*Einstein claimed that
he derived the Lorentz transformation equations from his two fundamental
postulates, and that such transformations in turn mathematically confirmed the
validity of his two fundamental postulates, all of his coordinate measurements,
and his twin concepts of the Relativity of Distance and the Relativity of
Simultaneity. However, it will be
demonstrated that such Lorentz transformations cannot be valid for any
such reasons because, *inter alia,

**A. Why did
Einstein need to adopt Lorentz’s transformations?**

In the previous Chapter 26, we have demonstrated that __relative motion__
(velocity) between two inertial frames was __not necessary__ for Einstein to
claim and attempt to explain his twin concepts of the Relativity of
Simultaneity (duration) and the Relativity of Distance (length). All that was really necessary was the
constant finite velocity of a light ray at *c*, relative distances and
positions, clocks, and Einstein’s arbitrary concept of the ‘common synchronous
time’ for each inertial system.[1] (see Figures 26.7 and 26.8)

Why then did Einstein insist upon
illustrating such twin concepts with __relative motion__ (velocity) between
two inertial reference frames? The
answer is quite simple. Without relative
motion (velocity) between two inertial reference frames there could be no
distorted __coordinates__ of time intervals and lengths (see Chapter 26),
and there would be __nothing to mathematically transform__. Without distorted time intervals and distance
intervals, and without distorted coordinates of different inertial reference
frames to transform, there would be no need for the Lorentz
transformations. Without the Lorentz
transformations, Einstein could not mathematically claim and demonstrate that a
light ray had an absolutely constant co-variant velocity of *c* for all
inertial observers (Chapters 21E & 24); nor could he mathematically claim
and demonstrate that all other physical phenomena were velocity dependent and
consistent with his second postulate concerning the co-variant velocity of
light. Why? Because only when such other phenomena were
transformed with Lorentz transformations did their law change and did such
changed laws became co-variant with respect to different inertial reference
frames. (Chapter 28)

In fact, without the Lorentz transformations there would be no Special Theory of Relativity. As Bertrand Russell concluded: “technically, the whole of the special theory is contained in the Lorentz transformations.” (Russell, 1927, p. 49) Thus, it becomes obvious that Einstein’s invalid twin concepts (the Relativity of Simultaneity and the Relativity of Length) were merely two stepping stones of false analogy, irrationalization and attempted justification on the path to his primary mathematical goal: the replacement of the Galilean transformation equations with the Lorentz transformation equations.

What was the real reason that Einstein was so eager to replace the Galilean transformation equations with the Lorentz transformation equations? Bertrand Russell also supplied us with the answer for this question:

“This [Lorentz]
transformation has the advantage that it __makes the velocity of light the
same with respect to any two bodies which are moving uniformly relatively to each
other__, and, more generally, that it makes __the laws__ of
electromagnetic phenomena (Maxwell’s equations) __the same__ with respect to
any two such bodies.[2] It was __for the sake of this advantage__
that it was originally introduced…”
(Russell, 1927, p. 49)

In other words,
Einstein desperately needed the Lorentz transformations in order to
mathematically solve his ‘difficulties’ (i.e. *c* – v and *c* + v),
and to demonstrate the mathematical plausibility of his second postulate: ‘that light has a definite constant velocity
of *c* for all spatially separated inertial observers, regardless of their
different linear speeds.’ (see Chapters
19, 20F, 21 and 22)

Relative motion, or rather relative
velocity, only became relevant for Einstein with respect to the Lorentz transformations…with
their built-in factors for relative velocity, v and v^{2}, and their
bizarre mathematical consequences. Once
the Lorentz transformations were rationalized and accepted, an uncountable
number of hypothetical or imagined physical consequences could follow with
mathematical precision, and relative velocity would become their
magnitudes. The only constraints on this
potentially unlimited invention of mathematical consequences were: 1) the ability to imagine and interpret
mathematical consequences, and 2) the
task of finding empirical phenomena or experiments which might somehow appear
to relate to or appear to ‘confirm’ such artificial consequences. (see Chapters 36, 37 and 38)

In Chapter 11 of his book *Relativity*, Einstein insisted that his
twin concepts, the Relativity of Simultaneity and the Relativity of Distance,
meant that the Galilean addition of velocities, and thus the Galilean transformation
equations themselves, were invalid.
(Einstein, *Relativity*, pp. 34 – 35) Einstein then posed a rhetorical question and
provided a pre-prepared answer:

“How have we to
modify [the classical ‘addition of velocities’]…in order to remove the apparent
disagreement between [the law of the propagation of light and the principle of
relativity]…such that __every ray of light possesses the velocity of
transmission c relative to the embankment and relative to the train__.[3]

“This question
leads to a quite definite positive answer, and to a perfectly definite __transformation
law__ for the space-time magnitudes of an event when changing over from one
body of reference to another.”[4] (*Id*.

This
“transformation law,” of course, turned out to be “the Lorentz transformation
equations,” which are described in Chapters 16 and 27B of our treatise, and in
Chapter 11 of Einstein’s book, *Relativity*. (Einstein, *Relativity*, p. 37) What a surprise!

