Hertzian Electromagnetic PotentialsAuthor(s): W. H. McCreaSource: Proceedings of the Royal Society of London. Series A, Mathematical and PhysicalSciences, Vol. 240, No. 1223 (Jul. 16, 1957), pp. 447-457Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/100050Accessed: 08/02/2010 20:22

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Hertzian electromagnetic potentials

BY W. H. MCCREA, F.R.S.

Royal Holloway College, University of London

(Received 1 February 1957)

The general properties of the Hertzian electromagnetic potentials and their gauge trans- formations, recently given by Nisbet, are re-derived in tensor form appropriate to the space-time of special relativity. There appears to be a considerable gain in conciseness and comprehensibility of the general results, although it is recognized that Nisbet's forms may be those suited to particular applications. Whittaker's scalar potentials are briefly con- sidered. The theory refers exclusively to flat space-time because the results require co- variant differentiation to be commutative, but it is formulated in terms of any curvilinear co-ordinates in such space-time.

1. INTRODUCTION

A comprehensive treatment of the Hertzian electromagnetic potentials was given in a recent interesting paper by Nisbet (I955). He worked entirely in terms of scalar and vector fields in Euclidean space (o) of three dimensions. This was probably natural, having regard to the applications which he had in view. However, the most concise and symmetrical formulation of general electromagnetic theory is that

appropriate to four-dimensional space-time. This seems to be particularly true of the part of the theory treated by Nisbet, where the three-dimensional formulation is bewildering in its complexity. The gain in brevity and ease of apprehension effected by the four-dimensional formulation is so great that it seems worth placing on record even though the results are equivalent to Nisbet's and may have to be translated into his form for special applications. Incidentally, this translation may well be deferred to the latest possible stage, since, among other complications, the

arrangement of algebraic signs of the translated terms offers an unfortunately considerable chance of error.

Although Nisbet does not employ a four-dimensional formulation, he does mention the relativistic covariance of his results and refers, too, to the work of

Laporte & Uhlenbeck (I931), also quoted by Sommerfeld (I935). The latter is a four-dimensional treatment, but it is based upon the theory of spinors which is not

required here. The present writer met the problems involved in reviewing (McCrea 1957) the work of the late Sir Edmund Whittaker and in seeking to discover the

general significance of Whittaker's (I904, 1951) expression of an electromagnetic field in terms of two scalar functions. The writer then became aware of Nisbet's work, which includes a discussion of Whittaker's contribution. Here the four- dimensional treatment seems to offer only a very little more insight; this is briefly mentioned in ? 8.

In ? 2 the general features of the four-dimensional treatment are summarized; in ?? 3 and 4 the tensors required are defined and their assumed properties stated. In ?? 5 and 6 are stated in tensor form the general properties of the potentials and of their gauge transformations, corresponding to Nisbet's results, and the four-

[ 447 ]

dimensional derivation of these properties is given in ? 7. This arrangement facili- tates the comparison with Nisbet's treatment, while making it possible, if desired, to read the present work without immediate reference to Nisbet's.

2. FOUR-DIMENSIONAL FORMULATION

The procedure need be only formalistic, since the aim is merely to give an alter- native expression of Nisbet's results. In particular, all required conditions of differ-

entiability are assumed to be satisfied. Also a solution of any of the differential

equations has the meaning, implied in the corresponding part of Nisbet's work, of a function or functions satisfying the equation in the relevant domain of the

variables, without any reference to boundary conditions at this stage. We consider the four-dimensional formulation appropriate to the space-time (S9)

of special relativity. The results are to be in tensor form, all tensor indices running through the range 1, 2, 3, 4. As usual, the fundamental tensor for any system of co-ordinates x1, X2, x3, X4 is denoted by gpq and the value of its determinant by g. First and second covariant derivatives will be denoted by brackets ( )p, ()pq. Covariant differentiation is commutative because 9 is flat; some of the results depend upon this and cannot be taken over without modification into space-time of non- zero curvature.

