A metric free electrodynamics with electric and magnetic charges

ANNALS OF PHYSICS

112, 366-400 (1978)

A Metric Free Electrodynamics with Electric and Magnetic Charges DOMINIC

G.

B.

EDELEN

Center for the Application of Mathematics, Lehigh University, Bethlehem, Pennsylvania 18015 Received June 10, 1977

A formulation of electrodynamics with both electric and magnetic charges is obtained without introduction of a metric structure into the space-time arena. The field equations and their general solutions are constructed without constitutive assumptions through use of extensions of the calculus of exterior forms. The solutions, being a 2-form field density, and a 2-form field pseudodensity, are given explicitly in terms of a potential 1-form, a potential pseudo 1-form, and the action of a nonlocal homotopy operator H on the electric and magnetic current distributions. The theory is generally covariant and also admit a 4-parameter group of dyality transformations that reduce to a 2-parameter subgroup under the restriction of consistency with the vacuum ether relations. Complete symmetry is obtained between the electric and magnetic effects, both having or not having "strings" depending on the interpretation adopted. A bilinear variational principle is given that yields the field equations, their general solutions, conservation of electric and magnetic charge, and the forces that act on the electric and magnetic current distributions. The forces on each kind of charge are the sum of two forces; the first being the Lorentz force that arises from the free fields, and the second is tentatively identified with the radiation force. The solutions have significantly improved smoothness properties over previous theories and allow the passage of one kind of current through the rays to infinity (strings) generated by either kind of current. Explicit representations of the field vectors ~, ~r, ---) . , and D are also gaven, together with the forces.

1. I N T R O D U C T I O N

AND SUMMARY

Electrodynamics, as it is currently viewed a n d practiced, is directly a n d intrinsically associated with the metric geometry of M i n k o w s k i spacetime a n d the Lorentz group. Now, o n a conceptual basis, this is something o f an a n o m a l y , a n d has been so since the p u b l i c a t i o n of the series o f f u n d a m e n t a l b u t little noted papers by D. v a n D a n t z i g [1] in 1934. V a n Dantzig showed that Maxwell's equations a n d their constitutive equations could be stated in general covariant form in a 4-dimensional space-time without any assignment of a metric structure to this space-time. A careful e x a m i n a t i o n of v a n Dantzig's papers shows that this same conclusion can be obtained if Maxwell's equations are extended so as to include magnetic monopoles. T h e purpse o f this paper is to o b t a i n the metric free f o r m u l a t i o n of electrodynamics that provides for the presence of magnetic monopoles. The results provide considerable conceptual dise n t a n g l e m e n t of the u n d e r l y i n g physical a n d mathematical structures a n d lead to useful, a n d in some instances, to far-reaching implications. The importance of 366 0003-4916]78/1122--0366505.00/0 Copyright © 1978 by Academic Press, Inc. All rights of reproduction in any form reserved.

METRIC FREE ELECTRODYNAMICS

367

internally consistent formulations of electrodynamics with magnetic monopoles is taken for granted, and will not be commented upon in this introduction. Specific references to current work and to Dirac's foundation papers are deferred to the points in the text where they are specifically germane. Van Dantzign's considerations are significantly simplified by use o f Cartan's calculus of exterior forms, and become more accessible in this modern context (see [13] for a similar simplification of electrodynamics from a point of view that is quite different and very useful for comparison and contrast with the results given below). Section 2 delineates the underlying structure of the 4-dimensional coordinate manifold M4, and the natural bases for tangent vectors and exterior differential forms on M4. Several new structural results of the exterior calculus are developed in Section 2 since they are required in order to avoid the introduction of a metric (inner product) for vectors or forms. The first of these consists of "top down" generation o f basis elements for exterior forms and provides the necessary structure whereby divergencelike quantities O~K~ can be invariantly constructed without a metric. The second is introduction of the homotopy operator H and its adjoint H ÷. The operator H allows us to obtain complete first integrals of exterior equations while H + allows us to compute variations of a Lagrangian function that contains nonlocal terms that arise from the action of H on field quantities. Section 3 commences with a statement of Maxwell's equations in the presence of both electric and magnetic charges, where, behavior under parity and time reversal transformations requires that the magnetic current be represented in terms of a pseudovector density. The top down basis introduced in Section 2 is used to show that Maxwell's equations can be stated in terms of a pair of exterior equations that involve both 3-forms and pseudo 3-forms. These equations are equivalent to the physical statements of conservation of both electric charge and magnetic charge and are obtained without introduction of a metric structure and without specific constitutive assumptions for the field variables E, H, B, and /5. The pseudodensity character of magnetic charge does demand, however, that ~ and/3 combine to form a 4-dimensional field pseudotensor rather than a field tensor as is the case with the classical formulation. Since Maxwell's equations are obtained as a system of exterior differential equations, their general solution can be constructed by use of the homotopy operator H. This is accomplished in Section 4. The form-pseudoform pair formulation is still in evidence in these general solutions through the action o f H on forms and on pseudoforms and in the occurrence of an exact potential 2-form and an exact potential pseudo 2-form. The implications of the occurrence of the homotopy operator H in the general solutions is examined in Section 5. This operator is shown to introduce very strong nonlocality into the theory, for its action takes any distribution with compact support on M4 and "smears it out" along 4-dimensional radial rays to infinity with a decay o f R -3 (R ~ = x 2 -k y~ q- z 2 -k t 2 = separation function on M4 of any point in M4 with the origin). The operator H induces a " r a y " operator on functions on Ma that provides an particularly simple explicit representation of E, H, B, and /) in 595/I12/2-IO

368

DOMINIC G. B. EDELEN

terms of the electric and magnetic charge distributions, a 4-vector potential, and a 4-pseudovector potential. The properties of the field equations and their general solutions are taken up in Section 6. The results are shown to be similar to those reported in [7, 10, and 11] in that two potential 1-forms are naturally present and the theory is invariant under a 2-fold collection of gauge transformations. The field equations and their solutions do not involve any constitutive assumptions (vacuum ether relations, etc.) and are generally covariant, so there is no preferred group of transformations of M4 that is singled out by either the field equations or their solutions. The form-pseudoform structure of the equations and their solutions are shown, however, to lead to a natural 4-parameter group of transformations, 9 , that is generated by pseudolinear group PL(1, 1). This group, PL(1, 1), is the group of all constant nonsingular linear automorphisms of the 2-dimensional vector space of ordered pairs (n, p), where n is a scalar and p is a pseudoscalar. Included within the group are transformations that conjugate both electric and magnetic charge, conjugate electric charge separately, conjugate mangetic charge separately, homogeneously scale electric and magnetic charge jointly and separately, and yield the known exchange transformations of Maxwell's equations and several additional ones. The group 9 obviously contains a large number of transformations that are inconsistent with any specific system of constitutive relations. In this regard, ~ may be considered as a universal covering group for the analysis of any subgroup that is consistent with a given system of constitutive relations. This view is a consistent one, since N is obtained from Maxwell's equations and their solutions before any specific constitutive relations are imposed. The nonlocality that comes from the action of the homotopy operator H on the electric and magnetic currents has already been remarked upon. What is significant here, is that this nonlocality occurs in a completely symmetric form. Thus, there are rays to infinity (strings) associated with the distribution of electric current as well as those associated with the distribution of magnetic current. This inner symmetry of the theory is also reflected in the fact that the exchange transformations generated by the dyality group also exchange the electrically and magnetically generated strings. This is in sharp contrast with the Dirac theory [9]. A further contrast is shown to be present, for the solutions admit a natural decomposition into a part (source free) that does not depend directly on the smoothness properties of the electric and magnetic charge distributions, and a part (source) that does reflect the smoothness properties of the currents. Section 7 clothes this general structure in specifics by adjoining explicit constitutive relations. After the general situation is dealt with, the analysis is confined to the vacuum ether relations, whose material symmetry group is the Lorentz group. Restriction of the transformations of M4 to those that preserve the ether relations gives M4 a Minkowski structure, and the field equations, solutions and constitutive relations are then invariant under the Lorentz group. The ether relations are shown to yield specific differential relations between the two potential 4-vectors, whose integrability conditions result in inhomoegeneous wave equations for these potentials. The restriction of the general dyality group N by consistency with the vacuum ether

369

METRIC FREE ELECTRODYNAMICS

relations yields a 2-parameter group ~ V that is generated by the commutative product of a 1-parameter homothetic group, diag(e e, e~), and the 1-parameter group @+ whose matrix elements 0 satisfy Or diag(y, 1) 0 = diag(y, 1),

det(0) = 1,

y = zr0/E0 .

Restriction to ~ V thus excludes those dyality transformations that generate separate conjugation of electric charge and of magnetic charge, and separate scalings of these two kinds of charge. A general bilinear variational principle is given in Section 9. This variational principle is shown to give the field equations, their general solutions, and, of course, conservation of electric and magnetic charge. This variational principle is free of both metric and constitutive assumptions, but is not invariant under the action of the dyality group. This is reasonable, for the dyality group consists of a set of transformations that transform the Euler-Lagrange equations (field equations and their solutions) into Euler-Lagrange equations without reference to any underlying transformations of 3/4 • Noetherian theorems are thus not applicable for dyality transformations. Explicit occurrence of the electric and magnetic currents in the variational principle are used in Section 10 to obtain the forces that act on the electric and magnetic current distributions. This is achieved by transformation of the variational principle to a Lagrangian coordinate description and then varying the orbits of the electric and magnetic charges in such a way as to preserve conservation of electric and magnetic charge; that is, we vary the orbits of the current distribution. The forces on the electric and magnetic charges thus obtained are shown to be orthogonal to the 4-velocity vectors of the electric and the magnetic charges, and are each made up o f the sum of two terms. The first term in each is the classical Lorentz force if this force is computed from the source-free parts of the field tensors. The second parts are nonlocal in nature and are tentatively identified with the radiation forces since the theory is exact rather than approximate. There is thus a significant difference between this theory and the original Dirac theory [9]. In particular, electrical particles can pass through the strings generated by the magnetic particles, and magnetic particles can pass through the strings generated by electric particles.

2. MATHEMATICAL PRELIMINARIES Most of the results obtained in this paper come about in a simple and direct manner by use of E. Cartan's calculus of exterior differential forms, together with certain extensions that are given in this section. We assume that the reader is familiar with the standard structure of the exterior calculus, so that we may use the following symbols without further remarks: A for the exterior product of forms, ] for the inner multiplication of a vector with a form, d for the exterior derivative o f a form.

