PHANTOM TANSFORMATIONS
Kinohito KULOTSUKI
PHANTOM TANSFORMATIONS

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  ABSTRACT

  There is a history on the Lorentz transformations. The Galilean transformations create surplus terms for d’Alembertian but the Lorentz transformations do not. This is the reason of the finish on the history of the Lorentz transformations. But I noticed there was a contradiction on the analysis of the relation between the Galilean transformations and d’Alembertian. If I would make the contradiction disappear, then the Galilean transformations should have no surplus terms for d’Alembertian. So the Lorentz transformations shall lose any reason for existence.

1. PREFACE

Lorentz did not create independently two ideas constructing the Lorentz transformations. There was a history of some forerunners creating different formulae on the Lorentz transformations [1]. I have thought that the idea of local time started from Poincaré (1900). However the formula, as meaning the time changing as local time, had been created yet by Voigt (1887). And there are several variations of Lorentz (1895), Larmor (1897), Lorentz (1899), Larmor (1900), Poincaré (1900), Lorentz (1904) which follows to Poincaré (1905) and reaches to Einstein (1905) [1]. It seems that the changes from Voigt (1887) to Lorentz (1904) are as like as trials and errors. Though, the formulae of Poincaré (1905) and Einstein (1905) do not have any symptom of after changing. On recent several texts [3], it is explained that the style of the Lorentz transformations was determined by the background as follow. For it, d’Alembertian relates deeply.

 2. D’ALEMBERTIAN

Someone explains that d’Alembertian is the extension on the 4-dimensions for the Laplacian, this may not be a reason of creating d’Alembertian. The forerunner of d’Alembertian existed as a form of the 3-D wave equation (1) before d’Alembertian as an operator. Where, “a” is velocity of the moving wave. If we would treat light or electromagnetic wave, then “a” should be changed by “c” of light velocity.

   ∂u2/∂t2=a2(∂u2/∂x2∂u2/∂y2∂u2/∂z2)                             (1)

D’Alembertian () is defined as (2) with transforming (1). If a function would locate in the position of u, then the function is considered as a wave function.

   □u[(1/a2)2/∂t22/∂x22/∂y22/∂z2]u0                     (2)

 3. GALILEAN INVARIANCE

When the directions of force, velocity and acceleration are limited along x axis, the Newton equation of motion should be (3). And now the Galilean transformation becomes (4). Where m is mass of a body, f is force, and v is velocity of the moving coordinate system for the stationary coordinate system.

   fmd2x/dt2                                                     (3)

   x’xvt                                                       (4)

  We transform (4) to (5), and substitute x for (3), so get (6).

   xx’vt                                                       (5)

   fmd2(x’vt)/dt2m(d/dt){dx’/dtv}md2x’/dt2                   (6)

Comparing (3) with (6), we notice that x changes x’. The Newton equation of motion remains its form between the coordinate value x before the Galilean transformation and the coordinate value x’ after one. When such situation is here, it means “the Newton equation of motion has the Galilean invariance”. Or it is expressed as “the Newton equation of motion is Galilean invariant”.

Generally d’Alembertian treats 4 variables of (t, x, y, z), so we consider the Galilean transformations as next form (7). D’Alembertian as an operator is a formula (8).

x’xvt,   y’y,   z’z,   t’t                                (7)

□=(1/a2)2/∂t22/∂x22/∂y22/∂z2                             (8)

We transform first formula of (7) to xx’vt. It becomes xx’vt’ by t’t. And we can see x as a function of f (x’, t’) having two variables of x’ and t’. So we will operate f as (9) by a partial differentiation. About ∂x’/∂t and ∂t’/∂t, we get the formulae (10) by (7).

(1/a2)(2/∂t2)[f](1/a2)(∂/∂t)(∂/∂t)[f]

(1/a2)(∂/∂t) {(∂x’/∂t) (∂/∂x’)(∂t’/∂t) (∂/∂t’)}[f]         (9)

∂x’/∂t=-v,   ∂t’/∂t1                                          (10)

We substitute (10) for (9), and make it be in order, so get (11) by deleting the function f. Where, and after here, let the terms multiplied after a double-byte be in the denominator.

   (1/a2)(2/∂t2)[f](1/a2) {v (∂/∂x’)(∂/∂t’)} {v (∂/∂x’)(∂/∂t’)}[f]

(1/a2) {v2 (2/∂x’2)2v(2∂x’∂t’)(2/∂t’2)}[f]

(1/a2)(2/∂t2)(v2/a2) (2/∂x’2)(2v /a2) (2∂x’∂t’)(1/a2) (2/∂t’2)     (11)

 We get (12) by (7) as preparing to check 2/∂x2. We get (13) by (12).

