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).
First review the orginal(1843) quaternion structure for definiteness, Hamilton's creation, the real quaternion ring H.
These facts make H a potent catapult for the imagination concerned with R^{4} .
Proposition : q^{2} = −1 ⇔ q* = −q and q q* = 1 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 } .
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 R^{4} from 1 : It was Hermann Minkowski in 1908 who presented M as spacetime, using 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.
Proposition 1: Suppose m = u + v r + w s where r and s are motors satisfying 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.
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.
Proposition 2: {q ∈ M : q q* = 1 } = { r exp(ar) : r is a motor and a ∈ R } Recall the motor plane {z: z = x + y j } with norm zz* = xx – yy has a countercircle {z: zz* = −1 }. Proposition 2 says that the countersphere is the union of the countercircles of the motor planes in M .
