By 1905, theoretical physics appeared
to some scientists to be in a state of complete disarray. Vestiges of

**A.
The theoretical background for Einstein’s mathematical ‘difficulties.’**

Anyone who has read Albert Einstein’s early writings on physics must conclude that he was greatly disturbed by several problems (or ‘difficulties,’ as he called them), which he perceived to exist in physics during the early part of the first decade of the twentieth century. In this Chapter 19 we shall discuss and analyze those perceived ‘difficulties’ that apparently led Einstein to construct his Special Theory of Relativity.

Despite Lange’s abstract invention of relatively moving inertial reference frames in 1885, there still remained in physics vestiges of Newton’s concept of ‘absolute space’ at rest. The most obvious example of this was the concept of stationary ether, which was generally considered to be an absolute frame of reference at rest in space from which measurements might be made.[1] Faraday, Maxwell and many other physicists had incorrectly based their theories on the existence of this ‘specially favored’ reference frame.

The paradoxical null results of
Michelson’s experiments, and many similar light experiments during the latter
part of the 19^{th} century, which attempted to detect and measure the
absolute motion (velocity, direction and distance traveled) of the Earth with
respect to the ether at rest, must have been particularly troubling for
Einstein. At the turn of the century,
the paradox of the M & M null results in particular remained a disturbing
enigma for the scientific community. The
absolute theories of Lorentz and Fitzgerald, which arbitrarily attempted to explain
away such paradoxical null results in order to defend and save the concept of
stationary ether as an absolute reference frame, would not have been acceptable
because they were so *ad hoc*. For
Einstein, the concept of stationary ether must be abandoned (or at least
ignored), measurements of space needed to be drastically changed and redefined,
and the baffling mystery of the M & M null results (and their theoretically
missing time intervals) must be more plausibly explained.

In the physics of Einstein’s early
days, there were similar vestiges of

As previously mentioned in Chapters 14 and 17, the classical Galilean transformation equations did not describe the theoretical mathematical increase in the electromagnetic mass (actually resistance) of a charged body or particle when its velocity is increased. Einstein believed that this mathematical ‘difficulty’ needed to be rectified by a modified set of transformation equations that would describe this theoretical phenomenon. (Goldberg, pp. 131 – 141; D’Abro, 1927, pp. 157 – 160) There were also other problems and paradoxes in physics that disturbed Einstein. (see Chart 19.3)

Yet, by far the most important ‘difficulty’ which
baffled Einstein and the entire scientific community in 1905 was the apparent __contradiction__
between two revered __mathematical__ principles of physics: the Galilean transformation equations of
mechanics which mathematically describe Galileo’s Relativity, on the one hand,
and Maxwell’s electromagnetic equations which inter alia implied that the velocity
of light had a constant velocity of c en vacuo, on the other hand.

What exactly was this apparent contradiction or paradox? The Galilean transformation equations, which
describe the velocity of material objects relative to inertial frames of
reference, “give rise to a certain __addition law of velocities__, the
relation commonly used in everyday life.”
(E. B., 1969, Vol. 19, p. 96)

“For example,
if a man walks along a moving train with a velocity of 2 mph relative to the
train, in the direction of motion of the train, and if the train moves with a
velocity of 60 mph along the tracks, then the velocity of the man relative to
the tracks will be, it is said, 62 mph.”
(*Id*.

Based on the
above examples, Rohrlich logically concluded:
“The speed of an object is a __relative quantity__ depending on the
frame __relative__ to which it is measured.”[3]
(Rohrlich, p. 52; also see Figure
7.1)

But what happens if, in place of the walking man, there is a light ray traveling through the train? (see Figure 19.1B) “[I]t would be expected that [the light ray’s] velocity also would be different if it is measured relative to the train or relative to the tracks.”[4] (E. B., 1969, Vol. 19, p. 96) Rohrlich agreed:

“Elementary
reasoning according to Newtonian mechanics requires that if the speed of light
is *c* as measured in a particular reference frame then it cannot also be
the same number *c* relative to a different frame.”[5] (Rohrlich, p. 52)

“However, experiments show that the speed of light is the same __in__
all inertial systems of reference. Consequently,
the Galilean principle of relativity…and the constancy of the light velocity in
an inertial frame of reference seem to be empirically incompatible.” (see E. B., 1969, Vol. 19, p. 96) In other words, the mechanics law for the
addition of velocities empirically does not apply to the velocity of
light. What could be the answer to all
of these paradoxes?[6]

Remember that Maxwell interpreted his electromagnetic
equations to mean that a ‘disturbance’ was created in the material ether, and
that this disturbance of matter was instantly transmitted in the form of
electromagnetic waves (light) in all possible directions through the stationary
ether at the constant velocity of c (300,000 km/s).
(Chapter 6A) Maxwell also “pictured
light waves as travelling __on__…some medium [such as water waves move on
water]. That medium was called the
[luminiferous] ether. [Thus] c is the speed of light __relative__
to the ether.” (Rohrlich, p. 52)

