(PDF) How Einstein and Minkowski missed but could have discovered superluminal Lorentz transformations and six-dimensional spacetime. Possible relation to quantum theory and consciousness

!

1!

How Einstein and Minkowski missed but could have discovered

superluminal Lorentz transformations and six-dimensional spacetime.

Possible relation to quantum theory and consciousness.

Jan Pilotti

a

)

Affiliated to Institute for Foundational Studies Hermann Minkowski, Montreal Canada https://minkowskiinstitute.com/

Abstract: The history and rationale for the discovery of superluminal Lorentz transformations

are first briefly provided. The analysis of how Einstein, Minkowski and others did not find the

possibility of v>c shows that we must be cautious in our assumptions not to implicitly exclude

v>c. It is also shown how they, especially Minkowski, could have discovered superluminal LT

and a six-dimensional spacetime with three space and three timelike dimensions, which allows

real-valued superluminal LT. Is this just a mathematical possibility? Does this contradict the

causality order? Is this related to entanglement? As no objective physical becoming seems

possible in Minkowski 4D spacetime, because all events of spacetime exist equally, this seems

to be an apparently insurmountable contradiction between the physical theory of relativity (and

the experiments that support it) and our everyday experience of change and the flow of time.

Can this be better handled with more timelike dimensions? Is there a relation to Quantum

theory and consciousness?

Keywords Einstein’s mistake, superluminal Lorentz transformations, six-dimensional spacetime, causality,

entanglement, becoming, quantum theory, consciousness

Note: Great parts of the article published 2020 as a conference report

1

in Spacetime 1909 – 2019, REF 1. In

contents these parts marked with R1

Contents

I. Introduction 3

A. The birth of the dogma [R1] 3

II. Experimental search for tachyons (R1) 5

III. Superluminal systems? (R1) 5

A. Rindler’s derivation of LT 1960 (R1 minor clarifications) 6

B. Superluminal Lorentz transformations in 2D. And 6D spacetime? (R1) 7

C. Superluminal frames 1922 and 1969-1981using

!"#$%&'&!"$

(R1+ minor adds) 8

D. Why do not most derivations of LT also give SLT for (

)

(

*+

? (R1) 10

IV. Derivations of Lorentz transformations missing (

)

(

*+

? 10

a

) pilotti.jan@gmail.com, Sweden

!

2!

A. Light clock (R1) 10

B. Einstein 1905 (Major new analysis and clarification of R1), 11

C. Einstein 1907 15

D. Einstein 1912(-1914) 17

E. Minkowski 1907-1909 (R1 minor clarification) 18

V. Mathematical interlude Euclidian vs hyperbolic rotation (R1) 21

VI. How Minkowski could have found (

,

(

*-

and 6D spacetime 23

A. Minkowski’s “rotation” derivation of LT (ajor clarification of R1) 23

B. Minkowski’s geometrical derivation of LT (major clarification of R1) 25

VII. Superluminality and six dimensional spacetime

- just a mathematical possibility? 28

A. Causality paradoxes? 28

B. “Spooky action at distance”, entanglement and superluminality 30

C. But why six dimensions? 31

Appendix A Landau-Lifshitz derivation of LT 1971 part one (R1) 31

Appendix B Alternative to Einstein’s derivation 1907 33

Appendix C More derivations of Lorentz transformations missing (

)

(

*+

(R1) 34

C.1. Cunningham 1914 34

C.2 Max Born 1920 36

C.3. Pauli 1920 37

C.4. Synge 1955/1963 37

C.5 Landau and Lifshitz 1971 part two 39

C.6. Rindler’s later derivations, 1977, 2006 39

C.7. Two types of errors in derivations of LT excluding (

)

(

*+&&&

41

Appendix D Proof of a part of Minkowski’s argument in Space and time 1908. 41

Appendix E Weyl’s proposal about consciousness and relativity 43

E.1 The state of the art in research on consciousness 44

Appendix F A mathematical conjecture (R1) 47

Appendix G Work in progress

Is becoming possible in six-dimensional Minkowskian spacetime? 48

References to Appendix G 50-51

Appendix H

Tachyons and tachyonian systems. Generalization of SR and an idea of six-dimensional space-time.

Pilotti J unpublished notes Jan1971 51

Acknowledgment 57

References to main article and Appendices A-E 57

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3!

I. INTRODUCTION

The dogma that nothing can have a velocity higher than that of light has for over hundred years

been a restrain to physics theories. And still when experimental violation of Bell’s inequality

shows that local realism is unattainable many chose non-realism instead of non-locality.

First, I will briefly remind about the birth of Einstein’s dogma and later critique leading to

speculations of tachyons. But more importantly show that Einstein’s postulates are not only

compatible with but correctly handled also shows and suggest superluminal Lorentz

transformations in two, four and six dimensional spacetime. This result shows that those

derivations which not find this possibility must make some mistake. I have in an earlier work

[Ref 1] analyzed the derivations made by Einstein 1905, Born, Pauli, Landau and Lifshitz and

Rindler 1977, 2006

b

showing that they make implicit assumptions which excludes

superluminality. And also showed that derivations by Cunningham, Synge, Rindler 1960,

Landau and Lifshitz, Bornb and Minkowski missed superluminality by not fully using the

mathematical possibility in their derivations. In this work I will extend, clarify and make some

correction of my analysis of Einstein’s derivation 1905 and also analyze his derivations 1907

and 1912. I will also include the analysis of Minkowski as he was the first and closest to find

superluminality and six-dimensional spacetime, and in this work this is shown in more detail.

Though I have searched rather extensively I have not seen any analysis showing Einstein’s

mistake in missing v>c. If I missed that I am grateful for any references. But I think the so

strong dogma that superluminality is impossible have blocked the view for seeing the implicate

assumptions excluding superluminality. But armed with the fact that also the superluminal

Lorentz transformations follows from Einstein’ postulates, rightly handled, this knowledge

made it easier to find the mistakes. Therefore, I here include the rationale and a simple

derivation of superluminal Lorentz transformation (SLT) and also a short history of the

discovery of SLT.

A. The birth of the dogma

Before Einstein scholars discussed superluminal particles, as e.g. Arnold Sommerfeld who

published a paper “Ueber Lichtgeschwindigkeits und Ueber-Lichtgeschwindigkeits-

Elektronen”

2

just a couple of months before Albert Einstein’s seminal paper 1905.

3

As is well known from two basic postulates,

!

!See!appendix!C!

!

4!

P I The laws of physics have the same form in all inertial reference frames,

P II The velocity of light in vacuum c is the same whether the ray be emitted by a stationary or

by a mowing body.

Einstein derived the Lorentz transformation

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"#$! "%&'

(

)%&*

+*!,!!!!!!!'#$'%&"

+*

(

)%&*

+*,!!!-#$-,!!!!.#$./!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0)1

!

He also derived for an electron accelerated in an electrical field, the kinetic energy W

!!!!!!!2$

3

456"$7

3

89

:

;&6&$7+*

<

=

>

)

(

)%&*

+*%)

?

@

A

!,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0B1

and concluded “Thus, when v=c, W becomes infinite. Velocities greater than that of light

have - as in our previous results

c

- no possibility of existence.” (see Ref. 3, pp.63-64). He also

argued that the expression for kinetic energy must apply also for ponderable masses.

Strangely enough it took almost 60 years

d

,

4

5

,

6

before Bilaniuk et al.

7

clarified that Einstein’s

conclusion is not fully correct. Einstein had derived Eq. (2) for an accelerated electron/

ponderable masses. Thus, it only follows that it is impossible to accelerate a material object to

v=c and even more to C

&

C

D+

. However, acceleration is not the only means to achieve speed.

Light does not start with low velocity and accelerate to achieve v=c. Light is “born” with v=c.

Thus, the postulates Einstein uses do not lead to a theory that can exclude phenomena, which

from start have velocities higher than that of light. It was speculated that there could be

particles with C

&

C

D+

and Feinberg

8

dubbed them “tachyons” from tachy=fast, and he also

discussed possible features even in a quantum theory for tachyons.

c

The previous result mentioned is in Ref. 3 (p.48), when Einstein discussing length contraction with .

/0)$1+$

and states “ For velocities greater than that of light our deliberations become meaningless; we shall, however, find

in what follows, that velocity of light in our theory plays the part, physically of an infinitely great velocity.”

1907 Einstein also used the argument that existence of the spreading of an effect with v>c could change the order

of cause and effect in different systems, see Sec. VII A.

d

With very few exceptions such as Strum 1923 (Ref. 4, see also Ref. 5) discussing faster-than-light processes and

Somigliana 1922 (Ref. 6) discussing superluminal frames see Sec. III C. Tolman discussed causality problem for

v>c 1917 [Ref. 47 pp. 59-60]

!

5!

II. EXPERIMENTAL SEARCH FOR TACHYONS

The first experimental search for tachyons was made in mid 1960: ties by Alväger et al. at the

Nobel Institute for Physics in Stockholm

9

,

10

and also published in 1968 with Kreisler.

11

No

direct evidence for tachyons was found and have not yet been found, even if it is argued that

some indirect experimental indication of superluminal phenomena exists.

12

The first approach (see Refs. 7, 8, 11 also Ref. 13

13

) was to start from the ordinary E=mc2

even for C

&

C

D+

, assuming that tachyons have an imaginary rest mass

7;$EF;!!F!;GH!,

and

derive

I$ 7;+*

(

)%&*

+*$7;+*

E

(

&*

+*%)$EF;+*

E

(

&*

+*%)$F;+*

(

&*

+*%)!,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0J1

thus, E real valued for C

&

C

D+

, and possible to

measure.

As ordinary matter always must have C

&

C

K+

so tachyons always must have C

&

C

D+

according to this formula and then tachyon rest

mass is not a measurable quantity and can be

imaginary.

Possible, but a bit ad hoc?

III. SUPERLUMINAL SYSTEMS?

There is an approach that seems to me

14

to be more in the spirit of Einstein’s postulate that all

IS are equally good. If we have a group of tachyons with the same velocity

&

L with C

&

L

!

C

D+,

these can be thought of as a new IS, albeit moving very fast, as from start Einstein’s postulates

say nothing about a limit for the relative velocity of IS, and the postulates can verbatim be the

same. In their own system tachyons are assumed to be at rest and thus ought to have real-valued

coordinates. And if at all these tachyons shall exist in our ordinary systems it seems plausible

that they must have real-valued coordinates also in our systems. Therefore, it seems plausible

that there should exist real-valued transformations between subluminal and superluminal

systems. And mathematically these can be derived as follows [Ref 14, here see Sec III B and

Sec. III C.]

!

6!

A. Rindler’s derivation of LT 1960

Wolfgang Rindler

15

,

16

[Ref.15, pp.16-21; Ref. 16, pp.14-18]

looks at any event

M!

and

!N!OPQRSTUVWQOR!PXPOY

!Z

whose

coordinates differ from those of

M!

by (dx, dy, dz, dt) resp. (dx’, dy’, dz’, dt’) in two different

systems, S and S’, in standard configuration.

From PII in the equivalent formulation as

PII’ The velocity of light in vacuum, c, is the same in all inertial frames

for the two events

M![\6!Z

one gets

!!!!!!!!!!!!!!!!!6"*]6-*]6.*%+*6'*$^!_6"#*]6-#*]6.#*%+*6'#*$^!

, (4)

as this means both events are on the same light beam. (Eq. (4) also derived by Einstein 1907

17

see Sec. IV C.). From this and that, (if transformations between coordinates are differentiable),

the transformation for differentials at any fixed

M

are linear and homogenous (as always)

Rindler derives as “it can easily be shown” [ Ref. 15, pp. 16-17, 21; Ref.16, pp. 15-16] that

!!!!!!!!!!!!!!!!!!!!!!!!!!!!6"*]6-*]6.*%+*6'*$`

a

6"#*]6-#*]6.#*%+*6'#*

b

,!!!!!!!!!!!!!!!!!!!!0c1

where K is independent of the differentials. (Eq. (5) also derived by Einstein 1912-1914

18

see

Sec. IV D. And by Cunningham 1914

19

, though instead of K with

d

0

",-,.,'

1

,

which is

commented on in Ref 1. pp. 127-128 and here in appendix C.1)

This also shows that the transformations must be linear [Ref. 15, pp.17,74; Ref. 16, pp.14-

15,73] thus, the finite coordinate differences satisfy the same transformation equations as the

differentials.

Rindler then argues that K at

M!

is independent of the choice of standard coordinates in S and

S’ and further

“Since the orientations of the rectangular axes in S and S’ can be arbitrary for the present

argument, and since inertial frames are isotropic, the relation of S and S’ relative each other and

to the event

M

is now completely symmetric whence we must have …”[Ref. 15, p.17; see also

Ref. 16, p. 16]

!!!!!!!!!!!!!!!!!!!6"#*]6-#*]6.#*%+*6'#*$`

0

6"*]6-*]6.*%+*6'*

1

/!!!!!!!!!!!!!!!!!!!!!!!!!!!0e1

Thus

!!!!!!!!!!!!!!!!!!!!!!!6"*]6-*]6.*%+*6'*$`*

0

6"*]6-*]6.*%+*6'*

1

/!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0f1

!

7!

So

e

, K2=1 and K=±1. Rindler correctly argues that Eq. (5) must remain valid for

&g^!

and

thus K=1. But then argues that

`$%)

can be discarded. However, looking for the possibility

of systems with superluminal velocities, this argument is not valid.

B. Superluminal Lorentz transformations in 2D. And 6D spacetime?

In a hypothetical two-dimensional spacetime Rindler’s arguments with one event

M!

= (0,0) in

both S and S’ gives

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"*%+*'*$h

a

"#*%+*'#*

b

/!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0i1

As transformations are linear and if S’ moves with v in S, we have x’=a(x-vt) and t’=gx+ht.

Substituting these expressions for x’ and t’ into Eq. (8) and comparing the coefficients of x2, xt

and t2 gives a, g and h.

Using the + sign in Eq. (8) gives the ordinary LT

!!!!!!!!!!!!!!!!!!"#$h "%&'

(

)%&*

+*,!!!'#$h '%&"

+*

(

)%&*

+*,

C

&

C

K+,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0j1

where + sign is valid for standard configuration.

However, using the

k

sign in Eq. (8) gives what can be called “Superluminal LT”, SLT in 2D

!!!!!!!!!!!!!!!!"#$h "%&'

(

&*

+*%),!!!!!!!!'#$h '%&"

+*

(

&*

+*%),!!!!!!!!!!

C

&

C

D+/!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0)^1

(An ambiguity in ± signs for choosing positive sense of x’- and t’- axises as the standard

configuration has no (clear) meaning for C

&

C

D+

as S and S’ can never be at rest relative to

each other and also as C

&

C

D+

is related to space-time interchange, see Eqs.(19), (63)-(65).)

Notable that even as for Eq. (10)

!!"*%+*'*$%!

a

"#*%+*'#*

b

!

it fulfils PII’ as

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"*%+*'*$^!!!_!"#*%+*'#*$^/!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0))1

But in the reality of 4D spacetime with

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"*]-*].*%+*'*$h!

a

"#*]-#*].#*%+*'#*

b

,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0)B1

the mathematical reason for not choosing the

k

sign is the law of inertia for quadratic forms

20

,

(also called “Sylvester’s law of inertia”), which states that for a real-valued transformation the

signature, that is, the number of positive coefficients and the number of negative coefficients

are both invariant. The left member has the signature

]]]%

and using the

k

sign in the right

e

"Another"way"to"derive"K=±1"see"Appendix"A."

!

