LXXVI. Quaternionic Form of Relativity.

L. Silberstein, Ph.D., University Lecturer in Natural Philosophy, Rome Communicated by Dr. G. F. C. Searle, F.R.S.

Phil. Mag. S. 6, Vol. 23, No. 137 (May 1912), 790-809.

It has been remarked by Cayley,^{1} as early as in 1854, that the rotations in a four-dimensional space may be effected by means of a pair of quaternions applied, one as a prefactor and the other as a postfactor, to the quaternion q whose components are the four coordinates of a space-point, say

q′ = aqb,  (1)

where in the case of pure rotation a and b must of course be either unitquaternions or at least such that T^{2} a. T^{2} b = 1; T denoting the tensor.

On the other hand, it is widely known that the so-called Lorentz-- transformation of the union of ordinary space (x, y, z) and time (t), which is the basis of the modern theory of Relativity, corresponds precisely to a (hyperbolic) rotation of the four-dimensional manifoldness (x, y, z, t), or of what Minkowski called the “world.”

Hence the obvious idea of representing explicitly the Lorentz-transformation in the quaternionic shape (1),— which, together with some allied questions, will be the subject of the present paper.

To solve this simple problem we have only to write down the well-known relativistic transformation, i.e., the formulæ of Einstein, then to develop the triple product in (1) and to compare the two.

1A. Cayley Phil. Mag. vol. vii. (1854), and Journ. f. reine u. angew. Mathem. vol. 50 (1855); or ‘Papers,’ vol. ii. Cayley limited himself to the elliptic, i.e. real, rotations, but the extension to the hyperbolic and parabolic cases was an obvious matter. For the whole subsequent literature of the subject, see the article of E. Study in the Encyclopédie d. Sc.

Math., tome i. vol. i. fascicule 3, p. 452; Paris and Leipzig, 1908. See also F. Klein and A.

Sommerfeld’s work Ueber d. Theorie des Kreisels, iv. pp. 939-943; Leipzig, 1910. It was in fact a general hint at Relativity made by these authors on p. 942 that, after I had a whole year tried in vain a great variety of quaternionic operations for relativistic purposes, suggested to me the choice of the particular form (1).

For our purpose it will be most convenient to put Einstein’s formulæ at once in vector form, eliminating thus the quite unessential choice of the axes of coordinates. Let the vector v = vu denote the uniform velocity of the system S′(x′, y′, z′, t′) relatively to the system S(x, y, z, t).^{2} Let O, O′ be a pair of points in S and S′, respectively, which coincide with one another for t = t′ = 0. Call r (= xi + yj + zk) the vector drawn in S from O as origin, and r′ the corresponding vector in S′, drawn from O′ as origin. Then the transformation in question may be stated as follows:—The component of r′ normal to the velocity v is equal to that of r, i.e.

r′ − (r′u) u = r − (ru) u,  (α)

whilst the component of r′ taken along the direction of motion is altered according to the formula

r′u = γ [(ru) − vt] ,  (β)

where γ = (1 − β^{2})^{−1/2}, β = v/c < 1, c = velocity of light.^{3} Finally, the time is transformed according to

t′ = γ [t − \frac{1}{c^{2}} (rv)] .

To get the resultant r′ take the sum of (α) and of (β) × u. Then write, for the sake of subsequent convenience,

ℓ = ıct, ı = \sqrt{−1},

and similarly ℓ′ = ıct′.

Thus, the relativistic formulæ will become

\begin{matrix}r′ & = r + (γ − 1)(ru)u + ıβγℓu \\ ℓ′ & = γ [ℓ − ıβ(ru)] , \end{matrix} \}  (2)

2u being a unit-vector in the direction of motion of S′ relatively to S and v the absolute magnitude of its velocity.

3In these and in all following formulæ (ru), generally (AB), means the modern scalar product of the vectors A, B, that is to say AB cos (A, B); hence (AB) is the negative scalar part of the complete Hamiltonian product, AB:

(AB) = − S AB.

On the other hand, the modern vector product V AB is identical with Hamilton’s V AB.

quite independent of any system of coordinate-axes.

Now, to obtain the required quaternionic representation (1) of the whole transformation (2), let us introduce the quaternion

q = r + ℓ = r + ıct,  (3)

and similarly

q′ = r′ + ℓ′ = r′ + ıct′.  (3′)

Then the problem will consist in finding a pair of quaternions a, b such that

r′ + ℓ′ = a (r + ℓ) b,

and will be solved by developing the right side of this equation.

Having done this, explicitly, and compared with (2), I found immediately that the quaternions a, b can differ from one another only by an ordinary scalar factor, and since this may be distributed equally among a, b (their tensors entering only by the product), we may as well take simply equal a, b, say, both = Q. In fact, then, the form (1) is much too general for our purpose. Thus, to spare the reader any superfluous complication, let us at once seek for

q′ = QqQ  (1a)

as the quaternionic equivalent of (2).