∂x’/∂x1,   ∂t’/∂x0                                            (12)

2/∂x2{(∂/∂x’) (∂x’/∂x)(∂/∂t’) (∂t’/∂x)}2

{(∂/∂x’)×1(∂/∂t’)×0}2

2/∂x’2                                                   (13)

Well we get (14) by (7). The formulae (15) are natural. The formulae (16) hold good by owing to (14) and (15).

∂y’/∂y1,   ∂z’/∂z1                                            (14)

∂y’/∂y’1,   ∂z’/∂z’1                                           (15)

∂/∂y∂/∂y’,   ∂/∂z∂/∂z’                                         (16)

We substitute (11), (13) and (16) for (8), and get (17) [2]. Where, ’ is d’Alembertian on new coordinate values after the transformations. Let us call TG “the Galilean tail of d’Alembertian”. Because TG appears, one declares “d’Alembertian is not Galilean invariant”.

□=□(v2/a2) (2/∂x’2)(2v /a2) (2∂x’∂t’)                        (17)

TG(v2/a2) (2/∂x’2)(2v /a2) (2∂x’∂t’)                            (18)

4. DETERMINING OF THE LORENTZ TRANSFORMATIONS

I note the starting idea of the Lorentz transformations as (19). Where, p, q, r, s are unknown coefficients. We can transform (19) to (20) with other conditions and known velocity v (as considering the velocities of coordinate origins O and O, and getting q/pv and q/s=-v ). And we get (21)(23) coming from (20).

   x’pxqt,  y’ y,  z’ z,  t’rxst                           (19)

   x’p(xv t),  y’ y,  z’ z,  t’rxpt                         (20)

   (1/c2)(∂2/∂t2) (1/c2){p2v2 (∂2/∂x’2)2p2v(∂2∂x’∂t’)p2(∂2/∂t’2)}       (21)

   -2/∂x2=-{p2 (∂2/∂x’ 2) 2pr(∂2∂x’∂t’)r2 (∂2/∂t’ 2)}               (22)

   -2/∂y2 =-2/∂y’2,   2/∂z2 =-2/∂z’2                          (23)

 We summarize these calculating results, and get (24). If we would introduce a hypothesis which makes appearing tail terms TL be identically zero, then we could determine p and r as (25).

   □=□TL

TL{(v2/c2)p2p2+1}(∂2/∂x’ 2) 2p{ pv /c2r }(∂2∂x’∂t’)

{ p2/c21/c2r2}(∂2/∂t’2)                                   (24)

   p1(1v2/c2)1/2,   r=-v{c2 (1v2/c2)1/2 }                    (25)

When we substitute (25) for (20), we get the Lorentz transformations (26).

x’=γ(xvt),   y’y,   z’z,   t’= γ(tvx/c2),   γ=1/(1v2/c2)1/2    (26)

Because the Lorentz transformations are gotten with making Lorentz tail TL on d’Alembertian be identically zero, of course it is natural that “d’Alembertian is Lorentz invariant”. Therefore, the Lorentz transformations make new coordinate’s d’Alembertian be invariant, and ensure the world where Maxwell electromagnetic theory, relating d’Alembertan logically, holds good. One has thought that this is the reason on determining last form of the Lorentz transformations.

5. PHANTOM TAILS

Well I try to make an order of the Galilean transformations (7) be reverse as (27). We can lead next operators (28) on (27). We substitute (28) for d’Alembertian (8), and get (29).

  t’t,   z’z,   y’y,   x’xvt                               (27)

/∂t’/∂t,   ∂/∂z’/∂z,   ∂/∂y’/∂y,   ∂/∂x’/∂x              (28)

□=□                                                        (29)

Where, the Galilean tail TG on d’Alembertian disappears. What happen?

The difference may be in a priority between t’t and x’xvt. On an analysis of (11), /∂t’/∂t coming from t’t was ignored, and (1/a2)2/∂t2 created the Galilean tail TG on d’Alembertian. Why is ∂/∂t’/∂t ignored? There should be no mathematical reason which have to give priority on “x’xvt” but “t’t”. If the analyzing results would be different whether we would give priority on t’t or on “x’xvt”, then the analyzing itself had to have mathematical contradiction. The problem may be out of the priority. Where is the problem?