8!

member gives

%%%]

. However, it is now easy to see a possible mathematical solution: by

adding two negative coefficients so

]]]%%%!

and thus on the right side both + sign,

giving

]]]%%%!!!!

and

k!

sign giving

%%%]]]

works. Could two extra “timelike”

dimensions exist? Six-dimensional spacetime? [See Ref. 14 and Ref.1]

C. Superluminal frames 1922 and 1969-1981

f

,

21

using

lmno$!h!lmo

1922 Somigliana6 used ds’2= - ds2 with transformation in 4D to describe LT for C

&

C

D+

including also imaginary numbers, (as Pavsic 1971, see below), but from a proposed more

general transformation leaving wave equation invariant.

“1969 Parker

22

used the “…extended principle of relativity …the totality of the laws of physics

has the same form relative to both superluminal and subluminal frames.” in two dimensions

g

,

23

and from Eq. (8)

"*%+*'*$h

a

"#*%+*'#*

b in the form 0

"%+'

10

"]+'

1

$h0"#%

+'p10"#]+'1

derived both the subluminal and real-valued superluminal transformations as in

Eqs. (9) and (10), although in the diagonal form

24

. He also notes that the

%!

sign gives LT for

C

&

C

D+

and interchanges spacelike and timelike intervals. But concludes “… not appear to be

possible to extend the one-dimensional theory to three [space] dimensions.”

1971 Yet a young student, Matej Pavsic

25

used also the

k!

sign in

"*]-*].*%+*'*$h

a

"#*]-#*].#*%+*'#*

b

!!,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0)J1

and obtained the SLT for 2D Eq. (10), but now also with

!!-n$hE-!!.n$hE.!!

in accordance

with the law of inertia for quadratic forms:

"#$h "%&'

(

&*

+*%),!!!!!'#$h '%&"

+*

(

&*

+*%),!!!!!!!!-n$hE-,!!!!!.n$hE.,

C

&

C

D+, 0)q1!!!!!

and which can be called a Generalized superluminal LT in 4D, GSLT(4). (For his physical

interpretation of the imaginary numbers see his 1981 articles.

26

,

27

).!

1971 Another young student Jan Pilotti14 from

lmno$!h!lmo

derived real-valued

superluminal transformations in 2D and had an idea to use the

k!

sign in 6D to get real-valued

superluminal transformations with three space and three timelike dimensions .

1972 Recami and Mignani published the same result as Pavsic using ds’2= ± ds2 describing also

superluminal LT in four-dimensional spacetime. They describe

28

,

29

how to understand and

f

It is beyond the scope of this article to give the full review of superluminal transformations but only showing

steps in discovery of real-valued six-dimensional transformation. For a very extensive review see Ref. 21.

g

Gilson 1968, Ref. 2, derived real-valued superluminal LT in 2D via complex variables.

!

9!

handle the occurrence of imaginary numbers, as for y’ and z’ in Eq. (14). See also later works

30

and a thorough discussion and interpretation of the imaginary numbers and its relation to

increasing the number of dimensions in SLT

31

.

1976 Mignani and Recami, using an idea by Demers

32

(who actually wrote about the theory of

three chromatic colors, where he suggested expressing t = (tx, ty, tz) as auxiliary variables

which cannot be separately measured and a quadratic form in six variables where r(x, y, z) and

t(tx, ty, tz) are exchangeable), formulated real-valued LT for both subluminal and superluminal

case using a six dimensional spacetime three space, three time dimensions, which we call

Generalized superluminal LT in 6D, (GSLT(6)).

33

1977 Cole

34

showed using ds’2= ± ds2 for either four fully complex variables or six real

variables that for + and C

&

C

K+

the extra dimensions where completely uncoupled.

However, using the

!k

sign gives formulas valid for C

&

C

D+

and the extra dimensions are

necessarily coupled. Here, we present only the six-dimensional case with (x, y, z, t1, t2, t3).

!!!!!!!!!!!!!"*]-*].*%+*

0

'r*]'*

*]'9

*

1

$h!0"#*]-#*].#* %+*

0

'r#*]'*

#*]'9

#*

1

1!/!!!!!0)c1

The + sign gives a possible GLT6 for C

&

C

K+

"#$]

0%1"%&'r

(

)%&*

+*,!'!r

#$]

0%1'r%&"

+*

(

)%&*

+*,sn$s,!!!tn$t!!!!!!!NOu!!'*

#$'*!,'9

#$'9!,!!!!!0)e1!!

and thus, the important result

!!!!!!!!!!!!!!!!!!!!!!!!!!"#* ]-#* ].#*%+*'r#*$!!"*]-*].*%+*!'r*!,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0)f1

still holds.

But the – sign gives a possible GSLT(6) for C

&

C

D+,

a real-valued SLT in 6D,

!"#$vr

0

vEw\&

1

"%&'r

(

&*

+*%),!!!!!!!!'!r

#$vr

0

vEw\&

1

'r%&"

+*

(

&*

+*%)!,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0)i1!!

!

!!!!!!!!!!!!!!!!!!-#$v*+'*,!!!!!.#$v9+'9!,!!!'*

#$v*-

+,!!!!!'9

#$v9.

+!,!!!!!!!!!!!!!!!!!!0)j1

where

!vr,v*!!![\6!v9!!!

can, separately, be ±1.

Thus, ambiguity in signs and the extra dimensions are coupled in space

_

time interchange.

Again, notably that in 2D, 4D and 6D for C

&

C

K+

lmno$lmo

is invariant. But for C

&

C

D+

lmno$%!lmo

, yet SLT fulfils PII’ as ds’2= 0

_

ds2= 0.

1981 Pavsic26indicates a connection between 6D spacetime and both quantum mechanics and

general relativity. He also shows26, 27 (in another way than arguing from the law of inertia for

!

10!

quadratic forms), that when a 6D spacetime is contracted to 4D, transformations must be

complex and interpret imaginary coordinates as that events observable to one observer are not

observable to the other.

D. Why do not most derivations of LT also give SLT for C

x

C

Dy

?

As shown above Sec III A Rindler’s derivation of LT 1960 gives

"*]-*].*%+*'*$h!0"#*]-#*].#*%+*'#*1

and at least mathematically, in analogy to

the 2D case, gives the choice to describe superluminal LT in 4D using imaginary

transformations or the possible extension to 6D spacetime with three space and three “timelike”

dimensions and real-valued transformations.

Below we will show how this choice is missed in derivations of LT by Einstein 1905, 1907 and

1912 and by Minkowski 1907 and 1908, but first in derivation with a light clock which

highlights the implicit assumption.

IV. DERIVATIONS OF LORENTZ TRANSFORMATIONS MISSING

h

C

x

C

Dy

A. Light clock

A photon reflected between two mirrors is a physically very clear way to derive LT. The time

between the photon leaving a mirror and coming back, the time

between “ticks” T0, can be calculated, in the system S’ where

the light clock is at rest, knowing the distance L between the

mirrors and velocity of light c:

z;$B{|+

When the clock moves in S the photon in S has to travel a

longer path to reflect in the moving mirrors; thus, we can

directly deduce that the time between the two ticks measured

in S will be longer. We can also calculate the relation between

T0 and the time T, the time between ticks in S using the Pythagoras

theorem

i

, which gives

+*0z|B1*$&*0z|B1*]!{*!!!!!!!z*$z;*|0)%&*|+*1

and thus

!z$hz;|

}

)%&*|+*

h

I have also, in Ref.1, analysed derivations by Cunningham, Born, Pauli, Synge, Landau-Lifshitz and later Rindler

and showed how they missed (

)

(

*+

. See Appendix A and C.

i

Note that we use the same L for the distance between the mirrors both in S and S’ as it can be

shown, without using LT, that there is no change in lengths perpendicular to the direction of

movement. And we use Postulate II’ that c is invariant.

!

11!

(where of course we use + in standard configuration) and we recognize the formula for time

dilation and C

&

C

K+

as there is no meaning for imaginary time. From this the full LT can be

derived. But why not any choice for C

&

C

D+

as we got in Rindler’s 2D derivation?

If the mirrors could move with C

&

C

D+

in S the photon with v=c will never catch up the upper

mirror and will not be reflected and we will have no tick. Therefore, no real triangle to use the

Pythagoras theorem for. So, this procedure works only for C

&

C

K+

and implicitly rules out

C

&

C

D+

and cannot be used as an argument for only C

&

C

K+

. However, more important are the

derivations in Einstein’s and Minkowski’s original papers.

B. Einstein 1905

Compared with Rindler’s derivation [ Ref. 15 pp.13–21; Ref. 16, pp. 14-18], (here partly

described in Secs. III A, III B and Appendix B), Einstein’s derivation [Ref. 3, pp. 35–65,

especially pp.43–48] of LT in his original paper is more complicated and here I give no

complete analysis of the first part

j

,

35

of it. Einstein also argues

from light reflected in mirrors, where he uses light beams both

in the direction of movement between the systems and in a

perpendicular direction. So even here the light will not catch

the mirror if

&D+

and the velocity in the perpendicular

direction when viewed from the stationary system is

~

+*%&*!

which is imaginary for C

&

C

D+/

So actually Einstein implicitly

k

, without any

argument, presupposes C

&

C

K+

.Therefore, it could be thought that this approach also from

beginning excludes C

&

C

D+

. However, perhaps, surprisingly, not.

Einstein using the postulates of relativity derives formulas for the transformation between

K(x, y, z, t) and k

0•,€,•,‚!1

in the standard configuration (here I put together the formulas on

pp. 44-45 in Ref. 3)

! " #

$%&'

('

)

*%&+

('

,

-..../ " #

$%&'

('

0

+%&*

1

-...2 " #

3

$%&'

('4-.....5 " #

3

$%&'

('6.-...............................0781.

Here Einstein writes “… where

[

is a function

ƒ

0

&

1 at present unknown ...”

j

A thorough and clarifying analysis of the whole derivation, (although not touching upon the implicit assumptions

of v<c), is given by Miller Ref. 35.

k

If not allowing imaginary numbers. In 1907 and 1912 Einstein [Refs. 16,17] uses a simpler derivation without

mirrors see Secs. IV C and IV D.

!

12!

He then, without further arguments

l

, set

[$ƒ

0

&

1}

)%&*|+*!

, note again a choice with a

seeminglyk implicit assumption C

&

C

K+

, which gives

‚$ƒ

0

&

1

8

0

&

1„

'%&"

+*

…

,!!!!•$ƒ

0

&

1

8

0

&

10

"%&'

1

, €$ƒ

0

&

1

-,!!!!!•$ƒ

0

&

1

.!,!!!!!!!!!!!!0B)1

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!†SPWP!!8

0

&

1

$)

(

)%&*

+*!/!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0BB1

Einstein later finds one solution

ƒ

0

&

1

$)

giving the ordinary LT for C

&

C

K+

.

But if in (20) we set

[$%

}

&*|+*%)!

we get

‚$8#

0

&

1„

'%&"

+*

…

,!!!!•$8#

0

&

10

"%&'

1

, €$hE-,!!!!!•$hE.!,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0BJ1

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!†SPWP!\‡ˆ!8p

0

&

1

$)

(

&*

+*%)!/!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0Bq1

That is like Eq. (14) a GSLT(4) valid for C

&

C

D+

. So, Einstein’s miss of the possibility of C

&

C

D

+

must come after Eq. (20).

Einstein shows that with Eqs. (21) and (22)

•*]€*]•*$+*‚*

transforms

to

!!!!!!!"*]-*].*$+*'*

thus that the two postulates of relativity are compatible.

From Eqs. (20) Einstein could also have derived

!!!!!!!!!!!!!!!!!!!!!!!!!!!!•*]€*]•*%+*‚*$! [*

)%&*

+*0"*]-*].*%+*'*!1,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0Bc1

and comparing this with Rindler’s Eq. (5),

[*|0)%&*|+*1

is the same as Rindler’s K. Using

Rindler’s symmetry arguments

m

to derive K=±1 Einstein could have obtained

[*|0)%&*|+*1$`$h)

and derived LT and GSLT as e.g. Pavsic see Eq. (14). He could

also more directly have got the two solutions

[$h

}

)%&*|+*

which in Eq. (20) gives LT

valid for C

&

C

K+

but also

[$h

}

&*|+*%)

which in Eq. (20) gives GSLT as in Eqs. (23)

and (24), valid for C

&

C

D+

, including

n

€$hE-,!!!!!•$hE./

Instead Einstein looked at a third system K’ moving relative k with the same speed v such that

"

!Which"Miller"comments"“It seems as if he knew beforehand the correct form of the set of relativistic

transformations.” [Ref. 35 p. 200]. And further discussed in Ref. 35 pp. 202-204.!

m

"Another"way"to"derive"

2 % '/

""see"Appendix"A."

#

!Of course, Einstein could have dismissed this LT for (

)

(

*+

as it needs imaginary numbers, as in Eq. (23)

although it later [Refs. 26-31] was argued how this can be understood physically. Also see Sec.VI B where I show

how already Minkowski could have solved this with real-valued LT in six-dimensional spacetime.

!

13!

the origin of the coordinates of system K’ moves with the velocity vk = –v in k on the axis of

X.

o

Then he uses Eqs. (21) and (22) twice, first with v and then with – v and obtains

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!'#$ƒ

0

%&

1

8

0

%&

1‰

‚]&•

+*

Š

$ƒ

0

%&

1

ƒ

0

&

1

',!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0Be1

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"#$ƒ

0

%&

1

8

0

%&

1

!

0

•]&‚

1

$ƒ

0

%&

1

ƒ

0

&

1

!",!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0Bf1

!

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!-#$ƒ

0

%&

1

€$ƒ

0

%&

1

ƒ

0

&

1

-,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0Bi1

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!.#$ƒ

0

%&

1

•$ƒ

0

%&

1

ƒ

0

&

1

./!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0Bj1

Einstein then concludes:

Since the relations between x’, y,’ z’ and x, y, z do not contain t the systems K and K’

are at rest with respect to one another, and it is clear that the transformation from K to K’

must be the identical transformation. Thus φ(v)φ(−v) =1. [Ref. 3, p.47]

To find φ(v) also another equation is needed, and Einstein continues:

We now inquire into the signification of φ(v). We give our attention to that part of the axis

of Y of system k, which lies between

•$^,€$^,•$^!NOu!•$^,€$‹,•$^!

. This

part of the axis of Y is a rod moving perpendicularly to its axis with velocity v relatively to

system K. Its ends possess in K the co-ordinates

"r$&',!!!!-r$Œ

•0Ž1!,!!.r$^

and

!!!!!"*$&',!!!!-*$^!,!!!.*$^

.

The length of the rod measured in K is therefore l /φ(v)); and this gives us the meaning of

the function φ(v). From reasons of symmetry it is now evident that the length of a given

rod moving perpendicularly to its axis, measured in the stationary system, must depend

only on the velocity and not on the direction and the sense of the motion. The length of the

moving rod measured in the stationary system does not change, therefore, if v and - v are

o

NB In the English translation in [Ref. 3, p.47] a reprint from Methuen and

Company Ltd 1923, there is typo writing k instead of K’ ”… the origin of co-ordinates of

system k moves with the velocity –v on the axis of X.” See German original [Ref. 3 p. 901.]

!

14!

interchanged. Hence follows that

Œ

•0:1 $Œ

•0•:1!!

, or

ƒ

0

&

1

$ƒ

0

%&

1

!

. It follows from this

relation and the one previously found that

ƒ

0

&

1

$)/

[Ref. 3, p.47.] (Actually, it follows

ƒ

0

&

1

$h)

but just a change in direction of axis.)

Which in Eqs. (21) and (22) gives the ordinary LT for C

&

C

K+/!

But why does not Einstein also get the solutions for C

&

C

D+/

?

Equations (27), (28) and (29), relating x, y, z to x’, y’, z’, do not contain t and thus, as Einstein

states, the systems K and K’ are at rest with respect to one another. However, this gives, at

least, two

p

possibilities:

(a) that the transformation from K to K’ is the identical transformation, so φ(v)φ(−v) =1.

