Analysis and Criticism to Lorentz’s Transform

(Author: Xia Xingguang)

[Abstract:]This article pointed out emphatically that the formula for Lorentz’s  transform is not the final result due to the neglecting of prerequisite for exis ting of the formula itself。What is left finally is the formula of Transform only  for time，after taking this prerequisite into account. Meanwhile, the author put  his finger on that the presupposition of Lorentz’s transform, i.e. the same int erval is a direct conclusion from objectivity principle based on dialectics, inst ead of the mathematical tistimony on format logic. The tistimony of Einstein is o nly a simple logical repetition.

I    foreword

      It’s well known that the mathematic foundation of the Einstein’s special theory of relativity is the classical Lorentz’s transform. There would be no the special theories of relativity if there was no Lorentz’s transform, it may be said. Simply thinking from this view, Lorentz had made indelible contribution to the special theory of relativity. But then Lorentz never deemed himself as the discoverer of the special theory of relativity, with which he could not gave the space-time concept of absolute space and time up for his whole life. However, it appeared that space and time is relative according to the theory result of Lorentz’s transform. Well then to the end, is space and time relative or absolute? Possibly are they dialectic instead of neither relative or absolute? Correct answer can be getting radically as long as physical foundation of the Lorentz’s transform can be shown clearly.

II    The classical Lorentz’s transform

      For the sake of illuminating mistake existing in the Lorentz’s transform, it is needed to cite the classical Lorentz’s transform to explain. Refers to following figure:

    The classical Lorentz’s transform points out: We shall get the transform formula of relative theory, which is nicely based on fixedness of interval between the event. If we want to make it easier to describe utilization quantity τ= ict , well then it just is as what that the interval between two events shown in §1-2 can be considered as the distance between two world points corresponded in four-dimensional space. Thus we can say that the required Transforms must be the Transform that makes all the distances in four-dimensional space x，y，z，τ no change, which include merely parallel movement and rotation of reference frame whereas. Thereinto we have no interest in that the coordinate axis is parallel moved himself because this only makes the original point of space coordinate move once and change the reference point of time once. So the required Transforms should be mathematically expressed as rotation of four degree reference frame x，y，z，τ. All rotations in four-dimensional space can be divided into six rotations separately in six planes xy，yz，zx，xτ，τy，τz ( just as that all the rotations in three space can be divided into what is in three planes xy，yz，zx ). Thereinto the former three rotations merely transform space coordinates, which is quite as the usual space rotation. When we study the rotation in xτ, coordinate y and z is not changed. Takingψ as rotation angle, then the relation between present and original coordinate are decided by following two formulae:

        x = x’conψ –τ’sinψ，τ= x’sinψ +τ’conψ                        （1）

    Now what we want to discover is the formula which transforms one inertia reference system K into the other K’, with which K’ moves relating to K along axis X at the velocity V. In this circumstance, apparently only the space coordinate x and time coordinateτare changed. Thus this Transform must have the format of formula (1). Now what is left is to decide the rotation angle ψ which has merely something to do with the relative velocity. Let’s come to study the movement in K made by the original point of coordinate in reference system K’. At the moment, x’ = 0，and formula (1) can be written as:

    x = –τ’sinψ; τ=τ’conψ.                 (2)

Got by division

    x/τ= - tanψ                                                     (3)

We knowτ= ict，and x/t is apparently the velocity V of K’ to K. Thus，

                tanψ = iV/c                                                         (4)

Educed from it

                sinψ= (iV/c)/(1-V²/c²)^1/2，cosψ=1/(1-V²/c²)^1/2                   (5)

Put it into (2)， obtained：

                x = (x’ - iVτ’)/(1-V²/c²)^1/2，y = y’，z = z’，

                τ= (τ’ + iVx’/c)/(1-V²/c²)^1/2                                (6)