370

DOMINIC

G.

B. E D E L E N

The reader who is not overly familiar with the exterior calculus is referred to [2], [3], and [4] for a complete treatment, and to [5] for a concise summary from an operations point of view. We will, however, display the principle results both in terms of the exterior calculus notation and in terms of the possibly more familiar vector and tensor notation. The arena for our discussion is a 4-dimensional flat manifold, M4, that is the Cartesian product of ordinary 3-dimensional Euclidean space, E3, and the real line. It is sufficient for our purposes to work with a fixed, global coordinate cover {x~} of M4 with x~ = x ,

x2 :y,

xz =z,

x4 =t,

since the results for any other regular coordinate cover or covers are directly obtainable by the standard mapping properties of differential forms and vector fields. Accordingly, a natural basis for tangent vector fields on M~ is given by G = 8/3x%

a = 1, 2, 3, 4;

(2.1)

that is, any tangent vector field, ~ ( x ) , on M~ can be written uniquely as

The class of exterior forms of degree k on M4 is denoted by A k k = 0,..., 4. We use the standard (natural) bases, so that {dx ~} is a basis for A 1, {dx ~ A dx~} ] a
~o = d x I ^ d x ~ ^ d x 3 ^ d x ~ = d x ^ d y ^ d z ^ d t

(2.2)

is a basis f o r / P . I f ~/~ is a field of exterior forms of degree k on 21'/4, then ~ be written uniquely as = ~(x)

=

can

w ~ l . . . ~ ( x ) d x ~ ^ ... ^ d x %

in which case, we define the )t-dependent 1-parameter family of k-forms ~¢7~(A)by

~ ( a ) = ~r(Ax) = w~l...~(ax)

d x ~1 ^

... ^ d x ~ .

(2.3)

This notation proves useful later on when we have to define the basic homotopy operator and its adjoint operator. There is an alternative way of constructing bases for A 3 and A S that provides certain distinct computational advantages. This "top down" construction proceeds as follows. Since oJ is the natural basis for A 4, we can define four 3-forms/z~ by 7r~ = g~] oJ,

a = 1,2,3,4,

(2.4)

METRIC FREE ELECTRODYNAMICS

371

so that "ffl = dx2 A dx 3 A d x 4,

7r2 = - - d x 1 ^ d x 3 ^ d x 4,

~3 = dxl A dx 2 A d x ~,

7r4 = - - d x 1 A dx 2 ^ dx 3.

(2.5)

It is clear from (2.5) that the four 3-forms (Try} constitute a basis for A 3. However, the particular ordering of the differentials that occurs in the relations (2.5) induces the transformation law cqxt3

~ ' ( y ) = A(y, x) ~

,~(x)

(2.6)

for a coordinate map y~ = #~(x0 with A(y, x) = det(~y~/Sx~), in fact, (2.6) follows directly from (2.4) and the known transformation properties of w and the vectors {~}. A straightforward computation based on (2.4) shows [6] that (zr~} exhibit the following properties: d~'~ = O, dx ~

^ ~ = 8B~o.

(2.7)

(2.8)

Since {/z~} is a basis for A 3, any 3-form °k' can be written uniquely as = U~zr~,

(2.9)

in which case, (2.6) and ~ ' ( x ) = o U ~ ( x ) 7 r ~ ( x ) = ~ ' ( y ) z r ~ ' ( y ) - - - - - q/(y) induces the law of transformation U~'(y) = A ( y , x) -1 8y~ UB(x).

(2.10a)

Similarly, if q / = s U ~ rr~ is a pseudo 3-form (i.e., a 3-form with the transformation law q/(x) = sign(A( y, x)) Yg(y) for a coordinate map ~:(x ~) ~-~ (y~)), we obtain U~'(y) = A ( y , x) -a sign(A(y, x)) 8Y~ UB(x).

(2.10b)

This shows that a 3-form oR is represented in terms o f the basis {rr~} by a contravariant vector density, and a pseudo 3-form ql is represented in terms o f the basis (rr~} by a contravariant pseudovector density. The converse is also true. If Y/" = V~g~ is a vector (pseudovector) density, (2.4) shows that og = ¢ - j o, =

w=~

(2.11)

is a 3-form (pseudo 3-form). In fact, (2.11) establishes a 1-to-1 correspondence between vector densities ( pseudovector densities) and 3-forms (pseudo 3-forms).

372

DOMINIC G. B. EDELEN

I f ~ is an arbitrary 3-form or pseudo 3-form, (2.7) and (2.8) show that dq/ = d(U~rr~) = (~U0o~, where ~ U ~ stands for ~U~/~x ~. Thus, if ~ density, then

= K~

(2.12)

is any vector or pseudovector

d(oU ] ¢o) = d(K~rr~) = ( ~ K ~) ¢o.

(2.13)

This shows that the vector ( pseudovector) density oYd = K~g~ is conserved(i.e. O~K~ = O) if and only if the corresponding 3-form (pseudo 3-form) X"]o~ is closed, that is, d(~(" ] oJ) = 0. This result is what will allow us to obtain explicit formulation of the conditions of conservation of electric and magnetic charge. No metric structure has been introduced into M4 in order to obtain the "divergence like" operation ~ K ~, nor has an inner product structure been introduced for forms. The Hodge star operator [13, p. 2008] (inner product on forms) and a metric (inner produce on vectors) are thus avoided. Let ~' be an arbitrary 3-form and let ~¢ be an arbitrary 1-form, then ~¢ ^ q / i s a 4-form, and hence a multiple of oJ. This product is particularly easy to evaluate if we use the basis {Try}for A 3, for, in that case q/ = U~rr~ and (2.8) yields s¢ ^ o?/ = (A~ dx ~) ^ (U~rr~) = A~U ~ dx ~ ^ 7r~ = (A~U 0 ¢o.

(2.14)

This result is again reminiscent of what obtains by introduction of a metric structure although we have not done so. Starting with the basis {rr~}, we construct a collection of 2-forms rr~B by •r~ = ~ ] ,r~,

~,/3 = 1 , 2 , 3 , 4 .

(2.15)

The set {Tr~z I a < fl} forms a basis for A s and exhibits the following properties [5]: d~-~ = 0,

~r~ = --7re~, (2.16)

dx ~ ^ ,r~ = ~*r~ -- 3~rr~. An immediate consequence of these results is that dql =- d(U~rr~) = ~ ( U ~B - - U ~) ~rB = 2~U~BTr~

(2.17)

for any 2-form ~ that is written in terms of the basis (¢r~B} by q / = U~Bzr~, U ~ = - - U ~ . An argument identical to that used with the basis {Try} shows that a 2-form (pseudo 2-form) ql is represented in terms o f the basis {tz~ [ o~ < fl} by an antisymmetric, second order, contravariant tensor ( pseudotensor) density (U~B}. Further, for ~/" = V~Bg~ A gB, we have q/ =

~V" ] oJ =

V~Brr~B

and this establishes a 1-to-1 correspondence between ~ and q/.

(2.18)

373

METRIC FREE ELECTRODYNAMICS

If q/is any 2-form on M4, we may alternatively write q/=

U~B~B =

(2.19)

U ~ d x ~ A d x s.

Thus, on'using (2.5) and (2.14) to obtain L_ Ti'21 "~- d x 8 A d x 4,

rrsl = - - d x 2 ^ d x 4,

~'41 = dx~ ^ d x3,

¢r1~ = - - d x s A d x ~,

zr32 = d x ~ A d x 4,

71"42 = - - d x 1 A d x 3,

¢qs = dx2 A d x 4,

~r23 = - - d x 1 A d x 4,

¢r4a -=- d x 1 A d x 2,

¢q4 :

rr24 = d x 1 A d x s,

zr34 =

- - d x 2 A d x s,

- - d x 1 A d x ~,

(2.19) yields the relations U12 = -- U a4,

Uls = U 24,

G4 = -- U ~3,

Usa = - - U 14,

G4 = U 13,

U34 = - - U 12.

(2.20)

If we consider the quantities U ~Band U~B as entries of antisymmetric 4-by-4 matrices ((U~S)) and ((U~B)), respectively, then (2.19) define the mappings/1 and/,-1 by (0

_v((~:~)) =

- - U s4

o

U 24

- on,

- u~.\ ) Uaa|

0 (2.21) /~-I((UaB)) =

0

- - U14

Ul~||

0

that is/-'U* =- U . , F--1U. = U*. We note, however, that the entries U* are components of antisymmetric, second order, contravariant tensor (pseudotensor) densities, while the entries of U . are components of antisymmetric, second order, covariant tensors (pseudotensors). T h e m a p p i n g 1~ t h u s i n d u c e s a 1-to-1 c o r r e s p o n d e n c e b e t w e e n t h e s e f a m i l i e s o f c o v a r i a n t a n d c o n t r a v a r i a n t q u a n t i t i e s . Again, we note that we have been able to relate covariant and contravariant quantities without the introduction of a metric structure into M4 or into A k. The matrix representation of the components of a 2-form ~ by U*, for the antisymmetric matrix of coefficients relative to the basis {rr~}, and by U . , for the antisymmetric matrix of coefficients relative to the basis { d x ~ A dxZ}, is useful in another respect. If ~ and ¢¢/"are two 2-form, then q/^

"g/" = (U~B d x ~ ^ d x ~) A W"~rr.~ = U~BW"~(3~.~ -- 3 ~ . ~) oJ = (U., -- U..) W"w

= 2 tr(U,W*) w.