So the exit of the maze must be only here. It is a part which we can see x as a function of f (x’, t’) having two variables of x’ and t’, coming from xx’vt’. Is this description right? The reason leading (9) may be in the formulae of differential calculus on a composite function. I note most fitting formulae about our case in several patterns.

   uf(x ,y), xg(s, t), yh(s, t)                                   (30)

   u/∂s(∂u/∂x)(∂x/∂s)(∂u/∂y)(∂y/∂s)                             (31)

   u/∂t(∂u/∂x)(∂x/∂t)(∂u/∂y)(∂y/∂t)                             (32)

 I compare the condition written before (9) with (30), and note (33).

   xf (x’ ,t’), x’g(x, t), t’h (x, t)                                  (33)

 I may be able to treat h(x, t) of t’ function as x term coefficient being zero. So it is out of problem that I think t’h(t). But how can I treat g(x, t) of x’ function? Where, x is included in g(x,t). I substitute x’g(x,t) and t’h(t) for xf (x’ ,t’), so I get (34).

   xf (g(x,t) , h(t))                                              (34)

 I do not make (34) be in order about x. Because x is included in both right hand side and left hand side, I must make (34) be in order, and I must lead the formula not having x in right hand side. I will do this concretely. The conditions leading (9) with the Galilean transformations are (35).

   xx’vt’,   x’xvt,   t’t                                 (35)

 I will make (35) be in order by substituting x’xvt and t’t for xx’vt’.

   xxvtvtx                                              (36)

Function-like f(x’, t’) became x. This clearly should not be two variables function. Any function representing x with independent two variables has not created completely. When the Galilean tail on d’Alembertian was created, the formula (9) came from the formula (7) of the Galilean transformations. I will recompose the process of (7)(9) in detail. Like this, I will express logical flow as (#)(#) and logical breaking off as (#)(#). Then, I can express detail process of (7)(9) as follows with the signs of above formulae. 

   (7)(35)(34)(30)(32)(9)

 The process of (7)(9) with considering x as a function of two variables x’ and t’, since there was a gap of logical breaking off between (34) and (30), was perfect illusion.

We have thought that once we treat d’Alembertian with the Galilean transformations then d’Alembertian (’) after transformation have the Galilean tail TG as surplus terms against d’Alembertian () having nothing before transformation. But the Galilean tail TG has not existed. It is just a phantom tail.

Then, why must we create something like the Lorentz transformations? Since the Galilean tail on d’Alembertian yet does not exist!

We can apply above consideration to the management from (20) to (21) and (22) on the Lorentz transformations. Here also it seems that x is as a function of two variables x’ and t’. But x’ includes x, so the formula of x has not be in order, therefore we cannot think x as a function of x’ and t’. And it seems that t is as a function of two variables x’ and t’, though t’ includes t, so the formula of t has not be in order, it also is not right. Therefore even if we would suppose (20), then we could not directly lead the formulae of (21) and (22). We get (37) coming from (20) without the terms of y and z. We get (38) with asking x and t on (37). For example, we substitute (37) and t formula of (38) for x formula of (38), and make it be in order about x, so it becomes (40).

   x’p(xv t),   t’rxpt                                      (37)

   xp1x’v t,   tp1(t’ rx)                                 (38)

   xp1p(xv t)v p1(t’ rx)                                  (39)

    xv p1(t’ rx)v p1(t’ rx)

    x                                                          (40)

 Same event happened as what did on the Galilean transformations. Any real composite function has not been constructed in the formulae (20) supposed for creating the Lorentz transformations. Therefore we cannot continue the logic to the differential formulae of the composite functions as like as (21) and (22).

These considerations insist that not only the tail on d’Alembertian for the Lorentz transformations is mere illusion but also the Lorentz transformations themselves are not real entity. Namely the Lorentz transformations are perfect illusion as like as just phantom.

 REFERENCES

[1] History of the Lorentz transformations, on Wikipedia

http://en.wikipedia.org/wiki/History_of_Lorentz_transformations

[2] Taizo MUTA, IWNAMI lectures, “Electromagnetic Dynamics”, Chapter 7, Section 7-1, b) The Galilean Transformations, pp.126129. (Japanese book)

[3] Taizo MUTA, IWNAMI lectures, “Electromagnetic Dynamics”, Chapter 7, Section 7-1, c) The Lorentz transformations, pp.129130. (Japanese book)

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