Denote the unknown vector and scalar parts of Q by w and s respectively, i.e. write

Q = w + s  (4)

Then, developing the complete product of q, Q, by (3) and (4), and by the fundamental rules of Hamilton’s Calculus,

qQ = V rw + ℓw + sr − (rw) + sℓ,

and similarly

q′ = QqQ = V w V rw − w(rw) + 2sℓw + s^{2}r − 2s(wr) + (s^{2} − w^{2})ℓ
= (w^{2} + s^{2})r − 2(rw)w + 2sℓw + (s^{2} − w^{2})ℓ − 2s(rw),

whence, splitting into the vector and scalar parts,

\begin{matrix}r′ & = (w^{2} + s^{2}) r − 2 (rw) w + 2sℓw \\ ℓ′ & = (s^{2} − w^{2}) ℓ − 2s (rw) \end{matrix} \}  (5)

Comparing this with (2), we get at once, as the conditions to be fulfilled by w, s,

\begin{matrix}w^{2} + s^{2} = 1; s^{2} − w^{2} & = γ; 2sw = ıβγ \\ w & = wu. \end{matrix} \}  (6)

Hence w = ±\sqrt{(1 − γ)/2}, s = ±\sqrt{(1 + γ)/2}, where, to satisfy the third of the conditions (6), we must take both square roots with the upper or both with the lower sign; therefore

Q = ± (\sqrt{(1 + γ)/2} + u\sqrt{(1 − γ)/2}) ,

and since in (1a) the quaternion Q appears twice, the choice of the ± sign becomes indifferent.

Thus, we obtain finally the required quaternionic expression of the relativistic transformation

\begin{matrix}q′ & = QqQ \\ with  Q & = \frac{1}{\sqrt{2}} (\sqrt{1 + γ} + u\sqrt{1 − γ}) , \end{matrix} \}  (I.)

u being a unit vector in the direction of motion of S′ relatively to S.

Observe that γ = (1 − v^{2}/c^{2})^{−1/2} > 1, so that the vector of Q is imaginary, whilst its scalar is real.

The tensor of Q is 1; thus denoting its angle by α, i.e., writing

Q = cos α + u sin α = e^{αu},  (7)

we have, by (I.),

cos α = \sqrt{(1 + γ)/2}, sin α = \sqrt{(1 − γ)/2}.

Hence

sin 2α = \sqrt{1 − γ^{2}} = ıβγ = \frac{ıβ}{\sqrt{1 − β^{2}}}

or

2α = arctg (ıβ) = arctg (ı\frac{v}{c} ) .  (8)

Now this is precisely the (imaginary) angle of rotation in the plane t, x,^{4} of Minkowski’s four-dimensional world, corresponding to the transformation (2). Hence, by (I.) and (7), we may say that one half of this rotation is effected by Q as a prefactor and the other half by the same quaternion as a postfactor.^{5} This circumstance throws a peculiar light on each of our Q’s.

4The axis of x coinciding with u, and x itself being our (ru).

But what we are mainly concerned with is their union, which considered as an operator may be written

ω = Q[ ]Q,  (I.a)

the vacant place being destined for the operand.

We have just seen that this simple operator converts the quaternion q = r + ıct into its relativistic correspondent q′. Our q is equivalent to Minkowski’s “space-time-vector of the first kind” or to Sommerfeld’s “Vierervektor” x, y, z, ℓ. These authors call by this same name any such and only such tetrad of scalars (three real and the fourth imaginary) which transforms in the same way as x, y, z, ℓ,—adding where it is necessary the emphasizing epithet “Weltvector”^{6}. Similarly we could call our q and any covariant quaternion a “world-quaternion”; but possibly the less pretentious name physical quaternion will do as well. Also, at least in the beginning, no further specification of the “kind” is needed.

Thus ω = Q[ ]Q, defined by (I.), or by (7) and (8), is what I should like to call the relativistic transformer of any physical quaternion.

To get the inverse transformer ω^{−1}, viz. that which turns q′ into q, apply to both sides of the equation q′ = QqQ the inverse quaternion Q^{−1} as a preand a postfactor; then, remembering that Q^{−1}Q = QQ^{−1} = 1, the result will be

5At the first sight it might seem that, the axis of Q being u, this quaternion turns r round u, i.e. in the plane y, z normal to u, while in Minkowski’s representation the rotation is in the plane x, t. But this is only an apparent contradiction. In fact,

Qr = cos α.r + sin α. V ur + scalar, that is to say, Q as a prefactor turns the transversal component of r round u by the angle +α and stretches its longitudinal component; similarly Q as a postfactor, besides stretching the longitudinal component of r, turns its transversal component round u by the angle −α, thus undoing the rotatory effect of the prefactor. Hence, what remains in the final result is but a stretching of r’s longitudinal component and a change of ℓ or t, and this amounts precisely to the Minkowskian rotation in the plane x, t.