**B. How did Einstein ‘derive’ his Lorentz
transformation equations?**

In Section 3 of his
1905 Special Theory, Einstein invented a new thought experiment. He theoretically established two inertial
reference frames (coordinate systems), S and S', that coincided at the same
point of origin. (see Figure 27.1A) Each system was provided with a rigid
measuring rod, a number of identical clocks, a light source at one end and a
mirror at the other end. System S' then
accelerated to a uniform inertial velocity v relative to system S. (Figure 27.1B) The ‘common synchronous time’ t' for system
S' was then determined by light signals, and likewise a different ‘common
synchronous time’ t for system S was also determined by light signals.[5] Therefore, the coordinates x, y, z, t
“completely defines the place and time of __an event__ in…system” S, and the
coordinates x', y', z', t' completely define the place and time of __the same
event__ in system S', and vice-versa.
(Einstein, 1905d [Dover, 1952, p. 43 – 44]) Einstein then stated that: “our task is now to find the system of equations
connecting these quantities [x, y, z, t and x', y', z', t'].” (Id.,
p. 44)

Einstein then
theoretically emitted a light ray from the origin O' of system S' (E_{1}),
which light ray propagated from E_{1} at clock time t'_{0} to
mirror M' (E_{2}) at clock time t'_{1 }and reflected back from
mirror M' to the light source at clock time t'_{2} (E_{3}). (see Figure 27.1B) According to his second postulate concerning
the absolutely constant velocity of light at c and his algebraic definition of clock synchronization, t'_{1}
= ˝(t'_{0} + t'_{2}), Einstein determined that for an observer
in S' the velocity of light measured by synchronized clocks in system S' was c.[6] (Id.,
p. 44; see Miller, p. 196)

However, when an
observer in system S measured with coordinates the events E_{1}, E_{2}
and E_{3} in distant system S' uniformly moving at v relative to S, the
algebraic time interval of light propagation from coordinates 0 – 10 became c – v in S, and the algebraic time interval of light
propagation from coordinates 10 to 0 became c + v in S. These different
velocities of light were merely __relative velocities__ when measured in S,
which resulted because of the relative motion of S and S' (x = x' + vt and x' =
x – vt).[7] (see Miller, p. 197) In other words, the tip of the light ray was
displacing at v relative to S (because of the relative motion between S and
S'), but it was not displacing relative to S'.
The light ray was emitted and always propagated over the same distance
to and fro within S'.[8]

However, Einstein
interpreted his thought experiment much differently. He concluded that such light ray was measured
in system S' to __propagate at velocity ____c__, but “when measured in the stationary system [S,
such ray moved] with the velocity c – v…”[9] (Einstein, 1905d [Dover, 1952, p. 45]; see
Figure 27.1A) This asymmetric algebraic
result violated both Einstein’s principle of relativity (vis. that the laws of
physics are the same [algebraically co-variant] for every inertial observer),
and it also violated Einstein’s second fundamental postulate (vis. that the absolutely
constant velocity of a propagating light ray is c relative to all inertial observers, regardless of their different
linear speeds).[10]

On the other hand,
what other result could anyone expect, where one constant velocity c is compared to another constant velocity v in two
different reference frames and in two different directions? The results c – v and c + v were nothing more
than the constant transmission velocity of a light ray at c in system S' propagating in two different directions
with respect to the constant relative velocity v between two bodies, S and S',
as measured by coordinates in S. Thus, c – v and c + v
were merely __relative velocities__ of the light ray measured by S as theoretically
__increasing__ coordinates. (see Figure 27.1C) They were not different __transmission
velocities__ of the light ray, as Einstein was asserting.

At this juncture,
Miller cautions the reader that Einstein was still dealing with an
‘intermediate Galilean system,’ and that he “avoided discussing a __contraction__
of length because…Einstein could not deduce this contraction [effect] until
after he derived the [Lorentz] space and time transformation equations.” (Miller, pp. 196 – 197) This conclusion by Miller shows that Einstein
knew at this early point in his so-called ‘derivation’ exactly what Lorentz
transformation equations he was supposedly ‘deriving.’ As Folsing strongly implies in Chapter 27C to
follow, Einstein was not really deriving the Lorentz transformations from his
two fundamental postulates as he claimed, because at an early and ‘opaque’
point in his derivation he already knew what such transformations had to be. In other words, in Section 3 of his Special
Theory, Einstein was really ‘contriving’ his Lorentz transformations.

Einstein then
mathematically demonstrated and supposedly deduced from the above asymmetric
results that the transformation equations connecting each set of coordinates,
for S and S', must include the factor √1 – v^{2}/c^{2} as a
denominator. (Einstein, 1905d [Dover,
1952, pp. 45 – 46) For Einstein, this
was necessary in order that:

“any ray of light, measured in the moving system, is propagated with
the velocity c, if, as we have assumed,
this is the case in the stationary system…[so] that the principle of the
constancy of the velocity of light is compatible with the principle of
relativity.” (Id., p. 46)

In other words, such denominator was
necessary for Einstein so that when the law for the velocity of light at c is transformed from coordinates x, y, z, t to
coordinates x', y', z', t', or vice-versa, it will retain exactly the same
algebraic form c in both frames of
reference; or in brief, so that such law of velocity c will be “co-variant with respect to the Lorentz
transformations.”[11] (Einstein, Relativity, pp. 47, 48)

The Lorentz
transformation equations which Einstein ‘derived’ for the moving inertial
reference system S', were:

x' = __x – vt__

√1 – v^{2}/c^{2}

y'
= y

z'
= z

t'
= t – __v/____c__^{2 }^{.}_{ }__x__

√1 – v^{2}/c^{2}

(Id.,
p. 37; Einstein, 1905d [Dover, 1952, pp. 46, 48]; also see Miller, pp. 195 -
200) At this point, Miller
concluded: “It seems as if he [Einstein]
__knew beforehand__ the correct form of the set of relativistic
transformations…”[12] (Miller, p. 200) Note that, for the sake of simplicity, a good
deal of Einstein’s algebraic progression was omitted by the author in arriving
at such Lorentz transformations, because it is not relevant to our discussion.