By the late 1880’s,
Maxwell’s equations, and his interpretations of them, were accepted as valid by
much of the scientific community. But
then the question arose: if the velocity
of a light ray at c is a constant velocity with respect to the material medium of
stationary ether, then must not such light ray have a different velocity with
respect to each material body that is moving __linearly__ at velocity v
relative to the stationary ether? For
example, pursuant to the Galilean transformation equations of mechanics, the
velocity of a light ray at c should a priori be either c + v or c – v with respect to a material body (i.e. the Earth) moving linearly
at v relative to the light ray. Such
different velocity should also depend upon the __direction__ of such body’s
motion relative to the direction of propagation of the light ray. (

Thus, the scientific
community concluded that the velocity of a light ray was not always __constant__. However, this conclusion appeared to
contradict Maxwell’s equations, and the constancy of the velocity of light that
they implied. Michelson’s paradoxical
null results only added to the scientific confusion, because they implied that
the velocity of light emitted for a moving body (Earth) was always c in every direction. How could a light ray propagate over an
absolute __greater distance__ in the direction of the Earth’s solar orbital
motion, but no greater time interval was detected by Michelson’s
apparatus?

Finally, the scientific
community concluded that there was only one preferred reference frame “in which
the measured speed of light is exactly c; that is…ether…at rest.” (Resnick, 1968, p. 17; also see Rohrlich, p.
52) In all other inertial reference
frames which moved relative to the ether frame, they assumed and concluded that
the transmission velocity of a light ray must __vary__ “from c + v to c – v” depending upon the
linear direction of motion of the inertial reference frame relative to the
direction of the propagating light ray. [7]
(Resnick, 1968, p. 16) Thus, by the
1890’s it appeared that with respect to linearly moving bodies, __either__: 1) the varied mathematical computations (c + v and c – v) based on the
Galilean transformation equations of mechanics and relativity were incorrect;
or 2) Maxwell’s equations, which
mathematically required that the velocity of light transmitting en vacuo must always remain a __constant
____c__,
were incorrect. But how could this
be? Both concepts had proved their worth
over extended periods of time. [8]

Remember that in Chapter
6, we also discussed the fact that the fictitious concept of stationary ether
was actually __irrelevant__ to the validity of Maxwell’s equations. In the absence of ether all that remains is
empty space, so Maxwell’s equations really implied that the constant
transmission velocity of light is c __relative to its medium of the vacuum of empty
space__. However, these facts were
apparently not by realized by Einstein nor by the scientific community in 1905.[9]

**B. Einstein
described his mathematical ‘difficulties,’ including those with respect to the
classical addition of velocities.**

Einstein referred to the
aforementioned mathematical paradoxes and apparent contradictions as the
‘difficulties,’ and he concluded:

“There is hardly a simpler __law__ in physics
than that according to which light is __propagated__ in empty space. Every child knows that this __propagation__
takes place in straight lines with a velocity c = 300,000 km/sec…[10]

“Who would imagine that this simple law has plunged
the conscientiously thoughtful physicist into the greatest intellectual __difficulties__?”[11] (Einstein, Relativity, pp. 21 – 22)

In an attempt to
describe and explain the mathematical ‘difficulties’ that he imagined—the __apparent
conflicts and irreconcilability__ between the Galilean transformation
equations and Maxwell’s equations for the constancy of the velocity of light at
*c* previously discussed—Einstein concocted the following analogies and
examples.[12]

In Chapter 6 of his book, *Relativity*, entitled “The Theorem of the
Addition of Velocities Employed in Classical Mechanics,” Einstein constructed
the following __attempted analogy__ to Galileo’s 1632 sensory and empirical
concept of relativity. Einstein supposed
that a railway carriage was traveling along a straight railway embankment at a
constant velocity (v), and that a man inside the carriage walked the length of
the railway carriage at a constant velocity (w) in the direction of the
carriage’s uniform motion. (see Figure 19.1A) Einstein then asked the question: What will be the total velocity (W) of the
man relative to the embankment? He added
the two uniform velocities (v) and (w) together to arrive at the total, v + w =
W. Einstein then asserted that, “this
result…expresses the __theorem of the addition of velocities__ employed in
classical mechanics.” (Einstein, *Relativity*,
pp. 18 – 19) Let us now scrutinize the
above attempted analogy to Galileo’s Relativity and Einstein’s characterization
of it, to see if they are correct.

In his attempted analogy to Galileo’s Relativity, Einstein merely added
the __constant velocities__ of two material objects (the carriage and the
man) together in order to compute a third constant velocity (W) of the man
relative to the embankment. Remember
that we also used this type of simplistic computation; a) in Figure 7.1 in order to demonstrate that
the velocities of material objects can be added together or subtracted from one
another in order to compute their total velocity relative to another material
object; and b) in Chapter 14 with
respect to the translations and measurements computed by the Galilean transformation
equations. Regardless of the fact that
Einstein’s algebraic computation, v + w = W, was correct, his addition of __constant__
velocities has absolutely nothing to do with Galileo’s empirical concept of
relativity (as Einstein implied), nor any of its purposes. (see Chapters 5 and 13)

