**TECHNIQUES
& CONVENTIONS OF MATHEMATICAL MEASUREMENTS**

*During the
17 ^{th} century, the positions, relative distances, and motions of
bodies in space were defined and illustrated by Cartesian coordinates. When theoretically attached to the eyes of an
observer, these mathematical coordinate systems (plus a hypothetical clock)
were called ‘frames of reference.’
Observers at different positions have different perspectives of the
relationships between material objects.
The attempt to mathematically measure and relate different ‘perceptions’
of the same event was sometimes called ‘Relativity.’*

Ancient Egyptian land surveyors first developed the practical rules of space measurement and geometry by trial and error. Then, about 300 BCE, Greek scholar Euclid formalized the empirical concepts of geometry (based primarily on points, straight lines, triangles, and circles) into a system of mathematical axioms in order to measure land and construct buildings. It became known as “Euclidean Geometry.” [1] (Goldberg, pp. 7-9) Another reason for such measurements was to determine the position of a material object or a physical event (i.e. a battle) in order to locate and describe it with precision.[2]

The term ‘__position__’ may be
defined as a specific material point in space at a specific instant in
time. The concept of ‘position’ implies
the measurement of __distance__ from a tangible body of reference (such as a
post on the surface of the Earth) to a material object or physical event. (see Figure 2.1A) “To establish the distance between two points
on a rigid body [i.e. Earth]…is the basis of all measurement of length.” (Einstein, *Relativity*, p. 6) With regard to the measurement of material
objects, __physical__
meaning, vis. that a yardstick had the same physical length for all observers
regardless of their relative positions, visual perspectives or states of
motion.[3] (Einstein, Princeton, 1922, pp. 25, 26) “Every description of events in space
involves the use of a rigid body to which such events have to be referred.”[4] (Einstein, *Relativity*, p. 9) Such rigid body is often called a ‘body of
reference.’

During the
early 17^{th} century, French philosopher René Descartes (1596-1650) devised a simple system to
abstractly describe and illustrate on a piece of paper the position of a point
or an event in space. He used a grid of
rectilinear intersecting lines. Each
line on the grid was assigned a different number, and the point where any two
lines intersected was called a ‘coordinate.’[7] (see Figure 3.1A) The origin or zero point on the grid played
the part of a body of reference. It was
the point of reference to which all coordinates referred. (Goldberg, p. 73)

Descartes
marked the position of a material object or physical event on this grid by
determining its three spatial distances or dimensions (width, length, and
height) from the zero point of reference.
These dimensions are represented by three perpendicular spatial
coordinate axes: x, y, and z. [8]
He depicted the distance between points (positions) by other lines. (Harrison, p. 126) This abstract graphical method of location
and measurement became known as ‘__Cartesian coordinates__.’[9]

Cartesian
coordinates can be used to measure __two__-dimensional distances between two
points on a plane (i.e. on a football field), usually by means of the
Pythagorean Theorem. (Figure 3.1A) Likewise, Cartesian coordinates can be used
to measure __three__-dimensional distances in space by determining the
magnitudes of the three spatial coordinate axes (x, y, and z) relative to the
zero point, and then computing the distance from the zero point to the material
object or physical event (P), again by means of the Pythagorean Theorem. [10]
(Einstein, ‘*Relativity*,’ pp. 6-9; see Figure 3.1B)

When the
origin (zero point) of a coordinate system is theoretically connected to the
eyes of a fictional observer, then the coordinate system can mathematically be
called the ‘frame of reference’ for such observer.[11]
(Young, p. 78) According to mathematical
theory, every observer in the universe has an imaginary system of Cartesian
coordinates (with a set of x, y, z axes) rigidly attached to his eyes. This fiction of a personal coordinate system
theoretically enables the observer to geometrically determine the position and
distance of any other body or event in space relative to his eyes. (*Id*.

