EINSTEIN’S RELATIVISTIC COMPOSITION OF VELOCITIES

*Based on the Lorentz
transformation equations, Einstein derived a relativistic formula for the
kinematic addition of two material velocities in the same direction, which
algebraically never could exceed the velocity c. Einstein then claimed that a variation of
this formula mathematically confirmed his second postulate concerning the
absolutely constant propagation velocity of a light ray at c relative to every
inertial reference frame. Later he
claimed that the 1851 Experiment of Fizeau empirically confirmed the validity
of these formulae. But it turns out that
none of such claims have any empirical validity. *

**A. The paradoxes that led to Einstein’s relativistic
composition of velocities.**

The classical or Newtonian law for the addition of velocities directly added or subtracted the different velocities of different reference frames (material bodies). (see Figure 7.1) This intuitive result was derivable from the Galilean transformation equations. (Goldberg, p. 100; see Chapter 14) However, the phenomenon of light did not appear to follow these simple kinematic laws.

The most baffling paradox of light
concerned Maxwell’s equations, where the __constant__ transmission velocity
of a light ray propagating in a vacuum (at 300,000 km/s) was represented by the
algebraic symbol *c*. According to the conventional scientific and
mathematical wisdom of 1905:

“If the speed
of light is *c* as measured __in__ a
particular [inertial] reference frame then it __cannot__ also be the same
number *c* __relative to__ another
[inertial reference] frame.”[1] (Rohrlich, p. 52)

Theoretically,
this scenario violated Maxwell’s equations, because the velocity of light was
no longer a __constant c__ with
respect to the other reference frame (Frame 2) moving at a different velocity v
than the first (measuring) reference frame (Frame 1). What could be the answer to this apparent paradox?

The empirical answer to this so-called paradox is again found in Chapters
6 and 21. Most importantly, Maxwell’s __theory__
was that light had a constant transmission velocity of *c* __relative to its medium of a vacuum__; __not__ relative to
a material reference frame.[2] (see Chapter 6A) Maxwell’s constant __transmission__ velocity
of a light ray at *c* in a vacuum is an
absolute or abstract velocity which will always be measured to be *c* (by the two-way method) __in__ any
inertial reference frame in the Cosmos.[3] (see Figure 29.1A) But a light ray transmitting at such constant
velocity of *c* will also __propagate__
over changing distance/time intervals at velocity *c* ± v __relative to__ any other inertial reference frame in the
Cosmos, depending upon the direction of such other reference frame moving
linearly at v __relative to__ the first reference frame, and moving linearly
__relative to__ the tip of the light ray.[4] (see Figure 29.1B) Einstein and the rest of the scientific community
were just not thinking.[5]

For some unfathomable reason, neither
Einstein nor the scientific community realized the real answer to the above so-called
paradox. So, in 1905, in an attempt to
reconcile the above paradox, which he called a crisis in physics, Einstein invented
*ad hoc* the first impossible part of
his second postulate which stated that the velocity of a __propagating__
light ray always has a constant velocity of *c*
__relative to any inertial reference frame__ in the Cosmos, regardless of
its linear velocity relative to the tip of the light ray.[6] (see Chapters 20F and 21E) Then, in order to mathematically demonstrate
how this bizarre and impossible concept might be possible, Einstein derived *ad hoc* from the Lorentz transformations
a new relativistic addition (composition) of velocities law which algebraically
states that relative to “all inertial frames of reference the speed of light
will have the same value” of *c*,
regardless of the linear velocity v of the inertial frame. (Miller, p. 261, Resnick, 1992, p. 477) For the above empirical reasons, we now know
that such *ad hoc* postulate and such *ad hoc* relativistic composition of
velocities law are not and cannot be empirically valid. (see the Preamble)

There was also a second baffling paradox in 1905 concerning Maxwell’s equations. Again, according to the conventional scientific and mathematical wisdom:

“If the source
of the light [say a lamp on a stationary railway embankment]…is used as a
reference frame then the speed of light should have one value __relative to__
the source at rest and another value relative to a moving source.” (Rohrlich, p. 52)

However, the M & M experiment (which was devised in part to prove the above reasoning) failed to do so. Instead such experiment asserted that the velocity of light was the same in all possible directions, regardless of the motion or direction of motion of its source. (see Chapter 9)

The empirical answer to this paradox
is found in Chapters 21 and 22. The __transmission__
velocity of *c* is an __inherent
property__ of a ray of light in a vacuum, and such absolute velocity will __instantly__
occur whenever a light ray comes into existence. Thus, the tip of a light ray will instantly
propagate away from its source body at the transmission velocity of *c* relative to its medium of a vacuum,
regardless of the velocity of the source body, or its direction of motion. However, __relative to__ the material
source body moving linearly at v, the tip of such light ray will propagate over
changing distance/time intervals at the __relative velocity__ of *c* ± v.