In Einstein’s attempted
analogy to Galileo’s Relativity, the relatively stationary embankment played
the part of the relatively stationary dock on the inertially moving Earth. The railway carriage played the part of the
ship moving at a constant uniform velocity away from the relatively stationary
dock. (Figure 5.3) But the uniformly walking man was __not
accelerating__ on the inertially moving carriage, so he played no part in
Galileo’s Relativity. Remember that one
major purpose of Galileo’s Relativity was to __empirically__ demonstrate
that accelerated motions occurring on one inertially moving body are
empirically the same as accelerated motions occurring on a different spatially
separated inertial body with a different velocity.[13] However, there is no way that a man walking
at a __uniform velocity__ on one inertially moving carriage could ever __empirically__
demonstrate this mechanical covariance (or equivalence) of accelerations
occurring on different spatially separated inertially moving bodies. Nor could the uniformly walking man
demonstrate the empirical invariance of __sensorally__
demonstrate that the man could not tell if the train was at rest or uniformly
moving in a straight line.

Galileo’s Relativity
involved __two__ different accelerated events on two different inertial
bodies (Earth and a sailing ship) separated by a distance, not one
non-accelerated event (a uniformly walking man) on one inertially moving body
(a carriage). Galileo’s concept of
relativity was also purely an __empirical__ and __sensory__ concept, not
a mathematical one. A mathematical
‘theorem of the addition of velocities’ of a passenger’s uniform velocity added
to the ship’s uniform velocity relative to the uniform inertial velocity of the
Earth had no purpose within the concept of Galileo’s Relativity. The magnitudes of uniform or inertial
velocities were never specified (even abstractly) in Galileo’s Relativity,
because they were __irrelevant__ to its purposes. For the same reasons, they were never added
together. (see Figure 19.4 and Chapter
5) Thus, Einstein’s attempted analogy
had absolutely no relevance to Galileo’s Relativity or its purposes.

The only ‘velocity’ that
was ever relevant to any form of relativity in the 19^{th} century
occurred when the Galilean transformation equations were invented after 1885
and were applied to Lange’s relative inertial reference frames in an attempt to
__mathematically__ describe and measure relative motions. (Chapters 13 and 14) With these equations there was only an
abstract algebraic computation of the uniform translation or distance traveled
(vt) by the inertially moving frame away from the relatively stationary frame,
so that there could be a one-to-one mathematical point translation
(transformation) of accelerated motions from the ‘stationary frame’ to the
‘moving frame,’ and vice-versa. This
algebraic translation of position, of course, could never demonstrate the
empirical or mechanical covariance of accelerations in different reference
frames (inertial bodies), nor could it demonstrate the sensory equivalence of
the different inertial reference frames.

Even with the Galilean
transformation equations, one constant uniform velocity was rarely, if ever,
added to another constant __uniform velocity__ in order to compute a third
constant uniform velocity, the way Einstein did in his attempted analogy. What purpose would this computation serve? [14] It would only be relevant to descriptions or __measurements__
of motions and positions of reference bodies.
But, again, these purposes had nothing to do with Galileo’s Relativity.

Normally, the only
addition or computation of velocities in Lange’s model of inertial reference
frames was made in order to describe or measure __accelerations__ on one
inertial reference frame (at one position) and to mathematically __translate__
such accelerations to the other inertial reference frame (at a different
position). Such identical accelerations
would obviously always remain mathematically invariant (the same) in either
reference frame. This dubious
mathematical exercise and its mathematical result was sometimes confusingly misnamed
Galilean Relativity, but it had nothing to do with Galileo’s empirical and
sensory concepts of relativity. (see
Goldberg, p. 374) Einstein’s above
attempted analogy to Galileo’s Relativity also had little or no relevance to
the Galilean transformation equations or their purposes, either.

It becomes obvious that Einstein was dealing with and confusing three completely different and separate concepts: 1) Galileo’s empirical and sensory concept of relativity, 2) Lange’s abstract model of relative uniform motions along with its Galilean transformation equations, which described and measured accelerations, and produced the abstract algebraic concept of Galilean Relativity, and 3) the simplistic computation or addition of velocities of material objects. (see Chart 13.2 and Figure 7.1) Einstein’s above analogy was really only relevant to the third concept, but he attempted to characterize it as being relevant to the first two concepts as well.[15]

Why then did Einstein
invent this irrelevant analogy to Galileo’s Relativity in both its empirical
and mathematical forms? Because he
needed the empirical concept of Galileo’s Relativity for his Special Theory in
order to claim that his Special Theory had an __empirical foundation__. Because he also needed the Galilean
transformation equations in order to __blame__ for the variations of light’s
velocity from *c* to *c* – v and *c* + v, and as transformation
equations which he could later modify to create the Lorentz
transformations. Also, because in the
next Chapter 7 of *Relativity* he would theoretically substitute a ray of
light transmitting at the __constant velocity__ of *c* for the uniform
velocity of the walking man in yet another false analogy to Galileo’s Relativity,
in order to attempt to relate electromagnetism and the velocity of light to
Galileo’s Relativity. An accelerating
man or some other accelerating material object in Einstein’s first analogy
would not satisfy Einstein’s relativistic agenda, because it would not be
analogous to the __constant velocity__ of light at *c*.