Then putτ= ict，τ’ = ict’ inside，finally elicited

                x = (x’ + Vt’)/(1-V²/c²)^1/2，y = y’， z = z’，

                t = (t’+ Vx’/c2)/(1-V²/c²)^1/2                                (7)

        These namely are the required Transform formulae, which are called as Lorentz’s transform formula and is the foundation of debate in future. (Refer to [1] of pages 14-15)

      As it is aware this groups of connection formulae is namely the well-known “Lorentz’s transform formulae” which are also the mathematical foundation of Einstein’s special theory of relativity. Indeed according to these connection formulae, what can only be obtained are the time coordinates（r’）and space coordinates（t’） in movement system. “Lorentz’s constriction” can be produced during movement. ----It just is this conclusion that directly affected time space concept of modern physics and thence conduced the subjective idealism thought a broadly popularized in natural science.

III    Physical Genius of Lorentz’s transform

      As it is described above, the mathematical foundation of special theory of relativity is formula (7). But we have to point out seriously that Einstein ignored the presupposition of educing this formula when he used this formula, which was a big mistake! Perhaps Lorentz himself made this mistake. Anyhow no matter whose fault was there is no excuse to bring this mistake into the 21cetury any longer!

      Why saying Lorentz himself made mistake in Lorentz’s transform is because he, at the result of this transform, ignored the presupposition under which the Transform formula (7) comes into existence. Originally at the beginning of this Transform, Lorentz pointed out clearly: “we come to study the movement of coordinate original point （O’）of reference system K’ in reference system K. At this time, x’ = 0.”----Since what we study is the movement of the coordinate original point (O’) of K’ in K, not only there is x’ = 0，but also there is y’ = 0，z’ = 0。Speaking for it, these connections are all the presuppositions for existence of formula (7). As long as taking these presuppositions for existing of connection formulae into account, formula (7) can be corrected further as this:

     x = Vt’/(1-V²/c²)^1/2                                                 (8)

And,

                t = t’/(1-V²/c²)^1/2                                        (9)

      When we pay attention to V = x/t again, formula (7) can also melted into formula (9). And then we can educe what the final result of Lorentz’s transform Formula is:

    t = t’/(1- V²/c²)^1/2                                                 (10)

      This shows that the former connection formulae of Lorentz’s transform formula (7) are somewhat superabundance!!

      Since only the above formula is the final result of Lorentz’s transform, what is the physical foundation of Lorentz’s transform withal? We come to make this point out underside. In previous figure, ifτ is not considered as the “imaginary number axis” first, the figure at once shows “Transform of rotating axis” under the coordinates of real number. Thus following can be simply educed.

      At once from the above figure according to “theorem of draw section string”:

      OM₂² = Ox² + Oτ²                                                 (11)

And,

                OM₂² = Ox’² + Oτ’²                                    (12)

In figure, r=OM₂. Therefore,

                r² = Ox² + Oτ²                                                 (13)

Also,

                r² = Ox’² + Oτ’²                                        (14)

The presupposition here is    Oτ= ict，Oτ’ = ict’, so we got,

Oτ² = - c²t²                                                     (15)

And,

                Oτ’² = - c²t’²                                   (16)

      We then put formula (15) and (16) into formula (13) and (14) respectively, and can obtain:

      r² = Ox² - c²t²             (17)

And,

r² = Ox’² - c²t’²                                                  (18)

      By introducing L² = Ox² and L’² = Ox’², then the above formula can be corrected as:

r² + c²t² = L²                                                     (19)

  And,

r² + c²t’² = L’²                                                     (20)

If the both sides of formulae (19) and (20) are detracted by 2r²，we educe:

c²t² - r² = L² - 2r²                                                 (21)

And,

                c²t’² - r² = L’² -2r²                                        (22)

Then by introducing

           Δs² = L² - 2r²                                                     (23)

And,

                Δs’² = L’² - 2r²                                      (24)

Thus formulae (21) and (22) can be corrected respectively as :

Δs² = c²t² - r²                                                  (25)

And,
                Δs’² = c²t’² - r²                                    (26)

      It is not difficult to find out that formulae (25) and (26) have the similar mathematical formats with the “interval square” defined by special theory of relativity, with which only the distance square in formula (25) is r² instead of r’ ². However there should be r’ ≡r because we have observed that r’ and r are the different expressions of space distance which is formatted by the two relative movement between the moving observer and the resting one.