Neither Einstein nor the scientific
community realized the real answers to the above second paradox. As a result, in 1905, Einstein (in an attempt
to reconcile the above perceived paradox) invented the second part of his
second postulate which states that the velocity *c* of a light ray “is independent of the state of motion of the
emitting body.”[7] (Einstein, 1905d [*ad
hoc* guess by Einstein turned out to be the correct result, but it does not
describe the real reason for the result.
Then, in order to mathematically demonstrate how such postulate might be
feasible, Einstein’s derived *ad hoc*
from the Lorentz transformations, a second new relativistic addition
(composition) of velocities law which algebraically predicted that the velocity
of light was *c*, regardless of the
velocity of its source body. But,
Einstein’s second relativistic addition of velocities formula was only an __artificial__
way to arrive at the correct empirical result, because with the real empirical reason
for such second paradox there is no composition of velocities involved.[8]

The sole purpose for __both__ of these new ‘composition of velocities
laws’ was to resolve the so-called paradoxes previously described in this section. As we have explained, they were completely *ad hoc* and unnecessary. There never was a real problem with the
velocity of light which needed any resolution.[9] Such paradoxes only existed in the minds of
Einstein and the rest of the scientific community.

With the above paragraphs as the
introduction to this chapter, let us now discuss the substance of Einstein’s two
*ad hoc* formulae for the relativistic
composition (sometimes called transformations) of velocities.

**B.
Einstein’s relativistic formulae for the composition of velocities.**

In Section 5 of his Special Theory, entitled “The Composition of
Velocities,” Einstein derived two new addition or composition of velocities laws
from the Lorentz transformation equations, which would theoretically and relativistically
__transform__ (or change) the velocities between two inertial frames of
reference, S and S'.[10] As we described in Section A of this chapter,
such relativistic transformation of velocities was completely *ad hoc*, and like Einstein’s second
postulate it was also completely empirically unnecessary, impossible and
invalid.

In Section 5, Einstein only considered uniform velocities which were __parallel__
to the direction of the relative motion v between S and S' in the x – x'
direction. (see Resnick, 1968, p. 82) If the relative velocity is in the same
direction along the common x axis of the two frames, then there will be symmetry
of reciprocity in either direction merely by interchanging the plus and minus
signs for the relative velocity.
However, if the relative velocity is in some other direction (i.e. a
different angle), then the “symmetry is lost.”
(Miller, p. 260)

At the beginning of Section 5, Einstein asked the question: What is the velocity v of a point moving at velocity w in system S' as measured by coordinates in system S, where S' is moving away from S at the uniform relative velocity of v? His answer was:

“It
is worthy of remark that *v *and *w* enter into the expression for
the resultant velocity in a __symmetrical__ manner. If *w* also has the direction of the
axis of x,

we get

^{2}.”

(Einstein, 1905d [Dover, 1952, p. 50]; also see Resnick, 1968, pp. 79 – 80)

This
was Einstein’s first formula for the composition of two __material__
velocities moving inertially in the same x direction.[11]

In this relativistic formula, V is the ‘relativistic addition of velocities’ with respect to two bodies S and S' moving uniformly in the same parallel direction, as measured by coordinates in S. Goldberg explained how the denominator in Einstein’s composition of velocities produced the desired results.

“The
quantity *wv/c*^{2} can be
understood as the product of w*/c* ^{.}
*v/c*, that is, as the product of the fraction
of the speed of light to the speed of the object as measured in one frame of
reference and the fraction of the speed of light which the relative speed of
the two frames of reference represents.
If either of those ratios is small, the product will also be small and
the result of this velocity addition law becomes formally close to the result
of the classical velocity addition law.”
(Goldberg, pp. 100 – 101)

Based on his new relativistic
transformation equation for material velocities, Einstein conjectured
that: “from a composition of two [material]
velocities which are less than *c*, there always results a velocity less
than *c*.”[12] (Einstein, 1905d [Dover, 1952, p. 51]) This algebraic result is neither mysterious nor
amazing; it is purely mathematical.[13] It is very similar to what algebraically
happens when the *ad hoc* Lorentz
transformation equations are applied to two inertial frames of reference. When the Galilean factor for distance
traveled (x ± vt) is divided by Lorentz’s contraction factor √1 – v^{2}/*c*^{2},
the factor √1 – v^{2}/*c*^{2} (along with certain
dubious interpretations) automatically changes the magnitude of the distance traveled
depending on the magnitude of v^{2}.
Lorentz and Einstein both mathematically manipulated this changed
distance and then characterized the false physical result as a
‘contraction.’ The maximum so-called
‘contraction’ automatically and mathematically occurs at *c*, similarly to
the result that occurs with Einstein’s new formula for the composition of material
velocities.[14] (see Figure 16.2A)

Based
on these mathematical results, Einstein further conjectured in § 4 of his
Special Theory: “that the velocity of
light in our theory plays the part, __physically__, of an infinitely great
velocity.”[15] (*Id*., p. 48) For the reasons cited above, this *ad hoc* conclusion by Einstein is also empirically
meaningless.