= 2Uv, W"oJ

(2.22)

374

DOMINIC G. B. EDELEN

This result is instrumental in computing invariants formed from 2-forms, as we shall see in later sections; in fact, (2.22) simplifies the construction of the Lagrangian densities for the variational principles underlying electrodynamics and many other disciplines. We now come to a different body of results that prove to be instrumental in obtaining general solutions of the equations of electrodynamics. Introduce the "radius" vector field, Y', on M4 by W = x~¢~, so that ~ = xa/ax q- yS/8y on k-forms g2 by means of

+ z8/8z + ta/at. H(,.Q) =

(2.23)

The h o m o t o p y operator H, is defined

,~ ] fi(,,)t) ~k-1 d/~,

where the notation ~ ( ) 0 is defined by (2.3). For instance, if o~ = then 5F ] -- = (x~g~) ] (FB(Ax) rro) = x~F~(;~x) rr~, and we have

(2.24)

F~(x) %

is a 3-form,

H(~) = (x~ ~; FB(P~x)h~ d2t) rr~. The operator H is usually introduced in the exterior calculus in order to establish Poincar6's lemma [3, 4, 6]: any closed form on a starlike region is an exact form. This is stated by the fundamental identity

= dH(Q)+H(dQ),

(2.25)

for then dr2 = 0 implies that D = dH(g2). The operator H has a much more important use, however; it allows us to integrate exterior differential equations directly. Suppose that we are given a (k + 1)-form 27 and we wish to determine all k-forms f2 that satisfy d/2 = 27.

(2.26)

I f we simply substitute (2.26) into the second term on the righ-hand side of (2.25), we obtain = d{H(D) + d~} -{- H(27), where the additional term ddcb comes about for any (k - - 2)-form • because d 2 ~ 0. However, since • is arbitrary, we obtain g? = dW + H(27)

(2.27)

as the general solution of (2.26), where W is an arbitrary (k - - 1)-form. As we shall

METRIC FREE ELECTRODYNAMICS

375

see, (2.27) will allow us to obtain evaluations of the field 2-forms of the electromagnetic field in terms of the sum of 2-forms of a potential nature and the operator H applied to the 3-forms of electric and magnetic current density and pseudodensity, respectively. Explicit calculations of the action of H on 3-forms will be given in later sections. Clearly, H maps k-forms into (k -- 1)-forms and is a linear operator. It thus has an adjoint H ÷ in the sense that

u{S

A H ( £ 2 ) - H+(S) ^ O} = 0

(2.28)

holds for every Z' E A k and every £2 e A 5-k. It is an easy matter to establish [6] that

H+(Z) = (--1)k+l f: f J ~ (~) h-'k+a' dA (2.29) = (_l)k+l f ~ &r l 2(h) Ak-1 dA. J1

Thus, if ~"

= F ~ - ~ is a

3-form, we obtain

H+(~-) = (x~ f~Fo(Zx) ~ d,~) ~.~. This operator, which maps k-forms into (k -- 1)-forms, will allow us to use a variational principle in order to obtain exact expressions for "radiation reaction" terms in the field momentum and in the forces that act on electric magnetic currents.

3. ELECTROMAGNETISM WITHOUT METRIC STRUCTURE Maxwell's equations with macroscopic distributions of free electric and magnetic charges and currents are the starting point for this discussion. Let q denote the density of free electric charge and J denote the 3-dimensional vector density of free electric current. The known transformation properties of Maxwell's equations under parity transformations P(2 ~ --2) and time reversal T(t ~-~ --t) demand [7] that magnetic monopoles be represented in terms of a pseudodensity and 3-dimensional magnetic current be represented in terms of a pseudovector density. Thus, let g denote the pseudensity of free magnetic charge and G denote the 3-dimensional pseudovector density of free magnetic current. If we use the standard coordinate cover {x, y, z, t} = {x~} o f M4 and the standard meanings for E, H, B, a n d / 3 in the M.K.S. system, then [8]

-~

x ~ -

3,/3 = 3,

~ • / 3 = q,

(3.1)

× E -

3,9 = ~,

¢ • B = g,

(3.2)

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DOMINIC G. B. EDELEN

hold throughout M4. This minus signs on the left-hand side of the first of (3.2) are included so that we secure conservation of free magnetic charge, • G q- O.g = 0,

(3.3)

as well as conservation of free electric charge, ~' • J -[- ~ q = 0.

(3.4)

The classic treatment, as given above, obtains the conservation of free electric and magnetic charge from the field equations. For our purposes, it is easier, and perhaps preferable in a conceptual sense, to obtain the field equations from the statements of conservation of free electric and magnetic charge. To this end, we define the vector density J of free electric current on M4 by (3.5) where {g~} = {~/9x ~} is the natural basis for the tangent space of M~ relative to the coordinate cover {x~}. Similarly, define the pseudovector density ff of free magnetic current on M4 by = GI*xl+ G~g2 -q- G3Jz + gg4 = G~J~ •

(3.6)

If we use (2.13), we see that d ( J J to) = (O~J~) to = (~' • 3 + 3,q) to. Accordingly, we conclude that free electric charge and free magnetic charge are conserved if and only if the 3-forms J J to and the pseudo 3-form f~ J to are closed. The basic conservation statements are thus expressed in M4 without the introduction of a metric structure. Since 344 is globally starlike with respect to any of its points, the Poincar6 lemma holds on M~ : any closed form (pseudoform) is an exact form (pseudoform). Thus, d ( J J to) = 0, d(~ J to) = 0 imply that a 2-form ~ and a pseudo 2-form ~- exist on M4 such that J J to = dig ~,

~ J to = d ~ .

(3.7)

These deciptively simple relations between forms and pseudoforms are actually Maxwell's equations. In order to see this, we proceed as follows. We first use (2.11) to obtain :

J to = : ~ = , ,

~¢ ] to = c ~ = ~ .

(3.8)

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METRIC FREE ELECTRODYNAMICS

Similarly, use of the basis {rr~B}to represent ~ and ~ by = H~Brr~,

o~ = F~,rr~,

(3.9)

so that {H ~} is an antisymmetric, second order, contravariant tensor density and F ~e is an antisymmetric, second order, contravariant pseudotensor density. Equations (3.7) through (3.9) then combine with (2.17) to yield the relations J~ = 2c~aHe%

G~ = 2~3~Fe~'.

(3.10)

It is now simply a matter of comparing the first of (3.10) with (3.1) and the second of (3.10) with (3.2) in order to replicate Maxwell's equations by the identifications

((o

2((m0) =

o

0

2((F~0) =

-&

D~

o

Ex

B~

o

= 2H*,

(3.11)

= 2F*.

(3.12)

We also note for later use that the mapping/1 given by (2.21) yields

2((H~a)) =

0

--D, 0

H~ Hz

= 2H.,

(3.13)

= 2F,

(3.14)

0

2((<~)) =

o

-&

0

--E~

--E~

with J Y = H~B dx ~ ^ dx ~ and ~ = F~B dx ~ ^ dx ~. A combination of the above results gives the following fully symmetric formulation of electrodynamics. Maxwell's equations f o r the electromagnetic field can be given in an M 4 without metric structure by

j

j o~ = d~e,

~ j o~ = d ~ ,

0.15)

378

DOMINIC G. B. EDELEN

and these equations are the direct consequences of the conservation of free electric charge, d ( ~ ] to) = 0

(3.16)

d(q~ ] to) =- O.

(3.17)

and of free magnetic charge

4. GENERAL SOLUTIONS OF MAXWELL'S EQUATIONS An abuse of language has grown up with respect to the meaning of a solution or a general solution of Maxwell's equations. What is commonly meant is a solution of Maxwell's equations (3.1) and (3.2) for given J, q, fir, and g and given constitutive relations whereby the field vectors/] a n d / 3 are related to the field vectors ~ and/7. Now, mathematically, the system (3.1), (3.2) constitutes a well defined system of first order partial differential equations for the determination of ~ , / 7 , / 3 , and/3 in terms of J, q, fir, and g. Viewed from this perspective, we define a solution of Maxwell's equations to be any collection of field vectors E, H, B, a n d / 3 that satisfy (3.1) and (3.2) for given J, q, fir, and g; that is, a solution of Maxwell's equations without constitutive relations. At first glance, the prejudice of common exposure might lead the reader to believe that such considerations are utter nonsense, for whoever solves Maxwell's equations without some kind of constitutive relations (ether relations or some equivalent statement). This is indeed a prejudice, for general solutions of Maxwell's equations can be obtained quite easily, as we now proceed to show. We saw in the last section that Maxwell's equations are given by J J t o = d ~ 4 t °,

qCJto=d~

(4.1)

under the identifications (3.5), (3.6), (3.11), and (3.12). Since J and ff are assumed given, we view the system (4.1) as a system of exterior differential equations for the determination of the 2-form ~ and the pseudo 2-form ~-. Such systems are, however, exactly the kinds of exterior differential equations considered in Section 2; that is, the two exterior differential equations in the system (4.1) are of the form given by (2.26). The general solution of each of them is accordingly given by (2.27), from which we obtain the following results. The general solution of the Maxwell system J ] to : d ~ , ff ] to : d ~ is given by Jt ° :

--d~ff + H ( J ] to),

=--dd

+ H(ff ] to),

(4.2) (4.3)

where ~ff & an arbitrary 1-form and zZ is an arbitrary pseudo 1-form. (The minus sign in front of the term d d in (4.3) has been introduced in order to simplify comparison with classic results, as will become evident almost immediately.)

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Before obtaining the implied representations o f ~ , ~q,/3, and/3, some remarks would seem to be in order in view of the result that d is a pseudo 1-form. If we restrict the problem to be a classical one for which ~ = 0 (i.e., no magnetic monopoles), then the second of (4.1) becomes do~- ~ 0 and (4.3) becomes ~- = - - d ~¢. However, the identical vanishin of f¢ eliminates the requirement that ~ be a pseudo 2-form with the consequence that we may take ~ ' to be a 1-form rather than a pseudo 1-form. This is the classical result that/~ a n d / 3 can be represented in terms of a "4-vector" potential. On the other hand, as soon as f¢ is nonzero anywhere in 3/4, its pseudovector density character becomes manifest. This, in turn, forces us to take ~ - to be a pseudo 2-form, in which case the transformation properties of E a n d / 3 exhibit the pseudostructure that is implied by the identification (3.12). We note, however, that the matrix representation of Y in terms of F , , as given by (3.14), gives the correct behavior under parity transformations P(~ ~ --~) (i.e, ~ ~ --/~, ~ ~ / ~ ) and under time reversal T(r ~ --t) (i.e., E ~ ff~, B ~ --B) provided J is a pseudo 2-form. It thus emerges that electrodynamics with just one monopole demands that ff~ and transform in such a manner that (3.12) gives a pseudotensor density law of transformation for F*, and that ~¢ is a pseudo 1-form. It follows from this that a study of electrodynamics without the inclusion of monopoles can lead to self consistent but spurious results.