Unfortunately,
the term frame of reference (or reference frame) is often used indiscriminately
and interchangeably to mean several very different things. For example, it can mean a system of
coordinates, the visual perspective of a human observer, a distant object that
the observer or measurer is referring to for measurements, or something with a
unique or specific velocity. In Special
Relativity, a frame of reference is the mathematical representation of a
material body of reference in __motion__ with a specific velocity, which
moving body can contain an infinite number of spatially separated observers or
measurers, each with his own clock. All
of these different meanings can become very confusing.

The frame of reference of an observer may be stationary relative to the position of an object or an event, or it may involve a relative change of position—by the object, by the event, and/or by the observer. This relative change of position, especially if it is continuous, is usually referred to as ‘motion.’ (see Memo 3.3) When a time scale or other method for measuring an instant or interval of time (i.e. a clock) is added to the coordinate system, the motion or relative change of position of a frame over time can be mathematically described. [12]

How can we
graphically illustrate the motion of a material point or body on a system of
coordinates? First, we must determine
the successive positions of the body at successive specified times. (Born, pp. 16 – 17) To do this, we must change the x-axis of the
coordinate system to represent the distance (d) traveled by the body, and we
must change its y-axis to represent an interval of time elapsed (t). (*Id*., p. 18) We then plot each successive position of the
moving body as a point for each successive interval of time. By connecting the points a line is created
which represents and illustrates the motion of the moving body. In this way we can abstractly illustrate
various different types of motion. (*Id*.,
pp. 16 – 17; Figure 3.2)

Coordinate systems can be used to analyze the
motions of matter in the abstract or with regard to their causes; in other
words, by kinematics or dynamics. ‘__Kinematics__’
is the study of abstract geometrical motions of bodies, without consideration
of their causes (i.e. a force). ‘__Dynamics__’
is the study of motion with regard to its causes, i.e. the magnitudes of force,
mass, and time which result in the acceleration, velocity, or momentum of a
body. (Folsing, p. 178)

The motions
of bodies may be characterized by their __magnitudes__. (see Memo 3.3) The ‘speed’ of a body is the __rate__ of
its motion…the distance a body moves from one position to another, divided by
the time elapsed. The ‘velocity’ of a body
(or a ray of light) expresses its speed in a __certain__ direction. [13]
(Goldberg, p. 33) When the rate of speed
of an object is uniform or constant and in the same rectilinear direction, this
can be called a ‘uniform rectilinear velocity.’[14]
(see Figure 3.2B) A uniform rectilinear velocity may result
from a continuous force, such as where a locomotive is needed to propel a train
down a straight track at a constant speed in order to overcome friction and the
Earth’s force of gravity. However, in
empty space, far from gravity and friction, this continuous application of
force is not necessary to maintain a substantially uniform rectilinear
velocity. This type of uniform
rectilinear velocity of a celestial or other body in space (without the
application of force) is known as ‘inertial’ motion. (see Figure 4.1A) Because Special Relativity is based upon
‘inertial motion’ (with or without the application of force), inertial motion will
become very important for our later discussions.

When either the rate of the speed or the direction (orientation) of a body’s uniform velocity is changed (vis., by another force), this changed motion is known as ‘acceleration.’[15] Accelerations may be ‘uniform,’ such as a train uniformly increasing its speed down a straight track (see Figure 3.2C), or the uniform circular orbit of a body (Figure 3.4C1), or the constantly increasing gravitational acceleration of a body falling toward Earth.[16] (see Figure 4.1B) Accelerations may also be varied or arbitrary, such as the herky-jerky motions of a roller coaster. (see Figure 3.2D)

Motions of
bodies may also be characterized by their __orientations__: their directions of motion. For example, ‘rectilinear’ or ‘translational’
motion is where a body moves or ‘translates’ in a straight line from one
position to another.[17]
(see Figure 3.4A) ‘Curvilinear’ motion is where a body moves
from one position to another in an arc.
(see Figure 3.4B) ‘Orbital’ motion is where one body moves
around another body in a circular, elliptical or arbitrary path. (see Figure 3.4C) ‘Rotational’ motion is where all points on a
body rotate or revolve around the body’s own axis. (see Figure 3.4D) The motion of the Earth through space
exhibits a combination of all of the above described motions, orientations and
trajectories at the same time: relative
to its own axis, and relative to other planets, the Moon, the Sun, other stars,
the core of the Milky Way Galaxy, and other galaxies.