What was the empirical foundation for these limiting
mathematical conjectures? In 1905, there
was probably only one type of experimental result that could be claimed to form
an empirical basis for them. That was
the highly speculative experimental and theoretical work with ‘electromagnetic
mass’ by Kaufmann, Abraham, Lorentz and others, where it mathematically appeared
that *c* it might be the ultimate speed limit for electromagnetic
mass. (see Chapter 17) In later chapters, we will discuss why the
velocity of light at *c* may actually be the empirical limiting velocity
for a material particle or body on Earth.[16] But this theoretical or empirical limitation
does not occur because of the Lorentz transformation equations, nor because of
Einstein’s relativistic formula for the composition of material velocities, nor
because of a body’s relative kinematic velocity.

There are also other possible directions of relative motion other than along the x axis. If the direction of the relative velocity is perpendicular or transverse to the x axis (for example, in the y direction), then the relativistic composition of material velocities will be:[17]

(Goldberg, pp. 475 – 476) Resnick pointed
out that this “result is not simple because neither observer is a proper one.”
[18]
(Resnick, 1968, p. 83) The same is true for the perpendicular z direction.
(*Id*.

If one frame is __accelerating__
relative to the other, “we can obtain the relativistic acceleration
transformation equations…by time differentiation of the velocity transformation
equations…with a_{x} = du_{x}/dt and a_{x}' = du_{x}'/dt'
as the x and x' components of the acceleration.
We obtain

. ^{”}

(Resnick, 1968, p. 84) Einstein used relativistic acceleration transformation equations for his accelerating electron in Section 10 of his Special Theory in order to derive the magnitudes of Longitudinal Mass and Transverse Mass.[20] (see Chapter 31)

Let us now turn to the possible composition of the
velocity of light at *c* in conjunction with a material velocity w. At this point, Einstein conjectured:

“It
follows, further, that the velocity of light *c* __cannot be altered__
by composition with a velocity less than that of light.[21] For this case we obtain

^{.” }

(Einstein, 1905d [Dover, 1952, p. 51])

This
formula is merely a variation of Einstein’s first formula for the composition
of velocities where v in the numerator is replaced by *c* (the transmission velocity of light in a vacuum), and v is eliminated
in the denominator. Einstein’s second
formula for the composition of velocities was also intended to be a
transformation equation which *ad hoc*
changed the classical composition of velocities.

The basic reasons for this variation of Einstein’s relativistic formula for the composition of material velocities were twofold:

1. To mathematically justify the
first part of Einstein’s second postulate:
that all inertial observers will measure the same absolutely constant
velocity of *c* for light rays propagating through their frames,
regardless of such observer’s own linear velocity w.[22] However, this attempt by Einstein fails for
the empirical reasons set forth in Chapters 21 and 22.[23] The real reason that the transmission
velocity of light at *c in vacuo*
cannot be altered by composition with a material velocity, vis. that the __transmission__
velocity of light at *c* is an inherent
and invariant property of the phenomenon of light, never involves any
composition of velocities.

2. To
mathematically justify why the velocity of a light ray at *c* is
independent of the velocity w of its emitting material body.[24] As previously mentioned, Einstein’s second
velocity transformation formula did not provide reasons for the above
phenomena. It merely provided a
mathematical description of light’s __absolutely__ constant velocity of *c*.

The algebraic results of this second formula for the
composition of *c* with other velocities are also neither mysterious nor
amazing. Such relativistic results are
likewise purely mathematical, and they occur for similar algebraic reasons as we
described with respect to Einstein’s first transformation formula for the
composition of material velocities. In
fact, they should be mathematically expected.
(see Miller, p. 261) As Resnick
also pointed out: “We know that an __assumption__
[postulate] __used to derive__ the transformation formulas was exactly this
result: that is, that all observers
measure the same speed *c* for light.”[25] (Resnick, 1968, p. 80) In other words, Resnick is asserting that the
first part of Einstein’s second postulate concerning the absolutely constant velocity
of light (in circular fashion) mathematically results in itself, in the
algebraic formula last above described. [26] So much for the mysteries and manipulations
of mathematics.

Rohrlich described both of Einstein’s formulae for the relativistic
composition of velocities as a __direct__ consequence of the Lorentz
transformations (along with Length Contraction and Time Dilation). He also characterized them as necessary for
the logical consistency of Einstein’s Special Theory.[27] (Rohrlich, p. 71) Why?

“[Because] when
the speed of a light signal is __always c__
then any motion of an observer in the same or in the opposite direction

“[This result]
is contained in the Principle of the Constancy of the Speed of Light that __stands
at the head of the theory__. And this
principle has been incorporated into the above formulae. But it shows how the composition of speeds
became modified as a result of __assuming that principle__.[30] It is a check on internal consistency. It also assures us that the Michelson-Morley
result is __fully accounted__ for.”[31] (*Id*.

