 + ai &amp; 0 &amp; 0\\ <br /> 0 &amp; 1 + bj &amp; 0\\ <br /> 0 &amp; 0 &amp; 1 +ck<br /> \end{vmatrix}

The space region (where one or more of a,b,c must not be equal to unity):

\begin{vmatrix}<br /> a + 0i &amp; 0 &amp; 0\\ <br /> 0 &amp; b + 0j &amp; 0\\ <br /> 0 &amp; 0 &amp; c +0k<br /> \end{vmatrix}

StevenO's picture

StevenO

Wed, 03/25/2009 - 18:44

In RST2 a motion would be described as s + it instead of s/t. However, Einstein held the opinion, which also shows the intrinsic relation that is the basis of RST(x) that the scale factor between the space dimensions and the fourth (time?) dimension should be ict. So is'nt it better in RST2 and this matrix to write s + ict ?

The Universe unlocks its secrets in geometry

davelook

Thu, 03/26/2009 - 21:56

Hi Bruce,

I hope you can incorporate the following discovery into the matrix.

As mentioned on the other RS forum, there is a purely geometric way to derive the 3 lepton masses, starting from (e^(2i/3))^1/3.

With this geometric scale, "unity" is 313.8563498 MeV in the conventional mass/energy scale.

If you take the fundamental weak force particles, the W boson (80.398 GeV) and Z boson (91.1876 GeV), you get the cosine of the "weak mixing angle", or, arccos(W/Z)= 28.154687 deg. Let's assume that the weak mixing angle is really just (arctan(3/2))/2=28.154966 (see the arxiv ref)

tan(45-28.154966)=.30277563773

(What's really cool is this is one of the metallic means, because 1/.30277563773=3.30277563773, as pointed out in

http://arxiv.org/abs/hep-ph/0609131 for more on metallic means, see www.mi.sanu.ac.yu/vismath/spinadel/index.html )

If we say e=.30277563773, then 1/(e^2/4pi) = 137.07807878 (yet just another way to derive the fine structure constant?)

tan(28.154966)=(e-1)/(e+1)

Let's convert the W and Z boson masses into "natural" values by dividing by the unity constant: W=256.161776, Z=290.5392867. If W is the adjacent, and Z the hypotenuse, the opposite is 137.0920187 (given the error bars for the W and Z masses this could very well be exactly 137.07807878), or as pointed out in the above arxiv paper (but without the benefit of natural units): W^2 + B^2 = Z^2.

Truly, The Universe unlocks its secrets in geometry !
bperet's picture

bperet

Fri, 03/27/2009 - 18:32

StevenO wrote:

the scale factor between the space dimensions and the fourth (time?) dimension should be ict. So is'nt it better in RS2 and this matrix to write s + ict ?

I treat the 4th and 5th "dimensions" as complex quantities, so I don't have to convert a temporal quantity into a spatial one by using "c"--it's built in as the "real" aspect of the complex quantity. (Space=real, Time=imaginary). The complex motion must be normalized for the reference frame of the observer, which alters the real aspect (space) and has properties similar to Einstein's 4th spatial dimension. By treating it as a complex effect upon space by temporal motion, part of which may be interpreted as clock time, but it is more like Einstein's concept of a torsion tensor that distorts space--only difference is, I've accounted for the source of the torsion--the temporal motions of the time region and cosmic sector.

I have concluded that "clock time" and "causality" are two, different things. The way we sequence events in our consciousness and memory, which we call a "chronological sequence"--cause and effect--does not appear to work the same way as "clock time" does in mechanical and electrical equations. Haven't quite figured it out yet, but I will in "time." :-)

bperet's picture

bperet

Fri, 03/27/2009 - 18:44

davelook wrote:

Let's convert the W and Z boson masses into "natural" values by dividing by the unity constant: W=256.161776, Z=290.5392867. If W is the adjacent, and Z the hypotenuse, the opposite is 137.0920187 (given the error bars for the W and Z masses this could very well be exactly 137.07807878), or as pointed out in the above arxiv paper (but without the benefit of natural units): W^2 + B^2 = Z^2.

I am still quite interested in your find that the masses are related by complex roots... I do want to follow up on it. It has all the characteristics of a "projection"--just like computers use to create images--except rather than projecting a geometric relationship between points, it is projecting a "mass" relationship--which means that what we are calling "mass" MAY BE geometric points--but locations in time, rather than in space.

(Still in process of setting up, debugging and defending the new server... it wasn't up 24 hours before the logs started recording thousands of breakin attempts every hour. The amount of resources wasted on "spamming" is almost beyond imagination. Probably still a couple weeks from getting some research time.)