It
is obvious from the above discussion that Einstein ‘derived’ or ‘contrived’ his
Lorentz transformations solely based on measurements from a theoretical ether, inertial
frames of reference, coordinates, synchronized clocks, light signals, relative
velocity, his second postulate concerning the absolutely constant velocity of
light at c, and algebra. They are not empirical transformations
deduced from observation or experiments.
Einstein’s first postulate concerning his principle of relativity played
no active part in such process; it merely served as his interim goal. Once in possession of his Lorentz
transformations, Einstein mathematically demonstrated his concept of
contraction of matter depending upon its relative velocity, and that his
Lorentz transformations were __reciprocal__ with respect to S and S'. (Einstein, 1905d [Dover, 1952, pp. 46 – 48];
see Miller, p. 202)

Nevertheless,
the facts that Einstein mathematically ‘obtained’ this Lorentz transformation,
that his two invalid fundamental postulates are mathematically compatible, or
that the Lorentz transformations are mathematically reciprocal between two
inertial reference frames, does not mean that such Lorentz transformations have
any empirical validity with respect to physical phenomena or physics. As we have demonstrated, and will further
demonstrate over and over in this treatise, such Lorentz transformations are
completely ad hoc, artificial, and have
absolutely no physical meaning whatsoever.
(see Chapters 15, 16, 27 and 28)
They only __algebraically__ serve to artificially __eliminate the distance/time
intervals and the velocity v__ between the two co-moving reference frames S
and S' so that the coordinate magnitudes of physical phenomena will
mathematically be co-variant, or the same, when translated from one to the
other, and in the process they radically change the coordinate magnitudes of
such phenomena and the laws of physics.
(see Einstein, Relativity, pp. 47
– 48; Hoffmann, p. 87; Feynman, 1964, pp. 25-1 and 25-11; Russell, 1927, p. 49)

The most fundamental
problem with Einstein’s so-called mathematical ‘derivation’ of the above
Lorentz transformation equations was that his second postulate for the absolute
constancy of the velocity of light… “that light is always __propagated__ in
empty space with a definite velocity c”…is
a __totally incorrect__ concept when applied to linearly moving bodies. This was a major primary false premise in his
‘derivation.’ As we explained in great
detail in Chapter 21, a light ray always __transmits__ at Maxwell’s constant
velocity of c with respect to its medium
of empty space or a vacuum, but relative to an inertial frame linearly moving
at velocity v with respect to such light ray it __propagates__ over __changing
distances__ at the relative velocity of c
± v during changing time intervals of ct
± vt.

A second major false
premise in Einstein’s so-called mathematical derivation of the Lorentz
transformations was his assumption (based on the ether theory) that light propagating
in S' has a greater distance to propagate in the direction of motion when
measured from S.

A third major false
premise in Einstein’s so-called derivation of the Lorentz transformations was
Einstein’s algebraic measurement of the propagation of light in system S' as
viewed from system S. (see Figures 27.1A and 27.1B) The constant transmission velocity of light
at c did not change to c ± v in his examples for two reasons: 1) the mirror in S' did not physically
displace from the light source in S' nor from the tip of the propagating light
ray, nor did such light source physically change its position relative to the
propagating light ray, as the measurement in S asserts. They were both relatively stationary (see
Chapter 12); 2) the algebraic
measurements of c – v and c + v from system S were merely relative velocities of
propagation because of such relative motion of the two reference frames.

A fourth false
premise was Einstein’s belief that any principle of relativity could apply to
light. Because the __transmission__
velocity of light at c was stipulated by
Maxwell to be constant relative to its medium of a vacuum, such velocity is
automatically __invariant__ at any point in any empty space.[13] Therefore,
any principle of relativity, any concept of co-variance, and any set of
transformation equations were automatically irrelevant to such constant,
abstract, absolute, and invariant velocity of light at c.

A fifth false premise
in Einstein’s ‘derivation’ was that an inertial reference frame has a ‘common
time’ for every spatially separated point and for every spatially separated
inertial observer with a synchronized clock in such frame. (see Chapter 25) These artificial concepts and Einstein’s arbitrary
use of them also aided him to mathematically ‘derive’ his Lorentz
transformations.[14]

A final contradictory
problem with Einstein’s derivation is that in order for his postulate of the
absolutely constant transmission velocity of light at c to empirically work (regardless of the linear
motions of any observer), all bodies and all observers in the universe would
have to remain absolutely stationary…forever.
But, of course, this does not empirically occur. Even in Einstein’s Section 3 thought experiment,
system S' and such light ray propagating to and fro in system S' were __moving
linearly__ at v relative to system S (and its observers with synchronized
clocks), so very naturally the propagation velocity of the light ray moving away
from S in S' was measured in S to be c –
v.[15] (Figure 27.1)

For all of the above
reasons, Einstein’s Lorentz transformation equations were not only irrelevant
and unnecessary…they were also __invalid__ and __meaningless__.

**C.
Did Einstein really ‘derive’ his Lorentz transformations?**

We now turn to the question: How
did Einstein really obtain his Lorentz transformations? In his 1916 book, *Relativity*, Einstein
claimed that he “obtained” the Lorentz transformation by uniting his two
postulates: his radically expanded
‘principle of relativity’ and his absolutely constant transmission velocity of
light at *c* *in vacuo*.
(Einstein, *Relativity*, p. 47)
The method that Einstein allegedly used to obtain or ‘derive’ his
Lorentz transformation equations is set forth in his 1905 paper (Einstein,
1905d [Dover, 1952, pp. 43 – 48]), and briefly described in the previous
Section B of this chapter.

With regard to Einstein’s deductions and calculations by which he supposedly derived the Lorentz transformations in his 1905 treatise, his recent biographer Albrecht Folsing comments as follows:

“Einstein must have begun with some idea of what he wanted to
deduce. At an __opaque point__ in his
deduction he introduces, without any warning or explanation, a slight
mathematical operation whose purpose becomes obvious only if __the desired
result is already known__.[16] This __underhand device__, by means of
which he rather __forcibly__ ‘computes his way’ to the Lorentz
transformations, deprives the deduction
of some of its elegance and stringency. [17] Thus it is __scarcely credible__ that
Einstein actually arrived at his formulas in the way he presents them in his
paper.” (Folsing, p. 188)

Miller mirrored these conclusions: “It is difficult to imagine that Einstein
first derived the relativistic transformations by the method described in the
1905 paper; [18] in
fact, he never used this method again.”
(Miller, p. 204)

