19. ADDITION OF SPEEDS
 
19.1 Addition of speeds in a vacuum
 
   The addition of speeds as Einstein presents it, goes against human experience and reason. Accepting this way of addition would mean rejecting all that has been learnt and affirmed about addition throughout the centuries.
   In order to understand the problem of addition it is important to see what Einstein said about it [6].
   Quotation: "Let a railway wagon be moving along a track at a constant speed . Let a man walk along the wagon at a speed in the direction of the wagon's motion. By which speed relatively to the railway embankment is the man moving during his walk? It seems that is only one possible answer results from this way of thinking:
   If the man stopped after one second, he would, relative to the embankment, have moved forward for a certain distance which is equal to the speed of the wagon. Actually, relative to the wagon, that is, in relation to the embankment, he would also have traveled forward by a pace the distance , which corresponds to the speed of his walk. Thus, relative to the embankment, in the given second the man travels in all the distance

(19.1)

Later on, we will see that this way of thinking which is in accordance with classical mechanics, expresses the addition theorem, cannot be retained, and that this law, we had just now written, does not represent the truth". End of quotation.
   As can be seen from the quotation, Einstein has a different attitude to the addition of speeds even with the simplest and most obvious forms of motion.
   Further on Einstein says:
   Quotation: "Instead of the man walking in the wagon, we will introduce a point which relatively to the coordinate system will move according to equation

(19.2)

   From the first and forth equation of the Galilean transformation and can be expressed by means of and , so we obtain

(19.3)

   This equation expresses nothing but the law of point motion relative to the system (a man relative to the railway embankment) which we will mark as , so we have

(A)

(19.4)

   This considered case we can likewise thoroughly study also on the base of the theory of relativity. Then in equation

we should express and by means of and using the first and fourth equation of the Lorentz transformation. [As we said, according to the second principle of the theory of special relativity have to be that is . Because of that in equation have to be at all events. Beside that, the Lorentz transformation has been derived by using corresponding equations for a case of light propagation, but not for a case of mechanical motion. Therefore, this transformation could not be derived, at all, by using equation of mechanical motion, and have nothing in common with mechanical motion. The exception is only spherical acoustic wave. Therefore that equations of Lorentz transformation can not be applied on mechanical motion, except in case of spherical wave motion, where instead of the speed of light, the speed of sound should be taken. So, it should always be born in mind that Lorentz transformed coordinates refer to the coordinates of the light wave position or a ray in the coordinate systems and , and by no means to an arbitrary position of a point in these systems. (Remark M.P.).] Then we obtain, instead of Eq. (A), equation

(B)

(19.5)

which, according to the theory of relativity, corresponds to the theorem on addition of speeds having the same direction. The question now is, which of these two theorems corresponds to experience. In this context we learn something from a very important test performed half a century ago by the genius physicist Fizeau but which was later repeated by some of the best physicists experimentalist, so that result of the test is unquestionable." End of quotation.
   In the passage quoted a shorter procedure of derivation of Eq. (B) about addition of speeds is given. Considering the great importance of this equation it is necessary for the sake of clarity to present the whole procedure.
   The first and fourth equation of the Lorentz transformation, where and are expressed by means of and , as we know are

   Using these equations and the Eq. (19.2) given in the previous quotation

we obtain

and from there finally

(19.6)

where is a sum of a speeds.
   According to Einstein, the sum of speeds can not be higher than the speed of light in vacuum. For example, if we take that , and also that , then according to the Eq. (B), that is (19.6), their sum is

(19.7)

which is contrary to everyday experience. That it is so, we can check and see in the following example.
   Let a light pulse of short duration be sent to a mirror formed by two adjoining sides of a cube. The mirror of that shape divides the light pulse into two parts and two light pulses are created. In this way they are directed in two opposite directions. In one second each of these two pulses will travel 300000 km. Bearing in mind that they move in opposite directions the distance between them will be 600000 km. From this it certainly follows that they went away from each other at the speed of 600000 km/s, i.e. their relative speed was 600000 km/s. In other words, the sum of their speeds was 600000 km/s, and not 300000 km/s as Einstein claims in his equation for addition of speeds.
   In a similar way, we can show the falsehood of Einstein's claim that the subtraction of the speed of light and some other speed equals the speed of light.
   Einstein's equations for addition and subtraction of speeds can be derived in a different, simpler way from which it becomes evident what they really represent.
   Eq. (19.6) is obtained by direct division of the first by the forth equation of the Lorentz or some other transformations as follows

(19.8)

because , and also .
   The difference of the speeds

(19.9)

is also obtained by direct division of the first by the fourth equation of the Lorentz or some other transformations, but under the condition that and are given as a function of and .

