Anyone who has explored the realm of the science beyond what is taught in the classroom will undoubtedly run across the term scalar, without any consistency of application. Scalar waves, scalar motion, scalar this, scalar that... it appears the term is popular to describe something that the author does not quite understand himself. So, let us start with a clear definition of what the term "scalar" means:
Scalar
"a quantity possessing only magnitude."
Quantity
"an exact or specific amount of measure"
Magnitude
"greatness of size or extent"
From the definitions, a scalar is simply the "specific amount of greatness." Sounds nebulous, but it fairly precise and a good definition of "scalar." First, consider the term amount. It comes from the old trading days, where people would barter for one "exact or specific amount of measure" for another. "I'll trade you this sack of sugar for two bags of flour." Amounts were the counting numbers. There are three attributes of the counting numbers that make them unique:
1. There is no zero. Suppose I came up to you, and said, "I'll trade you nothing for your new Lexus." Sound like a good deal? If so, please contact the author ASAP. If not, then you understand why zero is not included in the counting numbers. Since they are based in measures, and measures are used in trade, you can only trade what you have and if you have zero of it, then it cannot be used in trade.
2. There are no fractional parts. "I'll trade you two and a half necklaces for three-quarters of your mule." Possible, but pointless.
3. There are no negative amounts. With counting numbers, there must be something to count, and there is no such thing has having "-4" cats in the house.
Now that the idea of quantity is understood to be the whole or counting numbers, consider the term magnitude. How does a magnitude differ from an amount? In simplest terms, and amount is actually an amount of something. You can't have just six. You need six somethings. Amounts qualify other concepts.
But what about magnitude? The magnitude refers to the "greatness of size or extent", which means that it is the quantity specified in the amount of measure, the "six" in "six somethings." The "somethings" is not included in the magnitude, because it doesn't matter what it is, only how many there are.
And there you have the definition of scalar: "A quantity possessing only magnitude", which is one of the non-zero, non-negative, non-fractional, whole counting numbers, without any identification of what they are a quantity of. The minimum scalar magnitude is therefore one and the maximum is infinity.
Some people may say that zero and negative amounts are valid, but they are not part of the counting number system. If the computer at "Cars-R-Us" says they have "-2" brake pads in stock for you, are you going to walk home with anything? A promise won't stop your car. Until you have them, for all practical purposes, "promises" don't exist, and cannot be counted as an item up for trade.
Since we will be dealing totally with the natural systems of reference in this work, we have to stick to what is "real", not "promises" created by the inventive mind of man. They don't exist in Nature. Can you have "-1" ocean?
II. Scalar Ratio
When two scalars are brought into relationship with each other, a ratio is the result.
Ratio
"the relation between two similar magnitudes with respect to the number of times the first contains the second."
As can be seen in the definition of the ratio, the ratio adds the concept of proportion to the concept of magnitude. This gives rise to three possibilities for the proportions of the ratio: one magnitude is either equal, greater or less than the other. This introduces the concept of a scalar orientation.
Orientation
"to adjust with relation to, or bring into relation to surroundings, circumstances, facts, etc."
The three possible scalar orientations for a ratio of scalar magnitudes A and B, as A:B, are: A=B, A>B and AA
A=B, Unit Scalar Speed, A/B = 1.0
A>B, High Scalar Speed, A/B > 1.0
Recall that the minimum scalar magnitude is unity, so the ratios of A/B or B/A will never become undefined since neither A nor B can be zero.
Note that in the low and high speed ranges, the possible combinations of scalar ratios are infinite. But, where A=B, only one ratio is possible: unity. The scalar speed structure thus shows a natural separation across a common "scalar speed boundary" of unity, which can be used as a reference point, a natural datum. This gives a "place" or "location" in which we can begin to define scalar motion, but it has a problem: given any ratio, there is no way to determine if you are observing A/B or B/A... another reference point is needed to determine the orientation of the ratio, itself. This can be found in the invariant property of the cross-ratio.
Invariant
"a quantity or expression that is constant throughout a certain range of conditions."
A cross-ratio is literally a "ratio of ratios", and is the only projective invariant in all strata of geometry. In a scalar sense, it relates two scalar speeds thru a ratio, and that ratio remains constant--giving an orientation to the ratios and producing scalar motion. One can also think of it as the ratio of slopes between two lines on a graph.
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There are now two things to consider as the basis of scalar motion: the cross-ratio and the ratio orientation.Note: A "projective invariant" simply means that you can't alter it, regardless of what perspective transformation you apply to it. A perspective transformation is the process of introducing assumptions to coordinates, to produce a reference system (such as a plane at infinity, and the "eye cone"). More on this later.
The cross-ratio introduces the concept of association. In a geometric sense, it is like two points joining to form a line, except here you join two ratios to form the cross-ratio. The result is a concept we call "dimension."
Dimension
"a magnitude that, independently or in conjunction with other such magnitudes, serves to define the location of an element within a given set."
There are only two possible ratio orientations, since there are only two scalar magnitudes involved in a specific relationship: A:B or B:A.
III. Scalar Motion
Scalar motion is another term that is often used with very little understanding of its meaning. Scalar has already been defined, so let us examine the term motion and its connection with the concept of a scalar:
Motion
"changing place or position."
Motion is a simple enough concept to understand, but when you consider it in the context of "scalar motion", it becomes like "military intelligence"--a contradiction in terms. How is it possible for quantity possessing "magnitude only" to change place or position, when both concepts are totally foreign to the idea of a "magnitude only" scalar? It can't, and there lies the problem with the term "scalar motion."
Exactly what is meant by the term, "motion," when associated with the concept of "magnitude?" The answer is found in how we express the concept of motion as speed--an inverse relation between some "quantity of spatial distance", s, and some "quantity of time," t, as s/t. In other words, speed is just a ratio of space to time and therefore motion, in a more generic sense, is simply a ratio of quantities.
It is important to understand that the concept of motion is a subset of ratio, because ratios deal with magnitudes and motion deals with quantity (magnitudes of something, namely space and time). In essence, we have two similar concepts: that of scalar ratio (generic) and that of scalar motion (specific to space and time).
Scalar Motion is therefore the projectively invariant cross-ratio, with specific aspects of space and time.