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Sketch on quaternionic Lorentz transformations

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Abstract

The Lorentz transformations treated as a composition of rotation and Lorentz boost are decomposed into a linear combination of two orthogonal transforms. In this way a two-term expression of the Lorentz transformations by means of quaternions is proposed. An analytical solution to the problem of finding eigenvectors is given. The conditions for the existence of eigenvectors are specified. A quartet of eigenvectors that occurs when rotational axis is orthogonal to velocity direction is obtained. The accompanying relativistic velocity addition is starting to discuss.
M. Kharinov International Conference Polynomial Computer Algebra’2019(PCA’2019) St.Petersburg, April 15-20, 2019 1
Sketch on
Sketch on quaternionic
quaternionic Lorentz transformations
Lorentz transformations
Mikhail Kharinov
St. Petersburg Institute for Informatics and Automation
of the Russian Academy of Sciences (SPIIRAS)
April 17, 2019
M. Kharinov International Conference Polynomial Computer Algebra’2019(PCA’2019) St.Petersburg, April 15-20, 2019 2
Introductory notes on quaternions and octonions
W.R. Hamilton had discovered quaternions in order best to describe four-dimensional
real Euclidian spacetime, supplied with a cross vector product of vectors
Hypercomplex numbers (quaternions and octonions) provide an easy multiplication
of three vectors*).
*) M. Kharinov, Product of three octonions, Springer Nature, Adv. in App. Cliff. Algebras, 29(1), 2019. 11–26, DOI:10.1007/s00006-018-0928-x.
Hypercomplex numbers support the transformation of space regardless of the
coordinate system, otherwise for all coordinate systems including both right and left.
All calculations in terms of quaternions can be expressed by means of the ordinary
cross vector product. The quaternions only provide conciseness.
The generalization of quaternion transformations for octavions is of scientific interest.
vu,., vu
M. Kharinov International Conference Polynomial Computer Algebra’2019(PCA’2019) St.Petersburg, April 15-20, 2019 3
The quaternionic Lorentz transformations
Rotation V of any vector u
, where is the multiplicative unity;
is the unit vector along the
rotational axis; is the rotational angle.
Lorentz boost L
where is the unit vector along the velocity
direction, such that ;
the cross attached to a linear operator means
the Hermitian conjugation: ;
is the rapidity;
Compositional Lorentz transformation L
where the upper lines denote an ordinary conjugation:
and parentheses denote the inner vector product.
Note
For brevity is only treated instead of
The priority USA result
Douglas B. Sweetser, 2010
https://www.youtube.com/watch?v=DrVm1JTM8X4
2
sin
2
cos,}{: 0
ibbbuuVV 0
i
 
,,,: uuuuVL LLL

1,,0, 0
nnin
,shsh: nuaauuLunuaauLL
vLuvuL
,,
n
.2sh2ch
0
nia

uiiuu 00
,2

uVLu L

., uVLuuVLu
LL
M. Kharinov International Conference Polynomial Computer Algebra’2019(PCA’2019) St.Petersburg, April 15-20, 2019 4
The expanded formulae for rotation V
and Lorentz boost L


  

.
2
cos,
2
sin,,
2
sin2cos
,sin,cos,,,,
00
0000
uuiiuuuV
uuiiuuuiiuuV
The expanded formulae for the rotation V
via conventional cross vector product:
Similar equivalent each other formulae for the Lorentz boost L:
where the additional latter formula expresses L
via it’s own eigenvectors.
By changing the variables:
the expressions for L
are converted to the
standard one:
where is the speed of light; is the time; is the coordinate value and is
the signed velocity value.
    