There was another much more convenient and more logical source for Einstein’s Lorentz transformation equations other than such derivation. That was Lorentz’s April 1904 treatise which first publicly introduced his Lorentz transformation equations. It is obvious to some skeptics, including the author, that Einstein read Lorentz’s April 1904 treatise shortly after it was published in April and June of 1904. By July 31, 1904, German scientists Kaufmann and Abraham had found Lorentz’s April 1904 treatise on their own and Abraham had replied to it in a published article. (see Miller, p. 70) If they could find Lorentz’s April 1904c transformation treatise and reply to it within 3 months, why couldn’t Einstein, who was also intensely interested in the subject?[19]

Once Einstein found and read Lorentz’s April 1904 treatise, he realized
that its Lorentz transformations had “the advantage that it makes the velocity
of light the same with respect to any two bodies which are moving uniformly
relatively to each other,” as Bertrand Russell so aptly described. (Russell, 1927, p. 49) This Lorentz transformation would mathematically
solve the paradox concerning the constancy of the velocity of light at *c*,
relative to linearly moving material bodies, which Einstein had been pondering
about for almost a decade. But Einstein,
quite naturally, did not want anyone to know that he had merely copied
Lorentz’s transformation equations and put a different spin on them. So he contrived an elaborate subterfuge of
inventing two postulates and a kinematic theory based on such transformations,
which would allow him to claim a ‘derivation’ of such Lorentz transformation
equations on his own. In fact, Lorentz
was led to conclude and assert the next year (in 1906) that: “Einstein simply postulates what __we__
have deduced, with some difficulty…”[20] (Folsing, p. 215) In other words, Einstein in 1905 plagiarized
Lorentz’s April 1904 treatise and used its concepts in his Special Theory as
his own, albeit with a different spin.

Einstein was forced to continue this subterfuge until his death in 1955,
because the alternative would have ruined his reputation and cast great doubt
on the crowning achievement of his life:
Special Relativity. In 1954,
Einstein acknowledged that in 1905 he “knew Lorentz’s important treatise of 1895.” But (in 1954) he claimed that in 1905 he did
not know of “Lorentz’s later work” after 1895, which would include Lorentz’s
extensive April 1904 treatise containing the concepts of and formulas for the
Lorentz transformations, Lorentz’s analogies of a sphere contracting into an
ellipsoid, and Lorentz’s __modified__
concept of ‘local time,’[21]
among others. (see Folsing, p. 168) These 1954 claims by Einstein are also
“scarcely credible” for the following additional reasons.

Folsing concludes that Einstein “must have known __a lot more__ than
only Lorentz’s *versuch* of 1895”[22] (Folsing, p. 168) For examples, Einstein poured over Poincaré’s
1902 book, *Science and Hypothesis*, for weeks (Pais, pp. 133 – 134),
which book mentions Lorentz’s work, the concepts of relative motion, relative
velocity, relative time, simultaneity, and Poincaré’s new and expanded
‘principle of relativity’. (Poincaré,
1902, pp. 90, 111 – 114, 242 – 244) In
early 1904, Lorentz published a long and important paper in a well-known German
mathematics and scientific journal, which paper was entitled “World Picture of
Maxwellian Theory. Electronics
Theory.” Such paper referred to
Lorentz’s upcoming 1904 transformation treatise, wherein he would
systematically deal with second-order electrodynamic effects. (Miller, pp. 66, 403) All of these subjects were of vital interest
to Einstein. On April 23, 1904 Lorentz’s
transformation treatise (written in Dutch), which contained his Lorentz
transformations, was published. (*Id*.,
p. 66) Einstein could read the Dutch
language. On May 27, 1904, an English
version of Lorentz’s April 1904 treatise was also published. (*Id*.

Abraham’s 1904 published response to
Lorentz’s April 1904 transformation treatise and Abraham’s follow-up paper
described and critiqued Lorentz’s April 1904 transformation treatise. (Miller, pp. 70 – 71) In 1904, Emil Cohen discussed Lorentz’s April
1904 modified version of ‘local time’ in scientific literature. (*Id*., pp. 171, 202) In September 1904, Poincaré addressed a group
of scientists at the St. Louis World’s Faire and described Lorentz’s April 1904
transformation theories in detail.[23] (Miller, p. 74; Pais, pp. 127 – 128) In December 1904, Lorentz published another
widely read article, which further discussed his prior April 1904 theories and
transformation equations. (Miller, p.
71) In January 1905, Abraham published a
widely read German textbook on Electromagnetic Theory, which further discussed
Lorentz’s April 1904 theories and transformation equations. (*Id*., p. 72)

Poincaré’s June 5, 1905 paper on the Dynamics of the Electron, which critiqued Lorentz’s April 1904 transformation treatise, made frequent and specific reference to Lorentz’s 1904c transformation paper, (Pais, pp. 128 – 129), and Einstein could read the French language. Numerous other scientists and writers discussed Lorentz’s April 1904 transformation treatise in scientific journals during this period.[24] Einstein would have had to be deaf, dumb, blind and unconscious not to have discovered and read Lorentz’s April 1904 transformation treatise (which contained his Lorentz transformations) before June 30, 1905. [25]

Lorentz was an inspirational mentor for Einstein, and Einstein read many
of Lorentz’s other works. Lorentz’s April
1904 transformations and other of his April 1904 concepts are strewn all over
Einstein’s 1905 Special Relativity paper.
These other concepts included, *inter
alia*, Lorentz’s analogies to deformable spheres contracting into ellipsoids
in the direction of motion (see Chapter 26A), Lorentz’s modified concept of
local time (see Chapters 16 and 26), and Lorentz’s concepts that electromagnetic
mass increases with velocity. (Chapter
32) Einstein never specifically denied
Lorentz’s 1906 accusation concerning plagiarism. In his 1907 ‘Jahrbuch’ paper, Einstein
acknowledged that in 1905 he knew about Lorentz’s modified concept of ‘local
time,’ which appeared for the first time in Lorentz’s April 1904 treatise. (Einstein, 1907 [Collected Papers, Vol. 2, p.
253]) How could Einstein __not__ have
found, read and copied Lorentz’s April 1904 transformation treatise before June
30, 1905?