(b) that the transformation from K to K’ also reversed the axises, so

ƒ

0

&

1

ƒ

0

%&

1

$%)

.

If with (b) we use the other relation Einstein derived,

ƒ

0

&

1

$ƒ

0

%&

1 we get

ƒ

0

&

1

$h!E

,

which in Eqs. (21) and (22) give GLT as in (14), as

E|

}

)%&*|+*$E|0

~

%)

}

&*|+*%)!!1$

h)|

}

&*|+*%)

. One can wonder about

ƒ

0

&

1

$h!E

, but a superficial reason is Einstein’s not

motivated choice

[$ƒ

0

&

1}

)%&*|+*

where we need

ƒ

0

&

1

$h!E

to get

[$

h

}

&*|+*%)

. Had he instead chosen

[$ƒ

0

&

1}

&*|+*%)

the reasoning above have given

GSLT for

ƒ

0

&

1

$h!)

and LT for

ƒ

0

&

1

$h!E/

And that

[$h

}

&*|+*%)

also is a solution is

clearer shown if, which Einstein could have done, we use the argument with the three systems,

K, k and K’, Fig.5 and measure on length or better coordinate in Y -direction directly on Eq.

(20), thus without making any further implicit assumption.

So, from Eq. (20) now explicitly writing

[$[0&1

! " #0&1

$%&'

('

)

*%&+

('

,

-..../ " #0&1

$%&'

('

0

+%&*

1

-...2 " #0&1

3

$%&'

('4-.....5 " #0&1

3

$%&'

('6.-.......................0981

and using it twice first with v then with -v we get

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!'#$#0%&1

$% &7

(7

‰

‚]&•

+*

Š

$[

0

%&

1

[

0

&

1

$% &7

(7',!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0J)1

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"#$#

0

%&

1

$% &7

(7!

0

•]&‚

1

$[

0

%&

1

[

0

&

1

$% &7

(7!",!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

0

JB

1

$

!It"is"easy"to"see"that"real-valued"

3

4

)

5"is"not"just"a"scale"factor"but"its"sign"defines"if"axises"in"two"systems"

have"positive"direction"in"same"direction,"

3

4

)

5

*67

"or"opposite"direction,"

3

4

)

5

86

."So,"another"option"to"

get"reversed"axis"between"K"and"K’,"i.e."

3

4

)

5

3

4

0)

5

%0/7

"is"

3

4

)

5

%03

4

0)

5""which"again"gives"

3

4

)

5

%'/&9

"

!

15!

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!-#$‘0•:1

(

r•:;

<;€$‘

0

•:

1

‘

0

:

1

r•:;

<;-,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0JJ1

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!.#$#0%&1

(

$%&7

(7•$‘

0

•:

1

‘

0

:

1

$%&7

(7./!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

(34)

So, K’ and K are at rest with respect to one another. But we have the two cases K and K’

identical resp. K and K’ reversed axis so we have both

[

0

%&

1

[

0

&

1

|0$% &7:(71.

= 1 and

[

0

%&

1

[

0

&

1

|0$% &7:(71.$%)

. And using Einstein’s argument of symmetry for length or

coordinate in Y -direction we also get

[

0

&

1

|

}

$% &7:(71.!"[

0

%&

1

|

}

$% &7:(71.!

thus

[

0

&

1

$

[

0

%&

1.

!

For

[

0

%&

1

[

0

&

1

|0$% &7:(71

= 1 we get

[*

0

&

1

$$% &7:(7

so

[

0

&

1

$h

}

$% &7:(7

in Eq.

(30) giving LT for C

&

C

K+

.

For

[

0

%&

1

[

0

&

1

|0$% &7:(71.!" %$.

we get

[*

0

&

1

$&7:(7%$

so

[

0

&

1

$h

}

&7:(7%$

in

(30) giving (with ambiguity in signs as no clear meaning of the orientation of axises)

! " ;

)

*%&+

('

,

3

&'

('%$-........./ " ;

0

+%&*

1

3

&'

('%$-......2 " ;<4-........5 " ;<6...............................................................09=1

Thus, a GSLT(4) for C

&

C

D+

as in Eq. (14). Here the occurrence of

E

in 4D GSLT has a deeper

significance see Refs. 26-31, especially Ref. 31 where this is discussed at length and also

argued its relation to increase the number of dimensions in GSLT.

Einstein finds one possible solution, giving LT for C

&

C

K+

, but does not show it is the only

possible case. And Einstein’s derivation makes assumptions which implicitly excludes C

&

C

D+

and thus cannot exclude superluminal Lorentz transformations which on the contrary, as shown

here and above in Sec. III, can be derived from the basic postulates of Einstein’s’ theory of

relativity, albeit in four dimensions including imaginary transformation. How this can be

extended to real-valued six-dimensional superluminal transformation see Sec. VI B. Einstein’s

argument regarding causality, see Sec. VII A.

C. Einstein 1907 In this work [Ref. 17] Einstein uses a simpler derivation. He argues from

homogeneity of space and time and symmetry that the LT are linear and that it is possible with

a standard configuration between S’ moving with v along x axis in S. So, we have

q

x’=a(x-vt), y’=by, z’=dz. And then from Postulate II’ “Since the propagation velocity of light

in empty space is c with respect to both reference systems, the two equations

q

"I"use"z’=dz"instead"of"Einstein’s"z’=cz"for"clarity."

!

16!

"*]-*].*$+*'*

and

"p*]-p*].#* $+*'p*

must be equivalent.” And “… we conclude

after a simple calculation that the transformation equations must be of the form

'#$ƒ

0

&

1

8

„

'%&"

+*

…

,!!!"#$ƒ

0

&

1

8

0

"%&'

1

,!!!!!!-#$ƒ

0

&

1

-,!!!!.#$ƒ

0

&

1

.!,!!!!!!!!!!!

†SPWP!!8$ r

(

r•:;

<;!’/!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0Je1!!!

[Ref. 17, pp. 259-260.] (See here Appendix B)

That is same result as in 1905 see above, Sec. IV B and Eqs. (21) and (22).

And again, Einstein uses the method with a third system as in Fig.5. Now he is more specific

about that for the three systems, now called S, S’ and S’’, S’’ is oriented relative S’ in the same

way as S’ is oriented relative S. Thus

ƒ

0

&

1

![\6!!ƒ

0

%&

1 must have the same sign (see footnote

p) so reversed axis between S and S’’, i.e.

!ƒ

0

&

1

ƒ

0

%&

1

$%)

is now excluded for C

&

C

K+

,

But not so for C

&

C

D+

, as we saw above, if

ƒ

0

&

1

![\6!!ƒ

0

%&

1 are same imaginary number

]E!‡“!%E!

, that is if S’’ is oriented relative S’ in the same way as S’ is oriented relative S,

(albeit in a physically generalized meaning of relative orientation) we get

ƒ

0

&

1

ƒ

0

%&

1

$%)

that is reversed axis between S and S’’. So again, we actually have

ƒ

0

&

1

$!ƒ

0

%&

1 together

with φ(v)φ(−v) =1 resp.

ƒ

0

&

1

ƒ

0

%&

1

$%)

which give LT resp. GSLT as above. But Einstein

only use φ(v)φ(−v) =1. So, Einstein’s argument still implicitly, as shown above, exclude C

&

C

D

+

.

And as shown in Appendix B Einstein from the premises above instead of Eq. (36) equally

could have derived for F<0 (we explicitly write F=F(v) (Einstein’s

ƒ

0

&

1

$

}

>

0

&

1 )

*?"

@

%>

0

&

1

3

&'

('%$

)

*%&+

('

,

-...+?"

@

%>

0

&

1

3

&'

('%$

0

+%&*

1

-4?" ;<

@

%>

0

&

1

4- 6?" ;<

@

%>

0

&

1

6.A

0

9B

1

.

Einstein’s method with three systems K, k and K’ , Fig 5, with coordinates for the three

systems K(x,y,z,t) ,

!”0•,€,•,‚!1

, K’(x’,y’,z’,t’), gives using Eq, (37) first with v and then -v

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!'#$

}

%•0%&1

(

&*

+*%)

‰

‚]&•

+*

Š

$%

}

%•

0

%&

1}

%•

0

&

1

!',!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0Ji1

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"#$

}

%•0%&1

(

&*

+*%) !

0

•]&‚

1

$%

}

%•

0

%&

1}

%•

0

&

1

!",!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0Jj1

!

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!-#$hE

}

%•0%&1€$%

}

%•

0

%&

1}

%•

0

&

1

-,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0q^1

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!.#$hE

}

%•0%&1•$%

}

%•

0

%&

1}

%•

0

&

1

./!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

(41)

Again, it shows that system K and K’ are at rest with respect to one another and thus two

!

17!

possibilities, identical or reversed axises so

%

}

%•

0

%&

1}

%•

0

&

1

$h)

.

Which together with symmetry of length or better coordinate in Y direction, giving }

%•0&1$

}

%•0%&1

, give (as here

>

0

&

1

C81

>

0

&

1

"%$.

(

ƒ

0

&

1

$

}

•0&1$hE

) which in Eq. (37)

gives GSLT for C

&

C

D+

as in Eq.(14). (Formally also get

>

0

&

1

"$

(

ƒ

0

&

1

$

}

•0&1$)!

which in Eq. (37) gives LT for C

&

C

K+

.)

Thus, again showing that Einstein make implicit assumptions which excludes GSLT.

D. Einstein 1912(-1914)

r

In this work [Ref.18] which also is simpler than his 1905 work

Einstein also came closer to discover LT for C

&

C

D+

. From postulate PII’, invariance of the

velocity of light he gets

!"*]-*].*%+*'*$^!!_!!"#*]-#*].#*%+*'#*$^,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

0

qB

1

and thus, that LT must make the equation

!–*0"*]-*].*%+*'*$

a

"#*]-#*].#*%+*'#*

b

!!,!!!

(43)

into an identity. [Ref. 18, pp. 31-32].

This is exactly

s

Rindlers Eq. (6) with K=

–*

. So, with Rindler’s symmetry arguments Einstein

could have obtained

–*$`$h)

and thus for

–*$)

got LT for C

&

C

K+

and for

–*$%)

got GSLT(4) for C

&

C

D+

as in Appendix B.

But Einstein continued, arguing that

–*

must be independent of x, y, z, t, and writes “For now

we will examine the substitution for the case where

–*$)

and we will show later that from a

physical point of view this is the only case of deserving of consideration.” [Ref. 18, p. 32] He

then introduced u=ict and derived algebraically the ordinary LT valid for C

&

C

K+

.

Einstein then come back to “… we arbitrarily set

–*$)

. Had we not done this, we would have

obtained

"#$–

0

&

1

"%&!'

(

)%&*

+*!!!!!!!sn$–

0

&

1

s,!!!tn$–

0

&

1

t!,!!!!!!'!!#$–

0

&

1

'!%&"

+*

(

)%&*

+*!!!0qq1,!!!

For, since

–

is independent of x and t while the special Lorentz transformation is defined by v

alone,

–

can only depend on v alone.” [Ref. 18, p.38].

That is the same result as Einstein’s Eqs. (21) and (22) Sec. IV B above with

ƒ

0

&

1

$–

0

&

1.

And from here Einstein exactly follows the procedure described above, Sec. IV B to derive

%

!The exact time discussed in Ref. 18

&

!As"Eq."(5)"shows"that"the"transformations"are"linear,"so"finite"coordinate"differences"satisfy"same"

transformations"as"differentials."

!

18!

–

0

&

1

–

0

%&

1

$)

and

–

0

&

1

$–

0

%&

1 and gets

–$h)

and argues that “ The special case v=0

shows that the choice of the positive sign is the only one that is possible.” [Ref. 18 p.39]

But this argument again misses reversed axises

!!

and

–

0

&

1

–

0

%&

1

$%)

and more important

also miss

–$hE

and thus miss superluminal GSLT as shown above. Einstein seemingly

presupposes that the Eq. (42) which leads to a proportionality in Eq. (43) must have a positive

proportionality

–*

>0 which seems self-evident, but as seen in Appendix B is not the case for

C

&

C

D+

. So also, here Einstein makes implicit assumptions excluding C

&

C

D+

.

E. Minkowski 1907-1909

“Herman Minkowski presented the idea of a four-dimensional spacetime in three lectures. In

the first, November 1907, he, without further arguments, just states

36

,

37

[Ref. 37, p.82]: “I will

take the important result in advance, namely, that velocities of matter equal or greater than the

velocity of light proves itself to be an absurdity ... .” Perhaps he also had in mind Einstein’s

claim 1905 that velocities greater than that of light are not possible, but Minkowski more

correctly only refer to velocities of (ordinary accelerated) matter, which was what Einstein

really had shown. In the second lecture December 1907 Minkowski uses that the fundamental

equations for electromagnetic processes “… in vectorial notation reveals an invariance (or

rather covariance) … when the coordinate system is rotated around the origin”.

38

[Ref. 37, p.

100] One type of rotation involves only space co-ordinates and the other also involves t.

Minkowski uses the co-ordinates 0

"r!,!"*!, "9!,"—

1

$

0

"!,-!,.!,E+'

1

/t

The first type of rotation

is just an ordinary Euclidean transformation, for example, rotation of the x1 and x2 axes around

the x3-axis at an angle

ƒ

"pr$"r˜U™ƒ]!"*™QOƒ, "p*$%"r™QOƒ]"*˜U™ƒ!,!"#9, $"9!,!"#—$"—/

(45)

He then introduced a purely imaginary quantity

Eš

and considered the substitution

"r#$"r,"#*$"*!, "p9$"9˜U™Eš]!"—™QOEš!, "p—$%"9™QOEš!]!"—˜U™Eš/

(46)

But the analogy with a rotation through an “imaginary angle”, as Synge

39

says, “… must be

handled with caution, and it is better to use the fact that

˜U™Eš$˜U™Sš!!!

and

!!!™QOEš$

E™QOSš

and rewrite …” Eq. (46) with only real numbers as a hyperbolic transformation

’!!"#$",!!!-!#$-!,!!!.#$.˜U™Sš%+'™QOSš!,!!!+'#$%.™QO›!š!]!+'˜U™Sš!/!!!!!!!!!!!!!!0qf1

!

Minkowski defines

œ$%E!'w!Eš$YNOSš!

so

%)K!œK!)

. And

t

In the 2nd lecture Minkowski uses c=1 and x4 =it, but in the third lecture uses c

why for clarity in comparing also here use x4=ict.

!

19!

˜U™Eš$˜U™Sš$)|!

}

)%œ*!!, ™QOEš$E™QOSš$Eœ|!

}

)%œ*!!

, where }

)%œ*

is

taken with positive sign.

This gives the Lorentz transformation with only real coefficients

!!!"#!!$"!, -#!$-!,!!!!!.#!$.%œ+'

}

)%œ*!!,!!!+'#!$%œ.]+'

}

)%œ*!/!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0qi1

This transformation is valid only for C

&

C

K!+!

as q=

&|+

. (

6"#$6-#$6.#$^

gives

&$&•$ž•

žŸ $œ+

. ) Therefore,

YNOSš$œ$:

.

!

Minkowski by selecting the “rotation” Eq. (46), which gives Eq. (47), made an implicit

assumption which excluded GSLT for C

&

C

D+

and we will show below that there is another

mathematical choice for “rotation” which gives GSLT for C

&

C

D+

.