A set of rectangular co-ordinates in &, regarded as a spatial section of 9, together with the corresponding time determination, will be denoted by x, y, z, t; these will be called special co-ordinates in 9. We take the metric in such co-ordinates to be

ds2 = - dx2 - dy2 - dz2 + dt2. (2.1)

In these co-ordinates the raising or lowering of a tensor index merely changes the

sign of the component if the index is 1, 2 or 3, and makes no changes if the index is 4.

By a transformation of special co-ordinates we mean a transformation of x, y, z to another set of rectangular co-ordinates without change of t, the coefficients in the transformation being independent of t. Under such a transformation, (a) an invariant inY9 is an invariant in ?, (b) if V, is a vector inS9, then the set of components (Vx, V2, V3) tranforms as a vector in & while the component V4 behaves as an invariant in ?, (c) if Wpq is a skew tensor (six-vector) inV9, then each of the sets of components (W, W24, W4, W3), (W3, W31, W2) transforms as a vector in d. If a valid tensor relation in y be translated into relations between scalars or vectors in g according to these rules, then the validity of the latter relations is assured. Conversely, if V is a given vector in g having components V1, V2, 3 in any given special co-ordinates, and v is an invariant in g, we may define a vector Vy in Y having components (Vl, V2, V3, v) in the given system of special co-ordinates. Also, if U, V are two given vectors in ?

having components (U1, U2, U3), (V,, V2, V3) in the special co-ordinates, we may define a skew tensor W1q in 92 such that in these co-ordinates (WL,4 W24, W34) = (U1, 2, U3) and (W2, W31, W12) = (VI, VT, V3). Such definitions will be expressed by writing

= (V, v); W= (U, V), W =-Wqp.

448 W. H. McCrea

(2.2)

Hertzian electromagnetic potentials

It must be emphasized that the components may be identified in the manner stated

only in special co-ordinates, the components of Vp, Wpq in any other co-ordinates

being then determined by the laws of tensor transformation. By such means, suit- able relations between scalars and vectors in S can be translated into tensor relations in Y.

We recall the existence of the Levi-Civita tensor yPqrs, the components of which in any co-ordinate system are + (- g)- if p, q, r, s is an even or odd permutation of 1, 2, 3, 4 and are zero otherwise; thus (-g)zyPqrs= epqrs, the usual permutation symbol. The covariant derivatives of this tensor are zero. If Wpq is any skew tensor, the dual tensor *Wpq is defined by

Wpq = 'pqrsWrs; (2-3)

that is, a contravariant (skew) tensor is derived from Wpq in accordance with (2-3) and its indices can then be lowered in the usual way to give *Wp , * WPq, WVq. It is convenient to extend the notation to apply also if dummy suffixes are present; thus if Wpq = Xpqt yt, say, defines a skew tensor we may write

* Wpq= *XPqt yt = *Xpqt (2-4)

The dual of the dual is the original tensor with its sign reversed (on account of the occurrence of (-g)-l rather than g-~ in the definition). As is clearly permissible, we regard a tensor as being defined by either an expression for the tensor itself or an expression for its dual.

Finally, we note the following identities:

(a) If Wpq is any skew tensor then

(Wq)pq = 0. (2.5)

(b) If a tensor Fpq is defined in terms of a vector A, by

*Fpq = yPqrs(As)r, (2-6)

then (F1)q, = (A)qp - 2Ap, (2.7)

where F]2 is the tensor form of the d'Alembertian operator defined by

2 2grs( )rs (2-8)

In (2 6) there is summation over r, s; but if p, q are fixed (p q), then r, s are

uniquely determined so that p, q, r, s is an even permutation of 1, 2, 3, 4, and (2-6) becomes (_ g ) *Fpq

= (As)r- (At)s = As, r- Ar,

s (2.9)

where the comma denotes a simple derivative. Hence from (2.3)

Fpq =Aq, -A, q or ,pq- (Aq)p -g (Ap)r,

whence, on forming the covariant derivative and contracting,

(Fp )q (2. ()pq _g(Ap)rq.

This is (2 7), it being noted that the order of covariant differentiation is permutable.