  
,),(
2
),(
2
),(),(}{
,sh,ch,sh,ch,,,
,
2
sh,
2
ch,
2
ch,
2
sh,
2
sh2
0
0
0
0
00
00000
000
niu
ni
eniu
ni
ennuiiuuuL
iununnuiuinnuiiuuLu
nuiunnuiuiuuL

,th,,,, 0cc ,nznutiu
  
22 cc,c ,,n, 1sh11ch



,
1
,
1
,
}{ 00 22 c
v
c
cvc
c,
tnnz
n
,
znt
izntiuuL
c
,n
t
z
M. Kharinov International Conference Polynomial Computer Algebra’2019(PCA’2019) St.Petersburg, April 15-20, 2019 5
Claims on the eigenvalue-eigenvector problem
Rotation V
has two eigenvectors: multiplicative unity i0 and rotational axis
.
Lorentz boost L
has four linearly independent eigenvectors: two vectors orthogonal
to n
and i0 ±n. What about L
=VL
that seems to be non-trivial, since in limiting
cases has a striking difference in the number of eigenvectors? So,
V 2 eigenvectors
L 4 eigenvectors
L
=VL ? Eigenvectors
An attractive goal is to determine the condition for the existence of a quartet
of eigenvectors c0
, c1
, c2
, c3 for L
=VL..
Only target nontrivial and variative values of θ
and
when both are not fixed for a
given axis of rotation
and a direction of velocity n, are discussed as meaningful.
For reliability, each result is to be obtained at least twice in different ways.
M. Kharinov International Conference Polynomial Computer Algebra’2019(PCA’2019) St.Petersburg, April 15-20, 2019 6
Eigenvalues and eigenvectors in the general case
The eigenvalues are found from the characteristic equation for
L
in the
form of the polynomial of the 4th degree:
where and
= 0 for
any θ
and
when (
,n) = 0, i.e. in the special case of orthogonality of rotational axis
and velocity n:
= 0 (
,n) = 0.
Important trivial and easy provable statements:
The problem on eigenvalues and eigenvectors has a solution.
If ξ
is an eigenvalue, then 1/ξ
is also an eigenvalue.
If the eigenvalue is different from 1, then it corresponds to a lightwise eigenvector
having a zero pseudo-length: ξ≠1
For
different from 0, all eigenvalues ξ
are different from 1:
0 ξ≠1.
For
different from 0 a full set of target eigenvector quartet is unavailable.

  
,011
2
210121 2
2
0
234
β
.:, 1cVcLcccVL
LL
 
 
,0cos11ch,,0cos11ch 2
n
.0,
cc
M. Kharinov International Conference Polynomial Computer Algebra’2019(PCA’2019) St.Petersburg, April 15-20, 2019 7
Eigenvectors in the general case
In general case the solution of characteristic equation is provided by means of
parameterization with x:
where x>1 is determined from
Two mutually inverse ξ
are found from
For each of the mutually inverse eigenvalue
ξ
the eigenvector of the form i0 d
is found, where
(d, i0
)=0, (d, d)=1. In this case, the components of d
are determined from the formulae:

 
.0121
,:,
234
1
cVcLcccVL LL

,01
2
212 22
xx
:01
22
2
xx
4
D
1
1
22
2
xx
4
x
16
D
22 22
D
.012
2
x




.
1cos
sin
,,,,,
,
shch
,,
1
1
sh 1ch
,,
dnndnd
ndnd
M. Kharinov International Conference Polynomial Computer Algebra’2019(PCA’2019) St.Petersburg, April 15-20, 2019 8
Nota-
tion Eigenvector Eigen-
value
Case of (
, n)=0. The eigenvector quartet and existence condition



1
1
2
th
2
ctg,
1
sinh
cosh1
cos
sin
,
sinh
cosh
1cos
sin
,
3
02
01
00
c
nnic
nnic
nnic
Table A: Eigenvector quartet
 
.0cos11ch
,011
2
2: 2
Existence condition
2
sh
2
tg1
2
ch
2
cos
2
th
2
sin

.
2
sh
2
ch
,sh
,
2
sin
2
cos
,
0
0
nia
unuaauL
ib
bbuuV

.011
2
21
,:,
2
2
1
cVcLcccVL LL
M. Kharinov International Conference Polynomial Computer Algebra’2019(PCA’2019) St.Petersburg, April 15-20, 2019 9
Vector expansion via the eigenvector quartet for the case of (
, n)=0
   
  
 

1,
0,
2
sin
2
th
1,
0,0,0,
0,0,
2
th
2
sin
12,0,
33
32
2
2
22
312111
3020
2
2
1000
cс
сссс
сссссс
сссссссс
Table B:
Pseudo-scalar inner products with
the scheme of none-zeroes
Presentation of any vector u as the linear combination of the eigenvectors