**B. ****Trajectories,
Perspectives, Perceptions, & Transformations**

What is a
trajectory? It is usually described as
the __path__ of a moving object, such as a baseball in flight. The trajectory of an object may be more
specifically described as the continuance over space and during time of the
three-dimensional coordinate positions of such object as perceived or measured
by an observer.

The
trajectory of any moving terrestrial body will __appear__ to be different
for each observer, depending upon such observer’s unique position and visual
perspective relative to such moving body.
For example, a baseball slugger views the curved trajectory of his

Does this
mean that the physical laws of motion for the ball, during its flight from the
batter to the bleachers, are different for each individual observer? Of course not. There is only __one motion__ of the
baseball, and this motion does __not vary__ because of different observers
watching or measuring it. Only each
observer’s unique visual perspective, perception and __coordinate description__
of such motion (its trajectory relative to the observer’s eyes and his unique
position) varies. (see Rohrlich, pp. 20 – 21)

These simple
concepts and facts will become critical when we consider and analyze Special
Relativity in Part II of this treatise. Why? Because Einstein claimed *ad ho*c that his dubious methods of measurement from one moving body
to or from another distant moving body caused the __length__ of a meter rod
to become shorter or contracted (possibly to zero length) on the distant moving
body, and also caused the __duration__ of the time intervals on such distant
moving body to shorten (possibly to zero duration). (see Figure 3.8 and Chapters
26 & 28) Later Einstein even claimed
that such dubious perceptions and measurements caused the magnitude of the __mass__
of the distant moving body to increase (possibly to infinity).[18] (see Chapter 31)

Now let us
return to the discussion of perspectives, perceptions, and trajectories. What if the observers are also moving
relative to the trajectory of the moving baseball? Each moving observer at different changing
positions relative to the moving baseball merely has different visual
perspectives, different changing perceptions and __different coordinate
descriptions__ of the same event…the same motion of the baseball. (Rohrlich, p. 20) Algebraic[19]
equations that mathematically __relate__ two different coordinate
descriptions of the same (or an identical) event are called ‘__transformation__
equations.’ (Goldberg, p. 74) The different coordinate descriptions of the
same event that we have been discussing obviously result from the different
visual perspectives and perceptions of various observers or the same moving
observer. On the other hand, some
different coordinate descriptions that we will later discuss in Special
Relativity result from Einstein’s creation of __distorted__ transformation
equations and Einstein’s misapplication of such distorted transformation equations
to physical phenomena.[20]

As a further example of such different perspectives, perceptions and different coordinate descriptions of the same event, assume that a stone is dropped from a uniformly and rectilinearly moving railway carriage onto the straight railway embankment below. In 1916, Einstein asserted that:

“the stone traverses a straight line relative to a system of co-ordinates
rigidly attached to the carriage, but relative to a system of co-ordinates
rigidly attached to the ground (embankment) it describes a parabola. With the aid of this example it is clearly
seen that there is no such thing as an independently existing trajectory (lit.
‘path-curve’) but only a trajectory __relative to a particular body of
reference__.”[21] (Einstein, *Relativity*, p. 11; see Figure 3.7)

The falling stone in Einstein’s above description is merely an example of the stone’s combined inertial motion and gravitational fall in a parabolic motion (see Figure 4.1), expressed in terms of each observer’s different visual perspectives, perceptions and Descartes’ coordinates. The so-called “straight line” vertical coordinate trajectory described in Einstein’s above example is best referred to as an illusion. [22] As rationalized by Born, the “stone falls…along a vertical [path] that is moving with the [carriage]. (Born, p. 69)