“As I can find in Minkowski’s lectures the result C

&

C

K+!

is explicitly discussed first in the last

lecture, the famous “Raum und Zeit” , 21 September 1908.: “I want to show first how to move

from the currently adopted mechanics through purely mathematical reasoning to modified ideas

about space and time.“

40

,

41

[Ref. 40 in Ref.37, p.57; Ref 41o, p.75. ]. In this lecture Minkowski

used a graphic method and argued using analytical geometry. He points out that the equations

of Newton’s mechanics have a two-fold invariance: spatial change of position and uniform

translatory motion. If the origin in space and time is fixed the first group is again Euclidean

rotations, as e.g. Eq. (45), which leaves

"*]-*].*

invariant. “The second group, however,

indicates that, also without altering the expressions of the laws of mechanics, we may replace x,

y, z, t by x−αt, y−βt, z−γt, t, where α, β, γ are any constants. The time axis can then be given a

completely arbitrary direction in the upper half of the world t > 0.” [Ref. 40 in Ref. 37, p.59;

Ref. 41, p.77]. He says that these groups is taken separately but , “it is the composed complete

group as a whole that gives us to think.”[Ref 37 p.58] and continues “What has now the

requirement of orthogonality in space to do with this complete freedom (of choice) of the

direction of the time axis upwards?” [ Ref. 40 in Ref 35, p.59; Ref. 41, p.77]. To establish this

Minkowski takes a positive parameter c and considers the graphical representation of

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!+*'*%"*%-*%.*$)!/!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0qj1

for t>0 and looks for those homogenous linear transformations

U¡!

0

¢!,s!,t!,Y!!

1

!QOYU!

0

¢n,sn!,tn,Yp!!

1

!

for which the expression of the graph in 0

"n,-n!,.n,'p!

1 is of

the same form as Eq. (49). That is, those homogenous linear transformations

for which

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!+*'*%"*%-*%.*$)g!+*'#*%"#*%-#*%.#*$)/!!!!!!!!!!!!!!!!!!0c^1

!

20!

(Minkowski does not explicitly write, but if proposition Eq (50) is valid for a homogenous

linear transformations T, this implies, as can be easily shown

u

, that for T also

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!+*'*%"*%-*%.*$+*'p*%"#*%-#*%.#*!!/!!!!!!!!!!!!!!!!!!!!!!!!!!!!0c)1

Thus, the same relation which together with linearity gives LT for C

&

C

K!+

. ) Euclidian

rotations in 3D space about the origin fulfil this; thus, Minkowski continues, enough to look at

the upper branch of the hyperbola

+*'*%"*$)

in the graph with its asymptotes, see Figs. 6

and 7.

v

“If we now regard OC’ and OA’ as axes for new coordinates x’, t’ with the scale units OC’=1

and OA’=1/c, then that branch of the hyperbola again acquires the expression

+*'p*%"#*$)

,

t’>0 and the transition from 0

¢!,s!,t!,Y!!

1

!YU!

0

¢n,sn!,tn,Yp!!

1 is one of the transformations in

question.” [Ref. 40 in Ref 37, p.60; Ref. 41, p.77-78]. (That is the transformations satisfies Eqs.

(50) and (51) and is thus an ordinary LT related to the frame with the world-line OA’ as a new

time axis t’ and for which a world point on OA’ is at rest.) Minkowski then argues why that

natural phenomena are invariant for these transformations rather than for GT. He then “…

introduce this fundamental axiom:

With appropriate setting of space and time the substance existing at any worldpoint can always

be regarded as being at rest.” [Italic in original. Ref.40 in Ref 37, p.62; Ref. 41, p.80]

And continues [Ref. 40 in Ref 37, p.62-63; Ref. 41, p.80]:

u

See Appendix D

v

NB. In the transparency Fig. 6 Minkowski uses c=1 but in text c.

!

21!

“This axiom means that at every worldpoint

w

the expression

+*6'*%6"*%6-*%6.*

is always positive, which is equivalent to saying that any velocity v is always smaller than c.”

However, what is actually proven is only that if we have a substance with 0≤ C

&

C

K+

it will

also have 0≤ C

&p

C

K+

in any systems constructed as in Fig. 7. , because for those systems

+*6'*%6"*%6-*%6.*

is invariant, and of course positive in the system where the

substance is at rest. However, also in this derivation Minkowski made an implicit assumption

which excluded C

&

C

D+

, as will be shown below.

V. MATHEMATICAL INTERLUDE Euclidian vs. hyperbolic rotation

A ”rotation through an imaginary angle” is not quite similar to ordinary rotations in the

Euclidian plane, Eq. (45), where both axises rotate in the same sense and can have all directions

in the plane. Instead, in the hyperbolic transformation Eq. (47) the axises move towards or from

each other like a pair of scissors

42

,

43

when changing the “pseudo angle

š

”, and the axises

cannot reach all directions [Ref. 1, pp.118-119].

In the 2D Euclidian space, there are two different transformations using rotation, for which x2

+y2 are invariant. One in which the old and the new systems of coordinates are congruent Fig.

8, and one in which these two systems are not congruent Fig. 9, with the important

consequence that there are “transformations between which a continuous transition is not

possible.”

44

w

V. Petkov’s note: “Minkowski means at every worldpoint along the worldline of the

substance” [Ref. 37, p. 62].

!

22!

So also, in hyperbolic transformations. We can in 2D make different choices for this “rotation”

between which there are no continuous transitions. For

"p$"!˜U™Sš%-™QOSš!,!!!!!!!!!!!!!!!!!!!-p$%"!™QOS!š]-˜U™Sš!,!!!!!!!!!!!0cB1!!!!!!!!!!!!!!!!!!!!!!!

the x’-axis and y’-axis have the equations

-$!"YNOSš!!!

resp.

-$

!£

¤¥¦§¨!!!!

and can only be in regions IV and II resp. I and III as for

%©K

šK]©

YNOSš!!

change continuously but are limited between -1 and 1. The

positive x’ axis and the positive y’ axis can only be in region IV resp. region I

as for

š$^

x’=x and y’=y, and the transformations are continuous for

š

,

Fig.10.

There are two types of other hyperbolic transformations or

“pseudo rotations” which are not reached by continuous transitions from Eq. (52). First only

changes in signs on x’ and/or y’, so the axises are still in the same regions, but the positive

direction of axises changed. The other type is an

"n_!-n

interchange like

"p$%"!™QOS!š]-˜U™Sš!

-p$!!!!"!˜U™Sš%-™QOSš

(53)

so now x’ and y’ axises are in regions I and III resp. IV and II, Fig.11. Also, here changes in

the sign of x’ and/or y’ lead to other transformations which are not reached by continuous

transitions even if the axises are still in the same regions.

!

23!

VI. HOW MINKOWSKI COULD HAVE FOUND C

x

C

Dy

AND 6D SPACETIME

A. Minkowski’s “rotation” derivation of LT

Besides

.p$.˜U™Sš%+'™QOSš!,!!!+'p$%.™QOSš!]!+'˜U™Sš!,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0cq1

Figs.12a and 12b where positive ct’ is just in region I, actually implicitly presupposing C

&

C

K+,

Minkowski could also have chosen

.pp$%.™QOSš!]!+'˜U™Sš!!!!!!!+'pp!$.˜U™Sš%+'™QOSš!,!!!!!!!!0cc1

and

.## $%.™QOSš!]!+'˜U™Sš!!!!!!%+'##!$.˜U™Sš%+'™QOSš!!!!!0ce1!!!!!!!!

,

Figs 13.a and 13b.

In other words c

'_!.

interchange,

.nn$+'n![\6!!+'nn$.n

in Eq. (55) resp.

.nn$+'n!!!

!

and

+'nn$%!.n

in Eq. (56).

!

24!

Now for both Eqs. (55) and (56) for particles at rest in S’’ that is for

6.## $^

we get w=

ž•

žŸ

ç

•==ª; $!

Ÿ‘«¬!¨!!!!

and thus

'[\›!š$

-

valid only for C

ˆ

C

D+

.

Now

+‡v›š$ r

}

r•Ÿ‘«¬;¨$

(

>

(

<

(

>;

<;•r

and

!!!!!!!!!!™QOSš$ Ÿ‘«¬¨

}

r•Ÿ‘«¬;¨$

(

>

(

>

(

>;

<;•r

, (57)!

and Eqs. (55) and (56) in 2D give SLT for C

ˆ

C

D+/

Equation (55) gives

!.#p!$%

C

ˆ

C

ˆ

0

.%ˆ'

1

(

ˆ*

+*%)!!!!!'##!$%

C

ˆ

C

ˆ

„

'%ˆ

+*.

…

(

ˆ*

+*%)!!/

!

.

And using this for c<w<

©

, (that is for

^®šK©, ^®YNOSšK)1,!

gives

!.#p!$%

0

.%ˆ'

1

(

ˆ*

+*%)!!,!!!'#p!$%

„

'%ˆ

+*.

…

(

ˆ*

+*%)!!/!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0ci1!

.## g+'![\6!'## g•

!!!ˆ›¯\!ˆg]©

, so the positive ct’’ axis is in region IVa, as shown in

Fig. 13a (also seen directly from Eq. (55) with

š

=0.)

°Ou!¡UW!ˆK%+!!,0'›['!Ev!±‡“!%©K!!š®!^,!

%)KYNOSš®^1

we chose Eq. (56) which

gives

!.#p!$%

C

ˆ

C

ˆ

0

.%ˆ'

1

(

ˆ*

+*%)!!$

0

.%ˆ'

1

(

ˆ*

+*%)!!, '#p!$

C

ˆ

C

ˆ

„

'%ˆ

+*.

…

(

ˆ*

+*%) $%

„

'%ˆ

+*.

…

(

ˆ*

+*%)!/0cj1

When

ˆg%©!!!!!.## g+'![\6!'## g%•

as shown in Fig. 13b (also shown directly from Eq.

(56) with

š

=0), that is the positive ct’ axis is in region IIa. So even for C

ˆ

C

D+!

is possible as

Minkowski commented, about Newtons mechanics above:

“The time axis can then be given a completely arbitrary direction in the upper half of the world

t > 0.” [Ref. 40 in Ref.37 p.59; Ref. 41, p.77]. Which motivated us to the choose Eq. (56) for

ˆK%+!/

The physical difference between these “pseudo-rotations” in 2D is that with Eq.(54) the

positive ct’ axis can only be in the region I and C

&

C

K!+!

and with Eq. (55), a space

_

time

interchange so that the positive ct’ axis can only be in region IVa for w>c, and a space

_

time

interchange plus a change in sign for ct’, so for Eq. (56) the positive ct’ axis can only be in in

region IIa for

ˆK%+/

!

25!

Therefore, Minkowski using “rotation with imaginary angle” could, in 2D, also have used the

other rotations in Eqs. (55) and (56), giving SLT for C

ˆ

C

D!+

. How to extend this to 4D and

even to 6D is shown below.

B. Minkowski’s geometrical derivation of LT

However, with the use of the geometrical method, Minkowski was closer to finding the

generalization to C

ˆ

C

D!+

.

Minkowski uses a geometrical construction, but only with positive t’ axis in region I, compare

Figs. 7, 12a. and 12b, and because the t’ -axis is the world line for a substance thus implicitly

presupposing C

&

C

K!+

for velocity v in S. However, geometrically it is equally valid also to

draw a positive t’’ axis in regions IVa or IIa., as shown in Figs 13a and 13b. And it is

conceivable that Minkowski would eventually have done so as he wrote

“The time axis can then be given a completely arbitrary direction in the upper half of the world

t > 0.” [Ref. 40 in Ref. 37, p.59; Ref. 41, p.77] even if this was first formulated for Newtonian

mechanics.

And choosing a positive t’’ axis in IVa resp. IIa and taking t’’ axis as world line for a

“substance” at rest in S’’, implies

ˆD+

resp.

ˆK%+

in S but

is still compatible

x

with Minkowski’s “… fundamental axiom:

With appropriate setting of space and time the substance existing at any worldpoint can always

be regarded as being at rest.” [Italic in original. Ref.40 in Ref. 37, p.62; Ref. 41, p.80.].

What transformations could this give? In Minkowski’s geometric construction (for C

&

C

K+!1,

as shown in Fig.7., it is easy to see that if the slope of the t’-axis is changed, the t’- and x’-

axises move like a pair of scissors and will with same positive directions coincide at the

asymptote ct=x for v=c and with opposite positive directions coincide at

!+'$%"!

for

&$%+!/

Therefore, at these limits mathematical singularities exist for the transformation related to

&$!h+

. However, geometrically there is no hinderance to continue the movement of the t’

axis and draw it under the asymptote, in IVa resp. IIa, Figs. 13a and 13b. (NB we follow

Minkowski so here using x, but in Figs. 12 and 13 we use z. Additionally w is used for

superluminal velocities.)

What happens with the x’’-axis? That is not entirely given

y

. For positive t’’ in IVa a possibility

is to choose that positive x’’-axis continues above the asymptote in region Ia so x’’- and t’’ -

x

To exclude this possibility for a “substance” with (

?

(

*&+

needs a physical argument which Minkowski at least

not explicitly presents.

y

"Pavsicj"argues"for"the"choice"of"x-axis"for"which"c"is"an"invariant,"which"is"a"more"physical"argument"giving"

same"result"as"our"geometrical."

!

26!

axises continue to move as a pair of scissors. Using Minkowski’s geometrical construction of

S’(x’, t’) with positive t’ -axis in Ia and positive x’ axis in IVa, we can mathematically for

w>c construct a new system S’’ (x’’, t’’) with positive t’’ -axis in IVa , and positive x’’ -axis

in Ia by an x-t interchange i.e. x’’=ct’ t’’=x’/c , t’’>0. And for

ˆK%+

with positive t’’-axis in

region IIa and positive x’’-axis in region Ib, that is

'nn$%£#

[\6!"nn$+'n!

, t’’>0.

Both these transformations

z

x’’=ct’ t’’= x’/c resp. x’’=ct’

'nn$!%"n|+

fulfil PII’ in 2D as

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!+*'pp*%"##* $^_!+*'#*%"#*$^!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0e^1

So possible transformations for C

ˆ

C

D+

are LT (x, t) for v<c to (x’, t’)’ followed by x’’=ct’

t’’=x’/c , for w>c or

"nn$+'n!'nn$!%"n|+

for w< -c, to S’’(x’’, t’’) . These composition of

transformations also satisfy

+*'*%"*$!+*'#*%"#*$"##*%+*'##*$%0+*'##*%!"##*1!!!!

(61)

And

+*'*%"*$%0+*'##*%!"##*1

as in Eqs (8) and (10), Sec. III B , gives “Superluminal

LT”, SLT

!!!!!!!!!!!!!!!!!!!!!"## $h "%ˆ'

(

ˆ*

+*%)!!,!!!!!!!!'## $h '%ˆ"

+*

(

ˆ*

+*%)!!!!!NOu!!

C

ˆ

C

D+!!/!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0eB1

Direct calculation using LT S to S’ with C

&

C

K+

followed by x-t interchange gives

..........................+@@ "(*D " ( ABEF

GH

C

DBEH

GH

!!!!!,!!!!!'## $!£

#$! £•:Ÿ

(

r•:;

<;!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0eJ1

thus

w=

EF

EA

½

FIIGH "IH

J

so w=

;

:!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0eq1

which gives

!!!"## $+ '%"

ˆ

(

)%+*

ˆ*!!!,!!!!!!'## $!!!"%+*

ˆ'

+

(

)%+*

ˆ*!,!!!!!g!!

!"## $+ '%"

ˆ

+

C

ˆ

C(

ˆ*

+*%)!,!!!'## $!!!"%+*

ˆ'

++

C

ˆ

C(

ˆ*

+*%)!g!!!!!!!!!!

!

z

PII’ is fulfilled by all four transformations

K## % 'LM#&&&&&M##%'FI

I&

related to the ambiguity to choose sign on x’’

and t’’ axises. But this construction with pseudo rotation at least makes the choice here reasonable as continuity

for x’’ and t’’ always pointing to upper plane.