Thereafter, in Chapter 7 of *Relativity*, Einstein theoretically
substituted a constantly transmitting ray of light at *c* for the
uniformly walking man his first analogy.[16] Einstein stipulated that “the tip of the ray
will be __transmitted__ with the velocity *c* relative to the
embankment,” and asserted that “the ray
of light plays the part of the man walking along relatively to the carriage” in
his prior analogy.[17]
(Einstein, *Relativity*, p. 22) The
carriage continued traveling down the stationary embankment at velocity (v) in
the same direction as the propagating light ray. (*Id*.

When Einstein stipulated that the ray of light had a constant
transmission velocity of *c* relative to the embankment, he was
incorrectly assuming that the embankment was absolutely stationary in space,
like the ether. But we know that the
embankment (on the Earth) moves at 30 km/s relative to the Sun; at about 225
km/s relative to the core of the Milky Way Galaxy; and at a myriad of other
velocities relative to other planets, stars, and galaxies. (Chapter 10A)
So the velocity of the light ray relative to the moving embankment in
Einstein’s example was actually propagating at *c* + v or *c* – v,
depending upon at which relative velocity (v), and in which __direction__,
one assumes the embankment on the Earth to be moving through the cosmos. (also see Figure 19.2)

Was Einstein’s substitution of the light ray at the constant transmission
velocity of *c* for the uniformly walking man a valid analogy in any
sense? The answer is, no. The __constant__ transmission velocity of
the light ray at *c* relative to the medium of empty space (the vacuum) is
immediately and by definition an ‘invariant’ quantity, and an ‘invariant
quantity’ by definition does not and *a priori* cannot vary from one material
reference frame to another. In other
words, an invariant quantity cannot empirically vary or be mechanically
‘covariant.’ On the other hand, the
motion of a material body (the walking man) is not immediately and by
definition an invariant quantity. The
motion of the man may accelerate and it could empirically vary or be
mechanically covariant from one reference frame to another.[18]

If the light ray was emitted in the moving carriage and if it really
played the part of the man in Einstein’s prior analogy, then its transmission
velocity at *c* should be __added__ to the velocity of the carriage v
to determine its total velocity W relative to the embankment: W = *c* + v. (see Chapter 7A) But *a priori* this would have violated
Einstein’s second postulate concerning the independence of the velocity of
light from the motion of its source body (the carriage).[19] (see Chapter 20F) So Einstein avoided these potential
contradictions by merely stipulating that:
“the tip of the ray will be transmitted with velocity *c* relative
to the embankment.”[20]
(Einstein, *Relativity*, p. 22)
Because of all of these obvious ‘errors’ by Einstein, one senses that he
might have been trying to mislead the reader with his contrived analogies.

Einstein then asked a __different__ question than in his prior analogy
with the walking man: “What is the __velocity
of propagation__ [w] of the ray of light relative to the carriage?”[21] Einstein determined, according to the
so-called ‘Galilean Addition of Velocities,’ that the velocity of the ray of
light (w) relative to the carriage is *c* - v.[22] (see Figure 19.1B) He then concluded that: “the __velocity of propagation__ of a ray
of light relative to the carriage thus comes out smaller than *c*.”[23] (Einstein, *Relativity*, p. 22) Thus, according to Einstein, the
application of the ‘Galilean Addition of Velocities’ to the above situation
would cause Maxwell’s __constant velocity for the transmission of light at c
to mathematically vary__ (to

Here Einstein was not really treating the ray of light like the walking man. The velocity of the walking man inside the moving carriage relative to the embankment was

some value of v
+ w = W. But the walking man’s constant
velocity of w relative to the carriage would be w = W – v. Why did Einstein never ask a question with
regard to this last velocity like he did in Chapter 6? Because if he had the result would have been __equivalent__
for the ray of light (w = *c* – v or *c* + v) and for the walking man
(w = W – v or W + v).[25]
In other words, it would have resulted in a very natural __relative velocity__
in both cases.[26] (see Figure 7.1)

Finally, Einstein concluded that his ‘addition of velocities’ result with
regard to the light ray (W = *c* – v) “comes into __conflict with the
principle of relativity__.” [27]
(Einstein, *Relativity*, p. 22)
Why? Einstein then explained his
conclusion:

“For, like
every other general law of nature, the __law of the transmission of light__ *in
vacuo* must, __according to the principle of relativity__, be the same
for…[both the moving carriage and the relatively stationary embankment]. But, from our above consideration, this would
appear to be impossible. If every ray of
light is __propagated__ relative to the [stationary] embankment with the
velocity *c* then for this reason it would appear that __another law of
propagation of light__[28]
must necessarily hold with respect to the [moving] carriage—__a result
contradictory to the principle of relativity__. [29] (Einstein, *Relativity*, pp. 22 – 23)