5. THE RAY OPERATOR AND EXPLICIT REPRESENTATIONS

The general solution of Maxwell's equations, in its abstract form = --dX" + H ( J ] ,o), ~ ---- - - d ~¢ + H(f# ]ra),

(5.1) (5.2)

now needs to be "cut down" so as to obtain explicit representations for the field vectors E, H,/3, and/3. The more difficult part of this procedure entails the terms that involve the homotopy operator H, so we start with these. We first represent j ] oJ and f¢ ] 6o in terms of the basis (~r~} by use of (2.11). This gives J ] o~ = J ~ ,

~ J o~ = a ~ .

(5.3)

The definition of the homotopy operator H given by (2.24) then yields H ( J ] oJ ) = (x ~ f01 JB(Ax)h a dA)7r~.

(5.4)

Thus, if we introduce the linear operator h on functions defined over 3"/4 by

h(J)(x) ~°Z I ~ JOx) ;~ d;~, Jo

(5. 5)

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DOMINIC G. B. EDELEN

then (5.3) through (5.5) give us the evaluations

2 H ( J ] o9) =

(5.6)

( x ~ h ( J ~) - - xBh(J~)) Ir~B ,

2 ~ ( f ¢ J ~,) = (x~h(C ~) - - x~h(a~)) , ~

(5.7)

.

It is obvious that the operator h will enter into the explicit evaluations of E, H, B, and/3 in a very prominent way. We therefore take up a somewhat detailed study of the geometric and physical implications of the action of this operator on functions pn M4 • The easiest way of doing this is to introduce 4-dimensional spherical coordinates {R, ~1, ~2, ~3} on M4 so that the natural volume element becomes R 8 dR ^ dO, where dO is the element of differential solid angle on the unit 4-sphere. Although we still refrain from introducing a metric structure in M4, R is related to our original coordinate cover {x, y, z, t} by R2 =x 2+y2_kz

~+t3.

The 4-dimensional spherical coordinate cover of M~ may thus be viewed as defining a geometric (but not necessarily physical) measure of the separation between any point and the origin of the coordinate system {x, y, z, t}. Since J(hx) = J()oc 1, ;~x2, Ax3, )~xa) = J(~tx, by, )tz, At), and the coordinates {x, y, z, t} are held fixed during the integration process involved in (5.5), use of the spherical coordinate system gives the significant simplification J(~tx) = ](hR, ~1, ~2, fiB). Substitution of this result into (5.5) gives us

1 h(J)(R, ~ , q~2, ~3) = fo ,[(AR, q~a, qb~, q~) )t2 d)~,

(5.8)

from which it follows that h performs a "radial" integration in M4. Since {R, ~a, ~2, ~3} are held constant during the integration process, the change of variables p = ~R gives R

h(J)(R, ~1, ~2, ~8) = -R-3 f0 J(P, ~1, dD2, ~)8)p9 dp.

(5.9)

Now, suppose that ](R, ~1, ~b2, ~a) vanishes for R > R0. We then obtain

h(J)(R, ¢a, ~2, d?z) = R-S

(f? J(P, q~a, ~ ,

/

(%)p2dp ;

that is, h(J) smears a nonzero J out radially in 4-dimensions with a R -3 fall off outside the support of J. It thus seems acceptable, both geometrically and physically, to refer to the operator h as the ray operator, and, when need arises, to refer to {h(J~)} as the ray generated by {J~}. The ray operator h and the rays generated by h(J ~) and h(G~) are, in many ways reminiscent of the "string" that Dirac introduced in his 1948 paper on magnetic

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monopoles [9]. This is easily seen for (5.9) shows that if J has a dirac function singularity in more than one coordinate, then h ( J ) is singular along the entire radial ray from the singularity out to infinity. A detailed discussion of this point becomes fruitful, however, only after we obtain expressions for the forces that the field will exert on electrically and magnetically charged particles. It should be clear at this point, however, that the action of H on ,,¢ J to and ~ J to introduces a definite nonlocal structure into the whole theory that is quite different from those that have been considdered previously (see, for instance [14], [15] with the customary integrals of the momentum energy tensor). N o w that we have the representations (5.6) and (5,7), the rest is now easy. The standard representation of 1-forms (pseudo 1-forms) gives 2 d.Yd = ( ~ K ~ - - ~9~K~)d x ~ ^ d x t~,

2 ds/-~

(O~At~ - - OeA~) d x ~ ^ d x B.

Thus, a substitution of (5.6), (5.7) and (5.9) into (5.1) and (5.2) yields 2a%a = (O~KB - - ~ K ~ ) dx ~ ^ d x ~ + (x~h(J ~) - - x~h(JO) lr~B,

(5.10) 2~

= --(O~AB - - O~A~) d x ~ ^ d x B + (x~h(G B) - - xBh(G~)) ¢r,~.

All that now remains is to use the mapping/" to write the expressions in (5.10) in terms of the same basis {dx ~ ^ d x B I ~ < fl) and then compare with ~ = H~B d x ~ h d x ~, o~" = F~o'dx ~ ^ dx ~ and the identifications (3.13) and (3.14). If we use the representations {A~} -----{A, A4}, K~ = {/~, K4}, and {x~} -----{P, t}, the following results are then obtained: = ¢ ×/( E = 9X, --

+

x

+ h(q)~ - th(2),

= ~ × ~ + h ( g ) ~ - - th(~).

(5.11) (5.12)

The relations (5.11) a n d (5.12) constitute a solution o f M a x w e l l ' s equations f o r every choice o f the scalar (pseudoscalar) potential K4(A4) and the vector ( p s e u d o v e c t o r ) potential ~(.4).

6. PROPERTIES OF THE FIELD EQUATIONS AND THEIR SOLUTIONS The first thing we note is that the general solution = - - d ~ + H ( J ] to),

~" = - - d ~ + H ( ~ J to)

(6.1)

of Maxwell's field equations ,~'J to -----d,~~',

~]to =d~

(6.2)

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DOMINIC G. B. EDELEN

involves both the 1-form s f and the pseudo 1-form d . The theory is thus similar, in many respects, to the two vector potential formulation that was first given by Cabibbo and Ferrari [10] and then extended by Han and Biedenharn [11] and Mignani [7]. Since d~q~ = 0 for any scalar or pseudoscalar quantity, the solutions (6.1) of Maxwell's fieM equations are invariant under the 2-fold gauge transformations d ~--~d + dq),

,~t" ~--~,;(7 + d7 t

(6.3)

where q) is a pseudoscalar-valued function on M4 and ~ is a scalar-valued function on M s . These results agree with those of the Cabibbo-Ferrari theory [10], p. 1148, with the exception that • is a pseudoscalar rather than a scalar. It should be noted that we do not obtain the "mixing transformations" of the Cabibbo-Ferrari theory. This is to be expected since it is the constitutive theory that provides relations between and o~-, and hence between ~ and J{~, and it is the relations between d and z((" that give rise to mixing transformations. Maxwell's equations together with the ether relations are commonly viewed as a theory that is invariant under the Lorentz group. Without the constitutive assumptions reflected in the other relations, such an association is meaningless. This follows directly from (6.1) and (6.2) which express Maxwell's field equations and their solutions in generally covariant form. Accordingly, Maxwell's fieM equations and their solutions are generally covariant and admit no preferred group of transformations on Ms • It is thus the constitutive assumptions rather than the field equations and their solutions that single out certain preferred groups of transformations on Ms • A full discussion of this aspect of the theory will be given in the next section that deals with the constitutive theory and its implications. In contrast with the above results, there is indeed a specific group of transformations that is naturally associated with Maxwell's equations and their solutions, although this group has nothing to do with groups of transformations on the coordinate manifold, M4. As is well known, Maxwell's equations without magnetic currents admit the operation of electric charge conjugation. The extension of this operation to theories that include magnetic currents is usually accomplished by use of the known "exchange relations" q ~+ g,

Ks ~ As,

g ~--~--q,

j ~---~~,

A4 ~-~ --/£4,

~s ~--~- - j ,

/( ~ 4,

A ~ --/(,

(6.4)

which are the point of departure for the Han-Biedenharn formulation [11]. These considerations are also the point of departure for the theory given here. We start by noting that the field equations, (6.2), can be combined into a single 2-component matrix equation

l l

_-al l

•

(6.5)

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and their solutions can be similarly organized by writing (6.6) Care must be exercised, however, for the first entries in these column matrix representations are ordinary forms while the second entries are pseudoforms. This distinction among the entries of the column matrices is essential and constitutes the basis for our considerations. In order to account for the differences in transformation properties among the entries, we let n denote the unit scalar-valued function 3//4 and p denote the unit pseudoscalar-valued function on M4. Let P L ( r l , r2) denote the pseudolinear (matrix) group; that is, the group of linear automorphisms of the vector space of (rl + r2)-entried column matrices of constants whose first r~ entries are scalars and whose remaining r2 entries are pseudoscalars. Thus, PL(rl, O)~ GL(rO, PL(O, r2) = GL(r~), while the matrix S that represents an arbitrary element of PL(r~, r2) has the natural block form S = (vTM ~) (N = scalar-valued matrix, P = pseudoscalar-valued matrix). This representation follows immediately from the properties of matrix multiplication and the multiplication table nln~ = na , npl = Pxn P2, PiP2 = n. The group property of PL(1, 1) and the linearity and transformation properties of the relations (6.5) and (6.6) now lead to the following immediate result. The group PL(1, 1) generates the group ~ o f dyality transformations by =

with S = (~ ~12)~ PL(1, 1), and the group ~ is a 4-parameter continuous group that maps the fieM equations and their solutions onto fieM equations and their solutions. Let P0(I, 1) denote the pseudo-orthogonal subgroup of PL(1, 1) (the subgroup whose elements S satisfy SrS = E) and let P0(1, 1)+ denote the component of this subgroup that is continuously connected to the identity E. It is then readily seen that the subgroup of -~ that is generated by P0(1, 1)+ reproduces the group of dyality transformations studied by Han and Biedenharn [11 ] and Mignani [7]. In particular, if S = ( 0 0~),one obtains the standard exchange relations (6.4). This is also evident from the vector form of the solutions that is given by (5.1 l) and (5.12). The dyality transformation that is generated by S = (-0~ 0 ) ~ P0(1, 1)+ yields