!

27!

!!!!!!!!!!!!!!!!!!!!"## $

C

ˆ

C

ˆˆ'%"

(

ˆ*

+*%),!!!!!'## $!

C

ˆ

C

ˆ!!ˆ"

+*%'

(

ˆ*

+*%)!!!/!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0ec1!

!

that is a SLT with w=

+*|&

in S, C

ˆ

C

D+/

In 4D PII’ gives as in Eq. (4) Sec. III A,

!!!!!!!!!!!!!!!!!+*6'*%6"*%6-*%6.*$^!_!+*6'#*%6"#*%6-#*%6.#*$^/

(66)

Linear transformations entails that the finite coordinate differences satisfy the same

transformation equations as the differentials and with one event

M

=(0,0,0,0) in both S and S’

we get

å

!!!!!!!!!!!!!!!!!!!+*'*%"*%-*%.*$^!_!+*'#*%"#*%-#*%.#*$^/

(67)

For PII’ to also be valid for C

ˆ

C

D+

in 4D we must have

!!!!!!!!!+*'*%"*%-*%.*$^!_!+*'##*%"##*%-##*%.##* $^/!!!!!!!!!!!!!!!!!!!!!!!0ei1

!

Thus,

-*].*$+*'*%"*!_-##* ].##* $!+*'##*%"##*/

(69)

We search for a transformation of y and z so also

!!+*'*%"*$%

a

!+*'##*%"##*

b is valid so

we get

-*].*$%0-##*].#p*1

with a solution y=

h

iy’’ and z=

h

iz’’ and

!!!!!!!+*'*%"*%-*%.*$%

a

!+*'##*%"##*!%-##*%.##*!

b

/!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0f^1

and thus a GSLT in 4D for C

&

C

D+!

, as Eq. (14),

!!!!!!!"## $h "%ˆ'

(

ˆ*

+*%)!,!!!!!!!'## $h '%ˆ"

+*

(

ˆ*

+*%)!-n#$hE-!!,!!!!!!.n#$hE./!!!!!!!!!!!!!!!!!!!!!!!!!!0f)1!!!!!!!!

!

which still fulfils

ä

Postulate II’, as

6v*$^!!_!!6vpp*$^/

å

PII’

N&4&!"$% 6& O !"P$% 65&N&&4&!"$%'!"P$&&5&N

MQRSTUVQWRMXVST&RQY&ZXSYRQ

&N&

the finite

coordinate differences satisfy the same transformation equations as the differentials

N

[

\"$% '\"@$

]

N

&&&4\"$% 6& O \"@$% 65&

.

ä

But not that ds2 is invariant. However, there is nothing in the postulates that demands this. Not even P I. A

transformation between two IS:s is not a law in one IS, but a relation between two IS:s. And as v=c is singularity

in the theory of special relativity, it is perhaps strange but not excluded that there is a change even in the relation

of ds2 when passing v=c. Also, that ds2 is invariant is a result if only use ordinary subluminal LT. See also Parker

who use

&&!"$% 0!"P$

and ” The extended principle of relativity states that the totality of the laws of physics has

the same form relative to the superluminal frames as it does to subluminal frames” [italic in orig. Ref. 22, p. 2287.]

!

28!

Minkowski could have discarded this transformation with imaginary numbers as not physical.

ö

However, Minkowski was an excellent mathematician and had done major work on quadratic

forms

aa

,

45

,

46

,

47

and explicitly wrote about the invariance of coefficients in real-valued

transformations of quadratic forms [Ref. 45 in Ref. 47 p.10] (here in Sec. III B called the law

of inertia for quadratic forms). Therefore it is conceivable that he would have seen how the

problem with imaginary numbers in 4D superluminal LT was due to the negative sign in Eq.

(70) and thus could be solved with 6D, three space and three “timelike” dimensions, as others

did in the 70-ties, (see Secs. III B and III C). So perhaps Minkowski had followed through on

his mission “… to move from the currently adopted mechanics through purely mathematical

reasoning to modified ideas about space and time.” [Ref.40 in Ref. 37 p.57; Ref. 41, p.75] and

found real-valued superluminal Lorentz transformations in 6D spacetime, if not his untimely

death from appendicitis, only 44 years old, 1909.

And what change could it have been for physics?

VII. SUPERLUMINALITY AND SIX DIMENSIONAL SPACETIME

- JUST A MATHEMATICAL POSSIBILITY?

That there are real-valued superluminal Lorentz transformations in six dimensional spacetime

not only compatible with, but also suggested from, the postulates of special relativity yet do not

prove that the extra dimensions and superluminal phenomena have physical existence.

However, what if something real is described by superluminality exists?

A. Causality paradoxes?

It has been argued and much debated that tachyons contradict the order of causality. Consider

any process by means of which an event P

0"²,'²1

causes an event R

0"³,'³1

,

!'²K!'³

, at a

speed U > c in S. In other inertial frames S’, with velocity v, C

&

C

K+

, relative to S, we

obtain

´'#$'³

#%'µ

#$0¤^•Ÿ_1•:4`^a`_5

<;

(

r•:;

<;$¶Ÿ•:b`

<;

(

r•:;

<;$´' r•:c

<;

(

r•:;

<;

, where

´'

=

0Y³%'²1D^/

(72)

ö

Although later Recami and Mignani and Pavsic [Refs. 26-31] argue there are possible interpretations to handle

this.

aa

In a prize-winning essay, only 18 years old, “Grundlagen für eine Theorie

quadratischen Formen mit ganzzahligen Koeffizienten“ (published 1884) Ref. 45 and in his doctoral thesis

“Untersuchungen über quadratische Formen “(1885) Ref. 46, both in Gesammelte Abhandlungen Ref. 47.

!

29!

Therefore, if

;

·K&K+!!!!´Y#K^!

in these inertial frames S’ and “… cause and effect are

reversed. But this is impossible in deterministic physics, for then the relevant physical law

would have different forms [in different frames] in violation of the relativity principle”. [Ref.

15, pp. 36-37]

However, in the same systems S’ tachyons/superluminal processes also have a negative energy

E’ as

!I#$¸•µ:

(

r•:;

<;$¹r•:d

<;

(

r•:;

<;

(for E>0 and supposing

º$I ·

;&&

still holds for U>c)

bb

, so it is

suggested a reinterpretation principle [Refs. 7, 13] (Even suggested by Strum, already 1923,

according to Malykin et. al. [Ref. 5]): negative-energy tachyons travelling backward in time

are to be reinterpreted as positive-energy tachyons mowing forward in time with opposite

momentum.

And Bilaniuk et al. argue that “… various observers must agree on the identity of the physical

laws but not on the description of specific events. Only the physical laws, and not the

description of any given phenomena must remain invariant as we pass from one frame of

reference to another.” [italic in orig. Ref. 7, p.720]

Yet as Recami writes “…each observer will always see only tachyons (and antitachyons)

moving with positive energy forward in time. … however, this success is obtained at the price

of releasing the old conviction that judgement about what is ‘cause’ and what is ‘effect’ is

independent of observer.” [Ref. 21, p. 64. italic in orig.]

It is beyond the scope and length of this article to go more into this debate

cc

,7,21,26,27,

48

,

49

,

50

, yet it

is illuminating what Einstein thought already 1907, about a situation giving a relation

equivalent to Eq. (72).

“This result signifies that we would have to consider as possible a transfer mechanism whose

use would produce an effect which precedes the cause (accompanied by an act of will, for

example). Even though, in my opinion, this result does not contain a contradiction from a

purely logical point of view, it conflicts so absolutely with the character of all our experiences,

bb

Formally it follows from

º$ »·

(

r•d;

<;

and

I$ » ;

(

r•d;

<;

even if U>c presupposes imaginary m

for real valued p and E.

cc

For a review of many possible causal paradoxes and their possible solutions (at least in

“microphysics” that is if tachyon exchange with ordinary matter is assumed to be spontaneous

and uncontrollable) see [Ref. 21, pp.64-77, Ref.48]. NB vs signal p.71

!

30!

that the impossibility of the assumption [ here U>c] is sufficiently proved by this result.”

51

[p.248, italic in original]

Interestingly,8Einstein did not see a purely logical contradiction, that effect precedes the cause,

but only that it contradicts all our experiences. So far. However, this does not prove that these

mechanisms cannot exist and cannot be discovered.

B. “Spooky action at distance”, entanglement and superluminality

Another phenomena Einstein didn’t believe to exist was, what he later called, “spooky action at

distance”, in his argument against the claimed completeness of orthodox quantum mechanics.

However, these phenomena, now called entanglement, exist and have been experimentally

verified.

“To conclude, the results of the Bell-experiments leave only two possibilities:

(i) the world is non-local - events happen which violate the principles of relativity.

(ii) objective reality does not exist - there is no matter of fact about distant events.

…the proposition that relativity is not fundamental, and that the world is nonlocal, seems the

lesser of two evils. This was certainly Bell’s position, and is even seen as inevitable by some

philosophers; Maudlin says … [see below]”

52

Wiseman also refer to Cushing

53

who argues that if Einstein had been aware of this choice, he

had chosen non-locality before non-reality.

But perhaps not even a choice. Stapp argues: “A locality property expressing the idea that

causal influences can propagate only forward in time, from earlier cause to later effect, and no

faster than light is shown to be mathematically incompatible with certain predictions of

quantum theory. The contradiction is proved without invoking any additional assumptions as

realism or hidden variables that contravene conventional quantum thinking.”

54

Stapp also refer

to the experimental confirmation of the quantum predications.

Tumulka strongly agree: ”If the claims I made are right, one should conclude that there is no

assumption of realism that enters the proof of Bell’s theorem next to the assumption of locality,

and thus that we do not have the choice between the two options of abandoning realism and

abandoning locality, but that we must abandon locality.”

55

And Maudlin writes:

“Violation of Bell’s inequality does require superluminal causal connections.

Violation of Bell’s inequality can be accomplished only if there is superluminal information

transmission.”

56

!

31!

So, entanglement seems to be that kind of phenomena which is suggesting superluminal

causality and thus a counterexample to Einstein’s argument above, Sec. VII A that we never

experience superluminality, and so superluminality is for all we know possible.

C. But why six dimensions?

Mathematically to allow real valued superluminal LT which hopefully are easier to interpret

physically than the occurrence of imaginary numbers in transformation

dd

in 4D. But do these

extra dimensions have any reality? This is not yet proven. But there are still some ideas to

explore. The 4D spacetime, or so-called block universe, which is argued to be necessary to

explain the experimentally verified kinematic relativistic effects of the special theory of

relativity

57

,

58

(length contraction, time dilation) contradict our experience of change, becoming

and the flow of time (see appendix F). In 3D space we have a 1D flow time. In 4D block

universe all events, past, present and future exist equally, so the time coordinate is more like an

index for existence of 3D spaces, past, present and future. The 6D spacetime can perhaps in

some analogue then be construed as 5D block universe with a 2D “time surface” indexing

possible 3D worlds, and a1D flow time. Of course, if this is possible is not known. The 6D idea

seems to have a further merit. 4D block universe seems deterministic having just one future for

each time, but quantum theory describes many possible futures. In 6D the 5D block universe

contain many possible 3D worlds for every 1D “flowtime point”. A!further!indication!of!

some!perhaps!deeper!connection!between!relativity!and!quantum!theory!is!that!the!de!

Broglie!velocity!w!=!c'/v,!which!seems!“quantum!theoretical”!comes!up,!see!Eqs.(63)-(65)!

when!it!is!shown!that!a!SLT!can!be!reached!by!LT!+!space!

_

time!interchange!with!!w!=!

c'/v!This!must!be!studied!more,!even!if!it!is!already!clear!that!de!Broglie!when!he!derived!

this!relation!started!from!a!relativistic!transformation!of!a!plane!wave

()

.![Ref.!59!p.3-9]

Appendices

Appendix A

dd

Although it is argued “It has long been known that special relativity can be extended to

Superluminal observers only by increasing the number of dimensions of the space-time or—

which is in a sense equivalent—by releasing the reality condition (i.e., introducing also

imaginary quantities)” Ref. 31 Abstract,

!

32!

Landau and Lifshitz 1971

60

part one. They also like Einstein Sec IV D, Cunningham19 and

Rindler Sec. III A Eq. (5) use that from PII’, the constancy of velocity of light, follows

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!6v*$[6vp!*!!!!

(A.1)

And they state that

[

can depend only on the absolute value of the relative velocity of the two

inertial systems,

!

as it cannot depend on the coordinates or the time which would be in

contradiction to homogeneity of space and time, and not depend on the direction of the

relative velocity, since that contradict the isotropy of space.

They consider three reference systems K, K1, K2 with V1 and V2 the velocities of systems, K1,

and K2 relative K, and V12 the absolute velocity of K2 relative K1. Then

!

!!!!6v*$[0¼r16vr*!!!!6v*$[

0

¼*

1

6v*

*!!!!!!6vr*$[0¼r*16v*

*

which gives

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! [

0

¼*

1

!!!![0¼r1!!!! $![

0

¼r*

1

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0½/B1

They argue as V12 depends not only of the absolute values of the vectors V1 and V2 but also on

the angle between them and that the angle is not on the left side of the formula,

[0¼1

must be a

constant and thus 1. Compared with Rindler’s derivation Eq. (7) which gave what corresponds

to

[0¼1

that is K=±1.

But L-L here implicitly miss another solution

ee

.

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!![

0

¼

1

$

¾

!!!)!±‡“

C

¼

C

K+

!%)!±‡“

C

¼

C

D+!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

(A.3)

If C

¼r

C

K+

and C

¼*

C

D+

then C

¼r*

C >c, and if both C

¼r

C

D+

and C

¼*

C

D+

then C

¼r*

C <c .This

follows from the addition law for velocities (see below), which of course at this point of

derivation of LT is not known, but it shows that the solution is consistent. Therefore L-L’s

argument is correct only if implicitly, without explicitly argument, rule out C

¼

C

D+

and thus

cannot be used to rule out C

¼

C

D+

and we again have

6vp*$h6v!*!!!!

.

Law of addition for velocities

ˆ$ ¿]&

)]¿&

+;

**

!Pilotti![Ref.!1!pp.131-132],!also!earlier!Pavsic!Ref.!25.!

!

33!

is equivalent with

+%ˆ

+]ˆ$

„

+%¿

+]¿

…„

+%&

+]&

…

which more directly gives the relation between subluminal and superluminal velocities

addition.

Appendix B

In Ref. 17 Einstein from homogeneity of space and time and symmetry derives that the LT are

linear and that it is possible with a standard configuration between S’ moving with v along x

axis in S. So, we have x’=a(x-vt), y’=by, z’=fz. And also t’=gx+ht. And from Postulate II’,

invariance of the velocity of light, we get that the two equations

"*]-*].*$+*'*

and

"p*]-p*].#* $+*'p*

must be equivalent. It is easy to show (see Rindler [ Ref. 14, pp. 16-

17, 21; Ref. 15, pp. 15-16] that it follows (here for comparing with Einstein Sec. IV C using F

instead of K and instead of

–*!!

in Sec. IV D.)

"p*]-p*].#*%+*'#*$•

(

"*]-*].*%+*'*1!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

(B1)

A we saw in Sec. IIIA Rindler used symmetry argument and got K here = F=±1.

Einstein go another way. But it is important to realize that we at this stage have no restriction

on F. It is easy to see that for any F, even negative, Eq. (B1) fulfils that equivalent criterion.