**C. Einstein decided to reconcile the theories of
relativity and of the velocity of light.**

It is obvious from his above statements, that Einstein’s application of
the material principle of Galileo’s Relativity, and his application of the
Galilean transformation equations of mechanics, to the constant transmission
velocity of light at *c*…created mathematical ‘difficulties’ for Einstein
that baffled both him and the scientific community. How could the transmission velocity of light
at *c* be both constant and yet appear to mathematically __vary__ with
respect to material objects moving at different linear velocities? What could be the answer to this mathematical
paradox, this ‘difficulty?’[30] At this point, Einstein concluded that:

“In view of
this __dilemma__ there appears to be nothing else for it than to __abandon__
either the principle of relativity or the simple law of the __propagation__
of light *in vacuo*.”[31] (Einstein, *Relativity*, p. 23)

Nevertheless, after
numerous rationalizations, Einstein ultimately decided not to abandon either
concept. Instead, he chose to radically
change the classical mechanics ‘principle of relativity’ (Chapter 24) and to __extend__
it to the phenomena of optics and electrodynamics, based on the following
dubious and incorrect ground:

“…classical mechanics…supplies us with the actual motions of the heavenly bodies…

“The __principle of relativity__ must therefore
apply with great accuracy in the domain of mechanics. [32]
But that a principle of such broad generality should hold with such exactness
in one domain of phenomena [mechanics], and yet should be invalid for another
[electrodynamics and optics], is a priori not very probable.” (Einstein, Relativity, p. 17)

Einstein then decided that his two postulates (his radically expanded
principle of relativity and his concept of light propagating at velocity *c*)
should apply both to mechanics and to the velocity of light, because they were
only “apparently irreconcilable:”

“These __two postulates__
suffice for the attainment of a simple and __consistent__ theory of the
electrodynamics of moving bodies based on Maxwell’s theory for stationary
bodies.”[33] (Einstein, 1905d [Dover, 1952, p. 38])

“*in reality
there is not the least incompatibility between the principle of relativity and
the law of propagation of light*, and that by systematically holding
fast to both these laws a logically rigid theory could be arrived at.[34] This theory has been called the

In order to defend Maxwell’s theory of the constant velocity of light at
velocity *c*, and to attain his ‘consistent’ theory of Special Relativity,
Einstein claimed that what was needed was a “modification” to the ‘Galilean
Addition of Velocities’ and the Galilean transformation equations. [35] (Einstein, *Relativity*, pp. 34 – 35)
Einstein’s “modification” of the Galilean transformations turned out to
be his Lorentz transformation equations for space (distance) and time
(intervals):

x'
= x – vt t' =
t – v/*c*^{2}.x

√1 – v^{2}/*c*^{2 } √1 – v^{2}/*c*^{2 }

^{ }

(*Id.*, p.
37)

Actually, Einstein must have
realized that the real culprit that was causing his mathematical ‘difficulties’
was the __relative motion__ between the stationary embankment and the
carriage moving linearly at v. If he
could somehow mathematically eliminate the __time intervals__ and the __space
intervals__ described by the Galilean transformations, then he could negate
the relative motion between inertial reference frames and with it the
troublesome addition of velocities so that the velocity of light could
algebraically remain a constant *c* for all inertial frames and
observers. Einstein called this
mathematical result: ‘co-variance.’ (*Id*., pp. 47 – 48) Thus, Einstein’s ‘modifications’ of equations
and his reconciling of theories was only his means of attaining this co-variant
end result.

Einstein’s above described rationalizations, modifications, and his
Lorentz transformations resulted in an elaborately constructed *ad hoc*
mathematical theory and produced many bizarre mathematical consequences. (Einstein, 1905d [Dover, 1952, pp. 38 –
65]) These radical Lorentz
transformation equations and their mathematical consequences, in turn,
purported to modify *ad hoc*,
artificial, invalid, unnecessary, unwarranted and completely irrelevant to
anything.

**D. Possible solutions for Einstein’s
mathematical difficulties.**

In 1968, Resnick (an
ardent believer in Einstein’s Special Theory) restated the ‘difficulties’
concerning the velocity of light, which faced the scientific community during the last part of
the 19^{th} century and the early part of the 20^{th}
century. (see Memo 19.3) In an ether frame at rest, an observer would a
priori “measure the
speed of light to be exactly c.” (Resnick, 1968, p. 16) This (theoretically impossible) measurement
would be exactly consistent with Maxwell’s equations for electromagnetism with
respect to the (non-existent) stationary ether frame. (Id., p. 17)
But, continued Resnick, in an inertial reference frame “moving at a
constant speed v with respect to this __ether frame__ an observer would
measure a __different speed__ for the light pulse, ranging from c + v to c – v depending on the __direction__
of relative motion, according to the Galilean __velocity__ transformation[37]…Hence,
the speed of light is certainly __not invariant under a Galilean
transformation__.”[38] (Id., p. 16)

This misanalysis by
Resnick concerning different __transmission__ velocities for a light ray
under a Galilean transformation was the theoretical position of the late 19^{th}
and early 20^{th} century scientific community, including
Einstein. Unfortunately, it continues to
be the theoretical position of the scientific community during the early 21^{st}
century.[39]