449

W. H. McCrea

(c) If WPq is any skew tensor, then

Ppqrs(*Wt) r - *qrs (wt ++2 Wq. (2-10)

Let the skew tensor Vpq be defined such that

* Vpq = fpqrs(wst)tr + 2* W . (211)

Proceeding as in obtaining (2.9) with a similar determination of p, q, r, s, (2.11) becomes

(-g) *VPq = (W')t-(W)t + D2W

= gt(Wsu, r + Wur, s+ Wrs, u) (2.12)

= gP(Wsp, r + Wpr,s + Wrs,p)t + gq(JVsqr, + Wr, Wrs,q)t, (2.13)

since the expression in brackets in (2.12) vanishes if u is equal to r, s, so that con- tributions arise only from u = p, q; there is not summation over p, q in (2.13). Again using the permutation properties of p, q, r, s we find that

Wp,r + Wpr,s +Wrs, = (- g) (* Wr,r + * WSP, + * WPq) = -(-g) (* wf),,

while similarly the second such quantity in f(213) is + (-i)9f*WPu),,. Thus 12-13) yields

* q = gtP(- * Wqu)u + gtq(* WPU)t

or *Vpq =- (*Wq)up + (*Wpu)uq,

which gives - Vr = ?Prqs(*WptU)

or FVpq = Vfqrs(*Wst)r, (2.14)

where full summation is restored, the last form resulting merely from appropriate rearrangement of the indices. Identifying the values of VPq from (2-11) and (2-14), and using the notation illustrated in (2-4), we obtain (2-10).

3. DEFINITION OF FIELD TENSORS

The same notation as Nisbet's for the parameters associated with an electro-

magnetic field in g will be used except for the employment here of units such that his factors 47r and c are absorbed. The physical meanings of the parameters emerge as we proceed in subsequent paragraphs. Here it is sufficient to define tensors, related to quantities in J in accordance with (2-2), as follows:

charge-current vector J = (-j,p), (3-1) field intensities Hpq = (D, H), (3-2)

Fpq = (E, B), (3-3)

polarization tensor Ppq = (- P, M), (3.4) vector potential Ap = (A, - ), (3.5)

Hertzian potential WP = (IIe, -Hm), (3-6)

first stream potential Qp = (- Qe, Qn), (3-7)

second stream potential Rp = (Re, -Rm). (38)

450

Hertzian electromagnetic potentials

All of the second-order tensors are skew. The allocation of signs is arbitrary; some follow the standard convention and others are Nisbet's choice. In his work, it might have been more natural to interchange the roles of Qm, Rm in (3 7) and (3 8), but we adhere to his actual usage.

4. POSTULATED FIELD RELATIONS

We shall refer to equations in Nisbet's paper by his numbering prefixed by N; it must be remembered that his factors 47T and c have to be omitted in making comparison with our formulation.

Maxwell's equations (N2-2), (N 21) are

(HPq)q = JP, (4.1)

(*FPq) = 0. (4-2) The constitutive relations (N 2-3) are

Hp + =pq =Fpq. (4.3)

The stream potentials are such that Hpq = Qpq is some particular integral of

(4-1) and *Fpq = *Rpq is some particular integral of (4-2); but these particular integrals do not necessarily also satisfy (4-3). These properties are expressed by (N 2-12), (N 2-24).

5. PROPERTIES OF THE POTENTIALS

The usual representation (N 2.4) of the field by a vector potential is given by

*Fpq = prAs(As)r (5.1)

The Lorentz condition (N 25) on AP is

(AP) = 0, (5.2) an(d the wave equation for Av is

2AJP + p + (Pp)q = 0. (5.3)

Using (3.1), (3.4) and (3.5) this translates into

12A = j +P+curlM, (N2-7)

-q2 = p-div P. (N 2-6)

The meaning of these relations is that for all Av the tensor Fpq derived from (5-1) yields a solution of Maxwell's equations (4-2), while, if AP is a solution of (5-3) satisfying the auxiliary condition (5-2), then the tensor Hpq related to this Fpq by (4-3) is a solution of Maxwell's equations (4.1).

