Alexander Macfarlane and the Ring of Hyperbolic Quaternions

It was 1891 when Alexander Macfarlane first put forward the idea of a mutated quaternion ring built on motor planes.The provocative idea generated a vigorous discussion sometimes called "The Great Vector Debate". The faults of the hyperbolic quaternions showed how important it is to have good algebraic formulations. The discussion in the 1890's spurred on the development of a vocabulary of linear algebra, vector analysis, differential geometry and relativity that we have now. Today we view hyperbolic quaternions as a false start in spacetime theory, a start that goes astray algebraically, but is useful for examining subsequent structures like biquaternions, Minkowski space, and coquaternions. Literature references for Macfarlane's work, response, and context are given on his Homepage (see link below).
Warning: The ring of hyperbolic quaternions has a non-associative multiplication.

First review the orginal(1843) quaternion structure for definiteness, Hamilton's creation, the real quaternion ring  H.
This linear 4 - algebra has basis  { 1, i , j , k }  whose elements satisfy

ij = k = −ji ,  jk = i = −kj  ,  ki = j = −ik   and    i2 = −1   , j2 = −1  ,  k2 = −1.

These facts make H a potent catapult for the imagination concerned with R4 .
We take a generic quaternion   q = t + xi + yj + zk
which has conjugate   q* = t – xi – yj – zk.
Then  q q* = t2 + x2 + y2 + z2   is called the  norm-square  or  modulus  of q. The fact that the modulus is non-negative gives   H   its Euclidean nature. In contrast, the modulus of biquaternions, hyperbolic quaternions, and coquaternions does not stay non-negative and the topology of these other quaternion-type rings is then, non-Euclidean, as the persistent student learns.

Proposition : q2 = −1 ⇔ q* = −q  and   q q* = 1
proof: Sufficiency is easy ; for necessity note that   q2 = −1 implies
  q−1 = −q   and   q2 (q*)2 = (−1)2. Then   1 = q q*   and   q* = q−1 = −q .

Say that   V = { q ∈ H : q* = −q }   is the “vector part of H” . Clearly  q = xi + yj + zk ∈ V  , so   V  has three dimensions. It is the sphere of radius 1 in   V   that corresponds to the square roots of   −1   in the above proposition. It is called the sphere of right versors in H . In this way the   i , j , k   lose their special place within   H   ; any triple in   V   of mutually perpendicular elements, correctly oriented, is equivalent to  { i , j , k }  .

The Innovation

Macfarlane knew about James Cockle's work with an alternative complex plane. The idea had been reviewed by William Kingdon Clifford. For his hyperbolic quaternions he traded in versors   i , j , k   for motors   i , j , k  . In his space M of hyperbolic quaternions the vector subspace V ⊂ M has a sphere   S   of motors. Then in   M   the surface   exp ( V )   evolves as a hyperboloid in R4 from 1 :

q = cosh a + r sinh a , a ∈ R , r ∈ S where r2 = +1 .
It is traditional to denote   exp V ⊂ M   by   H3   for it is a model of Lobachevski – Bolyai space. It is a three-dimensional metric space of the “hyperbolic variety”, in the terminology of Felix Klein, which has become standard. Another aspect of this model of H3 is its connection with Carl Weierstrauss. In 1885 Wilhelm Killing, one of his students, presented this model in a book on hyperbolic geometry. The expresion   cosh a + r sinh a   is sometimes called “Weierstrauss coordinates” and the model itself, the “Weierstrauss model”.It was part of the mathematical folklore in Berlin and with students from there who taught in Chicago and Austin, for instance.
It was Hermann Minkowski in 1908 who presented   M   as spacetime, using
qq* = t2 – x2 – y2 - z2 ,
a quadratic form, letting the objectionable non-associative multiplication fall away as if irrelevant. Oliver Heaviside published the first volume of Electromagnetic Theory in 1893; it delved deeply into quaternion theory in the opening chapter. This turn of mathematical physics, where Heaviside seems to consider a variety of vector and quaternion possibilities, shows how abstract algebra has been put to work by the need to build a coherent electric and magnetic theory. But Heaviside could not reach the mathematical audience that Macfarlane could. Heaviside even said, "The quaternion system is topsy-turvy; you never know how things will turn out!"

The 3D hyperboloid model of hyperbolic space is important as a model of velocity space in special relativity. Indeed, any point on the hyperboloid represents a frame of reference having a particular velocity with respect to the waiting frame which corresponds to the point 1 + 0i + 0j + 0k . Thus the hyperboloid represents the "standard moment future" (SMF) from the origin in spacetime.

Operator Products in Hyperbolic Quaternions

Proposition 1: Suppose   m = u + v r + w s   where   r   and   s   are motors satisfying
rs + sr = 0. Then   exp(ar) m exp(ar) = (u + vr) exp(2ar) + w (cosh 2a) s .
proof: As exp (ar) lies in the motor plane of r , which is commutative, it is sufficient to consider the operator on the term ws.
Since rsr = s, we have exp(ar) s exp(ar) = (cosh a + r sinh a) s (cosh a + r sinh a) =
s cosh2a + (rs + sr) cosh a sinh a + rsr sinh2a = (cosh 2a) s .

This proposition suggests a stretch by cosh 2a in any direction s orthogonal to r ; the untransformed space feels a contraction relative to this stretch. G.F. Fitzgerald (1892) is credited with the first perception of such a contraction in spacetime operator theory.

Exercise: | exp(ar) exp(bs) |2 = 1 - 2 sin2θ sinh2a sinh2b
  where  θ   is the angle between   r   and   s.
Hint: The scalar part of rs is cos θ .

Note that under some circumstances the quantity in the exercise vanishes.
In other words, the elements exp(ar), which necessarily lie on the unit sphere of the hyperbolic quaternions, may have products with norm zero. Such a product element fails to have a multiplicative inverse. Thus the set U of invertible hyperbolic quaternions is not closed under multiplication. This fact means that (U,·) is not a group, not even a groupoid. Hence, apart from Proposition 1 and the above comment, the hyperbolic quaternions have been banished to the periphery of transformation theory, yet they stand behind Minkowski space as quaternions stand behind vector analysis.

The Counter-sphere in Hyperbolic Quaternions

Proposition 2: {q ∈ M : q q* = -1 } = { r exp(ar) : r is a motor and a ∈ R }
proof: Suppose   q q* = -1 and q = t + p where p = xi + yj + zk.   Then -1 = (t + p)(t – p) = tt – pp.
There is a Hyperbolic angle a such that tt = sinh2 a and pp* = −cosh2a.
Let  r  = p / (cosh a) .
Then r2 = − r r* = − p p* / cosh2 a = 1  so that  r  is a motor.
Then   r exp(ar) = r(cosh a + r sinh a ) = sinh a + r cosh a = t + r cosh a = t + p = q .
The reverse inclusion merely involves computing the norm of r exp(ar) when r is a motor.

Recall the motor plane {z: z = x + y j } with norm zz* = xx – yy has a counter-circle {z: zz* = −1 }. Proposition 2 says that the counter-sphere is the union of the counter-circles of the motor planes in M .

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