The Larson Research Center

Since we can see the analogs of scalar and vector motion, in the tetraktys, one would expect that it also includes analogs of scalar and vector force and acceleration as well. Still, how can force and acceleration, which by definition are vectors, ever be scalar? The short answer is that, just as the motion generating the multi-dimensional spaces in the tetraktys can be reinterpreted as scalar magnitudes, even though it has always been defined as vector magnitudes in LST science, so too force and acceleration can be reinterpreted as scalar magnitudes, even though we aren’t accustomed to doing so in LST science.

Since, in terms of their dimensions, we can think of velocity and energy as inverses of one another, we should be able to use this fact to find the scalar analogs of the familiar vector magnitudes of force and acceleration. For example, whereas the equation of velocity is normally interpreted as a change of distance over time, v = ds/dt, a vector, the equation works equally well when we are thinking of the vector as a change in the position of a point itself, or in the change in the position of the midpoint of a line (the motion of an arrow), or a change in the position of the midpoint of a plane (the motion of a frisbee), or of a sphere (the motion of a cannon ball).

Likewise, when we reinterpret the equation of velocity to mean the scalar expansion of a point (change of size), instead of the vector translation of a point, the equation works equally well whether we are thinking of the scalar as a change in the size of the point itself (expanding balloon), or a change in only one or two dimensions of the point (telescoping pointer, ripples on a water surface). In other words, we can write the velocity equation to express the linear expansion of a line, the bi-linear expansion of a plane, or the tri-linear expansion of sphere. We could write the three linear equations as follows

1. v = ds/dt, the uni-linear form of the equation of velocity
   
2. v = ds²/dt, the bi-linear form of the equation, and
   
3. v = ds³/dt, the tri-linear form of the equation.
   

These three equations express one, two, and three dimensional scalar motion, as one, two, and three-dimensional expansions over time. Clearly, the midpoint of an n-dimensional object can only be moved one-dimensionally, so these equations are unusual, albeit consistent, formulations of the velocity equation. If we make the scalar motion a vibration over one unit, in lieu of a continuous expansion, the equations don’t change, but all the spaces of the tetraktys are formed, as shown in figure 1, below:

Figure 1. The Four Spaces of the Tetraktys Formed by Multi-dimensional Scalar Motion

Actually, all the spaces of the tetraktys are included in the expanding/contracting sphere. In a very real sense, then, we can say that the expanding/contracting sphere describes all the spaces of Euclidean geometry, the spaces of the octonion:

1. The point,
   
2. The point and the line,
   
3. The point, two lines and a plane,
   
4. The point, three lines, three planes and a cube.

But this is old news. The new point is that, if this is true, it follows that we can write the analogs of the scalar velocity equations, the inverse scalar velocity equations, in like manner:

1. v_i = dt/ds, the uni-linear form of the equation of inverse velocity
   
2. v_i = dt²/ds, the bi-linear form of the equation, and
   
3. v_i = dt³/ds, the tri-linear form of the equation.

Of course, to do this, we have to reinterpret the scalar inverse velocity equation as a vector inverse velocity equation, the opposite of what we had to do in the case of scalar velocity. This is the inverse conceptual problem. Whereas, with vector velocity, there is no multi-dimensional possibility for midpoint change of location, with scalar inverse velocity, there is only the multi-dimensional possibility for midpoint change of size. So, instead of having to go from vector to scalar analogs, as we do in the case of velocity, we have to go from scalar to vector analogs, in the case of inverse velocity.

To achieve this, one approach we can consider is to regard scalar inverse velocity equations as we do vectorial velocity equations; that is, the equations of vectorial velocity describe the time rate of change of position, regardless of the number of dimensions of the object changing locations, but, as we have seen, it can also be interpreted as the description of the time rate of change of the size of linear, area, or volume space units, and that all three of these velocities are contained in the changing volume, as one, vibrating, octonion, as illustrated in figure 1.

It certainly follows, then, that the analogous equation of scalar inverse velocity, the temporal vibrational velocity of volume, v_i = dt³/ds, describes the time rate of change of 1D and 2D time units, as well (inverse velocity equations 1, 2, 3 above). But, then, what are multi-dimensional time units? Since time has no direction in space, we don’t know what this means from a vector standpoint, but from a scalar standpoint, we can easily see that a scalar time expansion would be scalar in the sense of expanding in all directions of time, generating the temporal pseudoscalar of the octonion, assuming there are temporal directions. In other words, the directions of time are time-like, not space-like, but they are three-dimensional nonetheless.

In fact, we use multidimensional units of time regularly. For example, the change in the time rate of change of velocity is what we call acceleration, and the time/space dimensions of the acceleration equation are a = ds/dt². So, in terms of space/time dimensions, the inverse velocity equation of “area,” or bi-linear, time units is the inverse of the velocity equation of acceleration; that is, a = ds/dt² is the inverse of v_i = dt²/ds. If we interpret this equation as the inverse velocity of bi-linear time units, then it follows that v_i = dt/ds, with time/space dimensions of energy, is the equation for inverse velocity of uni-linear time units.

Though it sounds strange to do so, given the foregoing, it follows that we can consistenly refer to the space rate of change of time units, as inverse velocity, and the change of the space rate of change of time units, as inverse acceleration. Thus, the time/space dimensions of force, f = t/s², are actually the dimensions of inverse acceleration. Moreover, since we can conceive of scalar velocities as the 1, 2, and 3 dimensional spatial expansions of the octonion, we can also conceive of a change in these multi-dimensional velocities as accelerations of multi-dimensional spatial expansions. Therefore, we can write the acceleration equations as

1. a = ds/dt², the uni-linear form of the equation of acceleration
   
2. a = ds²/dt², the bi-linear form of the equation, and
   
3. a = ds³/dt², the tri-linear form of the equation.