Was there any empirical foundation
for Einstein’s second composition of velocities formula: “that the velocity of light *c* cannot
be altered by composition with a velocity less than that of *c*”? In his book, *Relativity*, Einstein
asserted that Michelson’s null results were empirical confirmation that the
velocity of light at *c* in Michelson’s apparatus when added to the solar
orbital velocity of the Earth at 30 km/s does not change the velocity of light
at *c* in the direction of such motion.[32] Einstein further asserted that the empirical
reason for such null results was the contraction of Michelson’s apparatus in
the direction of such motion.[33] (Einstein, *Relativity*, pp. 58 –
60) Since we now know the real reasons
for Michelson’s null results, which have nothing to do with contractions of
distance or time dilations (Chapters 9 – 12), we should also realize that
Michelson’s null results were not empirical confirmations for Einstein’s
relativistic composition of velocities formulae (which have such Length
Contractions and Time Dilations embedded within them).

Let us now ask a more general
question: How can Einstein’s
relativistic formulae for the composition of velocities, which are derived from
an absolute, empirically impossible and invalid postulate for the velocity of
light (see Chapter 21 and the Preamble), and which are derived from the *ad
hoc* and meaningless Lorentz transformations (see Chapter 27), have any
empirical validity or physical meaning for physics? The answer is:
they cannot.

Einstein needed to demonstrate how transformations of velocities could be derived from the Lorentz transformations before he preceded with his applications of such transformations in Part II of his Special Theory. The reason being that such transformations of velocities were necessary in order “to calculate how masses relate to each other in different inertial frames of reference, to calculate the relationship between energy and mass,…to calculate his time relationships in the clock paradox…” (Goldberg, p. 476), and to calculate Einstein’s various relativistic Doppler theories of light, etc., etc.[34] If Einstein’s invalid relativistic composition of velocities was necessary to calculate such other relativistic concepts, then how can such other relativistic concepts have any validity?

**C. Is the 1851 Experiment of Fizeau a confirmation
of Einstein’s composition of velocities formula?**

In 1907, Max von Laue deduced Fresnel’s ether drag
coefficient, v' = v(1 – 1n^{2}), from Einstein’s Special Theory, which
he considered “as a further illustration of Einstein’s addition theorem for
velocities.” (Miller, p. 263; Rindler,
p. 69) Based on von Laue’s deduction and
conclusion, Einstein devoted the entire Chapter 13 of his 1916 book *Relativity*
to Fizeau’s 1851 so-called ‘experimental verification’ of Fresnel’s
mathematical coefficient. Such chapter
was entitled: “Theorem of the Addition
of the Velocities. Experiment of
Fizeau.” (Einstein, *Relativity*,
pp. 43 – 46) As Rohrlich conjectured:

“Another
application of the above new law for the composition of speeds lies in the __explanation__
of an old experiment. The 1851 Fizeau
experiment…found that the speed of light in flowing water differed from that in
water at rest. He attributed that to a
partial drag of the ether by the water. It finds its __explanation__ by the
special theory of relativity in __the peculiar way in which speeds add__
according to the above formula. That
formula is in __full accord__ with the Fizeau result.”[35] (Rohrlich, p. 71 – 72)

In such Chapter 13 of *Relativity*,
Einstein compared his formula for the relativistic addition of velocities with
the results of Fizeau’s 1851 experiment and with Fresnel’s coefficient. Einstein then claimed and conjectured that
such comparisons resulted in an __elegant confirmation__ of and a “crucial
test in favor of the theory of relativity.”
(*Id*., p. 46) Many of
Einstein’s followers also cite Fizeau’s 1851 experiment (and repetitions
thereof) as experimental confirmation of Special Relativity, and of his
relativistic formula for the composition of velocities. (see Resnick, 1968, p. 37; Zhang, pp. 207 -
214) On the contrary, none of these
claims or conjectures are correct.

Einstein began Chapter 13 of *Relativity*
by conjecturing that, although his concepts of ‘length contraction’ and ‘time
dilation’ only manifest themselves at velocities near the velocity of light, he
would now easily derive from his formula for the ‘relativistic addition of
velocities’ __a quantity__ that could be spectacularly demonstrated and
“elegantly confirmed by experiment” at __low velocities__. (Einstein, *Relativity*, p. 43; also see
Miller, p. 263) Einstein was, of course,
referring to Fizeau’s 1851 experimental confirmation of Fresnel’s ether drag
coefficient. He was also referring to
Pieter Zeeman’s 1914 and 1915 repetitions of the Fizeau experiment, which he
claimed __exactly agreed__ with his formula for the relativistic composition
of material velocities.[36] (see Einstein, *Relativity*, p. 46;
Zhang, pp. 211 – 212, 281) Let us now
scrutinize Fizeau’s experiment and Einstein’s Chapter 13, to see if Einstein’s
above claims, predictions and assertions are confirmed by reality.