We get in Eq. (B1)

[*0"%&'1*]À*-*]6*.*%+*

0

w"]›'

1

*$•

0

"*]-*].*%+*'*

1

!!!!!!!!!!!0ÁB1

This is an identity valid for all (x,y,z,t). Choosing y=z=0 gives

0[*%+*w*1"*%B0[*&]+*w›1"']0[*&*%+*›*1'*$•0"*%+*'*1

. (B3)

So we get

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!![*%+*w*$•

(B4)

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!![*&]+*w›$^

(B5)

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!![*&*%+*›*$%•+*!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0Áe1!!

Equation (B5) gives

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!![—&*$+—w*›*!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0Áf1

Equations (B4) and (B6) in Eq. (B7) give

[—&*$+—w*›*$+*w*+*›*$

0

[*%•

10

[*&*]•+*

1

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0Ái1

And we get

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!![*•

0

+*%&*

1

$•*+*

(B9)

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!![*$•+*

+*%&*!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0Á)^1

!

34!

NB that F can be both positive and negative.

Of course, we can as Einstein just write

[!$h

~

•|

}

)%&*|+*

and Einstein use

ƒ$

~

•

.

But then

ƒ

is imaginary for F<0. Another clearer way is to separate the two cases and write

[!$h

~

•|

}

)%&*|+*

for F>0 and

[!$h

~

%•|

}

&*|+*%)

for F<0

For F>0 we get from (B6) h=±a

$h

~

•|

}

)%&*|+*

and from (B4) g=

Â&

~

•|+*

}

)%&*|+*

as from (B5) we must have gh<0 for v>0.

Also (B3) in (B2) give

!!À*-*]6*.*$•0-*].*1

so

À$h

~

•!!!!!!6$h

~

•!!!

So, for F>0 we have (same as Einstein’s Eq. (36) with

ƒ$

~

•

) just using upper sign when

standard configuration

*?"

J

>

3

$%&'

('

)

*%&+

('

,

-.....+?"

J

>

3

$%&'

('

0

+%&*

1

- 4?"

J

>..4- 6?"

J

>..6..............0K$$1...........

For F<0 get from (B6) h=±a

$h

~

%•|

}

&*|+*%)

and from (B4)

g==

Â&

~

%•|+*

}

&*|+*%)

as we must have gh<0 for v>c(>0).

Again, we get

!!À*-*]6*.*$•0-*].*1

so

À$h

~

•!!!6$h

~

•!!

or here as F<0 better write

À$hE

~

%•!!!6$hE

~

%•!!

so we get, with ambiguity of signs as no

clear meaning of standard configuration for C

&

C

D+

*?";

J

%>

3

&'

('%$

)

*%&+

('

,

-....+?" ;

J

%>

3

&'

('%$

0

+%&*

1

-...4.?";<

J

%>..4-...6?" ;<

J

%>..6...0K$71.....

Thus, inevitable imaginary transformations for transverse coordinates in 4D for C

&

C

D+

, which

is discussed at length in Ref. 31, and argued it is in a sense equivalent with higher dimensions.

Perhaps so but 6D might have some other advantages as speculated in Sec. VII C.

Appendix C

MORE DERIVATIONS OF LORENTZ TRANSFORMATIONS MISSING C

x

C

Dy

C.1. Cunningham19,

ff

1914 Cunningham has a clear and simpler argument, which we

recognise in first part of Rindler’s argument 1960 above, Sec III A. Thus, Cunningham derive

from Postulate PII

!!!!!!!!!!!!!!!!!!!6"*]6-*]6.*%+*6'*$^!_6"#*]6-#*]6.#*%+*6'#*$^!!!!!!!!!!!!!!!!!!!0Ã)1

which together with linearity of transformation gives

++

!I!am!indebted!to!dr.!Daniel!Coumbe!for!showing!me!Cunningham’s!work.!

!

35!

!!!!!!!!!!!!!!!!!!6"#*]6-#*]6.#*%+*6'#*$d

0

",-,.,'

10

6"*]6-*]6.*%+*6'*

1

!!!!!!!!!!!!!!0ÃB1

He differs from Rindler in that the proportionality factor is not supposed to be a constant K but

that

d0",-,.,'1

could explicitly be a function of the coordinates, but not of the differentials

and instead of Rindler’s argument from homogeneity and isotropy (which should imply K to be

constant?) sees the analogy to the conformal transformation in 3D

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!6"#*]6-#*]6.#* $d0",-,.106"*]6-*]6.*!1!!!!!!!!!!!!!!!!!!!!!!!!!!!!0ÃJ1

where

d0",-,.1

is the square of the linear magnification. Cunningham writes:

With this in view Minkowski introduce a new variable u=ict where

E$

~

%)

and writing

the relation [C2] in the form

!!!!!!!!!!!6"#*]6-#*]6.#*]6¿p*$d0",-,.,¿106"*]6-*]6.*!]6¿*1!!!!!!!!!!!!!!0Ãq1

interprets this geometrically as indicating that the transformation is conformal in the

imaginary space of four dimension in which the coordinates are (x, y, z, u). (Ref. 19 p.87)

Then Cunningham refers to a proof that for any number of dimensions all possible conformal

transformations fall into three classes

61

a)

!d$)

, b)

d$[\-!+‡\v'[\'Ä!)

. c)

Å!Q™!™UÆP!¡VO˜YQUO!

of (x, y, z, u)

gg

a)

d$)

gives ordinary LT. Cunningham use the same derivation as Minkowski with “rotation

with imaginary angle” but the derivation can of course be done as usual as in textbooks from

linearity (see e.g. Rindler above and [Ref 15.]).

b)

d$[\-!+‡\v'[\'Ä!)!

Here Cunningham sees

d

just as a scaling factor and thus, implicitly,

consider only

!dD^

. I think he don’t consider negative

d

as he compares with 3D where

dD^

(if not allowing imaginary coordinates). But for the Minkowski pseudo Euclidean space with

purely imaginary u=ict or with real coordinates

!!!!!!!!!!!6"#*]6-#*]6.#*%+*6'#*$d0",-,.,¿106"*]6-*]6.*!%+*6'*1!!!!!!!!!!!!!!!0Ãc1

of course, mathematically also

dK^

is possible. Physically this implies the possibility C

&

C

D+

as we saw above and in 2D, in 4D with also imaginary coefficients in transformation, or in

extension to 6D with only real transformations. To exclude this possibility needs an explicit

argument, which I don’t find in Cunningham’s book.

,,

!Cunningham!writes!they!“…!may!be!looked!upon!as!built!up!from!elementary!

transformations!each!of!which!is!a!generalised!inversion!in!a!4-dimension!hypersphere.”!

[Ref!19.!p.!89!]!see!also[Ref!61]!and!give!as!an!example!

"pÇ$È;£e

É;

!!!i=1-4!!and!

“*$

Ê

"Ç*

—

r/!

But!is!this!consistent!with!homogeneity!and!isotropy!as!e.g.!Rindler!state!is!a!

consequence!of!the!principle!of!relativity![Ref.!43,!2006!p.39-40]?!Yet!Cunningham!writes!

“It!can!be!shown!that!the!invariant!properties!of!electromagnetic!equations!established!

for!the!Einstein!transformation!are!equally!valid!for!this!transformation”![Ref.19!p.89]!

!

36!

C.2 Max Born 1920

62

a student of Minkowski, presents a more mathematically elaborated and

thus clarifying argument using Minkowski’s 1908 graphic method as above, where Born in 2D

uses all four hyperbolae calibration curves to construct the new x’, t’ system. But still, as

Minkowski, he draws the t’ axis in just one quarter of the possible areas between the

asymptotes and, without explicitly argument, thus implicitly exclude C

&

C

D+

.

He also used an algebraic method: For S’ moving in S with v in positive x direction and in

standard configuration we have y’=y, z’=z and

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"n$^!_"$&'!!g!!!!Ë"n$"%&'!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0Ãe1

Born writes:

“According to the principle of relativity however, both systems are fully equivalent. Thus

we may equally well apply the same argument to the motion of the origin of S relative to

S', except that now the relative velocity v has the reverse sign. Therefore x’ +vt’ must

be proportional to x, and, on account of the equivalence of both systems, the factor of

proportionality a will be the same in each case:

Ë"$"n!]!&'n

.” (C7)

[Ref. 62 p. 198, Ref. 62.a p.236]

From this and (51) follows

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Ë'n$Ë*%)

&"]'!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

0

Ãf

1!

And Born then determines

Ë

by using that for uniform movement of light c is same in S and S’

by the postulate PII thus

in S

+$£

Ÿ

and in S’

+$£=

Ÿ=$£•:Ÿ

f;ag

:£ÌŸ $ •:

f;ag

: Ìr!

and this gives

Ë*$)%:;

;

and thus

Ë$h

(

)%:;

;

where + sign is chosen as for standard configuration

&g^!!!"pg"

and we get the ordinary LT and thus

"p*]-p*].p*%+*'#*$"*]-*].*%+*'*

.

But in contrast to Cunningham’s derivation this just give the value

d$)

in (C5) and not the

other possibilities, so evidently Born loses these possibilities

hh

. Instead of taking that ”the

factor of proportionality will be the same in each case” we could have different factors

Ë"n$"%&'

and

!8"$"n!]!&'n

which now gives

Ë'n$ÍÎ&•r

:"]'

![\6!Ë8$)%:;

;

and thus more solutions. E.g.

Ë$B

(

)%:;

;!!8$r

*

(

)%:;

;!

which together with y’=y/2,

--

!If!Born!with”!the equivalence of both systems” also means same standard coordinates the

scaling factor if >0 and real must be 1.!

!

37!

z’=z/2 gives

"p*]-p*].p*%+*'#*$r

—

0

"*]-*].*%+*'*

1 which according to

Cunningham is an adequate LT transformation, which seems correct as just a change in scale.

And for

8$%Ë

we get

Ë$h

(

:;

;%)

which gives GLT for 2D or 4D with y’=iy, z’=iz or

6D real transformations as above. So, it seems also Born implicitly exclude the possibility of

C

&

C

D+

with arguments which is not in agreement with Cunningham.

C.3. Pauli 1920

63

Pauli’s derivation is like Cunningham’s simpler than Einstein’s 1905 but like

Einstein’s 1912 (Seq. IV D) as he starts with that from Einstein’s two postulates (actually

enough with PII the constancy of c as Rindler and Cunningham show Eq. (5) resp. Eq. (C1))

follows

6"*]6-*]6.*%+*6'*$^!_6"#*]6-#*]6.#*%+*6'#*$^

which together with linearity of transformation gives

ii

!!!!!!!!!!!!!!!!!!!!!!!!!!!6"#*]6-#*]6.#*%+*6'#*$Ï*06"*]6-*]6.*!%+*6'*1!!!!!!!!!!!!!!!!!0Ãi1

It is notable that Pauli, as Einstein, states

Ï$Ï

0

&

1 when Cunningham states the more general

d

0

",-,.,'

1 which correspond to

Ï*

.(Difference in view of homogeneity and isotropygg? )

Pauli continues with that from (53) follows that the transformation between S’ and S in

standard configuration must be

!!!!!!!!!!!!!!"#$Ï "%&'

}

)%8*!!!!!-p#$Ï-!!.p#$Ï.!!!!'## $Ï'% &

+*"

}

)%8*!!!!!!8!$&!

+!!!!!!!!!!!!!!!!!!!!!!!!!!0Ðj1

!

Pauli then exactly with same arguments as Einstein Seq. IV.B gets

Ï

0

&

1

Ï

0

%&

1

$)

and

Ï

0

&

1

$Ï

0

%&

1

/

From which now follows

Ï

0

&

1

$)

. (actually ±1)

Thus, with the same inconclusive argument as Einstein for ruling out

!!

C

&

C

D+

.

C.4. Synge 1955/196339 A very detailed and clear discussion of LT. As is well known all LT

can be broken down into a translation and a “rotation” in 4D, as initially Poincare [ see Ref. 37

p. 29, 31] pointed out, with two types of rotations as also Minkowski used 1907, Sec. IV.E.

Synge use x4=ict:

First an ordinary plane rotation

!!!!!!!!!!!!!!!!!"pr$"r˜U™ƒ]!"*™QOƒ!, "p*$%"r™QOƒ]"*˜U™ƒ!,!"#9, $"9!,"#—$"—

(C10)

Synge then also uses a purely imaginary quantity

Eš

and considered the substitution

!!!!!!!!!!!!"r#$"r,!"#*$"*!!,"p9$"9˜U™Eš]!"—™QOEš!, "p—$%"9™QOEš!]!"—˜U™Eš

(C11)

..

!It!is!clear!that!with!

Ï!

in!(C9)!it!must!be!

Ï*

in (C8) even if Pauli have

Ï!

in both.!

!

38!

which by analogy may be regard as rotation through an imaginary angle

Eš

but this analogy

must be handled with caution so better use that it gives

"r#$"r,!"#*$"*!!,"#9$"9˜U™Sš]!E"—™QOSš,!"#—$%E"9™QO›š!]!"—˜U™›š!!!!!!!0Ã)B1

where the real quantity

š

is called a “pseudoangle” and I think we also should use “pseudo

rotation” as it differs clearly from ordinary rotation, see Sec. V.

With (x1, x2, x3, x4) = (x, y, z, ict) we get for real coordinates

"!#$"!!,-#!$-!, .p!$.˜U™Sš%+'™QOSš!, +'#$%.™QO›š!]!+'˜U™›š!!!!!!!0Ã)J1

Synge then [Ref. 39, p.112] gives a kinematical meaning to

š

: For particles in rest in S’ we

have

6"#$6-#$6.#$^

and we get

ž•

žŸ $+'[\›!š!!!!!ž£

žŸ $^!!!!žÑ

žŸ

=0

thus S sees every particle of the frame of S’ moving parallel to z- axis with v=

!+'[\›!š!!!!!

Thus

'[\›!š$!:

!!!

and with

Ò$ r

(

r•:;

<;!!!!!!!!+‡v›š$Ò

and

™QOSš$Ò:

!

which gives the Lorentz transformation with only real coefficients

"#!!$"!, -#!$-!,!.#!$Ò

0

.%&'

1

!!'#!$Ò0'%:

"1

valid for C

&

C

K+

But as in Minkowski the choice of Eq.(C13) implicitly ruled out C

&

C

D+

as we can in 2D make another choice for “rotation” Eqs (53) see Sec. V

!!!!!!!!!!!!.#$%.™QO›š!]!+'˜U™›š!!!!!!!!!!!!!+'p!$.˜U™Sš%+'™QOSš!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0Ã)q1

Now for particles in rest in S’ that is

6"#$6-#$6.#$^

and we get

ž•

žŸ $!

Ÿ‘«¬!¨!!!!ž£

žŸ $^!!!!žÑ

žŸ

=0 and thus

'[\›!š$

:

!

39!

thus valid only for C

&

C

D+

and

+‡v›š$Ò:

and

™QOSš$Ò

with

Ò$ r

(

:;

<;•r

!

which thus give GSLT for C

&

C

D+

!!.#!$Ò

0

.%&'

1

!!'#!$Ò0'%:

;.1

. And in 4D we must also have x’=ix, y’=iy

so

lmpo$%6v*

, see Eqs. (67)-(71).

C.5 Landau and Lifshitz 197160 part two. They also look at rotation in tz plane leaving

+*'*%.*

invariant and state that transformation in most general form

jj

is given by

!!!!!!!!!!!!!!!!!!.p!$.˜U™Sš]+'™QOSš!, +'#$.™QO›š!]!+'˜U™›š

(C15)

(thus, same as Minkowski and Synge besides the sign on z and z’, that is orientation of z and z’

axis) where

š

is the “angel of rotation”. But this implicitly rule out C

&

C

D+

as we saw above

as we also have another choice for this “rotation” which is valid for C

&

C

D+

.

Also see their other argument missing C

&

C

D+

in Appendix A.

C.6. Rindler’s later derivations, 1977

64

, 200643

In Rindler’s later books he uses derivations that do not give the choice for C

&

C

D+

as in his

1960 derivation.

