# RS-103: Three-Dimensional Motion

Narrated slide show on three-dimensional motion, presented by Gopi Krishna. ### Three Scalar Speeds

Hi Gopi,

Congratulations on a clear and concise presentation of the Reciprocal System. There is one important point that I must take issue with, however.

In RS-103, the concept of three scalar speeds is explained, as depicted in the slide entitled "Reciprocal System 3D:"

s1/t1, s2/t2, s3/t3

The next few slides discuss how only one of these three speeds can be measured, or represented, in a conventional reference system, a point emphasized by Larson again and again in his works.

Yet, it is easy to get confused on this point, if we fail to carefully distinguish between space, defined in the conventional system, as distance between points, and space, defined in the Reciprocal System, as the inverse of time.

Conventional space, as distance between points, is speed multiplied by an interval of time, but reciprocal space, as the inverse of time, is not. As you explain so well in your presentation, the unit progression is an increase of one unit of space for each unit of time, but since space has three dimensions, while time has no dimensions (in space), the spatial expansion is in all directions from a given point in the progression.

However, representing the motion of the 3D progression in a fixed coordinate system is problematic, because any point in the system expands over time. Thus, the 1D vector depicted within the balloon is misleading, because the points used to measure the 1D motion cannot remain points over time, but must themselves expand.

Motion between two points as shown in your balloon analogy is motion between two inertial systems, not motion between two points in a scalar expansion. While scalar motion between inertial systems certainly exists, it is not limited to one dimension (otherwise we wouldn't be able to observe the 3D scalar expansion, as manifest in the receding galaxies).

In order to represent scalar motion between inertial systems in the Reciprocal System, we have to first develop inertial systems. The way Larson does this in his works is to "kill" the expansion in one dimension first (by linear vibration), then kill it in the two remaining dimensions by rotation of the vibration. This establishes an inertial frame of reference (well, the beginning of one anyway).

Once these systems are established (as discrete entities of scalar motion), it becomes possible to define both scalar motion and vector motion between them. In the case where scalar motion affects the distance between them, the gravitational limit and the effect of the progression determine whether their separation increases or decreases, but any vector motion is independent of these factors, making the interaction of the system's motions quite complicated.

When Larson talks of the three scalar speed ranges, he is referring to speeds between inertial systems, interacting vectorially and scalarly. Where vectorial speeds reach scalar magnitudes, in any given dimension, strange effects are observed between these systems, and classes of these systems, which is a marvelous concept to explore, but to explain these speed ranges as three independent scalar speeds, while tempting, is only self-defeating in the end, in my opinion.

I wish I could give you more specifics on how to approach explaining it better, but I'll have to think about it some more.

Regards,
Doug ### Scalar Speeds

Thus, the 1D vector depicted within the balloon is misleading, because the points used to measure the 1D motion cannot remain points over time, but must themselves expand.

The points seen as a spatial representation would remain points, while only their position changes. Hence, if you are looking at the definition of a point as a mark of position, it definitely changes. But the point itself does not expand.

Motion between two points as shown in your balloon analogy is motion between two inertial systems, not motion between two points in a scalar expansion.

However, every point can be taken to be fixed in an inertial system which places it at the origin, so it works well in that way. However, the inverse condition, that every point necessarily needs to be at the origin, is not necessary. So, the inertial systems relation is a special case of the point-to-point relation.

When Larson talks of the three scalar speed ranges, he is referring to speeds between inertial systems, interacting vectorially and scalarly.

Could you clarify how Larson mentions the three scalar speed ranges interacting vectorially? As far as I know, the scalar speed range is just that... a range of magnitudes of speeds. And the three independent scalar ranges are not independent in the sense that they do not interact with each other, but independent in the sense that change in one speed range does not necessarily imply a change in the other speed range.

Gopi ### Hi Gopi, Thus, the 1D vector

Hi Gopi,

Thus, the 1D vector depicted within the balloon is misleading, because the points used to measure the 1D motion cannot remain points over time, but must themselves expand.

The points seen as a spatial representation would remain points, while only their position changes. Hence, if you are looking at the definition of a point as a mark of position, it definitely changes. But the point itself does not expand.

I'm sorry. I can't accept that. You can assume that it is the case, but you can't maintain that it is so and be consistent. If there is nothing but motion, then you can't consistently assume that there also exists something that is not motion to use as a reference to measure that motion.

Motion between two points as shown in your balloon analogy is motion between two inertial systems, not motion between two points in a scalar expansion.

However, every point can be taken to be fixed in an inertial system which places it at the origin, so it works well in that way. However, the inverse condition, that every point necessarily needs to be at the origin, is not necessary. So, the inertial systems relation is a special case of the point-to-point relation.

Yes, but the point that I'm making here, and it's an important one to note in passing, is that, in the RST, we don't have inertial systems, initially. That's why the surface of an expanding balloon is an imperfect analogy. For example, the size of painted spots on the balloon would increase in size as the balloon expands, because all points on the surface of the balloon, no matter how small, must move away from every other point on the surface, as it expands. The only way that the points themselves would not expand is if they were separate from the surface, i.e., if they were placed on the surface somehow.

However, in a universe of nothing but motion, objects separate from motion do not exist. So, the crucial question becomes, "How do we describe points that do not themselves expand?" More on the answer to that question later, but first you write:

When Larson talks of the three scalar speed ranges, he is referring to speeds between inertial systems, interacting vectorially and scalarly.

Could you clarify how Larson mentions the three scalar speed ranges interacting vectorially? As far as I know, the scalar speed range is just that... a range of magnitudes of speeds. And the three independent scalar ranges are not independent in the sense that they do not interact with each other, but independent in the sense that change in one speed range does not necessarily imply a change in the other speed range.

I didn't mean to imply that the independent speed ranges interact, but that the speeds between inertial systems (matter) attain both vectorial and scalar magnitudes, which have to be understood as modifying their observed behavior in ways that cannot be understood without realizing that both types of motion are present. Larson's point, in the Universe of Motion, was that when vectorial speeds approach scalar speed limits, transistions from one range to the other begin to take place, producing very intriguing effects that baffle LST astronomers and astrophysicists.

However, to understand the three dimensions of scalar motion more precisely, we now know that we can use mathematics that Larson did not exploit. For instance, we can modify the fixed reference equation of motion from a 1D equation to a 3D scalar equation of motion:

ms = ds3/dt0

where ms is scalar expansion of space that is substituted for velocity v (and here d is the delta, not the derivative), because it's not a measure of the velocity of an object changing its location in a given direction, defined by three dimensions; that is, the s3 term is expanding (contracting) space, not increasing (decreasing) distance.

Of course, the inverse of this equation is,

mt = dt3/ds0

which defines scalar expansion of time, where the 0D space term is the scalar and the 3D time term is the pseudoscalar. Fortunately, the mathematics is clear. The scalar, in each case, is the point relative to which the expansion takes place in all directions, creating the expanding pseudoscalar, permitting us to avoid the confusion of the balloon analogy.

Of course, the challenge we have is determining how this equation can be correctly applied in the context of Larson's work.

Regards,

Doug