6H. Minkowski, Die Grundgleichungen für d. elektromagn. Vorgänge in bewegten Körpern, Götting. Nachrichten, 1908; Raum und Zeit, Physik. Zeitschrift, vol. x. (1909), also separatim. A. Sommerfeld, Zur Relativitätstheorie, i. and ii., Annalen d. Physik, vol.

xxxii., xxxiii. (1910).

See also the admirably clear and beautiful book Das Relativitätsprinzip by M. Laue (Braunschweig, 1911), where the whole work of Einstein, Minkowski, and Sommerfeld, together with the author’s own contributions, will be found fully developed.

q = Q^{−1}q′Q^{−1},

or

ω^{−1} = Q^{−1}[ ]Q^{−1},

and since Q is a unit quaternion, its inverse is also its conjugate, i.e. Hamilton’s KQ, which may be more conveniently written Q_{c}; hence

ω^{−1} = Q_{c}[ ]Q_{c},  (I.b)

where Q_{c} = cos α − u sin α. Thus, we see that the inverse transformer is got from the direct simply by changing the sign of the angle α or by inverting the direction of u,—as it must be.

Observe that, since the product of quaternions is distributive, the transformer ω has also the distributive property, i.e., A, B being any quaternions,^{7}

Q [A + B] Q = QAQ + QBQ,  (9)

and consequently, if ∂ be any scalar differentiator, also

Q [∂A] Q = ∂QAQ,

since Q, being constant, is not exposed to ∂’s action. Again, by the associative property of quaternionic products, the dot signifying a separator,

A.QBQ = AQ.BQ,

and so on. For our present purpose we scarcely need a full enumeration of ω’s properties.

In the above we have been concerned with q as an example, or in fact the very prototype, of a physical quaternion. Another example, which will be needed in the sequel, is the quaternionic equivalent of Sommerfeld’s “Viererdichte,” or Laue’s “Viererstrom,” say

C = ρ (ı + \frac{1}{c}p) ,  (10)

7I.e. generally complete quaternions but also, more especially, pure scalars or pure vectors; either simple- or bi-vectors, that is to say real or complex. The heavy type (and this merely to suit the general custom) shall be henceforth used only for pure vectors, both real and complex.

which we may accordingly call the current-quaternion. Here ρ means the volume-density of electricity and p the velocity of its motion relatively to the system S. To prove that C is a physical quaternion, write p = dr/dt, and consequently

C = ıρ\frac{dq}{dℓ} ,  (10a)

and notice that, the charges of corresponding volumes in S and S′ being equal (by a fundamental postulate), dℓ/ρ is itself an invariant of the Lorentztransformation.

The transformer (I.a) may, of course, be applied not only to quaternionic magnitudes, but also to operators, as, for example, to differentiators, which have the structure of a quaternion. If Ω be an operator of this kind, in the system S, and Ω′ its relativistic correspondent in S′, and if Ω′ = QΩQ, we shall say that the operator Ω has the character of a physical quaternion.

As a chief example of such an operator, which also will be needed for what follows, we shall consider here our quaternionic equivalent of Minkowski’s matrix called by him “lor” to the honour of Lorentz. This will simply be the Hamiltonian ∇ plus the scalar differentiator ∂/∂ℓ. Let us denote it by D,

D = \frac{∂}{∂ℓ} + ∇  (11)
= ∂/∂ℓ + i∂/∂x + j∂/∂y + k∂/∂z.

Comparing this with

q = ℓ + r = ℓ + ix + jy + kz,

we see at once that the operator D will transform precisely as q did, i.e.

D′ = Q D Q.  (12)

Thus D has the character of a physical quaternion.

To obtain the above representation of the relativistic formulæ (2) we have introduced the quaternion q = ℓ + r. Now, for this purpose we might as well have used its conjugate, i.e.,

q_{c} = ℓ − r,  (3a)

and the corresponding q′_{c} = l′ − r′.^{8} It may often be convenient to recur to q_{c} and it is therefore of some interest to know how it transforms. Now, a glance at (2) suffices to see that both of these formulæ remain unchanged if, having changed the signs of r, r′ (and leaving ℓ, ℓ′ as before), we change also the sign of u. Thus it is seen that

q′_{c} = Q_{c}q_{c}Q_{c},  say = ω_{c}q_{c}  (I.c)

where

Q_{c} = cos α − u sin α = e^{−αu}.

Now q_{c} has precisely the same office as q, that is to say, (I.) and (I.c) are but two expressions of one and the same thing, namely, of the Lorentztransformation. Hence q_{c} and any quaternion covariant with q_{c} is certainly a physical quaternion as well as q and its covariants.

Thus, the conjugate of a physical quaternion will again be a physical quaternion. If the original transformed as q, its conjugate will transform as q_{c}. If A is covariant with q, then A_{c} is covariant with q_{c}, and vice versa. Speaking of a physical quaternion we shall, when necessary, add the explanation cov. q or cov. q_{c}. But generally, for the sake of shortness, this will be omitted, and any letters, as A, B, a, b, &c., without the subscript _{c} will be used to denote quaternions covariant with q. Observe that, with the above (formal) extension of our original definition, two physical quaternions may be either covariant with one another or not; in the last case we may call them antivariant, one being cov. q, and the other cov. q_{c}. Thus, by the above convention, A, B_{c}, or a, b_{c} will denote pairs of antivariant quaternions, the first in each pair transforming as q, and the second as q_{c}.