      In the above figure there isΔs² ≠Δs’² owing to c²t² ≠c²t’² . This shows that it is unauthentic in all circumstances for the Einstein conclusion of “interval equality” proved by him according to the principle of fixedness for light velocity!

      Supposing Δs’² = 0, then c²t’² - r² = 0. ThusΔs’² is namely what the interval square relative to “resting clock” (t)obtained by using of “moving clock” (t’) by observer standing on the event of light velocity movement, and moreoverΔs² the interval square relative to himself movement of “movement clock” recorded by the use of “resting clock” by still observer. Although the movement distance (r) is the same physical quantity for the observers (k’ and k) standing at the two ends of the movement distance, i.e. r’= r, the clocks they used are definitely different. This shows that Δs and Δs’ represents merely two observers moving relatively, relativity of result observed to the same movement event . ----This is the point of new consideration to the concept of relative theory.

      Based on the above understanding, we can compel these apparently unequal “interval square” equal, i.e. Δs² =Δs’²。For convenience of understanding, we take here the movement distance described by the relatively moving observer as mark of r’, the movement distance in the eyes of relatively still observer r. And we observe that r≡r’, then there is c²t² - r² = c²t’² - r’². Saying from the significance of relativity, movement distance r’= 0 when observer standing on the movement event takes himself located position as the coordinate original point. So it is educed:

    c²t² - r² = c²t’²                                                 (27)

      This is just about what the “negative negativity” rule of dialect is concretely applied in physics.

      Now we divide the two sides of the above formula by c²t², and use V² = r²/t² to represent the relative velocity square of movement clock educed by resting clock（t）by still observer, and then can obtain directly:

      t = t’/(1 – V²/c²)^1/2                                          (28)

      Thus it can be seen that the physical foundation of Lorentz’s transform is therefore the objectivity principle instead of the alleged light velocity fixedness one.

IV Dummy Space and Dummy Velocity

      Now let’s go back to have a look at formulae (21) and (22). Supposing that two observers are overlapped at one position, movement clock（t’）and resting clock（t）are just turned out to be “combining two in one”, and the relative movement distance between two observers r = 0，there out these are educed:

    ct = L ； ct’ = L’                                                 (29)

      Noticing that at the moment t’ = t，so there is that L’ = L。Here we define L’ and L as “dummy space”.-----Its physical meaning is, when time running out, supposing that movement event ( or particle ) moves at the absolute velocity (c )of light , the dummy “ space distance” which could be went by the event during the time interval of t’ and t.
(A) Here one extremity is the relative movement at the velocity of light c between two observers. At the moment we can educe from formulae (21) and (22):

    L²/t² = c² + V² ； L’²/t’² = 2c²                                     (30)

      Thereinto, c = r’/t’；V = r/t。If we define ɡ’ = L’/t’ and ɡ = L/t as the “dummy velocity” in movement system and still system respectively we can obtain then

                             (31)

And，

        ɡ = (c² + V²)^1/2                                                  (32)

    This result shows that the observer standing in the event of movement can obtains what his max dummy velocity （ɡ’max）is times of light velocity c; and the observer still standing relatively outside the movement event can obtains that his dummy velocity（ɡ）increases along with increasing of event ( or particle) relative velocity V.