Why did Einstein describe this example of two different
observers with different visual perspectives who __perceive__ different
trajectories for the same motion? It
certainly could not have been for the reason that he suggested, because
Einstein’s Special Theory has nothing to do with different perceived
trajectories for the same motion.[23]

Einstein’s real reasons for
describing the above example so early in his book *Relativity* were
obviously quite different. His example
attempts to establish certain mindsets for his readers with respect to his
later Special Relativity concepts, analogies and rationalizations.[24]
First of all, Einstein is suggesting that the __relative motions__ (of the
passenger on the train, the falling stone and the observer on the embankment) by
themselves can affect each different observer’s perceptions, mathematical
descriptions and coordinate measurements of the same event.[25] Secondly, for his Special Theory, Einstein
needed two different observers (measurers), each with a different uniform
rectilinear relative motion, and each with a different frame of reference (coordinate
system).[26] His above example provided all of these
requirements.

But above all, what Einstein specifically suggested with his above
example was that different observers on different __inertial__ bodies (i.e. observer
No. 2 on the train and observer No. 1 on the Earth) with different inertial
velocities can make different coordinate measurements of the same motion of an
object entirely by reason of their __relative velocity__. (see Figure 18.1) In effect, Einstein’s above example was a
preview of Special Relativity; an early artificial analogy that attempted to
indoctrinate his readers with misleading ideas that might confuse them into
believing that his later meaningless relativistic concepts, i.e. the Relativity
of Length, were at least plausible.

In Einstein’s relativistic concept
(called the Relativity of Length), the engineer in the detached engine (which
is moving at a different uniform velocity than the carriage) theoretically
cannot __simultaneously__ plot the time coordinates for the coordinate
positions of the front of the carriage and of the rear of the carriage by the
hand and eye method. (see Figure 3.7; Resnick,
1992, pp. 480 – 481) Therefore, the
engineer must plot each time and position coordinate separately. According to Einstein, another reason for
this physical inability is that the light from the rear of the carriage takes a
longer time interval to reach the engineer’s eye than the light from the front
of the carriage.

Theoretically, these physical inabilities will cause mathematical
measurement problems for Einstein.
During the time interval between each plotting, the carriage and the
engine will have changed their relative positions (because of their different
velocities) so that the plotting of the second coordinate position will be
different than if the engineer and the carriage were relatively at rest. Thus, claimed Einstein, the physical length
of the carriage is entirely __relative__ for the engineer and it depends
upon the relative velocity of the two moving bodies (in other words, their
reference frames) and the time interval between such coordinate measurements.[27] (also see Figure 18.1)

Stated in a different way, the coordinate
descriptions for the length of the carriage will be different for the
engineer’s delayed visual coordinate estimates of length than for the observer
on the carriage who physically measures its length with a rigid meter rod. Therefore, there are two different coordinate
descriptions and measurements for the length of the same moving object. Einstein’s artificial ‘solution’ for this
concocted mathematical problem of measurement was to relate such different
coordinate measurements for the same event with his radical Lorentz
transformation equations.[28] The Lorentz transformation equations, in
turn: 1) distorted Einstein’s inexact
distant moving measurements so that they became mathematically identical to the
measurements of the observer on the carriage, and 2) algebraically produced all of the bizarre
mathematical consequences of Special Relativity, including the absolute
velocity of light at *c*, length
contraction, time dilation, increase in mass with velocity, etc., which
Einstein needed for his Special Theory.
“Technically, the whole of the special theory is contained in the
Lorentz transformations.” (Bertrand
Russell, 1927, p. 29)

It may be difficult for the reader to believe at this point in our story,
but these theoretical and technical inabilities to make __simultaneous__
mathematical coordinate measurements for both ends of a moving object in 1905
were the basic mathematical premise and rationalization for all of Einstein’s
contrived relativistic theories of measurement.[29] (see Chapter 28) They were also the primary mathematical
justification for his entire Special Theory.[30]