The above transformer ω_{c} = Q_{c}[ ]Q_{c}, which by (I.b) becomes simply identical with ω^{−1}, is, of course, distributive, quite in the same way as ω = Q[ ]Q. Thus the sum, or difference, of two mutually covariant (but not of antivariant) physical quaternions will again be a physical quaternion.

The reciprocal of a physical quaternion is also a physical quaternion. For we have

a^{−1} = a_{c} (T a)^{−2} ,

q′q′_{c} = qq_{c}.

8It can be proved immediately that (q_{c})′ = (q′)_{c}. Therefore both may be written simply q′_{c}.

Notice also that the invariance of q’s tensor, T q′ = T q, which follows immediately from (I.) (since Q is a unit quaternion), may be written:

while the tensor T a of a physical quaternion is already known to be an invariant. Notice that a and a^{−1} are mutually antivariant.

Now for the product of physical quaternions. Take any pair a, b of such quaternions. Leave aside ab which transforms in the unmanageable way a′b′ = QaQ^{2}bQ (a, b being torn asunder), and pass at once to the product of antivariant factors, which might perhaps be called the alternating product, say

L = a_{c}b.  (13)

Then L′ = Q_{c}a_{c}Q_{c}.QbQ, whence by the associative property, and remembering that Q_{c}Q = 1,

L′ = Q_{c}LQ.  (13′)

Thus, L is certainly not a physical quaternion of the kind already considered; but since it is transformed in such a simple way and since it has, as will be seen in the sequel, an almost immediate bearing upon relativistic Electromagnetism, it deserves to be considered a little more fully. Consider, then, the conjugate of L. Remember the elementary rule, by which the conjugate of the product of any number of quaternions is the product of their conjugates in the reversed order, i.e. in our case

L_{c} = b_{c}a.  (14)

Now, transforming this, we get in quite the same way as above

L′_{c} = Q_{c}L_{c}Q.  (14′)

Hence we see that

Q_{c}[ ]Q  (II.)

is the relativistic transformer of both R = a_{c}b and its conjugate L_{c}. Similarly,

Q[ ]Q_{c}  (II.a)

will be the transformer of both R = ab_{c} and its conjugate R_{c} = ba_{c}. Thus the behaviour of L and R is characteristically distinct from that of q or of q_{c}.

Without trying as yet to invent for these kinds of quaternions any particular names, let us provisionally call any quaternion which is transformed by (II.) or by (II.a) an L-quaternion and an R-quaternion, respectively.^{9}

9L, R, being initials of left, right, may remind us of the position of that of the two generating factors which (as a_{c} or b_{c}) has the subscript _{c}, i.e. which is cov. q_{c}.

Now, Q_{c}[ ]Q, being the transformer of both L and L_{c}, is also the transformer of their sum and of their difference, i.e. also of the scalar and of the vector parts of the quaternion L separately, s = S L and A = V L. Now, s being a scalar, we have

s′ = Q_{c}sQ = sQ_{c}Q = s,

i.e. s is an invariant. Then

A′ = Q_{c}AQ,

and since Q, Q_{c} are unit quaternions, the tensor of A is another invariant.

Thus, the scalar of any L-quaternion and the tensor of its vector are invariants, while the vector itself is transformed into

V L′ = Q_{c} [V L] Q  (III.)

Or use the form L = σ(cos ϵ + a sin ϵ), where a is the unit of A. Then σ cos ϵ and σ sin ϵ are invariants and consequently also σ and ϵ, so that another form of the last theorem will be:—

The tensor and the angle (or argument) of any L-quaternion are invariants, while its axis is transformed by Q_{c}[ ]Q.

In quite the same way it will be seen that S R is invariant and

V R′ = Q [V R] Q_{c},  (III.a)

or in other words:—

The tensor and the angle of any R-quaternion are invariants, while its axis is transformed by Q[ ]Q_{c}.

If we wish to return to the generating factors a_{c} &c., we can write the above properties

S a′_{c}b′ = S a_{c}b  (15)
V a′_{c}b′ = Q_{c} [V a_{c}b] Q  (16)

and similarly

S a′b′_{c} = S ab_{c}  (15a)
V a′b′_{c} = Q [V ab_{c}] Q_{c}  (16a)

But as a rule it is better to avoid any splitting of quaternions, if we are to expect simplicity and other advantages from the use of quaternionic language.

Now take the product of any number of L-quaternions, say L_{1}, L_{2}, L_{3} &c.; then we see by (II.) that all the internal Q’s and Q_{c}’s as it were neutralize one another, and what is left is only the Q_{c} at the beginning and the Q at the end of the whole chain. That is to say the product of any number of L-quaternions is again an L-quaternion. In quite the same way we see, by (II.a), that the product of any number of R-quaternions is again an R-quaternion.