This, however, is nothing more than the known charge conjugation properties of the field equations and their solutions, where, by charge conjugation, we mean both conjugation of electric charge and conjugation of magnetic charge. Charge conjugation is a dyality transformation that belongs to the subgroup generated by PO(1, 1)+. 5951II212-xi

384

DOMINIC G. B. EDELEN

Further, the dyality transformation that is generated by S = (-1 o) ~ P0(1, I)- yields

l- ,ct, (i.e., conjugation of only electric charge), while S = (o~ _0) ~ P0(1, 1)- generates

15I 15I ILl-ILl l l lgl (i.e., conjugation of only magnetic charge). Conjugation of only electric charge and conjugation of only magnetic charge are dyality transformations that are generated by elements of PO(1, 1)-. Since the dyality group is a 4-parameter group that is generated by PL(1, 1), it is significantly larger than the class of dyality transformations considered in [7] and [11]. This is already evident from the above results concerning separate conjugation of electric and magnetic charges. AS further examples, S = (~ 0), b v~ 0, generates a dyality transformation that multiplies sources, potentials and fields by the fixed constant b (i.e., doubling the sources and the potentials doubles the fields), while S = (~ 0), ab :/= O, generates a dyality transformation that increases the electric sources by the factor a and increases the magnetic sources by the factor b. Another example is S -----p(0 ~), which generates the exchange relations

b~9,

~q~ - L

:~d,

J~_~¢

to within the appropriate multiple of the pseudoscalar unit p. This last set of transformations is clearly inconsistent with any constitutitve theory for which /) is a linear function of E and ~ is a linear functon of/3. Now, our formulation of Maxwell's equations and their solutions has not involved constitutive assumptions. Accordingly, the dyality group obtained without constitutive assumptions should be significantly richer than that which obtains under imposition of specific constitutive assumptions. In fact, the dyality group -~ is a universal group that contains the restricted dyality group that is consistent with any given system of constitutive relations. This restriction process will be examined in the next section for the case of the classic ether relations, and provides additional insight into the implications and structure of the full dyality group. We note, in particular, that the full implication of general dyality transformations is far from understood, although the following results provide some additional understanding. The dyality transformation that is generated by S = (~012~) is given in component form by ~ nlH -- pl ~,

b ~ n~/3 + p ~ ,

~-~ --p2,~ + n ~ ,

/~ ~ p~b + n~/~;

(6.9)

K~ ~ nlK~ + plA~,

A~ ~--~p~K~ + n2A~ ;

(6.10)

y~ ~-+ n J ~ -}- p i g ~,

G~ ~-~p~Y~ + n2G~.

(6.11)

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385

The principle difference between the solutions given here and those usually reported is that ,~ff and ~ are obtained in a completely similar format, both with respect to the occurrence of the potential 1-forms ~ and z~' and with respect to the sources J and ft. This identical structural similarity comes about because of the properties of the homotopy operator, H, and its implied ray operator, h. In this respect, it is useful to write the solution in the form {o~t~,~ ) r = {~ff~,~ } r _}_ { ~ , ~ } r

(6.12)

{3(Y~, ~}~ = - - d { ~ , d } r

(6.13)

where

is the source free part of the solution and { ~ , ~ } r = H { J ] oJ, N ] co}r

(6.14)

is the source part of the solution. The occurrence of the operator H is the source part of the solution for both ~ and ~ is of particular significance. We have already seen that H gives rise to the ray operator h and this operator has the effect that it takes any function with compact support and smears it our radially to infinity in 4-dimensions with a R -8 decay, i.e., introduces intrinsically nonlocal effects. The source part (6.14) of the solution of Maxwell's equations thus contains the rays to infinity that are generated from the free electric current as well as those generated by the free magnetic current. This is amply shown by the vector component solutions (5.11) and (5.12). The theory and solutions presented here is thus completely symmetric in that it yields both electric and magnetic rays to infinity (strings), rather than just magnetic rays to infinity that entered so prominently in Dirac's treatment [9]. Why indeed, in view of the exchange symmetry (6.4) of Maxwell's equations, should a theory give only rays to infinity from magnetic sources, as has been the case up to the present. Of greater importance, however, is that the theory developed here gives a clear separation between the potentials ~ and ~¢' (the source free parts of the solution) and the rays to infinity (the source part of the solution). It is clear that and d can have very nice behavior even though the rays to infinity that are generated by J and f¢ can be singular if J and ~ are singular. In fact, it follows directly from (6.1) and H d + d H = identity, H ( H ( ' ) ) ~ O, that of" = --H(~t ~) + d 7 t,

d

= --H(~-) -k d~b,

(6.15)

where 7 t is an arbitrary scalar-valued function on M4 and ~b is an arbitrary pseudoscalar-valued function on M4. Thus, unless 7-t and q~ are singular, the forms ~f" and d are smoother than the fields J¢~ and o~-, respectively. These considerations would appear to dispell two disquieting aspects of the usual treatment of electrodynamics with monopoles: the potentials can remain well behaved even if the rays to infinity become singular, and the rays to infinity arise from the electric current as well as from the magnetic current.

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DOMINIC G. B. EDELEN

7. CONSTITUTIVE RELATIONS AND WAVE PROPERTIES A characteristic feature of metric free formulations of electrodynamics is that it leads to somewhat cumbersome constitutive relations. Let c0 and/z 0 denote the free space permitivity and permeability, respectively, so that eoTr0 = c -2 and c is the speed of light in vacuum. The standard representation of a homogeneous, polarizable and and magnetizable medium is given by the constitutive relations

b = ,0(E + P),

(7.1)

g = ~o1(9 + ~ ) ,

where/~ and )~ are the flee-space normalized polarization and magnetization vectors, respectively. An inspection of (3.11) and (3.14) shows that the relations (7.1) are equivalent to the matrix relations H* = %cZar(F, + M , ) a,

(7.2)

where the superior T denotes the transpose operation, a = diag(:kl, ± 1 , ± 1 , q:c-2), and M .

(7.3)

is the matrix of coefficients of the polarization-magnetization form

•//g = M~B dx ~' ^ dx ~, with --e~e

2 M , = ((M~)) = ( ( 0

0

--Mx 0

--P~ --oZ

(7.4) "

Now, {F~B) is a covariant pseudotensor, and hence {M~} must likewise be a covariant pseudotensor in order that the sum F~B + M ~ be invariantly defined; i.e., J g is a pseudo 2-form. On the other hand, H ~e is a contravariant tensor density. Accordingly, (7.1) and (7.2) can hold only if c0 is a pseudoscalar density, in which case ~0/~0 = c -2 implies that ~r0 is an antipseudoscalar density. We thus obtain the transformation laws '% = A( y, x ) - I sign A ( y, x) % ,

'tLo = A ( y, x) sign A( y, x) tZo

(7.5)

for Co and ~'0 under coordinate transformations y~ = f~'(x~). The intrinsic cumbersomeness of these constitutive relations is clear, for (7.2) relates the coefficient matrices of 2-forms, where the quantities H* determine relative to the basis {Tr~] ~
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387

Having noted the general form of the constitutive relations, we restrict attention from now on to the case of the vacuum ether relations, in view of their fundamental importance. We therefore have the relations H* = %c2aTF,a.

(7.5)

The material symmetry group of these constitutive relations [12] consists of the subgroup of GL(4) whose matrix representation L yields an identical satisfaction of 'H* = '%c2ar ' F , a

(7.6)

as a consequence of (7.5) and y~ = L ~ x ~. Thus, since % transforms according to (7.5) and 'H* = A( y, x)-i LH*LZ, ' F , = sign A( y, x) L r - I F , L -1, (7.6) holds if and only if LaL T = a. (7.7) It thus follows, in view of (7.3), that the material symmetry group o f the vacuum ether relations is the Lorentz group. This is, of course, well known. It is noted here solely to point up the fact that this group has nothing to do with Maxwell's equations, per se, since these equations are obtained and solved without the constitutive relations (7.5). However, since Maxwelrs equations are generally covariant, the invariance group of Maxwell's equations and the ether relations, together, is the Lorentz group. We also note that the Lorentz group has no a priori relation to a metric structure on 3/4 unless one makes the additional assumption that the material symmetry group of the ether relations is also the isometry group of the metric structure. This additional assumption is, however, exactly what distinguishes special relativity from classical mechanics. The obvious thing that remains is to obtain the implications of the vacuum ether relations. Since the ether relations read b = ~0~, /3 =/~0H, a substitution of the solutions given by (5.11) and (5.12) into these relations yields the conditions × K + h ( g ) f -- th(J) = ~o(VA4 - - ~v/l -~- ~ )< h(C,)),

(7.8)

x 71 + h(q) ~ -- th(G) = --/z0(¢K4 -- ~,/( -]- ~ × h(J)).

(7.9)

The ether relations thus establish a system of first order differential relations between / £ , / ( 4 , 4 , and A4. If we take the divergence and the curl of the relations (7.8) and (7.9), and use the Lorentz gauge conditions ~(a~A~) = O,

~(a~K~) = O,

(7.10)

we see that each component of d and each component of ~ satisfy an inhomogeneous wave equation; for example, V~A~ - - c - ~ , ~ A , = %~q - - ~ . ( ~

× h(~)),

VZK, -- c-~,2K4 = --~olg -- ~ " (7 × h(J)),

388

D O M I N I C G. B. EDELEN

with similar relations for A a n d / ( . Of course, in obtaining these relations, we have made use of the fact that d ( J ] w) = 0 and d H + H d = identity imply that the ray operator on both J~ and GB satisfies the relations JB

=

3h(J0 + x ° Oh(JB)

x B Oh(J°)

~X o

(7.11)

OX o

This is a lengthy and tedious procedure and is not overly germane to the discussion, for we already have general solutions of Maxwell's equations. What is really needed is the relations between ~ ' and ~g" that is induced by the vacuum ether relations. The question of the relation between d and y~r is relatively easy to settle. We already have the general relations ~ = --ddC + H ( J ] oJ). Thus, if we use the constitutive relations (7.5), we obtain the relations (7.12) and these may be viewed as a system of first order partial differential equations for the determination of the coefficients of ~Y(. As such, their integrability conditions, namely d~g" = 0 must be satisfied. Taking the exterior derivative of (7.12), and noting that d H + H d = identity, d ( ~ ] o~) = 0 imply d H ( J J o~) = J J ~o = J&r~ , we obtain the conditions Ja = 2aoC2~(a~SF~,a~'B).