(19.10)

   Let us analyze Eq. (19.6) and try to find out what it really represents. Let's start from the beginning.
   Lorentz derived the transformation of coordinates for the case of spherical light wave motion along the -axis in the two inertial systems and , where the system moves translatory at a speed along the -axis and without acceleration relatively to . For that, he starts from conditions and . Consequently his first and fourth equation are valid only under such conditions. On the basis of this condition, the principle of the constancy of the speed of light, the special theory of relativity was derived. Because of that, it is always and only

(19.11)

and

(19.12)

so it is also always and only

(19.13)

and

(19.14)

   This is for the case for the Lorentz transformation and transformation No. 5 which gives the same equation for the addition and subtraction of speeds.
   In case of transformation No. 2 we derived the following equation for the addition of speeds

(19.15)

and for the speed subtraction

(19.16)

   If in Eqs. (19.15) and (19.16) we make substitution we obtain

(19.17)

   The form of the equation for the addition speeds and for the subtraction speeds in the case of transformation No. 4, considerably differs from the previous. So, in case of the addition of speeds

(19.18)

and in case of the subtraction of speeds

(19.19)

But here, as well, by substituting we obtain the same, that is

(19.20)

and

(19.21)

   Thus, for different transformations there may be different equations for the addition of speeds, and for the subtraction of speeds but the result of the sum and difference must always be the same and equal to the speed of light.
   At the end we can conclude as follows. Einstein's equation about the addition of speeds, is really about the velocity of propagation of a light wave along the -axis in the system . That velocity of the light wave propagation is expressed by means of coordinates and and velocity of the system . The sum of the addition of speeds cannot be higher than the velocity of light no matter how high is, and has to be equal to the velocity of light only, since under that condition it is derived by means of the Lorentz transformation. Many people, without any justification, have used this equation as a proof that the velocity of light is the highest possible velocity in nature. On the basis of this equation they assert that even relative speed cannot be higher than the velocity of light.
   Einstein's equation describing the subtraction of speeds is really about the velocity of light wave propagation along the -axis in a moving coordinate system which is expressed by means of the coordinates and and velocity . The difference of speeds given by the equation about subtraction of speeds always is also equal to the velocity of light no matter how high speed is, because it is derived under the same condition as the previous. Let us repeat that this condition in fact is the condition that the velocity of light in both coordinate systems and has to be the same and equal to for the case of vacuum.
   Einstein's inconsistency and the weakness of the theory of relativity can also be seen in the case of the theorem of addition of speeds.
   As we know, according to that theorem, when adding and subtracting the speed of light with any other speed the result equals the speed of light. If this is true then it is inexplicable why Einstein wrote in his first paper on relativity [2], in which he derived the Theorem on addition of speeds, in the third formula

   With these two formulas, at the very beginning, Einstein refuted his Theorem on the addition of speeds in the course of its derivation. For, if what the theorem claims were true, then it would be senseless to use the expressions and , when in their place only should be taken. However, that cannot be done, because then it would be , which is not true and which would make Einstein's treatment of the relativity of length and time interval absurd.
   To this point we have been discussing light propagation in a vacuum, because it was for those conditions the Lorentz and other transformations were derived.
 
19.2 Addition of speeds in water
 
   How would the Lorentz and other transformations, as well as other equations for the addition and subtraction of speeds look in the case of some other environment? It is clear that if the transformations are to be derived, the other new environment will have to be homogenous and isotropic too.
   Let us suppose that the new environment is water. Let both inertial systems be in water, so the light wave and the coordinate system move through water. In order to be valid the Lorentz transformation would have to be and where and are the coordinates of the light wave position along the and -axes in the system and respectively and is the velocity of light in water. In order to exist an invariability of the equation for the light propagation in water it is indispensibly to be

(19.22)

In that case the first and fourth equation of the Lorentz transformation solved for and would have the form

(19.23)

Dividing with we obtain

(19.24)

If we make substitution in Eqs. (19.23) and (19.24)

then we have for the addition of speeds

(19.25)

and for subtraction of speeds

(19.26)