Einstein analyzed the 1851 Fizeau experiment as if it was a study in Galileo’s Relativity:

“The tube plays the part
of the railway embankment,…the liquid plays the part of the carriage,…and
finally, the light plays the part of the man walking along the carriage…” (Einstein, *Relativity*, p. 45)

See Chapter 7 of this treatise and Figure 7.3 for a detailed discussion, illustration and explanation of the 1851 Fizeau experiment.

In his Chapter 13, Einstein assumed that “light
travels in a motionless liquid with a particular velocity w.” (*Id*.*Id*.

In order to answer his own question as to the magnitude of velocity W, Einstein deduced the formula for the ‘classical addition of velocities’ from the Galilean transformations and obtained, W = v + w. This was the same formula that he used with his prior example of the man walking in the same direction as the carriage moving along a stationary embankment. (see Chapter 19)

The ultimate question which Einstein really wanted
answered was which theorem for the addition of velocities better __describes__
Fizeau’s empirical results: A. the
classical theorem for the addition of velocities (W = v + w); or B. his new relativistic formula for the
addition (composition) of material velocities,

.

(*Id*., pp. 44 – 45, 46)
Einstein then concluded that his new relativistic formula was the winner
for purposes of such description, and that “the agreement is, indeed, __very
exact__.”[37] (*Id*., p. 46) Let us now continue to examine and scrutinize
these and other claims by Einstein.

There were three basic questions posed by the mysterious empirical results of Fizeau’s Experiment: 1) Why is only a fraction of the velocity of the moving medium (water) transferred to the velocity of light propagating through it? Unlike the velocity of light in the moving medium of water, the propagation of sound waves in moving air is directly proportional to the velocity of a wind. (Gamow, 1961, p. 162) 2) Why is such fraction of the velocity of the moving water related to the square of the Index of Refraction in Fresnel’s coefficient? 3) Which equation for the addition of velocities best describes the empirical results of the Fizeau experiment? There are numerous theories that attempt to answer the first two questions. (For example, see Pavlovic, Sections 13, 14, 19 & 20, and the author’s suggestions in Chapter 7) Einstein had no answer for such two questions.

The third question is the only one
that is really relevant to this Chapter.
In *Relativity*, Einstein claimed that Fresnel’s empirical
equation, v' = v (1 – 1/n^{2}),^{ }which mathematically
describes^{ }the empirical result of Fizeau’s experiment,^{ }is
equally well described by his relativistic equation for the addition of
velocities. (Einstein, *Relativity*,
pp. 43-46) In footnote 1 on p. 46 of *Relativity*,
Einstein even suggested an algebraic manipulation that would make the
approximation between such equations an exact agreement:

“Fizeau found *W
*=* w + v* (1 – 1/*n*^{2}), where *n* = *c*/*w*
is the index of refraction of the liquid.
On the other hand, owing to the smallness of *vw*/*c*^{2}
as compared with 1, we can replace (*B*) [Einstein’s relativistic equation
for the addition of velocities] in the first place by *W* = *w* + *v*(1
– *vw*/*c*^{2}), or to the same order of approximation by *w*
+ *v*(1 – 1/*n*^{2}), which agrees with Fizeau’s result.”[38] (*Id*., p. 46)

Why does Einstein’s relativistic formula for the composition of velocities
__appear__ to so closely describe the empirical result obtained by Fizeau
and described by Fresnel’s coefficient? The answer is several-fold: First, both formulae include the same values,
vis., the velocity of light (*c*) *in vacuo*, the speed of light
(w) in motionless water, and the velocity (v) of the material substance (water)
relative to the tube. Secondly, the
Index of Refraction, when plotted on a graph for all hypothetical velocities
of light through different material mediums, __coincidentally__ results
in substantially the same graphic configuration as the Lorentz transformation,
1/√1 – v^{2}/*c*^{2}. (compare Figure 29.2 with Figure 16.2B) The two formulae also hypothetically result
in substantially similar quantities. (compare
Chart 29.3 with
Chart 17.3)

Thirdly, and the most important
reason why Einstein’s relativistic formula for the composition of velocities __appears__
to be so close to Fresnel’s coefficient, is the __low magnitude of v__ (the
velocity of the water in Fizeau’s experiment in both equations). Einstein asserted that “Fizeau found W = w +
v(1-1/n^{2}), where n = *c*/w is the index of refraction of the
liquid.” (Einstein, *Relativity*,
p. 46, F.N. 1) The approximate values in
the Fizeau experiment were: w, the speed
of light in the motionless water (226,000 km/s); *c*, the speed of light
in a vacuum (300,000 km/s); v, the velocity of the water through Fizeau’s tube
(7 m/s); and W, the velocity of light propagating relative to the tube. (*Id.*, p. 45)

When we apply the above values to Fizeau’s experiment, and to the
equation which Fizeau supposedly found (W = w + v(1 – 1/n^{2}), this
results in the following magnitude for W:
W = 226,000.003027422222 km/s.
When we apply the same values of the Fizeau experiment to Einstein’s
relativistic formula for the addition of velocities (W = v + w/1 + vw/*c*^{2})
we get the following magnitude for W:

W = 226,000.003027422169 km/s.

