Quaternions and Special Relativity

[grzz]

TL;DR Summary

Quaternions and Secial Relativity

Is it worthwhile for a retired highschool teacher of Physics to learn the STOR using quaternions?

 

[PeroK]

TL;DR Summary: Quaternions and Secial Relativity

Is it worthwhile for a retired highschool teacher of Physics to learn the STOR using quaternions?

Probably not.

 

[Orodruin]

It depends on how you define your worth. Do you have a particular penchant for quaternions and how they may apply to different settings? If not, then no.

 

[Ibix]

Slight tangent: what's the point of quaternions in SR?

 

[FactChecker]

After my retirement, I got interested in Geometric Algebra. GA theory includes many special cases like quaternions and can methodically condense many physics results into more concise equation forms.

 

[Sagittarius A-Star]

Is it worthwhile for a retired highschool teacher of Physics to learn the STOR using quaternions?

Quaternions can be used for SR, as the following quotes suggest. But I think it does not help to learn SR. More helpful are four-vector and more general four-tensor notation. Differential forms may give addition insight.

Time is said to have only one dimension, and space to have three dimensions. ... The mathematical quaternion partakes of both these elements; in technical language it may be said to be "time plus space", or "space plus time": and in this sense it has, or at least involves a reference to, four dimensions. ... And how the One of Time, of Space the Three, Might in the Chain of Symbols girdled be.
— William Rowan Hamilton (circa 1853)[45]
...
Neither matrices nor quaternions and ordinary vectors were banished from these ten [additional] chapters. For, in spite of the uncontested power of the modern Tensor Calculus, those older mathematical languages continue, in my opinion, to offer conspicuous advantages in the restricted field of special relativity.
— Ludwik Silberstein (1924)[48]

Source:
https://en.wikipedia.org/wiki/Quaternion#Quotations

 

[renormalize]

— Oliver Heaviside (1893)[47]
Neither matrices nor quaternions and ordinary vectors were banished from these ten [additional] chapters. For, in spite of the uncontested power of the modern Tensor Calculus, those older mathematical languages continue, in my opinion, to offer conspicuous advantages in the restricted field of special relativity.

Oops, that's a misattribution, since special relativity was first published in 1905. According to Wikipedia, the quote is actually due to Ludwik Silberstein in 1924. And this more modern quotation is telling:

... quaternions appear to exude an air of nineteenth century decay, as a rather unsuccessful species in the struggle-for-life of mathematical ideas. Mathematicians, admittedly, still keep a warm place in their hearts for the remarkable algebraic properties of quaternions but, alas, such enthusiasm means little to the harder-headed physical scientist.
— Simon L. Altmann (1986)

 

[Vanadium 50]

My question is "Why"?

Is there a single problem in SR that is easier to solve with quaternions?

 

[Sagittarius A-Star]

Oops, that's a misattribution, since special relativity was first published in 1905. According to Wikipedia, the quote is actually due to Ludwik Silberstein in 1924.

Thanks! I corrected the attribution in my citation of Wikipedia.

 

[grzz]

My interest is not to start learning SR using quaternions. But, having learnt a bit of SR using four-vectors, one may feel inclined to see how this can be done using quaternions.
From the above replies it seems that
quaternions were not laid to rest for ever.
For example, did quaternions find some use in computer software?
Thanks

 

[Filip Larsen]

For example, did quaternions find some use in computer software?

Yes, quaternions are quite often used in computer graphics and in general numerical code for spatial calculations as they have some advantages over e.g. 3x3 rotation matrices, e.g. compared to rotation matrices its cheaper to compose quaternions, but more expensive to rotate vectors. But it really depends on what you are going to use it for. It is fairly common for a program that need the benefits of both notations to actually employ both, especially since converting between them is fairly straight forward and somewhat cheap.

Quaternions are also popular in analysis of attitude dynamics and using them makes some derivations possible or at least easier, e.g. Wahba's problem is often solved using quaternions even if the input is given as rotation matrices.

 

[Histspec]

My interest is not to start learning SR using quaternions. But, having learnt a bit of SR using four-vectors, one may feel inclined to see how this can be done using quaternions.

Unfortunately, in order to produce hyperbolic rotations (i.e. Lorentz transformations), ordinary quaternions don't suffice. You have to use complexified quaternions, better known as Biquaternions.