Notice also that, a being any physical quaternion cov. q (not necessarily that implied in L or in R), aL and Ra are again physical quaternions,^{10} and so are also La_{c} and a_{c}R, namely

aL and Ra cov. q.  (IV.)
La_{c} and a_{c}R cov. q_{c}.  (IV.a)

Thus, the alternating product of any number of physical quaternions (ab_{c}de_{c}...) furnishes us either an L- or R- quaternion or again (biquaternions covariant with) the primary physical quaternions, and never anything more.^{11}

One remark more before leaving this subject. Suppose we are given the equation

bX = a,

in which a, b are cov. q. What is the relativistic transformer of X? To get it, write the given equation X = b^{−1}a and remember that b^{−1} is cov. q. Thus the transformer of X will be the same as for b_{c}a, i.e. Q_{c}[ ]Q. In other words, X will be an L-quaternion,

X = b^{−1}a cov. L.  (17)

This will, of course, be still the case if we have instead of b the above differential operator D, i.e.:

if D X = a, then X is cov. L,  (V.)

or the transformer of X is Q_{c}[ ]Q. For D has the structure of q, and the entire manipulation with the Q’s is done precisely as before, since Q, Q_{c}, being constant in space and time, are not exposed to D’s differentiating action. Similarly it is seen that

10Or more exactly biquaternions (in Hamilton’s sense of the word) transforming like the primary physical quaternions. Cf. p. 808, infra.

11So much as to the alternating products. And as regards the products of covariant factors, like ab, I have not, up to the present, been able to make out any of their possible applications to physical subjects, and shall therefore not consider them here at all.

if D_{c} Y = a_{c}, then Y is cov. R,  (V.a)

or the transformer of Y is Q[ ]Q_{c}. Here the meaning of D_{c} is of course, according to (11),

D_{c} = \frac{∂}{∂ℓ} − ∇.  (11′)

Notice that X and Y may be but are not necessarily full quaternions;^{12} they can be, for example, pure vectors, either real (or ordinary vectors) or complex, i.e. bivectors, if we are to retain Hamilton’s terminology.

Let us now pass to consider the fundamental electromagnetic equations “for the vacuum,” as they are recently called, i.e. the system of differential equations

\begin{matrix}\frac{∂E}{dt} + ρp = c. curl M,  div E = ρ \\ \frac{∂M}{dt} = −c. curl E,  div M = 0 \end{matrix} \}  (18)

where E, M are the electric and magnetic vectors of the field, respectively, ρ the volume-density of electricity and p the vectorial velocity of its motion, both ρ and p being given functions of space and time.

First, to condense these equations, put together the electric and the magnetic vectors to make up the electromagnetic bivector (or the bivector of the field)

F = M − ıE  (19)

and write again ℓ = ıct. Both curl and div being distributive, this will give us instead of the four vector equations (18) the two bivectorial equations^{13}

\frac{∂F}{∂ℓ} + curl F = \frac{1}{c}ρp; div F = −ıρ,

or, using Hamilton’s symbols,

\frac{∂F}{∂ℓ} + V ∇F = \frac{1}{c}ρp; S ∇F = ıρ.

12This has no influence on their transformational peculiarities as expressed in the above quaternionic form.

13The reader will find these equations together with the corresponding bivectorial form of the density of energy and the Poynting flux in my paper published in 1907 in the Annalen der Physik, vol. xxii., and (supplement) vol. xxiv. I was then unaware of their possible application to the present purpose. (The η of that paper is the above ıF.)

Now, remembering that V ∇F + S ∇F = ∇F and using the quaternionic differentiator D, explained by (11), the last two coalesce at once into the single equation

D F = C,  (V I.)

in which C is the current-quaternion, as defined by (10).

Thus the whole system of four equations (18), the fundamental equations of the electron theory, are represented by one quaternionic equation, (V I.).

This condensation is even more complete than in Minkowski’s matrixform, which consists of two equations, lor f = −s, lor f^{∗} = 0 (loc. cit. 12), one for the first pair of (18) and the other for the second pair, or in Sommerfeld’s equivalent four-dimensional vector form: Div f = P and Div f^{∗} = 0 (loc. cit., 5). Here P is the “Vierervektor” corresponding to the currentquaternion C, and f the “Sechservektor,” corresponding to the bivector F, while f^{∗} is the “supplement” (Ergänzung) of f, which is another “Sechservektor,” though very nearly related to f. Minkowski’s f is an alternating matrix of 4 × 4 elements. But let us return to our quaternionic differential equation (V I.).

C is a (given) physical quaternion cov. q. The operator D has also the structure of q. What is the relativistic transformer of F? By (V.) we see at once that it is

Q_{c}[ ]Q,

or that F is transformed like a (scalarless) L-quaternion. Thus, the answer is already contained in (V.). But to see clearly the true meaning of the process implied in the relativistic transformation, let us repeat again the whole reasoning somewhat more explicitly. We have, in the system S, as an expression of the laws of electromagnetic phenomena, the equation

D F = C.  (S)

Now, what the Principle of Relativity requires is the same form of the law in the system S′, i.e.

