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Sketch on quaternionic Lorentz transformations

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Abstract

Lorentz transformations are decomposed into a linear combination of two orthogonal transformations. In this way a two-term expression of 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 discussed.
Sketch on quaternionic Lorentz transformations
Mikhail Kharinov
Abstract. Lorentz transformations are decomposed into a linear combina-
tion of two orthogonal transformations. In this way a two-term expression of
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 oc-
curs when rotational axis is orthogonal to velocity direction is obtained. The
accompanying relativistic velocity addition is discussed.
Introduction
W.R. Hamilton had discovered quaternions in the 19 century in order best to de-
scribe real four-dimensional spacetime R4, supplied with a cross product [ν, n]of
spatial vectors νand n. The main advantage of quaternions is that they allow work-
ing with linear transformations of 4-dimensional Euclidian space without explicitly
introducing a standard orthonormal basis and matrix representation of a linear op-
erator. The use of quaternion multiplication provides concise calculations. So, the
rotation Vof a 3-dimensional space is elegantly described through multiplication
of quaternions as V{u}=bu¯
b, where: b=i0cos φ
2+νsin φ
2;¯
b=i0cos φ
2νsin φ
2;
i0is the multiplicative unity; νis the unit vector of unit length along rotational
axis, such that (ν, ν) = 1 and φis the rotational angle [1].
In the quaternion space, the Lorentz transformations are expressed only
slightly more complicated than the rotation V.
1. Lorentz transformations in terms of quaternions
The Lorentz transformations Lare defined as a linear transformation of the space
of quaternions u,vthat preserves the real inner product (u, ¯v)of one conjugated
vector ¯v= 2(v, i0)vby another vector u:(L{u},L{v})=(u, ¯v). The Lorentz
transformations Lis decomposed into the two simple transformations Vand Las
in [2], so that L{u}=±V L{u}or L{¯u}=±V L{¯u}. For brevity, only one option
2 Mikhail Kharinov
L{u}=V L{u}is treated, where Lis the Lorentz boost. It is screw-symmetrical:
(L{u}, v) = (L{v}, u)for any u, v.
L{u}is expressed in quaternions as:
L{u}= ¯aua n¯usinh θau¯a¯un sinh θ, (1)
where a=i0cosh θ
2+nsinh θ
2and θis the rapidity, such that the velocity vector
vdivided by scalar speed of light cis expressed as v/c=ntanh θ.
It is noteworthy that in [3, 4] the Lorentz boost is described by the trun-
cated formula (1). But this is achieved only due to extra dimensionality, which
complicates interpretation.
The dual expression (1) for Land simple quaternion multiplication rules [1, 5]
provide easy operation with the Lorentz transformations L=V L in a coordinate-
free way. It is a good exercise to obtain eigenvectors ckfor the transformation
L=V L:L{ck}=ξkck, where ξkare real eigenvalues and eigenvector sequence
number kstarts from 0 and does not exceed 3.
2. Eigenvectors in the general case
It is trivial that the eigenvectors of the transformation L=V L, corresponding to
eigenvalues other than 1, are pseudo-orthogonal to themselves and to the invariant
eigenvectors that correspond to ξ= 1, e.g. for ξ0̸= 1, ξ1̸= 1, ξ3=ξ4= 1 the
formulae (c0,¯c0)=(c1,¯c1)=(c0,¯c2)=(c0,¯c3)=(c1,¯c2)=(c1,¯c3)=0are valid.
A concomitant fact is that the eigenvalues are pairwise mutually inverse due to
the invariance of the equation for ξwith respect to the replacement of ξby ξ1.
For easy finding of real eigenvalues, it is convenient to present the general
equation for ξas follows:
(ξξ0)(ξξ1
0)[ξ2+ξ(2xαβ) + 1] = 0,(2)
where x, which must be outside the interval (0,1), is found from the equation:
x2xα+β
2+αβ
21 = 0,(3)
ξ0is found from the equation
ξ0+ξ1
0
2=x(4)
and α= (cosh θ+ 1)(1 + cosφ)0, β = (ν, n)2(cosh θ1)(1 cos φ)0.
Concerning the latters it should be noted that in the expressions for αand β, the
values of θand φare assumed to be non-trivial and variative, i.e. both are not
fixed for given rotational axis νand velocity direction n.
With positive αand βthe equation (3) for xhas at least one required solution
x > 1. In this case, a pair of mutually inverse eigenvalues other than 1 is available.
For each eigenvalue ξ̸= 1 the corresponding eigenvector is represented as i0d,
Sketch on quaternionic Lorentz transformations 3
where (d, i0) = 0,(d, d) = 1. In turn, the components of the vector dare calculated
by the formulae:
(d, ν) = (ν, n)cosh θ1
sinh θ
ξ+ 1
ξ1,(d, n) = ξcosh θ
sinh θ,
(d, [ν, n]) = [(d, n)(ν, n)(d, ν)] ξsin φ
ξcos φ1.(5)
3. The case of velocity, orthogonal to the rotational axis
All four eigenvectors are available in the special case of (ν, n) = β= 0. In this
case, the equation for xbecomes trivial:
(x1) x+ 1 α
2= 0.(6)
In the case of x=α
21the equation for finding ξ0and ξ1=ξ1
0is expressed
by the formula:
ξ2ξ(α2) + 1 = 0.(7)
As follows from the last formula (7), in order to successfully find the target values
of ξ0and ξ1=ξ1
0, the next condition must be satisfied:
sin φ
2
tanh θ
2
cos φ
2
cosh θ
21
tan φ
2
sinh θ
2
,(8)
where pair vertical lines denotes absolute value. In the case of x= 1, we get the
trivial equation for ξ:(ξ1)2= 0 and obtain a pair ξ3=ξ4= 1 of unit values of
ξcorresponding to a pair of invariant eigenvectors.