Very close
indeed, but __not exactly__ the same.
However, when we apply the same values of the Fizeau experiment to the
classical addition of velocities (W = v + w), we get the following magnitude
for W: W = 226,000.007 km/s. Not so close.

Thus, the answer to Einstein’s
ultimate question concerning W is that his relativistic formula does better
describe the result of the Fizeau experiment and the equation which Fizeau
found than does the classical addition of velocities equation. From the above comparison, Einstein asserted
the “conclusiveness of the [Fizeau] experiment as a crucial test in favor of
the theory of relativity…”[39] (*Id.*, p. 46) But, as previously mentioned, the magnitudes
for Fizeau’s equation and Einstein’s formula are __not exactly__ the
same…Einstein’s magnitude for W is __slightly less__ than Fizeau’s at the
very slow velocity of 7 meters/second for the water.

Let us now see if the closeness of
the magnitudes between Einstein’s formula and Fizeau’s equation holds for __all
possible velocities__ of the water. If
we substitute the velocity of 1% of *c* for the velocity of the water
(instead of 7 m/s), we get:

Einstein’s formula W = 227,287.7655 km/s

Fizeau’s equation W = 227,297.4666 km/s

At this increased velocity of the water, the magnitude of W (the velocity of the light in the moving water) in Fizeau’s equation is somewhat more (about 9.7 km/s) than in Einstein’s formula.

On the other hand, if we substitute the velocity of 50% of *c* for
the velocity of the water, we get:

Einstein’s formula W = 273,123.4866 km/s

Fizeau’s equation W = 290,873.3333 km/s

(see Chart 29.4) At this greatly increased velocity of the water, the magnitude of W in Fizeau’s equation is significantly more (about 17,750 km/s) than in Einstein’s formula.

At the velocity of 1% of *c*
for the water, Einstein’s relativistic formula produces a velocity W, which is
approximately 9.7 km/s less than Fizeau’s equation. However, at the velocity of 50% of *c*
for the water, Einstein’s relativistic formula produces a velocity W, which is
approximately 17,750 km/s less than Fizeau’s equation. At the velocity of 50% of *c* for the
water, the close correlation completely disappears. Then, at the velocity of 99% of *c* for
the water, the difference between the two equations becomes tremendous (about
55,738 km/s). (see Figure 29.5)

Thus, the apparent closeness of the result between Einstein’s formula and
Fizeau’s equation only holds for velocities of the water, which are __very low__
relative to *c*. Contrary to
Einstein’s aforementioned assertion, there is __nothing conclusive__ about
“the [Fizeau] experiment as a crucial test in favor of the theory of
relativity.” (Einstein, *Relativity*,
p. 46) The 1851 Experiment of Fizeau is
in no way an experimental confirmation for the validity of Special Relativity,
the Lorentz transformations, or Einstein’s relativistic composition of
velocities. (also see Pavlovic, Sections
13, 14, 19 & 20, with regard to similar conclusions)

In fact, Einstein’s Chapter 13 and the Experiment of Fizeau theoretically
demonstrate just the opposite. When
Einstein fudged his equation for the relativistic composition of velocities so
as to make such relativistic velocities __agree exactly__ with Fizeau’s
equation (Einstein, *Relativity*, p. 46, F.N. 1), he was in effect
asserting the __identity__ of his relativistic equation for the composition
of velocities with Fizeau’s and Fresnel’s linear equation for velocities. But as we see on Chart 29.4 and Figure 29.5, Fizeau’s
and Fresnel’s equation, when applied to very high velocities of the water,
produces linear velocities for light in the medium of moving water well in
excess of *c*.[40] How can Einstein’s assertion of identity
between his relativistic formula for the addition of velocities and Fizeau’s
and Fresnel’s linear equation for velocities be reconciled with Einstein’s
kinematic conclusion concerning the limiting velocity of matter at *c*? (see Chapter 29A) Einstein’s Chapter 13 was, in effect, nothing
more than an exercise in self-contradiction.

[1] The
measurement of the velocity of light __in__ the first reference frame was of
course measured by a (two way) to and fro propagation of a light ray between a
light source, a mirror, and a detector, all of which were relatively
stationary. (see Figure 29.1A)

[2] It is likely that Einstein and most scientists have never even read Maxwell’s original papers which contain his theories. Rather they began their analysis with Maxwell’s equations which did not disclose Maxwell’s theories, and which equations were repeatedly modified by Helmholtz, Heavyside, Hertz and Lorentz.

[3] The __transmission__
velocity of light may also be described as a constant velocity of *c* __relative to its medium__: the vacuum of empty space.