The representation by a Hertzian potential is

A p = r(P)q (5p-4) Hpq = Vpqr(*Wts% Qpq (5.5)

*FP- = qPvrs(Wt)t + *Rpq, (5.6) with the wave equation 2W + + Qpq Rpq a P,+ 3PQJ rQ Oq- _Z - ?. (5-7)

451

W. H. McCrea

With the definitions of ? 3 these translate, respectively, to give

A = I + curl m, q =-div He; (N2-9)

fD = curl curl e - curl m - Q, (N 2-20) H = curl H + grad div n- - + Q.; (N 221)

fE = grad div 11 - I - curl + Re, (N 2-18)

,B = curl Ae + curl curl 1m - Rm; (N 2-19)

2IIe = P + Q + R,, (N 2-22)

2Im -- M + Qm+ Rn. (N2-23)

The meaning of (5-5) to (5-7) is that for any skew tensor Wpq, and Rpq as defined, the tensor Fpq derived from (5-6) satisfies Maxwell's equations (4-2), while, with Qpq as defined, the condition (5-7) is sufficient for the tensor Hpq determined by (5-5) to satisfy Maxwell's equations (4-1) together with the constitutive relations

(4-3).

6, GAUGE TRANSFORMATIONS OF THE POTENTIALS

Following Nisbet's terminology, gauge transformations of the first, second and third kinds apply to invariants, vectors and second-order tensors, respectively. We are not concerned here with the transformation of the first kind, which is

merely the multiplication of an invariant by an exponential factor. A transforma- tion of the second or third kind is effected by means of an invariant or a vector, respectively, which is arbitrary so far as the definition of the transformation is

concerned, but which may be subjected to various conditions for particular applica- tions. The transformation of the third kind, in the present context, is due to Nisbet. In the definitions that follow, the original value of any variable is distinguished by the affix 0 from its transformed value.

Applied to the vector potential, a transformation of the second kind is defined by

A = A + (X)p (6-1)

where X is an arbitrary invariant. But, if Ap, A? satisfy the condition (5.2), then we require [2X = 0, (6-2)

and, if this is satisfied, the wave equation (5.3) is gauge-invariant. Applied to the stream potentials and the Hertzian potential, Nisbet's trans-

formations of the third kind are defined by

Qpq = Qopq + pqrs(G)r,, (6.3)

Rpq = ROpq+ *4rrs(Ls)r, (6.4)

Wpq = WOpq + Pqr(rs)r + *rpqrs(As)r (6.5)

where in the first instance Gp, Lp, Up, Ar are arbitrary vectors.

452

Hertzian electromagnetic potentials

If Qj, ROq are stream potentials, then so are Qpq, Rpq defined by (6.3) and (6.4). The electromagnetic field Hp2, Fpq derived in accordance with (5-5) and (5.6) from Wpq, Qpq, RBp given by (6-3) to (6.5) is the same as that derived from Wpq, Qq, ROq,

provided G - 2r (6.6)

Lp=D2Ap + ()p, (6-7)

where 6, 0 are arbitrary invariants. The wave equation (5.7) is also gauge-invariant under these conditions.

If the vector A. is such that _ (68) A? ~ - (6p, (?8)

where C is an arbitrary invariant, then the transformation (6.5) of Wpq induces a transformation in Ap of the form (6.1) with

X = - (AP) - C (6-9)

In this case it follows from (6-7) that L, must be zero, or the gradient of an invariant which makes zero contribution in (6.4).

As regards the expression of the results in Nisbet's form, (6-1) gives (N 3 1) with the sign of X reversed and (6.2) is (N 3.2). With the identifications expressed in the manner of (2-2) by

p= (G,-g), Lp=(-L,1), rp =(-r, ), A =(-A,A), (6.10)

(6.3) to (6.5) become, respectively, in special co-ordinates

Qe - Q0 + curl G, Q = Q? -G-gradg; (N 3-3)

R =R ?- L-grad,, Rm= Rm -curlL; (N3-4)

e = no + curl r- A-gradA, } (N 3-12)

nm = no - grad y-curl A.

Inserting the values (6.3) and (6.4) in (5-5) to (5-7) we obtain (N 3-8, 9), (N 3-6, 7) and (N 3-10, 11), respectively; the results are, of course, the same as are got by using the values of Qe, etc., from (N 33, 4) as quoted in (N 2-18) to (N 2-23). Equations (6-6) and (6.7) are equivalent to (N3-15, 16), and equations (6.8) and (6-9) to

(N 3-13, 14), with minor changes of notation for the invariants. Formulae like (5 5) to (5 7), (6.3) to (6 5), compared with their translations into

the three-dimensional expressions, demonstrate the relative brevity of the four- dimensional description which in turn renders the significance of the results much more immediately evident.