(7.13)

Since the entries of {F~} depend on the first derivatives of the entries of {A~}, (7.13) are seen to be a system of second order partial differential equations for the determination of {As} which we will obtain explicitly in the next paragraph. Assume, now, that the A's have been determined so as to secure satisfaction of (7.13), where these equations may be viewed as the residue relations of Maxwell's equations by the ether relations. We can then determine ~ by direct integration in the manner discussed in Section 2. Thus, since H H q l -~ 0 for all ~', (7.12) yields (7.14) The upshot of all of this is that everything is determined provided d is chosen so as to secure satisfaction of (7.13). We thus turn to the problem of evaluating the righthand side of (7.13). Now, ~

= F~,~ d x ~' ^ d x B =

--d~

+ H ( ~ J o.,),

and hence 2F~B = --(O~,Ae - - OoA~,) + L~,B,,,x°h(G'),

(7.15)

where L~Bo,x~h(G ~) is what obtains from writing H(f¢ ] w) = xoh(G ,) ~,~, in terms of the basis {dx ~ ^ d x ~ [ ~
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389

and in the pair (a, ,/)). A substitution of (7.15) into the right-hand side of (7.13) yields the relations %lc-~J~ = - - a ~ ( a B ~ A ~ ) + aB~,~(a~A~) + e~(a~L~o,a~x°h(G")).

(7.16)

Satisfaction of the Lorentz gauge condition ~(a~*A~) = 0 then yields the following determining inhomogeneous wave equations for the functions As : a~e~8(A~) = - - , o ¥-2JBa~, + ~(a 'L~o,x h(G )).

(7.17)

We note that (7.17) reduces to the classic results whenever there are no magnetic currents (f¢ = 0). It now remains to obtain the restriction of the general dyality group that results from "imposition of the constitutive relations /) = %E, /~ =/z0H. A substitution of these relations into the genral dyality transformation (6.9) generated by S = (~o~ ~) yields the relations

(7.18) Consistency of these relations demands that n2 = nl ,

P2 = --7PI,

7 =/~o/Eo •

(7.19)

Now, (7.5) shows that '(/z0/Eo) --~ A(y, x) z/~0/E0, while (7.7) shows that the material symmetry group of the vacuum constitutive relations has A( y, x)~ = 1. Accordingly 7 is a natural number for all transformations of 344 that preserve the vacuum constitutive relations. It is also clear that matrices of the form ( . ~ ~) with n ~ + 7p 2 :/: 0 form a group PV(1, 1) under matrix multiplication (the pseudovacuum group) and this group is continuously connected to the identity matrix. The dyality subgroup, ~ V , that is obtained by the restriction of consistency with the vacuum ether relations, is a 2-parameter group that is continuously connected to the identity. It is generated by all matrices S of the form (-v~ ~) with n 2 + 7p ~ > 0, 7 =/~0/% • This result shows that the dyality transformations that belong to ~ V are much richer than those obtained previously [7, 11]. The structure of the dyality group, ~ V, of vacuum electrodynamics is easily obtained through a reparametrization. Let ~7 denote the unit pseudoscalar, and let ~9+ be the subgroup of PL(1, 1) whose elements satisfy Of Diag(7 , 1) 0~ I-- Diag(7 , 1),

det(0~) = 1.

(7.20)

O~+ is thus a 1,parameter subgroup of PV(1, 1) whose elements have the form ( 0~, =

COS "r]0 7 -1/2 sin ~70] sin ~70 cos ~10 1"

~__~,1/2

(7.21)

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DOMINIC G. B. EDELEN

Any matrix S = (_~ ~) with n 2 + 7p ~ > 0 can be written as S ~ e~O~

(7.22)

with ~: = ½ ln(n 2 + 7p2),

tan n0 = 711~p/n.

(7.23)

Thus, ~ V is generated by the 2-parameter commutative product group whose 1-parameter factor groups consist of the homothetic group, {Diag(e e, ee)}, and the group @% We note, in particular, that O1+ = P0(1, 1), and this is the situation considered in [7 and 11]. The additional degree of freedom in NV, over that reported in [7 and 11 ], thus consists of multiplication by an arbitrary element Diag(e e, e e) of the 1-parameter homothetic group; that is, a homogeneous scaling of all charges. However, we make specific note that the reduction of the full dyality group by the vacuum ether relations excludes a number of important dyality transformations. Among these are separate conjugation of electric charge and of magnetic charge, and separate homogeneous scalings of the electric charge densities. It is thus the vacuum ether relations, not Maxwell's equations or their general solutions, that preclude these operations.

8. VARIATIONAL FORMULATION OF THE FIELD EQUATIONS

The construction of a variational principle starts with with the formation of 4-dimensional volume invariants (4-forms) from the basis field quantities. Since ~vC, o~¢~, and J ] to are forms while ~-, d , and ff J to are pseudoforms, we let ~7denote a constant pseudoscalar unit. The following list can then be assembled by use of (2.14), (2.22), and (3.11)-(3.14): o¢~ =

~-

A ~

=

2 tr(F,F*)

=

2/~-Eto,

~¢2 = ~/o~" A a4g = 2~/tr(F,H*) = (~/2)(/~ • 13 - - / 3 . H) to, = J/g A ~ = ~d

^ (J

= 2 t r ( H , H * ) = --2_~ •/)to, ] to) = ~ ( j

J ~ ) to = ~J~A~to,

-- d

^ ( ~ j to) = (~¢ j d )

= ~

^ (j

= ,~

j to) = ( j

(8.1)

to = 6~A~to,

j ~)

^ (re j to) = ~(r¢ j ~ )

to = J~l<~to, to = ~c~K~to.

This list clearly splits into two sublists; the first consisting of these invariants that involve the pseudoscalar unit, ~, and the second consisting of those invariants that do not involve pseudoscalar unit. There is a further distinction, for the invariants in the first sublist are bilinear in the naturally associated triplets ( ~ , if, d ) and

METRIC

FREE ELECTRODYNAMICS

391

(aft, J , OF), while the invariants in the second sublist are quadratic in the entries of each of the triplets separately. Since a bilinear structure is significantly simpler to deal with than a quadratic structure, we confine our attention to the first sublist. Now, the usual variational procedure that leads to Maxwell's equations in the presence of the ether relations is one in which the potential 1-form ~ ' is varied. Accordingly, we express ~ and ~" in terms of OF, J , ~', and ~¢ by the relations {dr°, o~}r = --d{OF, sJ} r 4- H { J , ~}r j co.

(8.2)

This yields the following sublist = ~l(--dd + H(f~ J co)) a (--d~d ~ + H ( J J co))

-- n d ^ ( / J

co),

~ = ~OF ^ (~¢ J co)

(8.3)

that involves only 0U, j r d , and ~. We thus consider the action functional $1(R4)

~---

I_ "q{(--d.#' + H ( ~ J o,)) A (--doF + H ( J J co)) 4

+ d ^ (/j

,o) + OF ^ (~¢ j co)}

(8.4)

where R4 is the closure of any arcwise connected, simply connected open set in 3//4. A particular property of the action functional $1(R4) should be noted before we proceed in computation of the variation of $1(R4). Since d H 4- H d = identity, we have J J oJ = d H ( J J co) + H d ( J J o)), ~ J co : dH(fY J co) + Hd(£¢ J co). If we substitute these relations into (8.4) and note that - - d d A H ( J J c o ) + d A d H ( J ] co) = - - d [ d A H ( J J co)l, Stokes' theorem [3] gives us

SI(R~) = Q ,q{dd A d,Yd" 4- H ( J J co) A H(N J co) 4

4- d A H d ( J J oJ) + OF A Hd(~¢ J co)}

4- (,,'a .q{s~" A H ( J J co) 4- OU A H(~ J co)},

(8.5)

R4

where ~R4 denotes the 3-dimensional boundary of the 4-dimensional region R4 • We can now proceed with the calculation of the variations that are induced in S(R4) by variations 3~¢ = 8A~ dx ~' of the pseudo 1-form ~¢ and by variations 3OF = 8K,~ dx ~' of the 1-form OF. A straightforward calculation based on (8.4) or on (8.5) yields 8S~(R,) = fR, -q{8,N' A H d ( J J co) + 8OF A Hd(f¢ J co)}

4- f

•I 0 R4

~?{SdA [dot'-- H ( J l co)] 4- 8OF A [d~¢-- H(~Jco)]}

(8.6)

392

D O M I N I C G. B. E D E L E N

Thus, 8S~(R4) = 0 for all variations 3a¢ and 8oU that vanish on 0R4 if and only if H d ( J ] to) = O,

Hd(~ ] to) = O.

(8.7)

However, dHdql -~ dqg for any qg, and we see that (8.7) are satisfied if and only if

g(£

l ~o) = 0,

d(~¢ ] o~) = 0;

(8.8)

that is, if an only if electric and magnetic charges are conserved. The action St(R4) is stationary with respect to all variations 8~¢ and 3 S that vanish on OR4 if and only if electric and magnetic charges are conserved. We thus have a variational principle that recovers the starting point of the theory, namely (8.8). The variational principle just obtained is not altogether satisfactory. Although it leads to the basic statements of conservation of electric and magnetic charges, it does not give the full system of field equations nor their solutions. This could have been anticipated, however, for we used the first integrals of the field equations, namely (8.5) in the construction of the variational principle. The ideal situation would be one in which the variational principle would yield both the field equations and their first integrals, as well as expressions for the forces that act on the electric and magnetic charge distributions. We now proceed to this larger task with the variational principle given above as a guide. We consider the action functional & ( R , ) = (_ ~ { - - d ~ ¢ A X ~ -

d~

^ ~ - + ~¢ ^ ( J ] to) + ~

^ ( ~ ] to)

4

~- H(f¢ ] oJ) a ~

+ H ( J ] to) ^ ~ -- Jet° A .@-}.