    (B) Here the other extremity is that K’ and K is overlapped. At the moment there it is not existed for relative movement of two observers, and the entirely same then for two dummy velocities, i.e. ɡ’= ɡ = c. As a result only the running out of time is authentic and the movement dummy. The appearance of the connection of dummy space and dummy velocity is happened indivisibly together with the so said instances. Here why the max. dummy velocity （ɡ’max）is equal to c is because we suppose that the embranchment along axis X of dummy velocity is equal to the absolute velocity c of light. So it is educed that the max. Value of dummy velocity is the function value when the nip angle between dummy velocity and axis X is 45 degree.

      We say that the ultimate reason of “dummy space” and “dummy velocity” rests with which there is substantial difference between “space connection formed by physical movement” and “living space connection”. Geometry is merely the mathematics abstract of the living space connection, and geometry abstract based on that Transform velocity of reciprocity is infinite with the condition of principle of super distance function. Therefore it is said that geometry of living space connection cannot unconditionally reflect the various kind of space connections formed during the process of practical movement by substance. However from the very natal moment our physics at once take mathematics (geometry) as the tool of physical presentation, with which such a mathematical tool is not at all wholly consistent with practical physics matter possibly.

V Origin of Lorentz’s Constriction

      In classical Lorentz’s transform (refer to [1] of pages 16-17), it is supposed that there is a still stick parallel with axis X in system K , and the measured length in system K isΔx = x₂ – x₁（x₂ and x₁ are coordinates of two ends of the stick in system K）. Now we are seeking the length of the stick in system K. For this purpose it is necessary for us to find out coordinates x₂’ and x₁’ of two ends of the stick in system K’ at the same instant t₂’. It is educed from formula (7)

x₁= (x₁’+ Vt’)/(1 – V²/c²)^1/2                                         (33)

And   

                x₂= (x₂’ + Vt’)/(1 – V²/c²)^1/2                                         (34)
The length of the stick in system K’ isΔx’ = x₂’ – x₁’；It is obtained by subtracting x₁ from x₂
Δx = Δx’/(1 – V²/c²)^1/2                                             (35)
“Inhesion length” of the stick is the length that it is in the reference frame which is relatively still to it. Let ι₀ =Δx represent this inhesion length, and ι the length in all other reference frames K’. Then,
ι=ι₀ (1 – V²/c²)^1/2                                             (36)
Therefore one stick is the longest in the reference frame K which it is relatively still to it. The length could at once be shortened to (1 – V²/c²)^1/2 times in the reference frames K’ moving with a velocity V . Relative theory calls this result as the Lorentz’s constriction.

      Apparently the parlance “to find out coordinates x₂’ and x₁’ of two ends of the stick in system K’ at the same instant” itself have serious logic falsehood. It is because the presupposition of relative theory is that time concept of absolute simultaneity doesn’t exist already. Therefore as observer standing in relatively resting coordinate system（K）it is impossible for him to stand in the coordinate system（K’）relatively moving at the same instant as well; Vice versa, i.e. the observer standing in the coordinate system（K’）relatively moving is impossible to stand in relatively resting coordinate system （K） at the same instant. If the observer in the system（K）could really find out coordinates x₂’ and x₁’ of two ends of the stick in system K’ at “the same instant”, betwixt of K and K’ is surely the time concept that has not broken away from the absolute simultaneity!