Regardless of exactly how or when Einstein ‘obtained’ his Lorentz
transformation equations, they contain exactly the same unique, absolute and
invalid numerator for ‘local time’ (t = t' + vx/*c*^{2}), exactly
the same unique *ad hoc* contraction factor √1-v^{2}/*c*^{2}
in the denominator of such equations, and they result in many of the same
bizarre mathematical consequences as Lorentz arrived at in his April 1904
transformation treatise. (Lorentz, 1904 [Dover, 1952, pp. 11 – 34])

**D. Are
Einstein’s Lorentz transformations valid for all or even for any
physical phenomena?**

In early 1955, Einstein claimed in a letter (to Carl Seelig) that his
Lorentz transformations (and thus his two fundamental postulates) “transcended
its connection with Maxwell’s equations and was concerned with the nature of
space and time __in general__.”
(Einstein, 1955 [Miller, p. 195])
In such letter, Einstein also claimed that: “the ‘Lorentz invariance’ is a general
condition for __any physical theory__.”[26] (*Id*.

We have already demonstrated that Lorentz’s *ad hoc* April 1904
transformations were premised upon the invalid concept of ether, and for this
reason they, too, were invalid and meaningless.
(see Chapters 15 and 16)
Therefore, if Einstein merely copied them and gave them a new theoretical
spin, such transformations must also remain *ad hoc*, invalid and
meaningless for Einstein’s Special Theory and for science in general.

Even assuming that Einstein might actually have derived the Lorentz
transformations from his two fundamental postulates, does this mean that such
transformation equations are generally valid for physical science? Of course not. We have already demonstrated in prior
Chapters that Einstein’s two fundamental postulates were also *ad hoc*,
invalid, and meaningless, so anything based or premised on them should suffer
the same fate.

Could the Lorentz transformations only be derived from Einstein’s
postulates and his other false assumptions?
Of course not. The Lorentz
transformation equations could easily be derived from a completely different
set of assumptions and could easily be
interpreted to mean something entirely different than Einstein’s
interpretations. For example, the
numerator in such equations, x' = x – vt, could be interpreted to mean not a
straight line between two points x and x', but rather a curved geodesic line
like the surface of the spherical Earth with x and x' separated by a curved
distance vt. This curved geodesic
distance could be plotted on a different set of flexible coordinates (i.e.
Gaussian coordinates) like Einstein did with his General Theory. The denominator of such equations, √1 –
v^{2}/*c*^{2}, could then be interpreted to describe the
quarter arc of a circle or the quarter geodesic distance of a sphere, √1
– x^{2}/y^{2}. (see Figure 15.6) If the denominator was interpreted to become
smaller than the number one depending upon the velocity v of the sphere (which
would be normal mathematical interpretation), then the entire set of Lorentz
transformations could quite logically be interpreted to mean that when the
numerator is divided by a smaller denominator the sphere gets larger and the
distance vt between x and x' expands (rather than contracts), and that the time
interval between x and x' also expands (which of course would not be
relativistically reciprocal).

If these different assumptions and interpretations were described as postulates, then the Lorentz transformation equations for both time and space could be ‘derived’ from them. The author is confident that any imaginative pure mathematician could also come up with other very different assumptions, and could also ‘obtain’ the same Lorentz transformation equations from them, albeit they might be interpreted quite differently.

In fact, by 1905, similar mathematical
scenarios had already occurred at least four times. In 1887, Woldemar Voigt invented an early
version of the Lorentz transformations and from them he supposedly derived the
classic Doppler shift.[27] Voigt also noted that space and time
transformations of this type result in invariant values for electromagnetic
equations. (Pais, p. 121) In 1899, Lorentz used assumptions based on
the M & M null results in order to construct his Lorentz transformations,
so that he could mathematically explain such null results in terms of the ether
theory.[28] (*Id*., p. 125) Joseph Larmor then independently obtained the
same transformations by 1900 in order to support his own proof of the
Fitzgerald contraction. (Pais, p.
126) Lorentz then reinvented or
resurrected his Lorentz transformations in April 1904 in order to mathematically
explain and justify the null results of first and second order electromagnetic
experiments at any velocity. (*Id*.;
Lorentz, 1904c [Dover, 1952, pp. 11 – 34])
Thereafter, Einstein discovered Lorentz’s April 1904 treatise (which
contained the so-called Lorentz transformations), and applied such Lorentz
transformations to his Special Theory along with new *ad hoc* interpretations. (see
Chapter 27C)

Einstein asserted that: “once in
possession of the Lorentz transformation…we can combine [*c*] with the principle
of relativity, and sum up the theory thus…General laws of nature are __co-variant__
with respect to Lorentz transformations.”[29] (Einstein, *Relativity*, pp. 47 –
48) Einstein then claimed that: “If a __general law of nature__ were to be
found that did not satisfy this condition, then at least one of the two
fundamental assumptions [postulates] of the theory would have been
disproved.” (*Id*., p. 48)

Well, a general law of nature has been found which empirically is __not__
algebraically “covariant with respect to Lorentz transformations.” That is Maxwell’s general law of the constant
__transmission__ velocity of light at *c* *in* *vacuo*,
relative to its medium of a vacuum in empty space. Empirically and by definition, Maxwell’s
constant transmission velocity of light at *c* relative to its medium of a
vacuum never varies…it is automatically __invariant__ and velocity *c* with
respect to any frame of reference (see Chapters 6, 9 and 21D).[30] For
this reason, the Lorentz transformations are unnecessary and irrelevant to the
constant transmission velocity of light at *c* relative to its medium of a
vacuum in empty space. Thus, by
Einstein’s own words, his second fundamental assumption (postulate) concerning
the absolutely constant velocity of light at *c* relative to linearly moving inertial bodies has been empirically
disproved.