A good introduction was given by the great mathematician John Lighton Synge:
"Quaternions, Lorentz transformations, and the Conway–Dirac–Eddington matrices", Communications of the Dublin Institute for Advanced Studies, 21

The most general Lorentz transformation in terms of angle $\phi$ and rapidity $\eta$ probably looks something like this:

$$\begin{matrix}q'=Qq\bar{Q}^{\ast}\\
\left[Q\bar{Q}=1,\ i=\sqrt{-1},\ \mathbf{U}^{2}=-1,\ \chi=\frac{1}{2}\left(\phi+i\eta\right)\right]\\
\hline \begin{aligned}q & =x_{0}+ie_{1}x_{1}+ie_{2}x_{2}+ie_{3}x_{3}\\
Q & =a+ib\\
& =\left(a_{0}+e_{1}a_{1}+e_{2}a_{2}+e_{3}a_{3}\right)+i\left(b_{0}+e_{1}b_{1}+e_{2}b_{2}+e_{3}b_{3}\right)\\
& =\left(a_{0}+ib_{0}\right)+e_{1}\left(a_{1}+ib_{1}\right)+e_{2}\left(a_{2}+ib_{2}\right)+e_{3}\left(a_{3}+ib_{3}\right)\\
& =\cos\chi+e_{1}u_{1}\sin\chi+e_{2}u_{2}\sin\chi+e_{3}u_{3}\sin\chi\\
& =\cos\chi+\mathbf{U}\sin\chi=e^{\chi\mathbf{\mathbf{U}}}\\
\bar{Q}^{\ast} & =\bar{a}-i\bar{b}\\
& =\left(a_{0}-e_{1}a_{1}-e_{2}a_{2}-e_{3}a_{3}\right)-i\left(b_{0}-e_{1}b_{1}-e_{2}b_{2}-e_{3}b_{3}\right)\\
& =\left(a_{0}-ib_{0}\right)-e_{1}\left(a_{1}-ib_{1}\right)-e_{2}\left(a_{2}-ib_{2}\right)-e_{3}\left(a_{3}-ib_{3}\right)\\
& =\cos\chi^{\ast}-e_{1}u_{1}\sin\chi^{\ast}-e_{2}u_{2}\sin\chi^{\ast}-e_{3}u_{3}\sin\chi^{\ast}\\
& =\cos\chi^{\ast}-\mathbf{U}\sin\chi^{\ast}=e^{-\chi^{\ast}\mathbf{U}}
\end{aligned}
\end{matrix}$$

or solved for the coordinates:

$$\begin{matrix}\begin{array}{lll}
x'_{0} & =x_{0}\left(a_{0}^{2}+a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+b_{0}^{2}+b_{1}^{2}+b_{2}^{2}+b_{3}^{2}\right) & =x_{0}\cosh\eta\\
& \qquad+x_{1}\left(2a_{1}b_{0}-2a_{0}b_{1}-2a_{3}b_{2}+2a_{2}b_{3}\right) & \qquad-x_{1}u_{1}\sinh\eta\\
& \qquad+x_{2}\left(2a_{2}b_{0}+2a_{3}b_{1}-2a_{0}b_{2}-2a_{1}b_{3}\right) & \qquad-x_{2}u_{2}\sinh\eta\\
& \qquad+x_{3}\left(2a_{3}b_{0}-2a_{2}b_{1}+2a_{1}b_{2}-2a_{0}b_{3}\right) & \qquad-x_{3}u_{3}\sinh\eta\\
x'_{1} & =x_{0}\left(2a_{1}b_{0}-2a_{0}b_{1}+2a_{3}b_{2}-2a_{2}b_{3}\right) & =-x_{0}u_{1}\sinh\eta\\
& \qquad+x_{1}\left(a_{0}^{2}+a_{1}^{2}-a_{2}^{2}-a_{3}^{2}+b_{0}^{2}+b_{1}^{2}-b_{2}^{2}-b_{3}^{2}\right) & \qquad+x_{1}\left(u_{1}^{2}\cosh\eta+\left(1-u_{1}^{2}\right)\cos\phi\right)\\
& \qquad+x_{2}\left(2a_{1}a_{2}-2a_{0}a_{3}+2b_{1}b_{2}-2b_{0}b_{3}\right) & \qquad+x_{2}\left(u_{1}u_{2}\left(\cosh\eta-\cos\phi\right)-u_{3}\sin\phi\right)\\
& \qquad+x_{3}\left(2a_{0}a_{2}+2a_{1}a_{3}+2b_{0}b_{2}+2b_{1}b_{3}\right) & \qquad+x_{3}\left(u_{1}u_{3}\left(\cosh\eta-\cos\phi\right)+u_{2}\sin\phi\right)\\
x'_{2} & =x_{0}\left(2a_{2}b_{0}-2a_{3}b_{1}-2a_{0}b_{2}+2a_{1}b_{3}\right) & =-x_{0}u_{2}\sinh\eta\\
& \qquad+x_{1}\left(2a_{1}a_{2}+2a_{0}a_{3}+2b_{1}b_{2}+2b_{0}b_{3}\right) & \qquad+x_{1}\left(u_{1}u_{2}\left(\cosh\eta-\cos\phi\right)+u_{3}\sin\phi\right)\\
& \qquad+x_{2}\left(a_{0}^{2}-a_{1}^{2}+a_{2}^{2}-a_{3}^{2}+b_{0}^{2}-b_{1}^{2}+b_{2}^{2}-b_{3}^{2}\right) & \qquad+x_{2}\left(u_{2}^{2}\cosh\eta+\left(1-u_{2}^{2}\right)\cos\phi\right)\\
& \qquad+x_{3}\left(-2a_{0}a_{1}+2a_{2}a_{3}-2b_{0}b_{1}+2b_{2}b_{3}\right) & \qquad+x_{3}\left(u_{2}u_{3}\left(\cosh\eta-\cos\phi\right)-u_{1}\sin\phi\right)\\
x'_{3} & =x_{0}\left(2a_{3}b_{0}+2a_{2}b_{1}-2a_{1}b_{2}-2a_{0}b_{3}\right) & =-x_{0}u_{3}\sinh\eta\\
& \qquad+x_{1}\left(-2a_{0}a_{2}+2a_{1}a_{3}-2b_{0}b_{2}+2b_{1}b_{3}\right) & \qquad+x_{1}\left(u_{1}u_{3}\left(\cosh\eta-\cos\phi\right)-u_{2}\sin\phi\right)\\
& \qquad+x_{2}\left(2a_{0}a_{1}+2a_{2}a_{3}+2b_{0}b_{1}+2b_{2}b_{3}\right) & \qquad+x_{2}\left(u_{2}u_{3}\left(\cosh\eta-\cos\phi\right)+u_{1}\sin\phi\right)\\
& \qquad+x_{3}\left(a_{0}^{2}-a_{1}^{2}-a_{2}^{2}+a_{3}^{2}+b_{0}^{2}-b_{1}^{2}-b_{2}^{2}+b_{3}^{2}\right) & \qquad+x_{3}\left(u_{3}^{2}\cosh\eta+\left(1-u_{3}^{2}\right)\cos\phi\right)
\end{array}\\
\text{where}\\
\left[a_{0}^{2}-b_{0}^{2}+a_{1}^{2}-b_{1}^{2}+a_{2}^{2}-b_{2}^{2}+a_{3}^{2}-b_{3}^{2}=1,\ a_{0}b_{0}+a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}=0,\ u_{1}^{2}+u_{2}^{2}+u_{3}^{2}=1\right]
\end{matrix}$$

Ordinary Lorentz boosts follow by setting

$$\begin{matrix}\text{3+1 boosts:}\ a_{1}=a_{2}=a_{3}=b_{0}=\phi=0\\
\text{2+1 boosts:}\ a_{1}=a_{2}=b_{0}=b_{3}=\phi=u_{3}=0\\
\text{1+1 boosts:}\ a_{1}=a_{2}=a_{3}=b_{0}=b_{2}=b_{3}=\phi=u_{2}=u_{3}=0
\end{matrix}$$

 

[FactChecker]

For example, did quaternions find some use in computer software?

Quaternions are used in keeping track of orientations of a moving object (like a simulated airplane) in 3-space. The alternative is Euler angles, which suffer from the problem of "gimbal lock". If an airplane points straight up, the Euler angles become multiply defined, whereas quaternions do not.

UPDATE: Quaternions are also used in the inertial reference systems of real fighter airplanes for the same reason.

 

[grzz]

Unfortunately, in order to produce hyperbolic rotations (i.e. Lorentz transformations), ordinary quaternions don't suffice. You have to use complexified quaternions, better known as Biquaternions.

A good introduction was given by the great mathematician John Lighton Synge:
"Quaternions, Lorentz transformations, and the Conway–Dirac–Eddington matrices"

Is it possible for somebody to send me p26 of the above book by J L Synge because this page in my copy is very faint.
Thanks

 

[Frabjous]

Is it possible for somebody to send me p26 of the above book by J L Synge because this page in my copy is very faint.
Thanks

Since your copy is from their website, you might try contacting the library at the Dublin Institute for Advanced Studies directly.