And, if this is so, then we can also write the corresponding equations of inverse acceleration as

1. a_i = dt/ds², the uni-linear form of the equation of inverse acceleration
   
2. a_i = dt²/ds², the bi-linear form of the equation, and
   
3. a_i = dt³/ds², the tri-linear form of the equation.

However, where time in the material sector has no direction, all of the time terms in the inverse acceleration equations reduce to t. Yet, since there is an actual dimensional difference in these three inverse accelerations, there must be a dimensional difference in their three force analogs, as well, though we don’t recognize it in the force equations per se. We say, instead, that:

1. The force of equation 1 is one-dimensional, or electrical (F = q₁*q₂/r²).
2. The force of equation 2 is two-dimensional, or magnetic (F = Q₁*Q₂/r²).
3. The force of equation 3 is three-dimensional, or gravitational (F = M₁*M₂/r²).

Recall that the definition of energy, in the LST, though regarded as a scalar magnitude without direction, has the dimensions of inverse velocity, and velocity does have direction, in the LST. How can the dimensions of a scalar have the same number of dimensions as a vector? The answer to this contradiction is handled in the LST by defining energy in terms of work, and work in terms of force, applied over distance. Thus, if a force produces a displacement, either a positive or negative displacement, the product of the two vectors is the scalar work, and, thus, the scalar energy is manifest (via the inner product of two vectors), and it has the dimensions of a scalar, in one sense, but the dimensions of a vector in another sense.

This is known as the work-energy theorem. The ability to do work requires energy, and it is defined by virtue of the vectorial motion of mass called kinetic energy. Whether that work is to accelerate or decelerate a particle’s motion, is unimportant. Hence, the magnitude of the vector of the motion has direction, but the magnitude of the energy difference does not, making energy a scalar, associated with motion, a vector.

However, the dimensions of force, are the dimensions of energy per unit space, t/s x 1/s = t/s², regardless of the LST definition of kinetic energy, in terms of vectorial motion. Therefore, if we define force in this manner, then it follows that force is a quantity of acceleration, expressed in terms of energy per unit area, regardless of the dimensions of the physical entity generating the force. Whether it be electrical fields, magnetic fields, gravitational fields, or inertial fields, generating the force, it is all the same. Force is always a measure of energy per unit area, with dimensions dt/ds².

For example, the energy per unit area of electrical energy (disregarding constants of proportionality) is given by

f = q₁q₂ x 1/r² = t/s * 1/s = t/s²,

because the dimensions of area, even though area is length squared, are scalar, just as the inner product of two vectors is scalar. Actually, though, both forms of the dimensions of area are correct, depending on the sense intended, and neither the scalar nor the vector aspects of area space are sufficient to describe the space value alone. This is what GA’s geometric product is all about. In this case, we are interested in the magnitude of the area, not in the lengths of its sides, so the proper number of mathematical dimensions is zero, the magnitude-only dimension of scalars.

However, we are again running into difficulties with the inconsistent use of dimensions, because a scalar times a scalar is a scalar, and yet force is a vector. What’s going on? The problem can be even more clearly discerned when we consider the dimensions of acceleration,

a = ds/dt².

Now we know that time is scalar, so why do we express it in terms of square units? Well, the calculus saves us in this case, with its concept of derivatives, but the mixed up concept of dimension is still confusing, and we can’t use derivatives in the case of space, because energy per unit of area (force) is a different concept than the time rate of change of velocity (acceleration), expressed in terms of delta space per second, per second - a change of a change.

When you think about it, what is needed is the concept of three scalar dimensions, where a number raised to a higher power is a different kind of number, such that its dimensions are scalar dimensions (i.e. 2⁰ = 1), but its magnitude is multidimensional (2¹ = linear magnitude, 2² = area magnitude, 2³ = volumetric magnitude.) In other words, scalar lines, scalar areas, and scalar volumes, are descriptions of magnitude-only scalar values. In this way, the equation of force could express discrete units of energy per unit area, and still be just as much a scalar concept as is the concept of bushels of corn per acre.

On this basis, the equation for a single charge would be

f = q x 1/s² = t/s²,

as well, because the force (quantity of acceleration), is a measure of energy, whether or not it is measured in terms of work (where a direction is specified). This is an analog to potential versus kinetic energy in LST science, but charge is not understood this way in LST science. Instead, it’s understood in terms of charge density, Q, which density can be expressed in units of Coulombs per unit of linear, area, or volume space. Coulombs, in turn are defined in terms of amperes of current per second, but amperes are defined in terms of force!

One way to clarify this confusing maze of definitions is to think of force in the same way we think of acceleration, since the two concepts are reciprocal; that is, whereas we think of acceleration is a change of speed per second, force can be thought of as a change of energy per space, regarding the unit of space in the equation as a scalar value.  So we have

a = dv/t,

for accleration, and we have

f = dE/s,

for force.This makes sense for a couple of reasons.  First, we can readily understand from this formulation that force and acceleration are conceptual reciprocals, we might say, and second,  we see that force and acceleration are forms of energy exchange.  To Increase, or decrease, the speed of a mass, relative to some reference point, at a given time rate of change, requires an exchange of energy at a certain space rate of change.  Of course, this is well known, but we never put it in these terms, because we really don’t know what a “space rate of change” is.  We know what a force is, but what is a “space rate of change?”

I’ll tell you what it is.  It is the inverse square law. More on this later.  I have to give a presentation on scalar science tomorrow and I need to work on it.