Explicit expressions for eigenvectors and eigenvalues are listed in the table 1,
wherein ξ0is the solution of (7) under the condition (8).
Notation Eigenvector Eigenvalue
c0i0n+ [ν, n]ξ0sin φ
ξ0cos φ1ξ0cosh θ
sinh θξ0
c1i0n+ [ν, n]sin φ
cos φξ01ξ0cosh θ
ξ0sinh θξ1
0
c2i0+n[ν, n] cot φ
2tanh θ
21
c3ν1
Table 1. Eigenvector quartet in the case of (ν, n) = 0
Any vector uis trivially decomposed into a linear combination of the listed
eigenvectors:
u=c0(u, ¯c1) + c1(u, ¯c0)
(c0,¯c1)+c2
(u, ¯c2)
(c2,¯c2)+c3
(u, ¯c3)
(c3,¯c3).(9)
Note that the expansion (9) is available only if the condition (8) is fulfilled.
4 Mikhail Kharinov
Conclusion
Case (ν, n) = 0 is the most important because it is this case that arises in rela-
tivistic addition of velocities, interpreted in terms of Lobachevsky theory [6, 7, 8].
However, according to the authoritative opinion of John Frederick Barrett, “The
hyperbolic theory is not at all new and was described by V. Varicak shortly after
Einstein’s initial work. But it has been ignored now for over 100 years by the main-
stream theory.” Perhaps, the task of obtaining of the eigenvectors for the Lorentz
transformations represented in quaternions, and also in octonions, will be useful
for further development in this direction.
In the following papers it will be shown that the decomposition (9) of any
vector into eigenvectors is available for relativistic addition of velocities. Perhaps,
professional physicists will give a plausible interpretation for this. In any case,
the quaternion technique of working with spatial transformations seems useful for
solving modern engineering problems.
References
[1] I.L. Kantor, A.S. Solodovnikov, Hypercomplex numbers: an elementary introduction
to algebras. Springer, 1989, 166 pp.
[2] P.A.M. Dirac, Application of quaternions to Lorentz transformations. Proceedings
of the Royal Irish Academy. Section A: Mathematical and Physical Sciences 50,
1944/1945, 261–270.
[3] G. Casanova, L’algèbre vectorielle. Paris, Presses de Universitaries de France, 1976,
127 pp.
[4] C. Baumgarten, The Simplest Form of the Lorentz Transformations, arXiv preprint,
arXiv:1801.01840v1 [physics.gen-ph], 21 Dec 2017, 12 pp.
[5] M. Kharinov, Product of three octonions, Springer Nature, Advances in Applied Clif-
ford Algebras, 29(1), 2019, 11–26, DOI:10.1007/s00006-018-0928-x.
[6] J.F. Barrett, Minkovski Space-Time and Hyperbolic Geometry, MASSEE International
Congress on Mathematics MICOM-2015, 2015, 14 pp., available at https://eprints.
soton.ac.uk/397637/2/J_F_Barrett_MICOM_2015_2018_revision_.pdf.
[7] J.F. Barrett, The Hyperbolic Theory of Special Relativity. A Reinterpretation of the
Special Theory in Hyperbolic Space, Southampton, 2006. 109 pp., available at https:
//arxiv.org/ftp/arxiv/papers/1102/1102.0462.pdf
[8] J.F. Barrett, Review of Problems of Dynamics in the Hyperbolic Theory of Special
Relativity, PIRT Conference, Imperial Coll., 2002, pp.17-30, available at https://
arxiv.org/ftp/arxiv/papers/1102/1102.0462.pdf.
Mikhail Kharinov
The Federal State Institution of Science
St. Petersburg Institute for Informatics and Automation
of the Russian Academy of Sciences (SPIIRAS)
St. Petersburg, Russia
e-mail: khar@iias.spb.su
ResearchGate has not been able to resolve any citations for this publication.
Product of three octonions
  • M Kharinov
M. Kharinov, Product of three octonions, Springer Nature, Advances in Applied Clifford Algebras, 29(1), 2019, 11-26, DOI:10.1007/s00006-018-0928-x.
The Hyperbolic Theory of Special Relativity. A Reinterpretation of the Special Theory in Hyperbolic Space
  • J F Barrett
J.F. Barrett, The Hyperbolic Theory of Special Relativity. A Reinterpretation of the Special Theory in Hyperbolic Space, Southampton, 2006. 109 pp., available at https: //arxiv.org/ftp/arxiv/papers/1102/1102.0462.pdf