D′ F′ = C′  (S′)

Suppose also that both of these equations have been fully confirmed by experience. How are F′ and F correlated? To adopt language adapted to the general case, use in the accented law or equation (S′) the transformer already known, i.e. in our present ease Q[ ]Q for both D and C; then it becomes

Q D QF′ = QCQ, or D QF′ = CQ,

or, by the non-accented equation (S),

D QF′ = D FQ.

Hence, rejecting an additive function of obvious properties, i.e. requiring that F′ shall vanish together with F,

QF′ = FQ,

or finally, Q being a unit-quaternion,

F′ = Q_{c}FQ,  (V II.)

which is the required correlation, identical with the above.^{14} Henceforth we shall have to admit, in the name of Relativity, bivectors transforming like this calling them, say, physical bivectors (or in Minkowski’s way, “world”- bivectors). Or we can make the L-quaternion (of which F is the vector part) the master, calling it, say, a (left) physical quaternion of the II. kind, and writing F as its special case

F = V L = V a_{c}b.  (20)

(The supplementary scalar, S a_{c}b, necessary to convert F into a full quaternion, would present no difficulties, since it has been proved to be an invariant.) The short name physical quaternion might then continue to stand for physical quaternion of the first kind, of which q is the standard.

q′ = QqQ.

14Our quaternionic formula (V II.) resembles entirely Minkowski’s

f′ = A^{−1}fA, in which A is a matrix of 4 × 4 elements, and A^{−1} its reciprocal; loc. cit.. §11. The reason of this analogy will easily be seen to depend on the circumstance that both the product of quaternions and the product of matrices have the associative property. But at any rate the multiplication by a quaternion, like Q or Q_{c}, is actually done in a much more simple way than the application of a matrix of 4 × 4 elements.

Observe also that the above analogy does not extend to the transformation of Minkowski’s vectors of the I. kind and our physical quaternions in fact, here the matrixform is

s′ = sA, with s = |s_{1}, s_{2}, s_{3}, s_{4}|, whereas the quaternionic form is

But leave aside questions of nomenclature and return to (V II.). To verify this short formula remember that, by (I.),

Q = \sqrt{(1 + γ)/2} + u\sqrt{(1 − γ)/2}, Q_{c} = \sqrt{(1 + γ)/2} − u\sqrt{(1 − γ)/2},

and expand the right side of (V II.). Then

F′ = (1 − γ)(Fu)u + γF + ıβγ V Fu,  (21)

or splitting into the real and imaginary parts and remembering (19),

\begin{matrix}E′ = (1 − γ)(Eu)u + γE + βγ V uM \\ M′ = (1 − γ)(Mu)u + γM − βγ V uE \end{matrix} \}  (21a)

Now, these equations give immediately for the components taken along u (the direction of motion)

E′_{1} = E_{1}; M′_{1} = M_{1},

and for the two other pairs of rectangular components (the right-handed system being used)

E′_{2} = γ(E_{2} − βM_{3}); M′_{2} = γ(M_{2} + βE_{3})
E′_{3} = γ(E_{3} + βM_{2}); M′_{3} = γ(M_{3} − βE_{2}),

which are precisely the well-known transformational formulæ, obtained for the first time by Einstein. Thus (V II.) is verified.

Again, Q, Q_{c}, being unit-quaternions, we see from (V II.) that, as already has been remarked, the tensor of F is an invariant,

T F′ = T F,  (V III.)

which may also be written, more conveniently,^{15} F′^{2} = F^{2}. Now, by (19), −F^{2} = M^{2} − E^{2} − 2ı(EM); thus we see that (V III.) contains both of the well-known invariants of Minkowski:

M^{2} − E^{2} and (EM)  (22)

Notice that what is called a pure electromagnetic wave is defined by M^{2} = E^{2}, (EM) = 0. Using the above form we can characterize a pure wave more simply by^{16}

T F = 0, or F^{2} = FF = 0.

15Remember that, F being a scalarless quaternion, its conjugate is simply −F.

16This remark will be found also in my paper of 1907, cited above.

Thus, by (V III.), a wave which is pure to the S-inhabitants, is also pure to the S’-inhabitants. But this example only by the way.

Instead of the above F, as defined by (19), we may as well take the complementary bivector

G = M + ıE.^{17}  (19a)

Then we shall get as the quaternionic equivalent of the electromagnetic equations (18), instead of and in quite the same way as (V I.),

D_{c} G = C_{c},  (V I.a)

where C_{c} is the conjugate current-quaternion ρ(ı−p/c) and D_{c} the conjugate differential operator ∂/∂ℓ − ∇, as already explained.

We now see, by (V.a), that G is transformed like an R-quaternion, i.e.