Einstein also asserted that:
“Every general law of nature must be so constituted that it is __transformed__
into a law of exactly the same [algebraic] form…” (*Id.*, p. 47) But an __invariant__ law, such as
Maxwell’s __constant__ transmission velocity of light at *c* relative
to its medium of a vacuum, cannot be algebraically changed or transformed (from
algebraic form *c* into algebraic form *c*) by any transformation
equation (Lorentz or Galilean). It is
already and always will be the __same form__ in all inertial reference
frames, and its constant emission and transmission magnitude of velocity always
remains invariant at *c* relative to its medium of empty space.[31] Maxwell’s invariant law of velocity *c* in a vacuum must never be transformed
or otherwise mathematically changed.

For all of the above reasons, Maxwell’s general law concerning the
constant __transmission__ velocity of light at *c* does not meet
Einstein’s test of ‘co-variance with respect to the Lorentz
transformations.’ Therefore, __both__
of Einstein’s two fundamental assumptions of his Special Theory have once again
been ‘disproved.’ His fundamental
assumption concerning the __propagation__ velocity of light at *c*
(contained in the first part of his second postulate) is not empirically valid. (Chapters 21 and 22) Einstein’s first postulate—that his expanded
principle of relativity and its Lorentz transformation equations apply to all
general laws of nature—is also not empirically valid. Why?
Because the inherent and invariant transmission velocity of a light ray
at *c* also empirically becomes a relative velocity of *c* – v and *c*
+ v when it propagates with respect to bodies moving linearly at v, but the
invariant transmission velocity of *c* (Maxwell’s law of nature) cannot be
algebraically and covariantly transformed by Lorentz transformations from one
inertial frame to all other inertial frames and still remain physically and
empirically valid.

The same is true of other physical phenomena such as length, mass, time, energy, electricity, etc. As we demonstrated in Chapter 26 and will further demonstrate in Chapter 28, length and time are not really relative quantities, the magnitudes of which are dependent upon the relative velocity of a body. On the contrary, the magnitudes of these quantities are physically and empirically completely independent of any velocity. We will also discover in Chapters 31 through 35 that the same is true of matter, material mass, force, momentum, energy, electricity, and all other physical phenomena, including quantum physics and atomic particles. These phenomena and the magnitudes of their quantities are also independent of any relative velocity.

Einstein only conjectured and asserted that such physical phenomena were
velocity dependent (and applied the Lorentz transformations to them) so that
they would all appear to be consistent with his absolutely constant propagation
velocity of light at *c* relative to linearly moving bodies. However, strangely enough, the only velocity
dependent phenomenon is such propagation velocity of light at *c* ± v relative to linearly moving
bodies, which Einstein artificially and mathematically made velocity __independent__
(impossibly physically invariant and algebraically co-variant) by his
misapplication of the Lorentz transformations.

The conclusion is obvious:
Einstein’s Lorentz transformation equations are completely contrived, *ad
hoc* and meaningless, not only for the velocity of light, but also with
respect to __all__ other physical phenomena.
The Lorentz transformations only mathematically __distort__ all of
these physical phenomena, and thus also much of empirical physics.

**E. General conclusions concerning Einstein’s
Lorentz transformations.**

Let us now briefly reiterate how we got from the ‘difficulties’ contained in Chapter 19 to Einstein’s Lorentz transformations, which are the subject of this chapter.

By misapplying the
Galilean transformation equations to Maxwell’s electromagnetic wave equations,
and thereby __translating__ velocity *c* from one inertial reference
frame to another reference frame (moving at v relative to the first frame), the
late 19^{th} century scientists artificially made Maxwell’s
electromagnetic wave equations for light…appear to be velocity dependent. As a mathematical result, the algebraic form
for Maxwell’s transmission velocity of light changed from an absolute or
constant velocity of *c* in a vacuum to relative velocities of *c* –
v or *c* + v, depending upon the direction of motion of the moving inertial
reference frame. This unintended
algebraic result was a fundamental mathematical and theoretical blunder. Why would anyone want to translate a constant
and absolute (invariant) velocity from one inertial reference frame to another?[32] Why would anyone believe that Maxwell’s
constant transmission velocity of light at *c* had __changed__ its
inherent velocity of *c* to *c* ±
v because of such mathematical translation?
These mistaken and illogical beliefs became known as the ‘difficulties.’

When
Lorentz and Einstein later misapplied the Lorentz transformations to the transmission
velocity of light at *c* in two different reference frames separated by
the relative velocity v, Maxwell’s equations remained the same in both
frames. Why? Because the Lorentz transformations algebraically
eliminated the relative velocity v between the two inertial reference frames
(see Hoffmann, 1983, p. 87), and along with it they automatically eliminated
the relative time interval and distance interval between the two reference
frames. The mathematical result was that
the magnitude of a propagating light ray’s velocity was changed to *c*
with respect to both inertial reference frames at the same instant (which
Einstein called ‘co-variant’), rather than the very natural relative velocities
of *c* ± v.

