# Chapter 29

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 [Dover, 1952, p. 39])  This 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/c2 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 – v2/c2, the factor √1 – v2/c2 (along with certain dubious interpretations) automatically changes the magnitude of the distance traveled depending on the magnitude of v2.  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.)  However, if the angle of such relative velocity is not 90º, then a new formula for the relativistic composition of material velocities along this angle must be computed with the assistance of trigonometry. [19]

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 ax = dux/dt and ax' = dux'/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.  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 must not change that speed; it cannot subtract or add to it.[28]  Speeds do not add or subtract as numbers do.  This point…was necessary in order to account for the Michelson-Morley result.[29]  (Id.)

“[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 – 1n2), 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.)  This empirical velocity w is, of course, a known quantity, and is used to determine the Index of Refraction.  (see Figure 7.7)  The velocity of the liquid in Fizeau’s tubes (v) was also known to be approximately 7 meters/second.  Einstein then asked what was the velocity W of the light in the liquid with respect to the tube T, “when the liquid…is flowing through the tube with a velocity v”?  (Id.)  Fizeau empirically observed in 1851 that the velocity W was greater (by 44% of the velocity of the liquid) in the direction of the flow, and that velocity W was less (by 44% of the velocity of the liquid) against the direction of the flow.  (Gamow, 1961, p. 164)

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/n2), 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/n2), where n = c/w is the index of refraction of the liquid.  On the other hand, owing to the smallness of vw/c2 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/c2), or to the same order of approximation by w + v(1 – 1/n2), 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 – v2/c2.  (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/n2), 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/n2), 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/c2) 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)

[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.)  “The question of what the velocity of light would be did not arise in the Lorentz analysis.”  (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/c2.”  (Feynman, 1963, p. 16-5)  “In reality, ‘half’ and ‘half’ does not make ‘one,’ it makes only ‘4/5.’”  (Id.)  On the contrary, Feynman’s ‘reality’ is of course a false reality, and he does “get in trouble” by making these incorrect conjectures.

[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 – u2/c2.”  (Feynman, 1963, p. 16-6)

[18] In this regard, Resnick conjectured that:  “If we choose a frame in which ux' = 0, however, then the transverse results become uz = uz'√1 – v2/c2 and uy = uy'√1 – v2/c2.  But no length contraction is involved for transverse space intervals, so what is the origin of the √1 – v2/c2 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.