[4] Strangely enough, the above so-called paradox also applies to material bodies. For example: ‘If the speed of a carriage is 100 km/h measured in a particular inertial frame of reference, then it cannot also be the same number (100 km/h) relative to another inertial reference frame.’ (see Figures 7.1, 21.6 and Rohrlich, p. 52)

[5] As the reader can readily see, the above paradox was largely one of semantics.

[6] This
false concept, of course, had nothing to do with Maxwell’s theories nor his
equations, because Einstein was measuring the velocity of a light ray at *c* relative to a material reference
frame, rather than the correct concept:
relative to the light ray’s medium of a vacuum.

[7] Before
June 1905, Einstein “intended to do __without__ the universally constant
velocity of light, inherent in [Maxwell’s] theory. The velocity of light was to be constant only
for an observer stationed next to the light source, whereas all observers
moving __relative to__ that source would measure a different value,
depending on their own relative velocity with regard to the source.” (Folsing, p. 172; see Chapter 21) Little did Einstein realize that this concept
(which he discarded) was closer to reality than the impossible concepts which
he finally adopted.

[8] See the real empirical reason in the next preceding above paragraph.

[9] Such problems only needed to be correctly understood.

[10]
Einstein most likely realized all of these composition of velocities
possibilities (and their possible empirical implications) when he first
discovered Lorentz’s transformations in Lorentz’s April 1904 treatise. Why didn’t Lorentz’s treatise contain all of
these composition of velocity formulae?
Because they “would have had limited significance for him.” (Goldberg, p. 101) Lorentz’s April 1904
treatise was not concerned with the velocity of light *per se*. It was primarily
attempting “to account for all ether drift experiments by assuming…the Lorentz
transformations.” (*Id*.*Id*.

[11]
Einstein’s new relativistic formula for two material velocities was
symmetrical, because the x – x' axis for the direction of uniform motion is
known as the “axis of symmetry.”
(French, p. 125) For the
derivation of Einstein’s relativistic transformations of velocities which are
perpendicular or __transverse__ to the direction of relative motion of S and
S', see Resnick, 1968, pp. 82 - 83.

[12] Feynman
agreed with Einstein and conjectured:
“The ‘summing’ of two velocities, is not just the algebraic sum of two
velocities (we know that it cannot be or we __get in trouble__), but is
‘corrected’ by 1 + *uv/c*^{2}.” (Feynman, 1963, p. 16-5) “In __reality__, ‘half’ and ‘half’ does
not make ‘one,’ it makes only ‘4/5.’” (*Id*.

[13] Because
such composition of velocities is completely based on faulty reasoning, a false
paradox, and a false premise derived *ad
hoc* from the empirically invalid Lorentz transformations, such relativistic
algebraic result conjectured by Einstein is also empirically meaningless.

[14]
Einstein’s relativistic formula for the composition of two material velocities
does not result in a physical contraction of length or a real dilation of
time. (see Chapter 28) Yet such composition of velocities is
ubiquitously used by scientists to determine such contractions and
dilations. Are these *ad hoc* relativistic processes consistent
with empirical physics? Are they
circular? Do they mean anything?

[15] Lorentz
had also reached a similar mathematical conclusion in his 1904 transformation
treatise, where he conjectured: “the
only restriction as regards the velocity will be that it be less than that of
light.” (Lorentz, 1904 [Dover, 1952, p.
13]) The fact that Einstein copied Lorentz’s
*ad hoc* conclusions does not enhance
their empirical validity.

[16] In
fact, the terrestrial limiting velocity of a particle of matter may be far less
than *c*.

[17] “Thus
[conjectured Feynman] a sidewise velocity is no longer *uy*', but *uy*'* *√1 – *u*^{2}/*c*^{2}.” (Feynman, 1963, p. 16-6)

[18] In this
regard, Resnick conjectured that: “If we
choose a frame in which u_{x}' = 0, however, then the transverse
results become u_{z} = u_{z}'√1 – v^{2}/*c*^{2} and u_{y} = u_{y}'√1
– v^{2}/*c*^{2}. But no length contraction is involved for transverse
space intervals, so what is the origin of the √1 – v^{2}/*c*^{2} factor? We need only point out that velocity, being a
ratio of length interval to time interval, involves the time coordinate too, so
that time dilation is involved. Indeed,
this special case of the transverse velocity transformation is a direct
time-dilation effect.” (Resnick, 1968,
p. 83)

[19] “There will also be a transformation for velocities in
the y-direction, or __for any angle__; these can be worked out as
needed.” (Feynman, 1963, p. 16-5) See Chapter 30 for Einstein’s relativistic
Doppler effect, his relativistic aberration of starlight effect, etc, when the
relativistic transformation of velocities is along different angles.

[20]
Apparently there is no end to the games that mathematicians can play with
Einstein’s relativistic equations. The
only problem is that such relativistic equations are all *ad hoc*, based on false assumptions, and empirically invalid.