The choices that the
scientific community __appeared__ to face during the late 19^{th}
and early 20^{th} century are stated as follows. 1) If the transmission velocity of a light
ray __did vary__ relative to different linear uniform velocities of each
inertial reference frame in the Cosmos, then Maxwell’s equations for
electromagnetic waves (light) at the constant transmission velocity of c must be flawed. Why?
Because according to the Galilean transformation equations, the
algebraic form of Maxwell’s electromagnetic law of physics (the constant
transmission velocity of light at c) would not be invariant or constant when
transformed (translated) from one inertial reference frame to another.[40] It would change from c to c – v or c + v.[41] On the other hand, 2) it was claimed that if the transmission
velocity of a light ray at c did __not vary__ relative to the different uniform velocities of
each inertial reference frame in the Cosmos, then the principle of Galileo’s
Relativity and the Galilean transformation equations must be flawed, because
based on them the velocity of light varies from c to c – v or c + v relative to different inertial reference
frames.

These dubious (or rather
spurious) choices posed a serious __dilemma__ for the scientific community
of the late 19^{th} and early 20^{th} century. Was the established principle of Galileo’s
Relativity (in its empirical or mathematical form) irreconcilable with the
established law of Maxwell’s equations (the constant transmission velocity of
light at velocity c)? Which cherished law of
physics was invalid? [42]

Conventional wisdom
suggested three possible solutions for these mathematical ‘difficulties.’ As described by Resnick: “The fact that the Galilean relativity
principle does apply to the Newtonian laws of mechanics but not to Maxwell’s
laws of electromagnetism requires us to choose the correct consequences from
amongst the following possibilities.”
(Resnick, 1968, p. 17)

“1. A
relativity principle exists for mechanics, but not for electrodynamics; in
electrodynamics there is a preferred inertial frame, that is the ether
frame.

“2. A
relativity principle exists both for mechanics and for electrodynamics, but the
laws of electrodynamics as given by Maxwell are not correct. (*Id*.

“3. A
relativity principle exists both for mechanics and for electrodynamics, but the
__laws of mechanics as given by __.”[43] (

In 1905, Einstein sought to resolve the ‘difficulties’ with the velocity of light that he imagined by adopting the third of the above possibilities. He called this attempted reconciliation and resolution his ‘Special Theory of Relativity.’

However, there is also a fourth possibility that is asserted by the author, but it was apparently never considered:

4.
A relativity principle exists for mechanics, but it is __irrelevant__
for optics and electrodynamics, and light __transmits__ at a constant
velocity of *c* with respect to its medium of the vacuum of empty
space. However, such constant velocity
of light at *c en vacuo* also becomes
part of a __relative velocity__ (*c* ± v) when light is __propagated__
over changing distances and time intervals relative to material bodies moving
linearly at v.

This fourth
possibility totally eliminates the paradox of the aforementioned ‘difficulties’
without any strained logic or contrived *ad hoc* hypothesis, such as a
contraction of matter or Special Relativity.
It is discussed and explained in detail in Chapters 21 through 24 of
this treatise.

As will become more and more obvious
as we proceed, the ‘difficulties’ which triggered Special Relativity, and
Einstein’s Special Theory itself, strangely enough, resulted in no small
measure because of the confusing, misleading and conflicting __definitions__
of scientific and mathematical terms, such as relativity, transformations,
frames of reference, covariance, inertial motion, mechanics, distance,
velocity, and especially the __constant__ velocity of light. They also resulted because of the misanalysis
and misapplication of many false concepts (such as ether) and of many __material__
concepts with respect to the non-material phenomena of EM and light.

[1] This, of course, was a fallacy of logical thinking on its face, because by convention other reference frames were composed of ponderable matter and had a unique specific velocity, but by definition the hypothetically material ether was stationary and not ponderable.

[2] As we shall discover in Chapter 25, these imprecise approximations of the time of occurrence of a distant event had been well known by mathematicians such as Poincaré for many decades, but they were consciously ignored for purposes of mathematical simplicity. (see Neffe, p. 129; Miller, p. 176)

[3]
Strangely enough, we shall discover in Chapter 21D that this conclusion is also
valid for a ray of light when it __propagates__ relative to a material body
which moves linearly with respect to the light ray.

[4] Rohrlich also reasoned that: “If the source of the light (a lamp, say) is used as a reference frame then the speed of light should have one value relative to a source at rest and another value relative to a moving source.” (Rohrlich, p. 52) On the other hand, we will explain in Chapters 21 and 22 why these expectations do not occur with a ray of light, regardless of the motion of its emitting body.

[5] However,
as we will paradoxically discover in Chapter 21, the correct statement is that
a ray of light can have the same __transmission__ velocity of *c* in or
through any different reference frame, but almost never the same __propagation__
velocity __relative to__ different reference frames.

[6] We shall discover the detailed answers in Chapters 21, 22 and 23.