(8.9)

This functional has the property that it reduces to the functional $1(R4)whenever { ~ , ,~,~'}T = --d{Jg, d } r ÷ H ( { J , (¢}r ] to) are satisfied throughout R4. The distinction between $1(R4) and S~(R~) is evident, for in the first, only d , f¢, ~ ' , and J occurred with variations of d and : ( only, while S~(R4) involves all of the field quantities ~-, d , f¢, ~o, • , j , and we may consider independent variations of d , ~ , ~ , and ~¢~. A straightforward computation and Stokes' theorem gives us the following consequences of the variations generated by 8~¢, 8 ~ , 3 ~ , 3~-: 8S~(R4) = f_ r/{Sd A [--d~V~' q- J J to] q- 8.Y-( a [--d~- q- ~ ] ~o] 4

q- 8a*ff A [H((# ] to) -- d d -- ~'] q- 3 ~ a [H(~ J to) -- d J(( -- 3/f]} -

(

J0 R¢

n { a d ^ ae ~ + 8.,~ ^ ~ } .

(8.10)

The action $2(R4) is stationary with respect to all variations Sag, &,~, 3af°, 8~', such that &vl and 83U vanish on 3R4, if and only if

METRIC FREE ELECTRODYNAMICS ag

= J

J ~o,

(8.11)

d.~ : f¢ J w, ~- = -ad

393

(8.12)

+/-/(~¢ ] ~),

---- --d~g -k H ( J / oJ).

(8.13) (8.14)

hoM throughout the interior of Ra. Since a substitution of (8.13) and (8.14) into (8.11) and (8.12) yields the relations d ( J J ~ o ) : 0, d(f¢] oJ)----0, the action principle gS~(R4) = 0 subsums the whole theory presented up to this point. Further, if $2(R4) is rendered stationary relative to the arbitrary variations g ~ and gW, we obtain { ~ , ~ } r = - - d { ~ , d } r + H ( { J , ff}r j co) throughout R4. Under these circumstances, we have seen that $2(R4) reduces to S~(R4). We thus have

(s~(n,)l~s2

-- ~s~

= o) = s l ( R 4 ) ,

(8.15)

which ties the two variational principles together very nicely. We further note that we do not require 3~ff and 3~" to vanish on ~R a ; the variations 3geS2 and g~S~ that occur in (8.15) are unconstrained variations. We take specific note of the fact that the variational principles given above are not invariant under dyality transformations. This is not altogether unexpected, for dyality transformations map Maxwell's equations and their solutions, namely, the EulerLagrange equations that come from the variotional principle, onto Maxwell's equations and their solutions, but are unrelated to deformation processes on the underlying manifold M4. Accordingly, Noetherian theorems are not applicable and there is no underlying reason to expect invariance of the variational principles under dyality transformations. It would, indeed, be of interest if a dyality-invariant variational principle could be obtained. We have tried in vain and invite the interested reader to take up the task since a positive result would appear to have far-reaching consequences. As a last remark, we note the relative simplicity of the variational principle given here. This is to be contrasted with what must be one in the customary treatment with singular strings and multiple-valued 4-potentials [16].

9. FIELD INDUCED MOMENTUM AND FORCE DENSITIES The one question still outstanding is that of the evaluation of the field induced momentum and force densities that are associated with the densities of electric and magnetic currents (i.e., the forces that act on charged particles in the presence of the electromagnetic field). Such an evaluation obviously necessitates a system of constitutive relations whereby the basic mechanical variables of velocity are related to certain of the electromagnetic source quantities. For the purposes of this discussion, we assume that J~ = qv~(xe),

G ~ = gu~(x~),

(9.1)

394

DOMINIC G. B. EDELEN

where ~ =- v~t~ is the 4-dimensional velocity field of the electric charge distribution and q/ = u~t~ is the 4-dimensional velocity field of the magnetic charge distribution. It is, of course, assumed that V vanishes outside the support of q(xS), that q/vanishes outside the support of g(xB), and that v4(x ~) = 1, u*(x ~) = 1 on the supports of q and g, respectively We now substitute the relations (9.1) into the action functional $2(R4). This gives S2(R4) = f_ r / { - - a d ^ ~

-- dJC ^ o~ + q~, ^ (~p ] to) ~- US/{" ^ (q/] to)

4

q- H(gq[ ] to) ^ Jt ° q- H(q~/" ] ~o) A o~ -- ~ ^ o~},

(9.2)

and hence the part S~m(R4) of the action $2(R4) that depends on the mechanical variables ~e" and q/is given by S~,~(R4) = fR, ~?{qd ^ (~" ] to) + H ( q V ] to) ^ o~

-}- go~f" ^ ( ~ ] o9) -4- H(gqg ] to) ^ ~ } .

(9.3)

In order to bring the velocity variables outside of the operator H, we use the adjoint operator H + that is defined by (2.24). Since this operator satisfies the identity (2.28), it follows that

I_ H(q'¢/']

to) A ~ - = I_ ~ ^

H(q3e ] t o ) z I_ H + ( Y ) ^

(qYP ]to),

(9.4)

and hence (9.3) is equivalent to S2,~(R4) = I_ ~{q(d + H+(o~)) ^ ( ~ ] ~o) + g ( ~ + H+(Xe)) ^ ( ~ ] to)} 4

(9.5) = fR4 r/{q"//- ] (~' q- H+(~')) + gq/] (2/,(" + H+(~))} to. If we use {~"/g,o~-}r : --d{~', ~¢}r q_ H { J ] to, f¢ ] to}T and H+H =-- O, (9.5) can then be written directly in terms of d , ~/P, sC, and q/as

$2,~(R4) = fn, ~?{q~l/" ] ( d -- H+(dd)) + g~' ] (Sf -- H+(dJ~))} to.

(9.6)

Our task is now that of constructing the variation of $2,~(R4) that is induced by variations in the orbits of the electric and magnetic charges. Since we need to vary the orbits, it is necessary to go over a Lagrangian coordinate description. For this purpose, we assume that R4 is a 4-dimensional region that is contained between the two hyperplanes t ---- to and t = tl. We may then describe the orbits of the electric charges by putting x ~ -----@~(Xa, X 2, X 3, t),

~4(X~, t) -~ t

(9.7)

395

METRIC FREE ELECTRODYNAMICS

on the support of q, where {X~} are the spatial coordinates of the electrically charged "particle" at t = to • Thus, {cb~(x~, t)} are the space-time coordinates of the particle at time t that crossed the hyperplane t = to at the point with spatial coordinates {X~}. The orbits of the magnetic charges is similarly described by putting X~ =

ttl~(Yi,

t),

~sa(yi,

t)

=

t.

(9.8)

The velocity fields of the "particles" are thus given by

v ~ = dqg~/dt,

u ~ = dW~/dt.

(9.9)

Here, d/dt means the derivative with respect to t with {Xi) or {Y~} held flexed; that is, the derivative following the particle. For simplicity, let us deal with the term JR, r/q(¢/" J ~¢) ~o in (9.4), for all of the other terms will follow the same pattern of argument. Since q vanishes outside its support, by definition, we have

fR ~lq(3e~ ] ~¢) oJ = | 4

Jsupt(q)

~)q(~ ] ~¢) ~o.

(9.10)

Now, {Xi, t} constitute a regular coordinate cover of supt(q) because the velocity field ~ on supt(q) is autonomous; that is, the functions ~ ( X i, t) are obtained by solveing the autonomous system

dq)./dt : v~(q5e),

(I)i(to)

=

X ~,

(/)4(to)= t o .

(9.11)

We can thus refer the evaluation of the integral on the right-hand side of (9.10) to the coordinate cover {X~, t} and obtain

fR ,~

( a((pl, ~2, ~a, 44) dqg~ A~( q )~) dX 1 d X 2 d X 3 dt. rlq(~f J d ) o~ = Js upt(q) ~Tq(~) ~(X*, X 2, X a, t) dt

(9.12)

We saw in the previous Section that stationarity of $2(R4) with respect to ~f, of', ~', and ~¢ demands conservation of electric and magnetic charge. Accordingly, we must demand that any variations in the orbits of the electric and magnetic charges be such that they preserve the electric and magnetic charge densities. This is accomplished by the requirements that cb4) ~(X1, X 2, X a, t) = q°(X~' X2' X3)'

a(~b 1 , ~b 2, ~b 3 ,

q(qb~)

~(t/tl, 7,2, ~U3, W4) g(WB) a( Y~, y2, y3, t) = go(Y*, y2, y3),

(9.13)

for these equations are nothing more than the Lagrangian, as opposed to the Eulerian, statements of charge conservation. Clearly, q0(X9 is the electric charge distribution

396

DOMINIC G. B. EDELEN

on supt(q) c~ (t = to) and go(X~) is the magnetic charge distribution on supt(g) n (t = to). Thus, since supt(q) = (supt(q) n (t = to)) X[to, fi] in the coordinate cover {X i, t}, (9.12) becomes re4-qq(Y" ] d ) ~ o = ffo~ r/ (fsup,(q,c,(t=,o) qo(X~)--titd#~ A~(¢B) dX~ dX2 dX 3) dt.

(9.14)

For simplicity, we set supt(q) r~ (t = to) = Qa,

supt(g) n (t = to) = Ga.

The upshot of all of this is that we can now write (9.4) in the equivalent form

Se~(R~) = fro~ ~/ lfo qo(X') d ~ (A~' -- H+(d~C)~')(~) dX~ dX~ dX~ 3

+ oa ( . go(Yi) ~dt/t~ (K~ -- H+(d, Yl)~)(tP~) dY 1 dY ~ dYaf, dt.

(9.15)

This shows that dW~ (A~ -- H+(dd)~)(7 t~) G = nqo(x') -Ti-

(9.16)

is the Lagrangian per unit volume of Q3 of the orbits of the electric charge and that dtp~ 5Co = ~/g0(Y0 ~ - (K~ -- H+(dJd)~)(gJ0

(9.17)

is the Lagrangian per unit volume of G 3 of the orbits of the magnetic charge. The standard definition of the field induced momentum per unit volume [i.e., P~ = ~.W/~(dCrp~/dt)] gives us the following immediate results:

Pq~ = ~lqo(X~)(A~ -- H+(dd)~)(~),

(9.18)

Pg~, = ~Tgo(Y~)(K~ -- n+(d~C)~)(Tt°).