 

[Histspec]

Is it possible for somebody to send me p26 of the above book by J L Synge because this page in my copy is very faint.
Thanks

On p. 26 we find the inverse of formula 5.17:

(5.18) $q=(1-J)q'(1+J^{\ast})$

and the most familiar form of the Lorentz transformation:

(5.19) $x'=\gamma(x-vt),\ y'=y,\ z'=z,\ t'=\gamma(t-vx),\ \gamma^{-2}=1-v^{2}$

which is shown to be included in 5.16 on p. 25 by setting $I=e_{1}$, thus

(5.20) $q'=(\cos\chi+e_{1}\sin\chi)q(\cos\chi^{\ast}-e_{1}\sin\chi^{\ast})$

and by substituting

(5.21) $\begin{aligned}q= & xe_{1}+ye_{2}+ze_{3}+it\\ q'= & x'e_{1}+y'e_{2}+z'e_{3}+it' \end{aligned}$

the formulas on p. 27 follow.

 

[grzz]

On p. 26 we find the inverse of formula 5.17:
...

A big THANK YOU!
I am trying to read J L Synge book, though sometimes I am finding it rather difficult to follow.

 

[mma]

TL;DR Summary: Quaternions and Secial Relativity

Is it worthwhile for a retired highschool teacher of Physics to learn the STOR using quaternions?

I don't know. But I can show, how quaternions are related to Lorentz transformations. Quaternions are just one character in the whole story. The story is about the $2\times 2$ complex matrices, $M_{2,2}(\mathbb C)$. They can be classified as follows.

• traceles Hermitian matrices ($m^*=m, \mathrm{trace}(m)=0$)
• traceles anti-hermitian matrices ($m^*=-m, \mathrm{trace}(m)=0$)
• real scalar matrices ($m= rI \quad (r\in\mathbb R)$)
• imaginary scalar matrices ($m= irI \quad (r\in\mathbb R)$)

The traceless Hermitians are the real linear combinations of the Pauli matrices. They form a 3-dimensional real vector space $\mathfrak E_3$, which is an Euclidean vector space with Euclidean norm $\|m\|^2=-\mathrm{det}(m)$

The traceless anti-Hermitians form the well-known algebra $su(2)$. $su(2)$ happens to be equal to $i\mathfrak E_3$

Quaternions are the traceless anti-Hermitians and the real scalar matrices together, i.e, $\mathbb H= su(2)\oplus \mathbb RI$. It is a 4-dimensional (real) Euclidean space with Euclidean norm $\|q\|^2=\mathrm{det}(q)$ Quaternions with determinant (norm) 1 form the group $SU(2)$ (with the matrix multiplication as group operation) which is the double cover of the 3-dimensional rotation group, $SO(3)$, i.e. $SU(2)$ is a 3-sphere in the Euclidean space of quaternions.

The traceless Hermitians and the real scalar matrices together, i.e. $\mathfrak E_3\oplus\mathbb RI$ form a Minkowski-space of signature (1,3) with pseudo-norm $\|m\|^2= \mathrm{det}(m)$)

The traceless anti-Hermitians and the imaginary scalar matrices together $su(2)\oplus i\mathbb RI$ form a Minkowski-space of signature (3,1) with pseudo-norm $\|m\|^2= \mathrm{det}(m)$)

All $2\times 2$ complex matrices with determinant 1 form the group $SL(2,\mathbb C)$, i.e. $SL(2,\mathbb C)\subseteq\mathbb H\oplus i\mathbb H$. $SL(2,\mathbb C)$ is the double cover of the restricted Lorentz group $SO^+(3,1)$. This group consists of the proper orthocronous Lorentz transformations. So, complex multiples of quaternions contain a group that doubly covers the group of all restricted Lorentz transformations. But the other participants of the story are also interesting.

 

[Vanadium 50]

Is there a single problem in SR that is easier to solve with quaternions?

I ask again.

Sure you can. You can also work entirely in Roman numerals. Doesn't make it a good idea.

 

[mma]

Is it possible for somebody to send me p26 of the above book by J L Synge because this page in my copy is very faint.
Thanks

Here is p26.

 

[grzz]

Here is p26
[/QUOTE]
A big THANK YOU to you too, mma.