Why are we even discussing the ancient concepts of coordinates, frames of reference, transformation equations, relativity, and the like in this chapter? Because they are all man-made conventions that were invented during centuries past in order to physically and mathematically describe and understand the positions, motions, relationships and measurements of material bodies in space and during time. [31] Antiquated as these mathematical conventions may be, they are all necessary for a basic understanding of Einstein’s Special Theory of Relativity as well as other theories and mathematical concepts that we will discuss in later chapters.

[1] *Elements*.” The geometry that it described “originated
from observational data and practical experience.” (Logunov, p. 5) “The widespread use of Euclidean geometry was
a measure of the truth of Euclidean axioms.”
(Goldberg, p. 68)

[2] The Earth was first measured (a process called “geodesy” by the early Greeks) depending upon the number of a man’s paces or by the length of a chain or rod. Later it was measured by exact units (meters and yards). Time was originally measured by night and day, the phases of the moon, the seasons, and sundials. Later it was measured by exact units (seconds and hours). (Born, pp. 5 – 6)

[3] In other
words, for __invariant__
__properties__ of that object. The
author asserts that this is a correct concept.
On the other hand, Einstein (in 1905) challenged this assumption and
conjectured (as a cornerstone for his Special Theory) that relative linear
motion changes the physical linear distances on one moving object when viewed
by an observer on another object moving linearly at a different velocity. (Einstein, 1905 [Dover, 1952, pp. 41 –
42]) However, as we shall discover in
Chapters 26 and 28, Einstein’s ‘Relativity of Distance’ concept was only an
illusion based on Einstein’s dubious techniques of measurement.

[4] In
Chapter 22 we will discuss why this statement by Einstein has no real meaning
with regard to the point of emission of a ray of light in space, and its
velocity of transmission at *c* relative to such point of emission.

[5]

[6] For
example, D’Abro concluded that this invalid postulate by __inevitable__ a belief
in some real absolute medium, space or the __ether__, from which rotation
and acceleration would have a real meaning.”
(D’Abro, 1950, p. 111)

[7] It was German scientist Leibniz who first called these intersections ‘coordinates.’

[8] Another example of coordinate axes are the lines of longitude and latitude on a map measured from arbitrary zero points of reference on the Earth’s surface and used to designate positions, places and events on the surface of the Earth.

[9] “In modern physics, coordinate systems are nothing but a useful fiction.” (Jammer, 1954, p. 98) The abstract representation of points and figures in space by coordinates and their analysis is called ‘analytic geometry.’ Later, geometrical structures were described by algebraic symbols and equations (a process now called ‘algebraic geometry’).

[10] Thus, Descartes extended the Pythagorean Theorem to three dimensions. (Harrison, p. 134) Such three-dimensional distance measurements can be made either directly (by rigid measuring rods) or indirectly by instruments or estimations.

[11] Mathematically, bodies and moving bodies are referred to as ‘frames.’ A ‘reference frame’ is a fictional mathematical construct, which is composed of a rigid coordinate system with a hypothetical observer or measurer making space and time measurements from its zero point of origin. (Rohrlich, p. 20; see Figure 3.2)

[12] Thus ‘motion’ can mathematically be defined as the change of the position of a rigid body during an interval of time, as measured by an observer at a particular position with a particular frame of reference. (see Oxford Dictionary of Physics, p. 309) Motion can also be described as the ‘translation’ of a body from one position along a straight line to another position. (French, p. 67)

[13] Speed
is called a ‘scalar’ quantity, because only the __magnitude__ of a body’s
speed is important. Velocity is called a
‘vector’ quantity because the __direction__ of the body is also important. (Goldberg, p. 33)

[14] Galileo was the first person to realize that “a uniform velocity is a constant rate of change of position.” (Born, p. 7)