Thus, the Lorentz transformations only algebraically and artificially
reversed and eliminated the above described original mathematical and
theoretical blunder, which occurred when the Galilean transformations were
misapplied to the constant velocity of light at *c*. As Hoffmann summarized this mathematical
scenario:

“if one applied the Galilean
transformation to the unprimed Maxwell equations, the resulting primed
equations had additional first-order terms involving *v/c* and
second-order terms involving *v ^{2}/c^{2}*…[But then] if
one applies the Lorentz transformation [to Maxwell’s equations]…the primed
equations, except for the primes, turns out to be

And as Feynman concludes:

“It is just a trick…When you unwrap
the whole thing, __you get back where you were before__.” (Feynman, 1964, p. 25-11)

Smolen also describes why
Einstein’s application of the Lorentz transformations to light was just a trick…”the
__trick__ that made relativity special.”
(see Smolin, pp. 228 – 229) Again,
as Russell further explained in 1927:

“Technically, the
whole of the special theory is contained in the Lorentz transformation. This transformation has the advantage that __it
makes the velocity of light the same with respect to any two bodies which are
moving uniformly relatively to each other__, and, more generally, that it
makes the laws of electromagnetic phenomena (Maxwell’s equations) the same with
respect to any two such bodies. It was
for the sake of this advantage that it was originally introduced…” (Russell, 1927, p. 49)

In other words, the algebraic change of c to c ± v
caused by the misapplication of the Galilean transformations to Maxwell’s
equations was mathematically __reversed__ by the mathematical __trick__
of applying the Lorentz transformation equations.[33] It is just that simple. However, in the process the Lorentz
transformations created many more consequential blunders and theoretical
distortions when they were thereafter deemed to apply to all of physics, and
were misapplied by Einstein and his followers to all other physical phenomena,
such as length, time, mass, and

To illustrate our aforementioned
conclusions, let us again cite the following passages from D’Abro’s 1950 book:

“In the theory of
relativity, Einstein asks us to agree that a certain finite invariant velocity
which turns out to be that of light in vacuo must be considered __invariant
for all Galilean [inertially moving] observers__.”[34] (D’Abro, 1950, pp. 161 – 162)

“Now it is obvious
at first sight that if our space and time measurements were such as classical
science believed them to be, it would be __impossible__ for a ray of light
to pass us with the same speed regardless of whether we were rushing towards it
or fleeing away from it.[35] A simple mathematical calculation shows us,
however, that we can make our results of measurement compatible with
[Einstein’s] postulate of invariance provided we recognise that our space and
time measurements are __slightly different__ from what classical science has
assumed. This is purely a mathematical
problem and can be solved by mathematical means.[36] It leads us, of course, to the
Lorentz-Einstein transformations, and from these transformations it is easy to
see that rods in relative motion must be shortened, durations of phenomena
extended, and the simultaneity of spatially separated events disrupted.”[37] (Id.,
p. 162)

The obvious ad hoc arbitrariness of such absolutely ‘invariant velocity,’ of such ‘purely
mathematical problem and solution,’ and of such ‘slightly different assumed
measurements’ speak volumes by themselves.
So do the following arbitrary and ad hoc conclusions by Feynman:

“All of the laws of
physics are invariant under the Lorentz transformation.” (Feynman, 1964, p. 25-11)

“Every law of
physics…must have the same invariance under the same transformation.” (

“The laws of physics
must be such that after a Lorentz transformation, __the new form of the laws
looks just like the old form__.”
(Feynman, 1964, p. 25-1)

In
light of the above discussions, why must we believe that all of these
mathematical tricks and conclusions by Feynman are true? Because, states Feynman, it was found by the
M & M experiments and by all other ether drift experiments that Maxwell’s
equations (and specifically that the __transmission__ velocity of light at c) was __invariant__ “in all __inertial systems__.” (see Feynman, 1964, p. 25-11). However, we now know from the discussions of many
prior chapters that inertial systems and the Lorentz transformations had
absolutely nothing to do with such invariance.
The constant transmission velocity of light at c relative to its medium of a vacuum in empty space is
inherently always velocity c and __invariant__ everywhere. (Chapters 6, 9, 21 and 22)

_______________ _{o} _______________

For all of the above reasons, Einstein’s Lorentz
transformations are completely ad hoc, arbitrary, invalid and meaningless for any
reason, and especially with respect to his Special Theory and with respect to
all physical phenomena (including the transmission velocity of light).

[1] This was the same synchronous and artificially simultaneous time for all spatially separated observers with synchronized clocks who are relatively at rest in an inertial system.

[2]
Empirically the laws of electromagnetic phenomena (including the transmission
velocity of light at *c*) are automatically the same with respect to all
inertial bodies (including stars, planets, moons, satellites and rockets),
regardless of such bodies’ relative motions, or any transformation equations
involved. (see Chapters 21 and 22) Therefore, Einstein’s goal of algebraically
making Maxwell’s equations (including velocity *c*) the same for all inertial bodies was both unnecessary and
meaningless.

[3] Again,
we must point out that every ray of light already does possess, and always has
possessed, such constant velocity of transmission *c* relative to its
point of emission in space, and relative to its medium of empty space. (Chapters 6A, 21D and 22) But such transmission velocity of a light ray
at *c* __relative__ to __distant material bodies with different linear
velocities__ of approach toward or separation from the light ray, is (and
will always be) the very natural __relative velocity__ of *c* + v or *c*
- v…the transmission velocity of the light ray at *c* plus or minus the
relative linear velocity v of the distant material body. (Chapters 21D and 22) When the Galilean transformation for distance
is misapplied to the __constant transmission velocity__ of a light ray in a
vacuum, *c*, with respect to a material body moving linearly at v, the
mathematical result must always be some magnitude of *c* + v or *c* –
v…a relative velocity. Einstein never
seemed to grasp these simple concepts.

[4] But if
the event or physical phenomenon is already __invariant__, such as the __constant__
transmission velocity of a light ray at *c* or the __constant__
velocity v of an inertial frame, then *a priori* no variation or algebraic
transformation can or should occur from one inertial frame to another. (Chapters 14 and 21)

[5] The
reader of 1905 had to be able to imagine “Einstein’s __mechanical__ models
of inertial reference systems.” (Miller,
pp. 196 – 197; see Figure
25.5B)

[6] All of
the above theoretically occurred in ‘stationary space’ (see Einstein, 1905d [

[7] In system S' the mirror M' was not physically displacing from the tip of the propagating light ray, and the light source was not physically changing its position relative to mirror M'. The light source and the mirror in S' were always relatively stationary. (see Chapter 12) The fact that system S' was physically displacing from system S would be irrelevant to such measurements in S'. .