G′ = QGQ_{c}.  (V II.a)

Again we may write, similarly to (20),

G = V R = V de_{c},  (20a)

d, e_{c} being a pair of physical quaternions covariant with q and q_{c} respectively. And since G is a physical bivector, just as much as F, we may again call R = de_{c} a (right) physical quaternion of the second kind.

Notice that, at least for the time being, we have no need of both F and G, since we require either F only or G only. (Possibly for the further development of Quaternionic Relativity the simultaneous use of F, G may turn out to be convenient or even necessary.)

As regards the relation of (20a) to (20), observe that generally we cannot write d = a, e = b; in fact, the reader will easily prove for himself that this would require (EM) = 0, i.e. E⊥M, and would not, consequently, be sufficiently general. The only essential thing here is that in (20) it is the first and in (20a) the second factor which has the subscript c. This is shown also by the symbols L (left), R (right).

Let us return to the quaternionic differential equation for the vacuum, in its first form, i.e.

D F = C.  (V I.)

17G is a complex vector “conjugate” to F, in the sense of the word used in the Theory of Functions. But to avoid confusion with the quaternionic notion of conjugate, I do not call it by this name and do not denote it by F_{c}.

Remember that D D_{c} = (T D)^{2} = \frac{∂^{2}}{∂x^{2}} +\frac{∂^{2}}{∂y^{2}} +\frac{∂^{2}}{∂z^{2}} +\frac{∂^{2}}{∂ℓ^{2}} is the four-dimensional Laplacian, or Cauchy’s □,

D D_{c} = □  (IX.)

Hence, if Φ be an auxiliary quaternion and if we put F = − V D_{c} Φ (since F is scalarless), or more simply if we write

F = − D_{c} Φ  (X.)

demanding at the same time that

S D_{c} Φ = 0,  (XI.)

then we get at once from (V I.)

□Φ = −C,  (XII.)

which is the well-known equation, obtained by Sommerfeld for his “Viererpotential.” But here, I daresay, it follows from (V I.) more immediately, than by the use of four-dimensional divergences and curls or “Rotations.”

The above Φ, which may be called the potential-quaternion, is easily proved to be a physical quaternion, namely, cov. q. For by its definition, (X.), and remembering that F is cov. L, we have immediately

Φ cov. D_{c}^{−1} F cov. D F cov. D L,

i.e., by (IV.), Φ cov. q,—q.e.d.^{18} Writing the potential-quaternion

Φ = ıϕ + A,  (23)

where ϕ is a real scalar and A a real vector, it is seen at once that ϕ is the ordinary “scalar potential” and A the ordinary “vector potential.” In fact, developing (X.) we have

F = V ∇A − \frac{∂A}{∂ℓ} + ı∇ϕ = M − ıE,

whence the usual formulæ

M = V ∇A = curl A,

18This is seen even more immediately from (XII.). For, since □ = (T D)^{2} is an invariant, Φ is transformed like C and, consequently, like q.

E = −∇ϕ − \frac{1}{c} \frac{∂A}{∂t}.

Also the condition (XI.) is expanded immediately into the usual equation

\frac{1}{c} \frac{∂ϕ}{∂t} + div A = 0.

Finally, notice that the “equation of continuity,” as it is commonly called, i.e.

\frac{∂ρ}{∂t} + div(ρp) = 0,

assumes the quaternionic form

S D_{c} C = 0.  (XIII.)

The scalar of D_{c} is, in fact, the same thing as Sommerfeld’s four-dimensional divergence Div.

Or we may write, equivalently,

S D C_{c} = 0.  (XIIIa.)

We know already that the electromagnetic bivector F is a (scalarless) L-quaternion. Hence, by (IV.), if we multiply it, on the left side, by any physical quaternion cov. q, the resulting product will again be transformed like q. Now, the current-quaternion C being precisely such a quaternion, consider the product

P = CF,  (24)

which, by the above, will again be transformed by Q[ ]Q. Develop it, by (10) and (19); then

P = ρ \{ıM + E + \frac{1}{c}pM − \frac{ı}{c}pE\}

or, remembering that the full product AB is V AB − (AB),

P = P_{e} + ıP_{m},  (25)

where P_{e}, P_{m} are the quaternions

P_{e} = ρ \{\frac{ı}{c}(pE) + E + \frac{1}{c} V pM\}  (25e)

P_{m} = ρ \{\frac{ı}{c}(pM) + M − \frac{1}{c} V pE\}  (25m)

The vector of P_{e} is the well-known ponderomotive force, per unit volume, and the scalar of P_{e} is ı/c times the activity of this force, while P_{m} is the magnetic analogue of P_{e}. Notice that the whole P, (25), though having with q the transformer Q[ ]Q in common, has not the structure of the standard q, inasmuch as it is a full biquaternion.^{19} (And how each of its constituents, P_{e}, P_{m}, which have the structure of q, are transformed, we do not as yet know,—though we shall know in a moment.)