(9.19)

Let £P,~ denote the Lagrangian per unit volume of Qz that results from purely mechanical properties of the electric charge and let £~a denote the Lagrangian per unit volume of Ga that results from purely mechanical properties of the magnetic charge. If we perform variations in the orbits of the electric and magnetic charges that vanish on the hypersurfaces t = to and t = tl, the vanishing of the induced variations in J'{~(£~oq + ~ ) dt and J'~ (scrag + 5co) dt give the Euler-Lagrange equations

d,( ~ddp~/dt )

~¢f~c,

d ~-¢¢,~ d-~ ( ~d~/dt )

3 ~ ~ - - f ~ -- 8 ~

= fqe, =

~¢~

dt ( ~dq)~/dt (9.20)

dt ( Od~/dt

397

METRIC FREE ELECTRODYNAMICS

where fq~ are the components of the force per unit volume of Q3 that acts on the electric charge and fg~ are the components o f the force per unit volume o f Ga that acts on the magnetic charge. These forces are given, however, relative to the natural basis induced by the coordinate covers {X ~, t} and {Y~, t}, respectively. It this follows from (9.16), (9.17) and (9.20) that F

d~ fq~ = ~/qo ~

{~[AB -- H+(d~/)~] -- e~[A~ -- H+(d~')~]}, (9.21)

d~ fo~ = :qg0 ~ - { O , [ K a

-- H+(dJU)a] -- O~[K, -- H+(dgU)~]}.

in the coordinate covers {X ~, t} and {Y~, t}, respectively. Since the variations in ~ and 7 s~ vanish on the hypersurfaces t = to and t = t l , we have 8

f,i'

~dt

=

f,"l ~

~¢~

'

dt ~(dcP~/dt)

8q)~dt =

('

o fa~3¢~dt

from the definition o f the variational process and (9.20). A combination o f (9.6) with (9.15)-(9.17) thus yields

~S.m(R~) =

f£1 ~

fo. qo td@~(~ (~[A~

-- H+(dd)B]

-- Oe[A, -- H+(d,.~¢)~]) 8(/)- 1 d X ~ d X z dX" dt

+ f,iI, fo. ,o tdTJ~ -

0 K

"+,J>.,

00[K~ -- g + ( d ~ ) ~ ] ) 8 ~ I d Y ~ d Y ~ d Y ~ dt,

-

(9.22)

where ~ denotes variation of the orbits. It is now a simple matter to reverse the whole process in order to recover an integral over R4 with respect to the coordinate cover {x~}. When (9.13) is used, and we note that d@~/dt{O~A, -- OoA~} d x ~ = --~/~ ] & d , (9.22) gives

~S.m(R~) = f~ {/o ^ ( 8 ~ J o~) + / . ^ (~e~ J o~)} 4

= -- I_ ~{[q~F" ] d ( d -- H + ( d d ) ) ] A ( 3 ¢ ~ ] w) ~R

4

+ [gO//] d(JY" -- H+(d~Y))] A (87I ~ ] w)}

(9.23)

where leq = fq~ d x ~ is the force density 1-form that acts on the electric charge distribution and fo = £~ dx~ is the force density l-form that acts on the magnetic charge distribution. We thus obtain the following final results: the electromagnetic f o r c e

398

DOMINIC G. B. EDELEN

density 1-forms that act on the electric and magnetic charge distributions are given by ,¢q -- --r/q~//~ ] d ( d -- H+(d~4)),

(9.24)

fo = --~gql ] d ( ~ -- H+(d~)).

(9.25)

The first thing we note is that these force densities satisfy the standard requirements of any 4-dimensional formulation of force distributions; namely ¢/- ] fiq = 0,

~ ] ~ = 0.

(9.26)

Further, since ?eq and ?eg are linear in the velicity variables ¢: and q/, respectively, they behave properly under transformations t = T(s) of the dynamical parameter. This same fact is reflected by (9.15); that is, S2,~(R) is invariant under t = T(s). Thus, if a Lorentz metric is introduced in M4, the forces transform properly under Lorentz transformations. Second, if there are no magnetic charges, that is, f¢ = 0, (5.2) yields ~- = - - d d . In this event, (3.14) shows that the 3-dimensional vector part offlq is --~q~/" ] d ~ = ~/q(E -5 ~ × /~). Since that is exactly the Lorentz force to within the factor ~7, we conclude that the numerical value of ~/is given by r/ = 1.

(9.27)

Third, the results given by (9.24) and (9.25) are exact rather than approximate consequences of the variational principle that yields the field equations and their solutions. On the other hand, the Lorentz force q(~ + ~ x /3) is correct only in the approximation wherein radiation forces are neglected. Since the term --q~/" ] d d leads to the Lorentz force law, the remaining term q~//~] H+(dd) can be tentatively identified with the radiation forces. The expressions (9.24) and (9.25) give exact expressions for the radiation forces that act on the electric and magnetic current distributions as a consequence of the electromagnetic radiation generated by the motions of the charges that comprise these currents: =/g~z + L R ,

(9.28)

fqL = --'qq¢: ] d~¢ = Lorentz force on electric charges,

(9.29)

flqR = ~Tq~//~] dH+(d~') = radiation force on electric charges;

(9.30)

L =LL +LR,

(9.31)

fgL = --~/gq/] d~¢ = Lorentz force on magnetic charges,

(9.32)

l:gR = ~gql ] dH+(d~) = radiation force on magnetic charges.

(9.33)

Fourth, (9.28)-(9.33) show that the forces depend on the fields only through their "source free" parts o~/ = --d:¢¢,

~¢:/= --d J("

(9.34)

METRIC FREE ELECTRODYNAMICS

399

of the fields o~ ---- --ck~'+ H(~ ] to),

d~f = - - d J d + H ( J ] to).

(9.35)

Accordingly, the forces/q and/g do not depend on the unphysical "rays to infinity" that are generated by the action of H on ~ J to and J J to. In fact, (9.28)-(9.35) yield /qL = ~lq"1/" J ~ ' / ,

/,,R = --~q'¢/" ] d H + J f ,

(9.36)

ffgZ = ~Tgql ] ~ / ,

/~,R = - - W ql J dH+'~/.

(9.37)

The electric current may thus pass through the rays to infinity that are generated by either q¢ ] to or J ] to, and similarly for the magnetic current. This result is in sharp contrast to the Dirac theory [9], in which the electric current is not permitted to pass through the strings (rays to infinity) generated by the magnetic current. We note in particular, however, that/qL = ~q¢/" ] o~" only if H(f¢ J to) = 0 and ffgt~ = ~qq/] 3¢~ only if H ( J J to) = 0, and hence 7aL = q(/~ -t- ~ × /3) only when H ( ~ ] to) -----0; that is, ffqz = qCP ] o~- only when the electric current does not intersect the rays to infinity generated by f9 ] to. Fifth, (9.28)-(9.35) can be written in the equivalent form /~ = --~/q¢/- ] d(~¢ + ,~R).

/~

= -~g~

1 d(~f +

~),

(9.39)

where dR = --H+(dd) = H+~

(9.40)

can be viewed as the radiation pseudopotential 1-form of the electric current, and JT'R =- --H+(d~,T-) = H + ~

(9.41)

can be viewed as the radiation potential 1-form of the magnetic current. The smoothness of the forces/q a n d / g are thus determined exclusively by the smoothness of d , d R , and J(', #fR, respectively. In order to see just what this entails, we use (2.29) and (3.14) to evaluate dR in the case ~ = 0. This yields d.

= {7 x h - ( ~ ) + th-(•)}

• d~ -- ( 7 . h-(~)) dr,

(9.42)

where h-(~) = --

@(Ar,At) A dA

(9.43)

is the "antiray operator" that is induced by the operator H +. If we use the vacuum ether relations E = e~zb,/3 =/~ot7, (9.42) becomes dR = {/*07 × h-(H) + eolth-(/3)} • d~ -- eol(~ • h-(D)) dt, 595/II2[2-z2

(9.44)

400

DOMINIC G. B. EDELEN

which is linear in the field vectors H and /). The operator h - thus introduces yet another source o f nonlocality. The radiation force is then given by ,¢qR --- --~lq "1/" J d d R ---- ~Tq{St(/L0~ × h-(H) + eolth-(D)) + ~o1~(~ • h-~(D))

-- ~ × ~ × (izo? × h - ( ~ ) + ,oth-(D)))" dx -- ~Tq(eo~O • ¢ ( r . h-(D)) -~ ~ . O,(izo? × h-~(H) q- eoath-(D))) dt

(9.45)

in terms o f h-(H), h-(/)) and their derivatives. U p o n appropriate rearrangement, some o f the terms in (9.45) are reminiscent o f the approximate expressions obtained classically for the radiation force. Clearly, a full understanding o f the m a k e u p o f I¢~R ---- --~Tqy/" J dH+(~/) requires exact solutions o f the field equations and the equations o f m o t i o n for the charges. This is a complex and lengthy problem and will be studied in detail elsewhere.

REFERENCES 1. D. VAN DANTZIG,Akad. Wetensch. ,4msterdam 37 (1934), 521, 526, 643, 825; Proc. Cambridge Philos. Soc. 30 (1934), 421. 2. W. ~L~BODZr~SKI, "Exterior Forms and Their Applications," Polish Scientific Publishers, Warsaw, 1970. 3. H. CARTAN,"Differential Forms," Houghton Mifflin, Boston, 1970. 4. S. SrERNanRG, "Lectures on Differential Geometry," Prentice-Hall, Englewood Cliffs, New Jersey, 1964. 5. D. G. B. EOELEN, "Lagrangian Mechanics of Nonconservative Nonholonomic Systems,'" Noordhoff, Leyden, 1977. 6. D. G. B. EDELEN,"Antiexact Differential Forms," to appear. 7. R. MIGNA~, Phys. Rev. D 13 (1976), 2437. 8. J. A. STRAYrON,"Electromagnetic Theory," McGraw-Hill, New York, 1941. 9. P. A. M. DIRAC,Proc. Roy Soc. London Set..4 133 (1948), 60. 10. N. CABmaoANDF. FERRARI,Nuovo Cimento 23 (1962), 1147. 11. M. Y. HAN AND L. C. BIEDEt,r,-IARN,Nuovo Cimento Set..4 2 (1971), 544. 12. A. J. M. SPENCER,Theory of invariants, in "Continuum Physics," Vol. 1, Academic Press, New York, 1971. 13. S. OHKURO,3". Math. Phys. 11 (1970), 2005. 14. P. HAVAS,Phys. Rev. 74 (1948), 456. 15. R. ROHRLICH,Nuovo Cimento 21 (1961), 811. 16. T. T. Wu, Phys. D 14 (1976), 437.