[7] These
conclusions were, of course, ludicrous on their face, for two reasons: 1) there is no ether frame from which to
absolutely measure a __different__ velocity of light…thus such measurement
is only a figment of one’s imagination; and
2) how can the speed of a moving body in empty space and __at a
distance__ change the velocity of __anything__ moving or transmitting
toward or away from it…a rocket, a planet, or a ray of light?

[8]
Strangely enough, as we shall soon discover in Chapters 21 and 22, __both__
concepts were __actually correct__.

[9] Why were such facts not realized? Because Einstein (like most other scientists even to this day) probably never read Maxwell’s original papers. Miller concluded that before 1905, Einstein’s familiarity with Maxwell’s theory of electromagnetism appears to be limited to a few chapters from an 1894 text by August Föppl on the Hertz-Heavyside version of Maxwell’s theory, which Einstein read just enough of in June 1902 in order to qualify as a patent clerk. (Miller, p. 142)

[10] Take
particular notice that Einstein refers to the ‘__propagation__ velocity’ of
light at *c*, rather than the correct
concepts: the __emission__ velocity
and the __transmission__ velocity of light at *c* relative to its medium. In fact, __there is no specific law for the propagation
velocity of a light ray__ over varying distances and relative to linearly
moving reference bodies. In Chapters 21
and 22 we shall demonstrate how and why Einstein’s

[11]
Einstein also described these mathematical paradoxes, which he called the
‘difficulties,’ with respect to inertial __reference bodies__: “…assuming that the Maxwell…equations hold
for a __reference body__ *K* [at rest], we then find that they do not
hold for __a reference body__ *K'* moving uniformly with respect to *K*,
__if we assume that the relations of the Galilean transformations exist
between the coordinates of K and K'__.” (Einstein,

[12] Many of Einstein’s analogies and analyses in his Special Theory are strained, artificial, irrelevant or invalid, and we shall point out these flaws as we proceed.

[13] In
other words, the variable magnitudes of F = ma, a = F/m, and m = F/a are
mechanically equivalent or covariant (and

[14] Of
course, in this mathematical model, the velocities of the two __inertial
frames__ could be abstractly added together:
v_{1} + v_{2} = v_{3}. But this would serve no purpose for the
concept of Galileo’s Relativity. Nor did
Einstein’s similar addition of two uniform velocities in order to produce a
third relative velocity serve any purpose with respect to Galileo’s Relativity
or even with respect to the Galilean transformations.

[15] Einstein arbitrarily switched back and forth between these three different concepts whenever it suited his relativistic agenda.

[16] This
time the carriage, the man, the embankment, and the light ray all theoretically
existed in a vacuum, because this is the medium relative to which light
transmits or propagates at *c*.
(see Einstein, *Relativity*, p. 22)

[17] Notice
that in this analogy Einstein used the word ‘transmitted’ instead of
‘propagated’ in order to describe the ‘velocity *c*’ of the light ray.

[18] Also, in Galileo’s Relativity, no man walking on the uniformly moving ship was ever required to maintain a uniform velocity. He could have stopped, walked slower, or run at varying speeds down the center of the ship or the carriage. And he could have moved this way and that, or in a serpentine motion, or he could have stopped and walked in the opposite direction. In other words, the velocity of a man on an inertial body in Galileo’s Relativity was never required to maintain a constant quantity.

[19] It
would also have violated the first part of Einstein’s second postulate (the
absolute constancy of light’s velocity at *c* *en* *vacuo*), and
likewise would have violated his conjecture that no physical computation of
velocities can exceed *c*. (see
Chapter 29)

[20]
However, again, since the embankment was really moving in common with the Earth
at v, Einstein’s stipulated value for the light ray relative to __both__ the
carriage __and__ the embankment should have been

*c* ± v ± v_{1}, depending upon relative
directions of motion.

[21] This
time Einstein arbitrarily referred to the velocity of __propagation__ rather
than the velocity of __transmission__ as he did with respect to the
embankment as if they were both the same concept (which they are not).

[22] We have referred to the Galilean addition of velocities as ‘so-called,’ because Galileo never created such a concept and Galileo’s Relativity does not contain such a concept.

[23]
Strangely enough, this last velocity of __propagation__ (*c* – v),
smaller than *c*, is correct, as we shall explain in Chapters 21 and 22.
Einstein was also unwittingly correct when he asserted that the __propagation__
velocity of light varies relative to linearly moving bodies, but he did not
realize this fact. (Chapter 21) However, none of such varying velocities of __propagation__
create a contradiction to Maxwell’s equations, nor to the constant __transmission__
velocity of a ray of light at *c* relative to its medium of empty space,
as we shall also explain in Chapters 21 and 22.

[24] By *c*,
Einstein was asserting that the transmission velocity of a light ray relative
to the stationary embankment was 300,000,000 m/s. By *c* – v, Einstein was asserting that
the transmission velocity of a light ray relative to the carriage moving in the
same direction at 30 m/s was 300,000,000 m/s minus 30 m/s, a total of
299,999,977 m/s, or less than *c*.
Likewise, by *c* + v, Einstein was asserting that the transmission
velocity of a light ray relative to the carriage approaching the light ray at
30 m/s was 300,000,000 m/s plus 30 m/s, a total of 300,000,030 m/s or more than
*c*. In other words, Maxwell’s
constant transmission velocity of light at *c* appeared to vary depending
upon the magnitude and direction of motion of material bodies.