7. PROOFS OF THE FORMULAE

The tensor *FPq defined by (5.1) or (2-6) clearly satisfies Maxwell's equations (4-2) identically for arbitrary Ap. If we then impose the condition (5-2) on A., (2.7) becomes (FPq) = - 2AP. (7.1)

Therefore, on substituting for Hpq from (4-3) in Maxwell's equations (4.1) we get immediately the wave equation (5.3).

453

29 Vol. 240. A.

If now the vector potential Ap is defined in terms of an arbitrary skew tensor

Wpq by (5-4), the condition (5.2) is automatically satisfied by virtue of (2-5). The tensor *Fpq derived from this Ap according to (5.1) gives the first term of the right- hand member of (5.6), but, following Nisbet, we may add explicitly any particular solution of (4-2) and so obtain (5-6) in the form stated.

Since (5 2) is satisfied, by a further application of the identity expressed by (7 1) we obtain immediately from (5-6)

(Fpq)q = 2 2( W2 )q + (RPq)q. (7.2)

Therefore, substituting from (4-3) in (4-1) we have from (7-2)

- o2(Wp) + (Rpq)q (Ppq)q = JP. (7-3)

But QPq is a particular integral of (4.1) and so

(QP)q = JP. (7-4)

Combining (7-3) and (7-4) we have

(02Wpq + ppq + QpP- Rp)q = 0, (7-5)

and a sufficient condition for this is (5-7). Using (5-6) and (5-7) in (4-3) we then obtain

HPq = FpI - Ppq = -- *qr8(Wst)tr + Rpq - pPq

= - *pqrs(W)r + ]2 Wpq + Qpq

= pqrs(*Wst)t + Qp, (7-6)

using (2-10) in the last step; this is (5.5). Proceeding to the gauge transformations, the properties of the transformation

(6-1) are immediate; we consider those of the transformations defined by (6-3) to (6.5). When Wpq is given by (6-5) let Hpq be derived from Wpq in accordance with

(5.5) and let Hq be similarly derived from Wpq with stream potential Qq. Then, if we introduce tensors defined by

rpq = rpqrs(rs)r, Apq = pqrs(As)r, (7-7)

(6-5) becomes Wpq = WOpq + r + *APQ, (7-8) and (5.5) gives

HPq = PQrs(* Ws t)tr QP

= HOq - Q op + Qpq + ypqrs(* t) - prs(A^)tr (7-9)

Now from (7.7) (At)t = gfsu(AUI)t = gsurul(m(Am)t = 0 (7-10)

since one factor is symmetric and the other is anti-symmetric in 1, t. Also, using (2-7) applied to *(*rpq)= - pq, we have

(* 8t)t = - (rt)s + 2s, (7-11)

and, further, Pwrs(rt)t = 0. (7-12)

454 W. H. McCrea

Finally, from (6-3), Qpq _ QOpq = apq(rs(s)r. (7-13)

Hertzian electromagnetic potentials 455

Therefore, from (7-10) to (7-13), the result (7-9) becomes

HPq = HOpq +, pqrs([E2rs + ̂s)r.

The last term is the tensor curl of a vector and vanishes if and only if the vector is the gradient of an invariant, i.e. say,

E2r + 8 = (6).

This is (6.6) and a similar procedure yields (6.7). Again from (7.8) we have

F-2Wpqa = [-2 Wopqf + [-2rpQ +E 2*Apq. (7-14)

From the definitions (7-7) and the properties of the operator ?2,

[]2rpq = -pqrs([2rs)r,

-2 *Apq = *D22APq = *fpqrs(E]2As),

and from (5 7), (6-3) and (6 4)

C]2 Wpq = _ ppq _ QOpq + ROpq

= _ ppq _ Qpq + Rpt + ylpqrs(s), _- *PqrS(Ls)r.