      As it is well known, special theory of relativity call it as the “inhesion time” of the object which time is shown by clock moving together with some given object system （K’）. By simulating the aforesaid process it can be written out the mutual connection among the inhesion time interval （Δt’）and the time interval （Δt）corresponding to the still system（K）:
Δt = Δt’/(1 – V²/c²)^1/2                                             (37)
In fact whether a stick with a length ofιo or other physical quantity , coordinates（x₁ and x₂）of two ends can be educed by the means of supposing that their coordinates at the original point of coordinates in system K’ are x₁’=0，x₂’ =0. Therefore attentions are only needed to pay to here:
    Δt = (t₂ –t₁)，Δt’= (t₂’ –t₁’)，Δt =Δt’/(1–V²/c²)^1/2
And，
                υx = Vx/(1 – V²/c²)^1/2，Vx = x/t =Δx/Δt = dx/dt，
So formulae (35) and (37) are just well the formulae (8) and (10) suffice the presupposition of Lorentz’s transform.
Indeed if according to formulae (36) and (37), the ruler（ι0）in moving system could be shortened and the inhesion time of clock slowed down.. Here the question is what is the through cause to make the ruler shortened and clock slowed down? In order to answer this question, we use the classical Lorentz’s transform (7) and definition of “four degree string section” to deduce reversibly. Result can at once obtained from formula (7);
x² + y² + z² +τ² = x’² + y’² + z’² +τ’²                                 (38)
Apparently the above said formula don’t represent that time space ruler degree shrink between two different coordinate systems with “not with dash” and ‘with dash”. Hence it is said that the real cause for Lorentz’s constriction don’t come from the relative movement in coordinate system K’.
Honestly as what has said above, we ever introduceτ= ict andτ’ = ict’ in Lorentz’s transform. Here we don’t consider at first the presupposition of x’ = 0, y’ = 0, z’ = 0; and y = z = 0, which makes Lorentz’s transform formula (7) come into existence. Instead absolute value |r| and |r’| of three degree radial arrow are applied to represent the three degree space coordinate of the above-mentioned formula which can at once be rewritten as follows by putting the value represented byτandτ’ in it:
c²t² – r² = c²t’² – r’²                                             (39)
Apparently this connection formula is namely the “interval equality” testified by Einstein and also the presupposition for special theory of relativity coming into existence.
The sound fact shows that there is no presupposition of interval equality no Lorentz’s transform, and also no Lorentz’s transform no Einstein’s special theory of relativity. However just as what we have afore-testified: not only “interval equality” testified by the fixedness principle of light velocity is fallacious in all circumstances, but also it is ultimately impossible for the conclusion about “interval equality” to be testified by mathematical deduction, with which can only be explained by physical facts. Therefore it is said that the physical facts used to explain “interval equality” can but be “objectivity principle” and moreover impossible to be “fixedness principle of light velocity”. Thus it can be seen that the so-called Lorentz’s constriction can only be from a simply logic repetition instead of the mathematical attestation of format logic!!
We shall not forget that the contents of objectivity principle are which object has impersonality content not depending on main body. ----This is a “integration proposition” under the dialectic logic. Thus it is said that “Lorentz’s constriction” testified by objectivity principle can only represent the relativity of result which “main body” observes “object”, instead of “time and space are all not impersonality fact but perceptual intuition format of human”. It is determined there out that there is no existence at all for Lorentz’s constriction in concept of relativism, and any viewpoint believing time and space could produce Lorentz’s constriction during movement is the thoughts of one hundred percent subjective idealism!

VI     Conclusion

      Based on the afore-said debate one conclusion can be obtained: Because it ignores presupposition of the formula itself coming into existence classical Lorentz’s transform is thus not the final Transform result. The final Transform result obtained is merely a Transform connection for time coordinate after the presupposition of the formula itself coming into existence is observed. By compared with broad sense relative theory of time space the result of Lorentz’s transform is proved merely as a straight deduction of objectivity principle. Therefore it is said that the physical foundation of Lorentz’s transform is objectivity principle instead of fixedness principle of light velocity. It is ultimately impossible to obtain the conclusion of Lorentz’s constriction from the final Transform formula. Why the special theory of relativity educes this absurdity conclusion, saying from impersonality, is because that Einstein failed to get rid of the time concept of absolute simultaneity; saying from subjectivity, that Einstein was deeply affected by thoughts of Mach’s relativism and did thus not know to use the epistemology of dialectic materialism to treat this matter.

Reference literature:

[1]《Theory of Field》written by Л.Л.ландау,Е.М.лифшиц, interpreted by Ren lang、Yuan Bingnan, by the People’s Education Publishing House, Beijing, Firstly issue, Aug. 1959.

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