[25] The reason why we cannot exactly specify the magnitude of any of these velocities is because we do not know the absolute velocity v of the Earth through the Cosmos, nor its absolute direction of motion, nor the absolute point of emission of the light ray, nor its absolute direction of propagation, and neither did Einstein. (see Chapters 10A and 22) Einstein’s abstract algebra masks all of these questions and unknown quantities.

[26] All of
these very natural __relative velocity__ results would be completely
consistent with Galileo’s Relativity and with the Galilean transformation
equations, so Einstein would have had no justification to modify them. For this reason, he avoided disclosing their
existence.

[27] Here,
Einstein was referring to his __expanded__ definition of the ‘principle of
relativity,’ the first postulate in his Special Theory, which incorrectly
applies to the velocity of light. (see
Chapters 20E and 24)

[28] Notice
that in this paragraph Einstein indiscriminately refers to the ‘law of the __transmission__
of light’ and the ‘law of __propagation__ of light’ as if they were the same
concept. But they are entirely different
concepts, as we shall discover in Chapters 21 and 22. Einstein obviously wanted the ray of light to
__propagate__ at velocity *c* relative to both the stationary
embankment and the linearly moving carriage.
But this would be an impossible absolute concept, as we shall soon
discover in Chapter 21E. Also, again,
there is no specific law for the propagation of light relative to linearly
moving objects like the carriage.

[29] In
essence, Einstein’s analysis amounted to the same thing as stating that the
linear motions of distant material bodies determine the __transmission__
velocity of light.

[30] It took the author several weeks of concentrated thought and analysis to finally discover and understand the correct answer.

[31] But
little did Einstein realize, there is no simple law of the __propagation__
of light *in vacuo* relative to linearly moving bodies, such as a moving
carriage. There is only Maxwell’s
constant __emission__ velocity of light at *c* relative to its point of
emission in space, and Maxwell’s constant __transmission__ velocity of light
at *c* __relative to the medium of the vacuum of empty space__ through
which light passes. (see Chapters 21and
22)

[32] This is
a non sequitur. No classical principle
of relativity (including Galileo’s Relativity) has anything to do with “the
actual motions of the heavenly bodies.”
Their motions are determined solely by their masses, their inertia,

[33] Maxwell’s theory of electricity was not about stationary bodies (electrostatics), and his theory of light only referred to stationary ether, not ponderable stationary bodies.

[34] This incorrect conclusion by Einstein was based upon a myriad of false analogies, false assumptions and the misanalysis of the situation by Einstein, which will become obvious as we proceed with other chapters.

[35] Einstein’s radical “modification” of the classical Addition of Velocities is discussed and explained in Chapter 29. His “modifications” of the Galilean transformations is discussed in Chapter 27.

[36] In light of all of Einstein’s aforementioned false analogies, misassumptions and misanalysis, one must also ask the question: Was this not Einstein’s secret plan in the first place?

[37] This impossible
and ridiculous analysis with respect to a non-existent ‘ether frame’ results in
an absolute concept, as will be explained in Chapter 21E. Resnick’s notion of an observer __measuring__
the velocity of a light ray in the way described is, of course, pure fantasy;
much less with a rigid meter rod and a clock, as Einstein has suggested. Also, the Galilean transformation was not
really a mathematical __velocity__ transformation equation (as Resnick
asserts). Rather it was a mathematical
equation which translated coordinates and which described an __acceleration__
in one inertial reference frame relative to another inertial reference frame

[38] The
correct analysis should be: the constant
transmission velocity of light at *c* is an invariant property of light,
and an invariant property cannot be translated so as to vary from one inertial
reference frame to another. If one
attempts to transform the invariant transmission velocity of light at *c*,
the __relative__ propagation velocities (*c* – v and *c* + v) of
the light ray will be the natural result.
(see Chapters 21 and 22)

[39] Such analysis is totally incorrect, as will be fully explained in Chapters 20 through 24 to follow.

[40]
Remember that Galileo’s Relativity empirically demonstrated and the so-called
Galilean transformations mathematically implied that, although the magnitudes
of accelerated motions may vary in different inertial frames (vis. they are
mechanically covariant), the algebraic form of ^{nd} law remains __invariant__
in all inertial frames of reference. The
early 20^{th} century scientific community must have asked: Why should the laws of electromagnetics be
any different?

[41] The scientists, of course, failed to come up with a theory for how the inherent transmission velocity of a light ray could vary because of the linear motion of distant bodies.

[42]
Einstein, like the other scientists of his day, asked the question: Could this __dilemma__ and these two
established laws of physics (vis. Galileo’s Relativity in any form and
Maxwell’s equations) be reconciled; or must one be abandoned? (see
Einstein, *Relativity*, p. 23)

[43] Dingle added to this possibility that there might be “some unknown effect of motion that had been neglected.” (Dingle, 1972, p. 162)