[21] In § 4 of his Special Theory, Einstein asserted that: “For velocities greater than that of light our deliberations become meaningless…” (Einstein, 1905d [Dover, 1952, p. 48])

[22] Feynman
conjectured: “If something is moving at
the speed of light inside the [space] ship, it will appear to be moving at the
speed of light from the point of view of the man on the ground too! This is good, for it is, in fact, what the
Einstein theory of relativity was __designed to do__ in the first
place…” (Feynman, 1963, p. 16-5) On the contrary, the man on the Earth moving
at v relative to the space ship, would theoretically measure the transmission
velocity of the light ray at c propagating inside the space ship to be c ± v
relative to him. In other words, a
relative velocity. (see Chapter 21 and
the Preamble) Einstein’s theory of
relativity was certainly not designed to measure a __relative velocity__ of
light.

[23] There
are really __two__ velocities of light:
its absolute transmission velocity of *c en vacuo*, and its relative propagation velocities of *c* ± v with respect to reference bodies
(frames). The constant transmission
velocity of light at *c* *en vacuo* is an inherent property of all EM
radiation, and without such constant velocity it would not be the same
phenomenon.

[24] The real justifications and empirical verifications for this phenomenon and for the second part of Einstein’s second postulate are again described in Chapter 22. (also see the last footnote)

[25] This statement contradicts Einstein. Einstein claimed that all of his relativistic formulas were derived from his two postulates, not the other way around.

[26] By implication, Resnick was also claiming that such formula is a mathematical confirmation for Einstein’s entire Special Theory, including the Lorentz transformations. More circular reasoning.

[27] Rohrlich went through all kinds of illogical and incorrect mathematical gyrations in order to attempt to prove the validity of Einstein’s relativistic compositions of velocities. (Rohrlich, p. 71) Unfortunately, this unscientific process of attempting to confirm Special Relativity at all costs, and regardless of logic and commonsense, is currently endemic within the scientific community.

[28] This
scenario, where the linear motion of an observer is artificially negated by
decree, is on its face an obviously __invalid__ concept.

[29] Here Rohrlich is attempting to have the unnecessary and invalid end justify the invalid means.

[30] This statement acknowledges that Einstein’s relativistic formulae for the compositions of velocities only results if one assumes the validity of the first part of Einstein’s invalid second postulate.

[31] On the contrary, we know from Chapters 9 – 12 the real reasons for the M & M paradoxes, and that Einstein’s relativistic composition of velocities had nothing to do with Michelson’s paradoxical null results.

[32] On the
contrary, the absolute transmission velocity of light at *c* in a vacuum has nothing to do with any relativistic formulae for
the composition of velocities. It is
merely an inherent and invariant property of light in a vacuum. Many of Einstein’s followers also cite
Michelson’s null results as experimental confirmation for various aspects of
Special Relativity. (see Resnick, 1968,
p. 37)

[33] Many of
Einstein’s followers later claimed that such contraction was not physical, but
rather was only a result of the relativistic way that relative motion is
measured. (see Chapter 28) But how can mere illusions of measurement
result in the physical contractions in Michelson’s experiments that are *a
priori* necessary for the reciprocal decreased time intervals for the light
rays to propagate at *c*? It
cannot. On the other hand, other
followers such as Rohrlich claimed that Length Contractions and Time Dilations
were no doubt __real effects__. (see
Rohrlich, pp. 68 – 71)

[34] The
question remains: If Einstein’s *ad hoc* transformations of velocities are
based on a false premise and are empirically invalid, where does that leave his
relativistic applications of such velocity transformations? The answer is: Nowhere!

[35]
However, as we shall soon discover, Einstein’s relativistic composition of
velocities formulas had absolutely __nothing to do__ with Fizeau’s
bewildering 1851 experiments, and vice-versa.
Not only is Einstein’s relativistic formula __not__ in full accord
with Fizeau’s result, it is certainly not an explanation of its results. Rather, the correct empirical explanation is
described in detail in Chapter 7.

[36] The
agreement was actually within __one percent__ of Zeeman’s experimental
results. (see Einstein, *Relativity*,
p. 46) The reason for this close
agreement was most likely because Zeeman used Special Relativity and the
Lorentz transformations in arriving at his results. (see Zhang, pp. 211 – 212)

[37]
However, it turns out that the agreement is __not__ very exact. Also, Einstein never explained __why__
Fizeau’s paradoxical result occurs. On
the other hand, the author’s empirical quantum explanation of ‘why,’ is set
forth in Chapter 7.

[38] Here,
Einstein is progressively changing his relativistic equation so that it will be
__identical__ to Fresnel’s and Fizeau’s.

[39] Many
scientists agreed with Einstein: “that
the mysterious empirical formula [of Fresnel and Fizeau] is a __direct result__
of the theory of relativity.” (For
example, see Gamow, 1961, p. 164.)

[40] If
instead of moving water we refer to high energy moving particles, there the __identity__
of Einstein’s formula with Fizeau’s formula can have some theoretical meaning
even in the twenty-first century. But it
is not a welcome meaning for relativists, nor for particle physics.