Hence, substituting in (7-14),

Dw2 Wp + Pp + QP q- RP = P2s(2rs + Gs)r + *P ( rs(2As - L) = 0,

if (6-6) and (6.7) are satisfied. Thus the gauge-invariance of (5.7) is established; this is noted by Nisbet (sentence preceding (N 3.15)).

Lastly, if Ap, AO are the vector potentials corresponding to Wpq, WOq in accord- ance with (5 4), using the form (7.8) with identities like (7*10) and (7-11) but having the roles of F, A interchanged, we have

A = (WpJ)q

-=(Wo)q + (rp)q + (*ApQ)q

=A - (A)qp + D12A

= -A + (X)p

where x =-(A)-C

provided Ap is such that [2Ap = - ()p. Thus we have obtained (6.8) and (6 9).

8. WHITTAKER'S POTENTIALS

Whittaker's theorem states that, in a region in which J = 0, PPq = 0, the Hertzian potential WPq may be taken in special co-ordinates in the form I/p,q, say, where

? = ^ . Y , (8.1)

29-2

the dots denoting zeros and ', 9 being functions of x, y, z, t. Thus the field is expres- sible entirely in terms of these two functions. There are, of course, three ways of

pairing off the rows and columns in (8.1) and it is immaterial which one we select; the form (8-1) corresponds to (N4.12).

Whittaker (1904) proved his result by finding explicit expressions for Y, 9 in the case where the field is produced by moving point-charges. But Nisbet points out that it holds good for any electromagnetic field in vacuo at points away from the sources. He sketches the proof of this generalization and there is nothing to add from this point of view. Once again, however, the four-dimensional statement of the result makes its significance more plain.

The theorem is in fact, if Wpq is a given skew tensor such that

D2Wq = 0, (8-2)

and if the tensor Y#q has the form determined by (8 1), then the equations

(WK - yP),7 (-)W = 0 (8.3)

are compatible partial differential equations for F, ( and an invariant X. As

stated, the proof is that sketched by Nisbet. Now, in a region where Jp = 0, Ppq = 0, we have from (4.3),Hpq = Fpq, andwe maytake Qpq = 0, Rpq = 0. Then (5-7)becomes (8-2) and from (5.6) and (8-3)

*Fpq = pqrs(Wst)tr _= qrs(/s)t. (8.4)

Thus the field is described in terms of r, W. From (8.3), using (2.5), we have

f(X)pr- = Pr(Wp- qj)qr = (W r)Q = 0

that is o2]x = 0. (8-5)

Using (8-2) and (8-5) in (8-3), we then obtain

(L2"i~2 ) = 0.

Thence it can be shown that without loss of generality we may take I2 YpVq = 0,

giving 22giv = 0, 2 = 0.

The form (8.1) is a canonical form for a general skew tensor, whereby the tensor is reduced to a wrench (Synge I956). Any skew tensor, and so any Hertzian potential, may be reduced to this form at any particular event in y by a suitable choice of co-ordinates. The point of Whittaker's theorem is that, for an electromagnetic field in free space, a Hertzian potential may be used for which the axis of the equivalent wrench has the same (space-like) direction at every event in the region of S concerned.

By employing non-vanishing stream potentials, Nisbet also formulates a general- ization of Whittaker's theorem to regions where Jp * 0, Ppq = 0.

456 W. H. McCrea

Hertzian electromagnetic potentials 457

REFERENCES

Laporte, 0. & Uhlenbeck, G. E, I931 Phys. Rev. 37, 1380. McCrea, W. H. I957 J. Lond. Math. Soc. 32, 234. Nisbet, A. I955 Proc. Roy. Soc. A, 231, 250. Sommerfeld, A. I935 In Frank & von Mises, Riemann-Webers Differentialgleichungen der

Physik, 2nd ed. 2, 790. Brunswick: Vieweg. Synge, J. L. I956 Relativity: the special theory, chap. ix, ?5. Amsterdam: North-Holland. Whittaker, E. T. 1904 Proc. Lond. Math. Soc. 1, 367. Whittaker, E. T. I95I History of the theories of aether and electricity: the classical theories,

p. 409. Edinburgh: Nelson.