1977 From standard configuration and linearity follows y’=By. Rindler then argues that

reversing directions of x and z axes in both S and S’ cannot effect y’=By but (just)

interchanges roles of S and S’ so we must have y=By’ that is with same B so B=±1 and -1

discarded by standard configuration and same for z. It also follows from standard

configuration and linearity that

"n$Ò

0

"%&'

1

!

0

½

1 and

"$Òp0"#]&'p1

(B) Then Rindler

argues that reversing direction of x and z in both S and S’ and thus change x and x’ to –x and –

x’ in (A) gives

"n$Ò!

0

"]&'

1

!

but also that by reversal of roles of S and S’ in B gives

"p$

Òp0"!]&'1

thus

Ò$Òp

. Then by second postulate x=ct and x’=ct’ which in (A) and (B)

using

Ò$Òp

gives

+'p$!Ò'0+%!&1, +'$!Ò'#

0

+]&

1 thus

Ò$ r

(

r•:;

<;

C

&

C

K+

thus

excluding C

&

C

D+

.

//

!It!is!the!most!general!form!for!which!ds!is!strictly!invariant!as!the!other!transformations!

makes!ds!pseudo!invariant!but!importantly!still!fulfils!PII.!

!

40!

But we have seen for 4D GLT

-n$E-!‡“!-n$%E-

and direct calculation gives

-$%E-n!‡“!-$E-n!

that is for C

&

C

D+

it is not valid that the same B. And also, for

!!!!!!!!!!!!!!!!!!!!!!!!!!!"#$h "%&'

(

&*

+*%)!!!!!!!!'#$h '%&"

+*

(

&*

+*%)!!!!!!

C

&

C

D+!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!0Ã)e1

direct calculation gives the inverse transformations

!!!!!!!!!!!!!!!!!!!!!!"!$•

Ì"#]&'p

(

&*

+*%)!!!!!!!!'!$•

Ì'#]&"p

+*

(

&*

+*%)!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

0

Ã)f

1

and it is thus not valid for C

&

C

D+

that in (A) and (B)

Ò$Òp

but instead

Ò$%Òp

. So evidently

Rindler’s argument somewhere implicitly exclude

Ò$%Òp

and C

&

C

D+

, and why will be better

seen in his next derivation.

In 2006 Rindler first argues that “The transformation between any pairs of inertial frames in

standard configuration with the same v, must be the same, by RP [relativity principle].” [Ref 43

p. 44] And then that under xz reversal, that is

"_%"#!!-_-#!._%.#!'_'p!

transformation equations must be unchanged so

"#$Ò

0

"%&'

1

!!!

gives

"$Ò0"#]&'p1

But

as we just saw this is not valid for C

&

C

D+

. Why is it that Rindler’s argument fails?

Rindler’s formulas with

Ò$)|Ë

are actually the same as Born’s formulas (51) and (51’)

Ë"n$"%&'!

Ë"$"n!]!&'n

with same

Ë

But as Born phrases it, in Sec. C.2, it seems clear that the equivalence of the systems according

to P I The laws of physics are the same in all inertial reference frames,

needs not to give same proportional factor. A LT is a relation between two IS, not a law in

one IS, and it could not be excluded that LT depends on some differences in the relation.

Evidently all accept the difference between moving along positive x -axes and moving along

negative x-axes i.e. the using of v resp. – v in LT and its reverse. Einstein shows v=c is a strong

singularity and we are not entitled to exclude that passing this limit, which is a strong change in

relation between S and S’, could effect the transformation. But in Rindler’s procedure changing

direction of axises, the relation between S and S’ seems to be exactly the same in both

directions.

But I think it can be seen that Rindler’s xz reversal argument, even if it sounds self-evident, is

!

41!

valid only for C

&

C

K+

. Because for C

&

C

D+

, as shown above in Eqs (63) –(65) a SLT is the

composition of LT for C

&

C

K+

between S and S’, for which xz reversal holds, but then also

x’’=ct’ t’’=x’/c in 2D and for 4D

kk

also y’’=iy’ z’’=iz for which it is not possible to argue for

only an xz reversal as it involves both x-t interchange and “imaginary space”.

C.7. Two types of errors in derivations of LT excluding C

x

C

Dy

A: Using a faulty argument (sometimes even from symmetry or even Postulate I) which are

only valid for C

&

C

K+

and implicitly exclude C

&

C

D+

(Einstein, Pauli, late Rindler, Born and

Landau and Lifshitz).

We are not entitled to use what seems clear and self-evident for ordinary subluminal systems

when we pass the strong singularity v=c. I can see nothing in the principle of relativity, which

forbids space-time interchanges at these singularities. As shown above, for 2D, 4D and 6D,

even including superluminal systems the P II The velocity of light in vacuum c is the same in

all inertial frames is still clearly valid as still

lmo$Ó!!_lmpo$Ó!

. PI is still also valid if it is

realised that a transformation between two systems is not a physical law in a single system but

a relation between two systems and it cannot be excluded that this relation can depend not only

on the magnitude and direction of speed but also if it is less or more than c as c is a strong

singularity. And for C

&

C

D+

by direct calculation that the inverse change sign.

B: Not fully use of the mathematical possibilities: negative

d

0

",-,.,'

1 (Cunningham, Rindler

1960) , two different “rotations” in non- Euclidean Minkowski space, (Minkowski,

Cunningham, Synge, Landau-Lifshitz) , or different relation to asymptotes in the geometrical

method (Minkowski, Born) thus perhaps more of a lapse.

Appendix D.

Proof of a part of Minkowski’s argument in Space and time 1908.

Minkowski looks

for homogenous linear transformations

zŒ

from S to S’ where an event

M!

has coordinates

kk

In 4D To LT in 2D add y’=y, z’=z and to spacetime interchange add y’’=iy’ z’’=iz’ and this

gives a GLT with

!ˆ$ ;

:!!!

.

In 6D To LT in 2D add

-n$-!,.n$.!!'*

#$'*!!!!!!!'9

#$'9

and to spacetime interchange

-pn$!+'*

#!,!!!!.np$+'9

#!!!'*

## $Ñ#

!!!!!!!'9

## $•#

which gives GLT with

ˆ$ ;

:!!!

.

!

42!

Ô$

0

",-,.,'

1

!!!E\!Õ![\6!Ô#$

0

"#,-#,.#,'#

1

!E\!Õp

0",-,.,'1$0z£Œ

0

"p,-p,.p,'p

1

,zÑŒ

0

"p,-p,.p,'p

1

,z•Œ

0

"p,-p,.p,'p

1

,zŸŒ

0

"p,-p,.p,'p

1)

Such as

A:

+*'*%"*%-*%.*$)!g!!+*'#*%"#*%-#*%.#*$)!!!!

We will show that for those same transformations

zŒ

B:

!+*'*%"*%-*%.*$!+*'p*%"#*%-#* %.#*

for

Ö!

0

",-,.,'

1

![\6!

0

"#,-#,.#,'#

1

!!“¯‹['¯6!'›“‡¿w›!!zŒ

0

",-,.,'

1

thus fulfilling the criteria for LT

(see Rindler Ref.15)

(The reverse that

Ág½!!!!Ev!!'“E&E[‹,

)

See Rindler REF. p.21 (Also the proof of linearity p. 16-17, 74 can be the same

Or the linearity can be proven as in Rindler Ref 43 p. 43 from Newton’s first law)

Proof :(NB for the relevant linear homogenous transformation the inverse is supposed to exist

Not needed??)

Any linear homogenous transformation T 0

",-,.,'

1

!!!g

0

"#,-#,.#,'#

1

transforms

+*'*%"*%-*%.*$)

to

±

0

"p,-p,.p,'p

1

$!½'p*]Á"p*]Ã-p*]×.p*]I"#'#]•-#'#]Ø.#'#]Ù-#.#]Ú"#.#]

Û"#-#$)

(II)

For all transformations

zŒ!

for which is valid

+*'*%"*%-*%.*$)!g!!+*'#*%"#*%-#*%.#*$)!!!!

also

±

0

"p,-p,.p,'p

1

$)

so we for

zŒ!

have both

!+*'#*%"#*%-#*%.#*$)!!

and

±

0

"p,-p,.p,'p

1

$)

(that is all (x’, y’, z’, t’) on the “surface” lies on “same” surface written in other form)

which then give restriction on

zŒ!

(or more strict for any point (x,y,z,t) on

+*'*%"*%-*%.*$)!

zŒ!

is such as that

both

!+*'#*%"#*%-#*%.#*$)!!

and

±

0

"p,-p,.p,'p

1

$)

or even stronger taking any point (x’,y’,z’,t’) such as

+*'#*%"#*%-#*%.#*$)!!!!

it has under

zŒ!!!!

a counterpart (x,y,z,t) such as

+*'*%"*%-*%.*$)!

!

43!

and thus for (x’,y’,z’,t’) also

±

0

"p,-p,.p,'p

1

$)

thus we can chose any point on

+*'#*%"#*%-#*%.#*$)!!

and put in

±

0

"p,-p,.p,'p

1

$)

For t’=1/c and x’=y’=z’=0 we get A=c2

(NB if both

!+*'#*%"#*%-#*%.#*$`!!

and

±

0

"p,-p,.p,'p

1

$`

for

'p$

~

Ü

and x’=y’=z’=0 we again get

½$+*

and also for all other coefficient )

Then for

'p$h

~

9

~

*

and

"p$hr

~

*

y’=z’=0 we get

9

*]Ár

*h

~

9

*

E=1 gives E=0 and B=-1 and in same way F=0 and G=0 and C=D=-1

Now

'p$h

~

9

~

*[\6!-p$.p$hr

*

gives

9

*%r

—%r

—hÙr

—$)

gives H=0

In same way M=N=0

(And f(x’,y’,z’,t’)=

+*'p*%"#*%-p*%.p*

for the transformations

zŒ!

For all K)

So for these transformations

+*'*%"*%-*%.*%)$±

0

"#,-#,.#,'#,

1

%)$+*'p*%"#*%-#*%.#* %)

(for all K

+*'*%"*%-*%.*%`$±

0

"#,-#,.#,'#,

1

%`$+*'p*%"#*%-#*%.#*%`

)

Thus for these transformations

+*'*%"*%-*%.*$+*'p*%"#*%-#*%.#*

Which is the relation which leads to LT see Rindler Ref. 15 p. 18-19

Appendix E.

Weyl’s proposal about consciousness and relativity

A theory with more dimensions than the usual four has the risk, as with string theory, to be just

a mathematical technicality. But here I think the remark of Herman Weyl is of importance:

The objective world simply is, it does not happen. Only to the gaze of my consciousness,

crawling upward along the life line of my body, does a certain section of this world come

to life as a fleeting image in space which continuously changes in time.

65

(my bold)

!

44!

Petkov comments this “… - that the flow time is mind-dependent – outlined by Weyl should

have been examined more rigorously.” and “At first glance this idea appears to be self-

contradictory since Weyl assumed that consciousness (leaving aside the question of what

consciousness itself is) moves in Minkowski spacetime where no motion is possible” [57] p.

150 italics in orig.]

and

“But when we want to understand how we can have the feeling that time flows in the

Minkowski four-dimensional world, it appears that Weyl’s proposal holds the greatest

promise for the resolution of the apparently insurmountable contradiction between the

physical theory of relativity (and the experiments that support it) and our everyday

experience. Moreover, so far, no one has found a way, which does not involve our

consciousness, to reconcile the spacetime world view and the fact that whatever we

perceive happens only at the present moment

66

,

ll

. [Ref. 66 p.110-11]

So, there is an interesting, albeit seemingly contradictory, possible connection between the

nature of spacetime and consciousness, and thus a need to question what consciousness is.

Let me just ultra-short describe

E.1 The state of the art in research on consciousness

mm

,

67

.

In 19th centaury the success in physics and chemistry inspired the new science of psychology.

But at that time the introspective method trying to find “the psychic atoms” and an analogue to

The Periodic Table in chemistry for one researcher (Edward Tichtener) brought about over

40000 such psychic qualities. And in a reaction to that behaviourism declared we can only

study the objective behaviour and not subjective consciousness. This taboo against research on

consciousness was broken as late as in the 1960:ies by Sperry

68

in the seventies by Libet

69

and

in 1980:ies by researcher as Francis Crick and Christof Koch

70

who wrote an article 1990

urging brain researchers to also study conscious experiences and try to find the neurological

correlates to consciousness (NCC) . 1994 a young unknown philosopher David Chalmers

71

focused the problem by describing the easy problem, mainly behaviour, which we can (more or

less) explain by brain and the hard problem, the subjective experience of e.g. the c1 of a piano

or the red of an apple, which have not even today been explained by the brain. Even those, as

""

!JP:"I"think"memory"is"better"explained"as"experiences"of"earlier"events"."See"Ref."67."

mm

This is discussed in more detail in my dialogue, On the best available evidence for the Survival of Human

Consciousness after Permanent Bodily Death on Researchgate Ref. 67.

!

45!

the now late Crick and also Koch, who was resp. are convinced about that consciousness will

be found in the brain admit that. “No one has produced any plausible explanation as to how the

experience of the redness of red could arise from the action of the brain.”

72

“In fact, there is nothing in the world that we would be acquainted with better than the

subjective experiences vividly present for us all the time. The problem, the absolute mystery, is

elsewhere: we do not know how to fit consciousness together with the world view of

science.”

73

[Ref 73. p. xxii]

Tononi and Koch

74

presented Tononi’s Integrated Information Theory (IIT) at Towards a

Science of Consciousness Conference 2014. They argue that even all correlations between

conscious experiences and events in brain found by new methods of studying living brains, “…

are not enough if we are to understand even basic facts, for example, why the cerebral cortex

gives rise to consciousness but the cerebellum does not, though it has even more neurons and

appears to be just as complicated”. [Ref. 74 abstract] They even stated that the problem is not

just hard but almost impossible if starts with the brain and IIT is a theory that starts with five

axioms about consciousness, which ought to be self-evident and derive five postulates for

structures, which can implement that. I think this is a very needed way. For the situation in

consciousness’ research is much like that in physics in end of 19th century when we had a huge

amount of spectroscopic data and even exact mathematical formulas for them but no theory and

could not even explain the stability of the atom before quantum theory and relativity. I think we

need an equal radical change in our conceptualisation of brain and consciousness. And as in

physics it must be based on observations and experiments. And I think this change is on its

way.

Alva Noë, professor in philosophy at Berkeley Institute for Cognitive and Brain Sciences,

writes:

“It would be astonishing to learn that you are not your brain. All the more so to be told that the

brain is not the thing inside you that makes you conscious because, in fact, there is no thing

inside you that makes you conscious.”

75

A simple observation 1:

I ask you to take an object in your hand and concentrate on your tactile experience of the

object.

Where is your tactile experience of the object, where is your tactile experience of the

object located?

I now ask you to put down the object in front of you but continue to look at it.

!

46!

Where is your visual experience of the object? Where is your visual experience of the

object located?

Take time to look.

If you say, “in my brain or in my eyes”, I must ask you how you then know that there is an

object out there in front of your nose? Georg von Békésy

76

[Ref. 76, p.220-21] claims this

“projection” to be learnt and that it has a survival value. Of course, a survival value! But

how learnt? The neurophysiologist Benjamin Libet

77

[Ref. 77 p. 79] said that this automatic

subjective referral of our sensory experiences to the space is mysterious.

I suggest we take our experiences at face value and that all sensory experiences are located in

space outside the brain. But why is the view so strongly held that consciousness is localised in

the brain (or at least created by the brain?) We have a lot of correlations, but correlations are

never a cause or an explanation. More important for the brain localisation view are our mental

experiences as thoughts, dreams and memory.