Similarly, the complementary electromagnetic bivector G being a (scalarless) R-quaternion, multiply it on the right side by C. Then the product GC will, by (IV.), again be transformed by Q[ ]Q, i.e., again like q. Develop it; then, by (10) and (19a),

GC = ρ \{ıM + \frac{1}{c}Mp − E + \frac{ı}{c}Ep\} ,

and this is precisely, with the same meanings of P_{e} and P_{m} as above, equal to

GC = −P_{e} + ıP_{m}.  (26)

This again is a full biquaternion.

Now, both biquaternions, P = CF and GC being transformed by the same Q[ ]Q, this will also be the transformer of their sum, and of their difference, i.e., by (25) and (26), of P_{m} and of P_{e}.

Thus we see that not only P but also its constituents P_{e} and P_{m}, taken separately, are cov. q; and since each of them has also the structure of q,^{20} both P_{e} and P_{m} are physical quaternions, cov. q.

They are given explicitly by (25e), (25m), and may, by the above, be written also

P_{e} = \frac{1}{2} \{CF − GC\}  (27e)
P_{m} = −\frac{ı}{2} \{CF + GC\}  (27m)

It is true that (at least on the ground of the fundamental electronic equations) only P_{e} has an immediate physical meaning, and not P_{m}. But this does not seem to me a disadvantage. On the contrary; since our stock of physical quaternions, as the reader will certainly have observed, is as yet not very big, it may be better to have one more.

19In Hamilton’s, of course, and not in Clifford’s meaning of the word.

20Namely an imaginary scalar and a real vector.

P_{e} corresponds to the “Viererkraft”^{21} and might consequently be called here the force-quaternion. It has a dynamic vector and an energetic scalar, as observed above. As to P_{m}, it is of no importance to give it (at least for the “vacuum”) any special name. On the other hand, the whole P, which may possibly turn out to be more convenient for the quaternionic treatment of Relativity, might be called the dynamical^{22} biquaternion, and be looked on as the standard of physical biquaternions, in the same manner as q, F have been the standards of physical quaternions and of physical bivectors, respectively.^{23}

Now, using the quaternionic differential equation (V I.), or C = D F, the formula (27e) for P_{e} may be written

2P_{e} = D F.F − G. D F,  (28)

and similarly (27m) for P_{m}, the dot being a separator, as regards the differentiating action of D. In (28) the force-quaternion P_{e}, is immediately expressed by the electromagnetic bivector F and its complementary G. Thus, the formula (28) is adapted for showing the properties of the Maxwellian stress and of the electromagnetic momentum along with the flux and the density of energy, in correspondence to the equivalent formula of Minkowski’s four-dimensional system.

But, since we already know everything about the behaviour of each constituent of P, i.e. of P_{e}, P_{m}, we may dismiss them altogether and use more conveniently the full dynamical biquaternion P, as defined by (24). Thus, using again the equation (V I.), we shall have, more simply,

P = D [F.F] ,  (XIV.)

where the purpose of the brackets is only to emphasize the circumstance that F.F plays the part of a dyad. This will lead us to the quaternionic treatment of questions regarding stress, and localization and flux of energy

But these fundamental dynamical questions will best be postponed and reserved for a future publication, in which also the quaternionic treatment of the electrodynamics of ponderable bodies and of some other relativistic subjects will be given.

21See Laue, loc. cit., 15.

22Notwithstanding that it is partially energetic.

23It is worth noticing again that F (plus an invariant and consequently unessential scalar) and P may be regarded as alternating products of 2 and of 3 physical quaternions, respectively. From this standpoint q, F, P and their respective companions might he considered as quaternionic entities of the 1st, 2nd, and 3rd degree, respectively.

November, 1911.

XVIII. Intelligence and Miscellaneous Articles. Phil. Mag. 24 (1912), 208. THE QUATERNIONIC FORM OF RELATIVITY To the Editors of the Philosophical Magazine.

GENTLEMEN,— The appearance of Prof. Silberstein’s paper entitled “The Quaternionic Form of Relativity” in the May issue of the Phil. Mag. is a welcome sign that continental mathematicians, who have already largely availed themselves of various systems of vector notation, are perhaps awakening to the suitability of quaternions in such a connexion. Minkowsky (see footnote p. 79, Nach. Gött. 1908) must have had such an idea, but decided in favour of the matrix notation. The quaternion form of the Hertz Heaviside equations was given by the present writer at the British Association Meeting at Dublin (1908), and had been given by him in lectures for some years previously. An application of quaternions to the Relativity Principle will be found in a paper, vol. xxix. Section A, No. 1, Proc. Irish Academy (read Feb. 1911). Other writers, such as Somerfeld, subsequent to Minkowsky, have used the four-dimensional vector. The obvious defect of this latter method is that it involves a second kind of vector having six components. Beyond this, however, the quaternion has the advantage of being asymmetrical, the timescalar occupying a different position from the space-vector. It is thus more in touch with real phenomena. For no matter what view we take of relativity, the physical methods of measuring time are quite different from those of measuring spaces, and the flexibility of the quaternion product allows us to put our results at once in a form suitable, if need be, for numerical computation.

Yours truly,

ARTHUR W. CONWAY