PHANTOM TANSFORMATIONS Kinohito KULOTSUKI PHANTOM TANSFORMATIONS |

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.
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.
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. ∂u 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/a
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. f＝md x’＝x－vt
(4) We
transform (4) to (5), and substitute x for (3), so get (6). x＝x’＋vt
(5) f＝md 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’＝x－vt, y’＝y, z’＝z, t’＝t (7) □＝(1/a We transform first formula of (7) to x＝x’＋vt. It becomes x＝x’＋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/a ＝(1/a ∂x’/∂t＝－v, ∂t’/∂t＝1
(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/a ＝(1/a (1/a We get (12) by (7) as preparing to check ∂ ∂x’/∂x＝1, ∂t’/∂x＝0
(12) ∂ ＝{(∂/∂x’)×1＋(∂/∂t’)×0} ＝∂ Well we get (14) by (7). The formulae (15) are natural.
The formulae (16) hold good by owing to (14) and (15). ∂y’/∂y＝1, ∂z’/∂z＝1
(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 T □＝□’＋(v T
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/p＝v and q/s＝－v ). And we get (21)～(23) coming from (20). x’＝px＋qt, y’＝ y, z’＝ z, t’＝rx＋st (19) x’＝p(x－v t),
y’＝ y, z’＝ z, t’＝rx＋pt (20) (1/c －∂ －∂ We summarize these calculating results, and get (24). If we would introduce a hypothesis which
makes appearing tail terms T □＝□’＋T T ＋{ p p＝1／(1－v When we substitute (25) for (20), we get the Lorentz
transformations (26). x’=γ(x－vt), y’＝y, z’＝z, t’= γ(t－vx/c Because the Lorentz transformations are gotten with
making Lorentz tail T
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’＝x－vt (27) ∂/∂t’＝∂/∂t, ∂/∂z’＝∂/∂z, ∂/∂y’＝∂/∂y, ∂/∂x’＝∂/∂x (28) □＝□’
(29) Where, the Galilean tail T The difference may be in a priority between t’＝t and x’＝x－vt. On an analysis of (11), ∂/∂t’＝∂/∂t coming from t’＝t was ignored, and (1/a 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 x＝x’＋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. u＝f(x ,y), x＝g(s, t), y＝h(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). x＝f (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 x＝f (x’ ,t’), so I get (34). x＝f (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). x＝x’＋vt’, x’＝x－vt, t’＝t (35) I will make (35) be in order by substituting x’＝x－vt and t’＝t for x＝x’＋vt’. x＝x－vt＋vt＝x
(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 T 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(x－v t), t’＝rx＋pt (37) x＝p x＝p ＝x－v p ＝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.
[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.126～129. (Japanese book) [3] Taizo MUTA, IWNAMI
lectures, “Electromagnetic Dynamics”, Chapter 7, Section 7-1, c) The Lorentz
transformations, pp.129～130. (Japanese book) |

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