Observation 2:

Your physical body is now at the event E2 at (r2, t2) and you can remember the event when

your physical body where at an event E1 at (r1, t1) such as that you still can see the place

ÝÞÄÝo/

But at time t2 you don’t see your physical body at r1. So where is the memory of

the event, when your physical body was at E1, located? Not in space outside brain! So where

can it be besides in the brain? But I claim the memory is the event E1 at (r1, t1) in spacetime.

So, as consciousness can spread in space so it can spread in time.

I also claim that a species who in Darwinian evolution learn to tap the eternal, perhaps even

infinite, spacetime “library” will have an advantage to those species who need to store all

information in the brain, because then the brain can be free to effect our chosen actions.

Riccardo Manzotti rightly argues that

In fact, current identity theories are based on two key claims, of which only one is

BRAINBOUND:

1. Consciousness is identical with a physical phenomenon (PHYSICAL).

2. Consciousness is inside the body or the brain (BRAINBOUND).

The two claims are to a large extent utterly independent. Crucially, only physical is

mandatory for physicalism because it requires that everything is physical

78

. [Ref 78, p. 49]

!

47!

And Manzotti claims, “ one’s experience of an object is the object one experiences” [Ref 78, p.

xii]. So, our sensory experiences are identical with matter in the now, or actually the extended

now also including events in the past light cone. Manzotti also explains that illusions, e.g. of

water in desert is not a wrong perception but a wrong interpretation: even hot sand can reflect

light and look like water. And he explains dreams and hallucinations as “kaleidoscopic

recombinations” of earlier experiences, that is events in the past spacetime. I fully agree but

will add that the mental experiences can be identical to possible events in 6D/5D block

universe, even to possible events not realised in past and possible future events67,

79

.

Of course much more work must be done before this can be seen as a working theory but I

claim it is a possible hypothesis worth studying more as there are no better alternatives (yet)

and no scientific fact against it, only “common-sense” beliefs.

Appendix F A mathematical conjecture

Heuristically, we can imagine that a magician in 2D Flatland would be able to by a trick take

out a flat cat from a flat hat. But we can’t imagine that even a magician in Flatland can get a 3D

cat from a flat hat for seemingly two reasons: First a 3D cat must exist in 3D, which does not

exist in 2D Flatland. But perhaps if 2D Flatland is imbedded in a 3D world? But it is still a

question if even a magician can create something 3D from 2D.

My conjecture is:

A N-dimensional structure can in no sense “create”, “produce”, “emerge” a (N+1)

dimensional structure.67,79

At a conference in Albena May 2018 it was pointed out what could be a counter example to this

conjecture. I have then got further argument against it but also some support.

80

So perhaps the

question is not yet settled, and I still think it is worth studying and of course welcome both

support and more elaborated criticism or even counter examples. But if the conjecture is true it

is of importance both for the question of dimensionality of spacetime and the nature of

consciousness. There are near-death experiences (NDE) where persons claim they “saw” their

life past, present and future at once, which I think is a possible experience of 4D spacetime.67

One also experienced how he could move to different events in this “4D landscape” which I

think support the existence of a flow time besides the 4D block universe. He also in his NDE

had the experience of feeling his brothers pain when he hit him, but which he didn’t

experienced at the time of the earlier event when he actually hit his brother, which I again think

can be seen as a support that experiences exist as different possible objective structures in a

!

48!

5D/6D block universe

81

, 67. So arguably an experience of 5D or 6D and if the conjecture is

correct this cannot be caused by a 4D brain and thus it could be concluded that our world is

more than 4D

nn

,

82

and that conscious is more the brain.67

Appendix G Work in progress

oo

Is becoming possible in six-dimensional Minkowskian spacetime?

(NB references for the abstract in its own list p. 49-51

Change, becoming and “the flow of time” are fundamental human experiences. And they are

also basic in our first encounter with the physical reality, and is expressed in Newtonian

mechanics, which state that time is absolute and same for all, and in Galilean transformation as

t=t’.

But in Einstein’s Special Theory of Relativity experimentally verified kinematical effects show

that space and time are relative, seen e.g. in time dilation and relativity of simultaneity. These

effects were given a physical explanation in Minkowski’s four- dimensional spacetime.

But as no objective physical becoming is possible in Minkowski 4D spacetime, because all

events of spacetime exist equally, this seems to be an “... apparently insurmountable

contradiction between the physical theory of relativity (and the experiments that support it) and

our everyday experience.” [1 p. 110-11][2, 16]

There is no consensus among philosophers about the nature of time. [see e.g. 3]

Recently Stefanov in Space and Time Philosophical problem [4, p. 29] comments that “The

controversy between the A- and B-theorists of time has still not come to an end.”

But Stefanov also present an elaborated-BA theory, based on Baker’s BA-theory [5] and

claims ”Thus, the relativistic picture of the world is reconciled with our clear experience of

time flow”. [4 p. 121] as

“The elaborated BA-theory of time elucidates the connection between time and consciousness

by taking seriously Weyl’s idea about consciousness to be crawling upward along the world-

lines (or better say along the world-tubes) of our bodies. ” [4 p. 121]

nn

Pavsic Ref. 81 has shown that when six-dimensional real spacetime is contracted to 4D, transformations must

be complex and interpret imaginary coordinates as that events observable to one observer is not observable to

another observer, which is difficult to understand if the world is only 4D. This argument for more than 4D is valid

only if it is empirically verified that these difference in observability exists.

"

oo

Abstract accepted (but for personal reason not presented) to Third Hermann Minkowski Meeting on the

Foundations of Spacetime Physics, Albena, 2023 https://minkowskiinstitute.org/old/meetings/2023/

"

!

49!

Still my concern is that even if this elaborated BA-theory is correct and is accepted the question

is if it is the only possible theory explaining all we know about time. For the temporal

asymmetry and the assertion “But the only possible answer is that a continuous series of

consecutive regions of three-dimensional subspaces is given to our consciousness” [4 p. 70]

does not by itself give an answer to the dimensionality of the world.

And still there is no consensus about the nature of time even among physicists:

Presentism: all that is real exist in the now, “in a moment of time”; past and future do not exist;

time flows. [6]

Growing block universe: past and present exist, but not future; time flows. [7]

Eternalism Block universe, four-dimensional spacetime,: past, present and future exist equally,

no physical objective time flow [2, 8, 9, 16]

Neither presentism nor eternalism: becoming and present real locally.[10]

The lack of consensus about the nature of time is strange and seems to me to indicate that there

is some fundamental aspect of reality which is neglected. An example of a fundamental neglect

and which seemingly is related to this dilemma of time is the Einsteinian dogma that there can

be no velocities higher than that of light. Einstein only proved that matter cannot be accelerated

to the velocity of light. But acceleration is not the only means to get speed, which light clearly

shows. It has been speculated to exist superluminal particles. These have been questioned as

leading to paradoxes (like grandfathers’ paradox), but which are arguably solved (at least in

microphysics) [11]. Superluminal causal processes do make the order of cause and effect

relative, which already Einstein 1907 pointed out. But he didn’t see it as a logical problem but

that it conflicts with all our experience so sufficiently proved impossible. [12 English p. 248]

But we now arguably have such phenomena in experimentally verified entanglement.[13]

Another approach to superluminality is that Einstein’s postulates are not only compatible

with but also suggest superluminal LT in two, four and six dimensions. I have analyzed and shown how

many derivations of LT missed that but also shown that Minkowski especially with the approach in his

last lecture 1908 could have discovered superluminal LT and six dimensional spacetime, with three

space and three “timelike” dimensions. I also argue that it is conceivable that he would have done so if

not his untimely death 1909 shortly after the 1908 lecture. [14] The possibility of extra timelike

dimensions seems to me to have a possible relation to the problem of 4D spacetime vs. becoming. A

possibility, which have been suggested are moving spotlight theories [4 pp.56- ] and with versions with

a second time dimension, supertime or metatime. Even if this is strongly criticized by Stefanov [4 p. 58]

and by Dainton as leading to “... a regress that is as absurd as it is infinite:” [15 p.23] I will first argue

!

50!

that an extra time dimension might not lead to infinite regress and secondly if it does it need not to be

absurd, when it is realized that becoming and the flow of time is related to consciousness. [Weyl and

Petkov in 16 and 2, 4]. And perhaps a sixdimensional Minkowskian spacetime construed as a 5D block

universe, a 2D “possibility surface” of many possible 3D worlds with a 1D absolute time is a way to

understand becoming in spacetime.

References to the Appendix G

1. V. Petkov, From Illusions to Reality: Time, Spacetime and the Nature of Reality (Understanding Reality Series

Book 1) Minkowski Institute Press. 2013

2. V. Petkov, Relativity and the Nature of Spacetime, Springer, 2005

3. K. Miller, “Presentism, Eternalism, and the Growing Block.” In: A Companion to the Philosophy of Time.

Edited by Heather Dyke and Adrian Bardon, John Wiley & Sons, Inc. 2013, p. 345-364

4.A. Stefanov, Space and Time: Philosophical Problems Minkowski Institute Press, Montreal 2020

5.L.R. Baker, “Temporal Reality.” In: Time and Identity. Topics in Contemporary Philosophy. Edited by Joseph

Keim Campbell, Michael O’Rourke, and Harry S. Silverstein. Cam- bridge, Massachusetts, London, England: The

MIT Press. 2010

6. L. Smolin, Time Reborn. From the Crisis in Physics to the Future of the Universe. Boston, New York Houghton

Mifflin Harcourt. 2013

7. G.F.R. Ellis, “Physics in the Real Universe: Time and Space-Time”. In [9] pp. 49-79

8. H. Minkowski, Space and Time 1908. New translation in Space and Time: Minkowski’s Papers on Relativity.

Ed. Petkov, V. Minkowski Institute Press Montreal 2012.

9. V. Petkov, ed. Relativity and the Dimensionality of the World. Fundamental Theories of Physics 153, Springer

2007

10. C. Rovelli, ”Neither Presentism nor Eternalism.” Foundations of Physics 49, 1325–1335 (2019).

11. E. Recami, “Tachyon Kinematics and Causality: A systematic Thorough Analysis of the Tachyon Causal

Paradoxes.” Foundations of Physics 17:3, 241-296 (1987)

12. A. Einstein, “Über die vom Relativitätsprinzip geforderte Trägheit der Energie.“ Annalen der Physik 23, 371-

384.(1907). In English at https://einsteinpapers.press.princeton.edu/vol2-trans/252

13. R. Tumulka, “The assumptions of Bell’ proof.” https://arxiv.org/abs/1501.04168v1

14. J. Pilotti “How Einstein and Minkowski missed real valued Lorentz transformations for v>c which

are possible in 2D and in extended special relativity to 6D spacetime (three space three

time)”. Fifth International Conference on the Nature and Ontology of Spacetime 14- 17

May 2018, Abstract http://www.minkowskiinstitute.org/conferences/2018/abstracts/Pilotti.pdf

!

51!

J. Pilotti, “How Minkowski could have discovered six dimensional spacetime.”

Second Hermann Minkowski Meeting on the Foundations of Spacetime Physics 13-16 May 2019, Abstract

http://www.minkowskiinstitute.org/meetings/2019/Abstracts/Pilotti.pdf

J. Pilotti, “How Minkowski Could Have Discovered Superluminal Lorentz Transformations and Six Dimensional

Spacetime.” In Spacetime 1909-2019 see 16. pp. 111-146.

15. B. Dainton, Time and space. 2nd ed. Routledge 2010

16. V. Petkov, “Can there be becoming in spacetime?” In Reinoud Jan Slagter and Zoltán Keresztes (Eds),

Spacetime 1909 – 2019. Selected peer-reviewed papers presented at the Second Hermann Minkowski Meeting on

the Foundations of Spacetime Physics, dedicated to the 110th anniversary of the publication of Minkowski’s paper

“Space and Time,” 13-16 May 2019, Albena, Bulgaria (Minkowski Institute Press, Montreal 2020). pp. 71-84.

https://www.minkowskiinstitute.org/old/mip/books/2019conf.html

Appendix H

Tachyons and tachyonian systems. Generalization of SR and an idea of six-dimensional

space-time. Pilotti J unpublished notes jan1971 Correction 6 march 2018

"(Translation"from"unpublished"notes"in"Swedish"January"1971""Jan"Pilotti"B.Sc*.

pp

"Included"as"it"shows"how"I"

discovered"the"possibility"of"superluminal"LT"in"2D"and"got"the"idea"that"it"could"be"extended"to"6D.)

Introduction*

In!Alva_ ger!et!al!1968!(1)!the!history!and!rational!for!tachyons!are!summarized.!In!short!

the!special!theory!of!relativity!doesn’t!exclude!particles!“born”!with!the!velocities!greater!

than!that!of!light!even!if!the!energy!and!rest-mass!relation!have!a!singularity!for!v=c!and!

this!precludes!ordinary!particles!to!reach!v=c!or!v>c!as!well!as!precludes!tachyon!to!have!

v<c!or!v=c.!

"

In(1)!they!use!

I$ »h ;

(

r•:;

<;!

even!for!v>c!and!rearrange!it!to!

I$ ßh ;

(

:;

<;•r

!where!m0=iμ!μ!real!

and!they!argued!that!it!is!no!problem!that!the!tachyons!(seems!to)!have!imaginary!rest-

mass!as!they!never!can!be!at!rest!in!our!systems,!so!it!is!not!directly!measurable.!Seems!

possible!and!perhaps!good!enough!for!first!attempt.!But!a!little!ad!hoc?!But!actually,!it!is!

not!very!clear!how!to!handle!tachyons!theoretically!or!experimentally!and!I!think!this!is!

due!to!the!singularity!at!v=c.!We!are!not!entitled!to!use!the!old!views!on!particles!when!

trying!to!visualize!tachyons.!

pp

At that time undergraduate student in theoretical physics at University of Stockholm Sweden

!

52!

Generalization*of*the*principle*of*relativity*

There!is!yet!another!approach!which!seems!to!be!more!in!line!with!the!spirit!of!the!special!

theory!of!relativity!and!therefore!perhaps!less!ad!hoc.!

If!tachyons!exist!I!think!it!is!conceivable!that!many!tachyons,!which!have!the!same!

velocity!v* C

&

C

D+

!relative!some!ordinary!inertial!system!(OIS)!S!,!would!all!be!at!rest!

relative!each!other!and!therefore!it!is!conceivable!to!think!of!a!tachyonian!rigid!system!

(TIS)!tSv*with!the!velocity!v*relative!S.!And!also!think!that!there!are!tachyons!with!all!the!

different!characteristics!as!ordinary!matter!besides!having!v*>c!relative!an!OIS.!So!it!will!

be!conceivable!that!there!is!tachyon!matter

00

"and!we!can!build!tachyon!clocks!(e.g.!a!light!

clock!where!light!is!reflected!between!two!mirrors).!

In!our!world,!“world!1”!and!for!C

x

C

K+

!we!have!the!ordinary!LT!between!S!and!S’!in!

standard!configuration!!

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"#$! "%&'

(

)%&*

+*!,!!!!!-#$-,!!!!.#$./!!!!!!!'#$'%&"

+*

(

)%&*

+*,!!!

In!a!world!which!only!consists!of!tachyons,!world!2,!we!can!as!a!first!trial!generalize!the!

ordinary!postulates!and!even!for!tachyon!world!postulate:*

I. the*special*principle*of*relativity*and!!

II. that*the*velocity*of*light*is*the*same*in*all*TIS*=*ct**

and!the!derivation!of!LT!within!the!tachyon!world,!world!2!can!be!done!exactly!as!in!

world!1!!

•#$à•:á

â

r•:;

<&i

;!!!!!!!€#$€!!!!!!•#$•!!!!!!!!!!‚#$á•:j

<i

;

â