33
3
3
22
2
2
10
0110 ,
,
,
,
,,,
cc
cu
c
cc
cu
c
cc
cuccuc
u
M. Kharinov International Conference Polynomial Computer Algebra’2019(PCA’2019) St.Petersburg, April 15-20, 2019 10
Nota-
tion Eigen-
vector Eigen-
value Notation Eigenvector Eigenvalue

1
1,
3
2
01
00
c
nc
enic
enic
Limiting cases under the (
, n)=0 condition
Table C: Eigenvector quartet for L
Table D: Eigenvector duplet for V
in the case of (
, n) =
= 0 in the case of (
, n) =θ
= 0

.011
2
21
,:,
2
2
1
cVcLcccVL LL

 
}.,{),,(}{),(),(),(}{
,),(
2
),(
2
,),,(),(}{
00
0
0
0
0
nVnunVnuuiiuuV
niu
ni
eniu
ni
ennuuuL
1
1
existtdoesn'
existtdoesn'
3
02
1
0
c
ic
c
c
Expansion of transformations using the eigenvectors
M. Kharinov International Conference Polynomial Computer Algebra’2019(PCA’2019) St.Petersburg, April 15-20, 2019 11
Case of (
, n)=0. The composition of Lorentz boosts L1 L2
=VL *).
*) In accordance with the papers of John Frederick Barrett and Abraham Albert Ungar
,:, 1
212121 cLcLccLLcVLLL
L





1,
1
,1 2
cth,
2
cth
,1 2
cth,
2
cth
value -Eigen
rEigenvecto
tion -Nota
213
2
21
1
21
2
2
2
21
2
21
1
10
2
nnc
nn
nn
n
nn
nn
ni
c
Table E: Invariant eigenvectors for L1
L2
The composition of Lorentz boosts L1 L2 =VL
 





,
,1
,
,
,1
,
2
cth
2
cth
2
ctg
1chsh,chshshsh
,shsh,chchch
2
21
21
2
21
21
21
212112211
212121
nn
nn
nn
nn
nnnnn
nn

,
2
sinh
2
ch
2
sh
2
ch
2
sh
2
ch
,sh,sh,sh
0
2
2
2
02
1
1
1
01
2222211111
nianiania
unuaauLunuaauLunuaauL
M. Kharinov International Conference Polynomial Computer Algebra’2019(PCA’2019) St.Petersburg, April 15-20, 2019 12
Conclusions
In this report the Lorentz transformations treated as a composition of rotation V
and
boost L
are decomposed into a linear combination of two orthogonal transforms.
In this way a two-term expression of the Lorentz transformations by means of
quaternions is proposed.
The analytical solution to the problem of finding eigenvectors is given.
The conditions for the existence of eigenvectors are specified.
The quartet of eigenvectors that occurs when rotational axis is orthogonal to velocity
direction is obtained.
The accompanying relativistic velocity addition is started to discuss.
Perhaps this report may somewhat complement the prior results of contemporaries:
John F. Barrett (England), Abraham A. Ungar (USA), Douglas B. Sweetser (USA)
and others.
... Section 5 is the main addition to the pilot results, briefly announced in [17]. This section independently solves the problem of obtaining a quartet of eigenvectors for the composition of two Lorentz boosts. ...
Article
Full-text available
In this paper the Lorentz transformation, considered as the composition of a rotation and a Lorentz boost, is decomposed into a linear combination of two orthogonal transforms. In this way a two-term expression of the Lorentz transformation by means of quaternions is proposed. An analytical solution to the problem of finding eigenvectors is given. The conditions for the existence of eigenvectors are specified. The quartet of eigenvectors, which occurs when the rotational axis is orthogonal to the velocity direction, is obtained for two cases: for the generic case of the Lorentz transformation and for the composition of the Lorentz boosts. It is shown that a quartet of eigenvectors exists for the composition of any Lorentz boosts. For the composition of boosts it is established that the half-sum (arithmetic mean) of the square roots of mutually inverse eigenvalues is found by combining half-rapidities of the original boosts according to the same cosine rule, which is used to combine the source rapidities into the resulting rapidity.
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