[8] Relatively stationary system S played the part of the stationary ether in this scenario

[9] This
result was obtained by an auxiliary set of space coordinates that “transforms
according to the Galilean transformations…”
(Miller, p. 196) The coordinate
measurement by S of a propagating light ray in S' moving away at v was theoretically
equivalent to mirror M in System S physically __displacing__ from a light
source in S and from the tip of the propagating light ray because of
theoretically __increasing__ coordinates (see Figures 27.1C and 9.2), which scenario was
the theoretical ether basis for the Michelson & Morley paradox, i.e. that
light must propagate a greater distance in the direction of motion v. (also see Figure 26.2) On the contrary, in Chapter 12 we have
explained that this stationary ether scenario did not exist, and that __no
greater distance__ of to and fro light propagation occurs where the light
source and the mirror remain relatively stationary, regardless of their
in-tandem inertial motion through space.

[10] Note
that in the process of arriving at these bogus conclusions, Einstein used two
different methods to measure the velocity of light. Velocity *c*
in system S' was measured by synchronized clock times, and the relative
velocities *c* ± v in system S were
measured by coordinates and algebra. If
both measurements had been made by synchronized clock times, the light ray
would have been measured to be *c* in
both inertial reference frames. (see Figure 27.1C)

[11] All of
such mathematical factors, equations and concepts may have been necessary for
Einstein’s Special Theory, but they were totally unnecessary for, and
irrelevant to, the empirical phenomena of light and its velocity of *c* in a vacuum. (see Chapters 21 and 22)

[12] We shall discuss the implications to be drawn from this conclusion by Miller (and others) in the next Section C of this chapter.

[13] In
other words, such transmission velocity of light at *c* __never varies__
in any vacuum or in any empty space.

[14] Again,
Einstein also arbitrarily used different methods of measurement in each
system. In system S' he used
synchronized clocks to measure the velocity of light at *c*, and in system S he used coordinates and algebra to measure the
velocity of light at *c* ± v.

[15] On the
other hand, it should have __also__ been measured to be *c* – v in the opposite direction (rather than *c*
+ v), because the direction of v was always away from S in the x' direction.

[16] Folsing is probably referring to the place where Einstein “avoided discussing a contraction of length because…[he] could not deduce this contraction until after he derived the [Lorentz] space and time transformation equations.” (see Miller, pp. 196 – 197)

[17] As the
author pointed out in Chapter 20E, Einstein’s expanded Principle of Relativity,
which is described at the __beginning__ of his Special Theory, also __impliedly
refers__ to the Lorentz transformations (‘derived’ __much later__ in § 3),
because such transformation equations are the __only “equations__ of
mechanics [which] hold good” within the context of his Special Theory. Folsing’s conclusion is consistent with the
author’s above interpretation and assertion.

[18] See Miller, pp. 196 – 197, where he concludes that Einstein avoided discussing a contraction until after he ‘derived’ his Lorentz transformations, which were necessary to the deduction of such contraction. (see Chapter 27B in this regard)

[19] “For example, Einstein’s official duties as a patent clerk would have involved visits to city and university libraries.” (Folsing, p. 169)

[20] Lorentz’s reference to “we” means Poincaré also.

[21]
Einstein also adopted Lorentz’s 1904 __modified__ concept of ‘local time’ for
his own 1905 Special Theory, which fact Einstein did later unwittingly
acknowledge in his December 1907 Jahrbuch article.

[22] The German word ‘versuch’ means ‘attempt’.

[23] Poincaré’s lectures were published and discussed in numerous newspapers and scientific journals.

[24] In 1940
“Einstein would have been able to read in *Annalen* Wilhelm Wien’s *Differential
Equations of the Electrodynamics of Moving Bodies*, which contained many
references to the latest literature, as well as a subsequent polemic between
Wien and Abraham, in which Wien not only quoted Lorentz’s work of the same year
but actually provided an outline of it.”
(Folsing, p. 169)

[25] Einstein’s Special Relativity paper was received by the publisher on June 30, 1905. (Pais, pp. 128 – 129)

[26] By the term ‘Lorentz invariance,’ Einstein meant that, “with respect to Lorentz transformations”, the general laws of nature have the same (co-variant) algebraic form in all inertial frames of reference. (see Miller, pp. 164, 227, 272)

[27] Did this mathematical process make the empirical Doppler shift any more valid? Of course not.

[28] Does this mean that the Lorentz transformations have any inherent physical meaning? Of course not, especially since we now know the real reasons for the M & M paradox. (Chapters 10 and 12)

[29] By 1916, Einstein had generalized the algebraic term ‘covariant’ for classical mechanics to also mean relativistically and physically ‘invariant.’ (see Miller, pp. 164, 227, 272)

[30] Even in
Einstein’s Special Theory, it is only the dimensions of material objects that
theoretically vary and contract in order that the __relative propagation__
velocity of light at *c* – v or *c* + v may theoretically and
algebraically remain absolutely *c* relative to distant linearly moving
bodies.

[31] The
transmission velocity of light at *c* cannot be both invariant in the
empirical sense, and co-variant (changed in algebraic form from *c* ± v to *c*) in Einstein’s algebraic sense, relative to linearly moving
material bodies.

[32] It is already defined as being constant, invariant and the same everywhere.

[33] If the application of the Lorentz transformations was just a mathematical trick, why do such mathematicians take such transformations and Special Relativity so seriously?

[34] In a
footnote, D’Abro concluded that such a “‘finite invariant velocity’…is __acceptable
mathematically__, and the only question that we shall have to consider is
whether it corresponds to __physical reality__.” (D’Abro, 1950, p. 162) The answer is: it does not.

[35] Remember, that this was also Bird’s impossible paradox described in Chapter 21E.

[36] Of
course, it is not a purely mathematical problem if we want the result to
‘correspond to __physical reality__.’

[37] In
other words, the theoretical end justifies the *ad hoc* and arbitrary means.