[15] Mathematicians often refer to ‘deceleration’ (a decrease in speed or velocity) as ‘acceleration in a negative direction.’ (see Figure 3.2E)

[16] Again, Galileo was the first person to realize that “a uniform acceleration is a constant rate of change of velocity.” (Born, p. 7)

[17] The
mathematical term ‘translation’ means:
‘Motion of a body in which __all the points__ in the body follow
parallel paths.” (Oxford Dictionary of
Physics, p. 507)

[18]
Einstein also claimed *ad hoc* that
(based on his measurements) a light ray had one velocity with respect to a stationary
object, and a greater or lesser velocity with respect to a body that was moving
linearly toward or away from the light ray.
(Einstein, *Relativity*, pp. 22
– 23)

[19] What do we mean by the words ‘algebra’ and ‘algebraic?’ See Memo 3.6.

[20] For example, Einstein misapplied the Galilean transformation equations to light and misinterpreted the results. (Chapter 19) Then he derived and applied the radical distorting Lorentz transformation equations to light in order to attempt to rectify and justify the mischief that he had created. Both of these transformation equations will become critical to our later discussions.

[21] But see the note and question at the bottom of Figure 3.7.

[22] From
the __perpendicular__ perspective of the eye of observer No. 1 standing on
the embankment, the combined (inertial and gravitational) parabolic motion of
the stone was obvious. (see Figure 3.7) But the observer No. 2 on the carriage __perceives__
a different trajectory. From the __vertical__
perspective of the eye of observer No. 2 on the uniformly moving carriage, the
stone moves uniformly forward substantially at the same velocity as such
observer’s eye (because of the horizontal inertia of the stone) as it also
falls downward toward the embankment. So
such observer’s moving eye watching the moving stone fall always sees the stone
directly below his co-moving eye __as if__ the stone was falling straight
down. The resulting “straight line”
vertical coordinate trajectory is a visual illusion. Einstein inferred that such different
trajectories were solely the result of __relative motion__, which of course
is nothing more than a continuous relative change of __position__.

[23] Instead, Special Relativity has to do with two different measurements of lengths and time intervals for one motion of an object. (see Chapters 26 and 28)

[24] It should be realized at this early juncture that Einstein seldom made any statement concerning physics unless it furthered his agenda for one of his theories.

[25] But if a slow motion movie was taken of the events from either frame of reference, then everyone would realize exactly what was happening.

[26] Throughout his theories, Einstein equates the concepts of reference frames and coordinate systems. For Einstein, they are identical.

[27] In
short, the engineer (in his moving coordinate system) cannot physically, by
Einstein’s __sequential__ coordinate method of measurement, measure the
correct (stationary) physical length of the moving carriage in its differently
moving coordinate system.

[28] The
Lorentz transformation equations were invented by Dutch physicist H. A. Lorentz
during the period 1899 – 1904 in an *ad hoc* attempt to mathematically
rationalize the Michelson & Morley paradox and other paradoxes caused by
the ether theory. We will discuss these
subjects in detail in Chapters 9 – 12, 15 and 16.

[29] In
addition, Einstein asserted that such inability to __simultaneously__
measure the instant of time for such coordinate positions also meant that __time__
was entirely __relative__ for the engineer.
This concocted reciprocal concept was called the Relativity of
Simultaneity, or the Relativity of Time.
(see Chapters 26 and 28)

[30] One
might ask: What do Einstein’s physical
inabilities to make exact measurements by hand and eye coordinate plottings in
1905 have to do with physics in the 21^{st} century? The answer is: nothing.

[31] It should be emphasized that the aforementioned concepts of coordinates, frames of reference, transformations and relativity are merely abstract mathematical conventions invented for purposes of mathematical description and theoretical measurement. Whereas, the concepts of positions, a body of reference, rigid measuring rods, trajectories, changes of position over time (motions), distances (lengths) and perspectives are all observed, empirical and physically real.