Three-vector and scalar field identities and uniqueness theorems in ...

set the divergence of Eq. (29) equal to a suitable (electric)scalar source charge density ρ e scaled by the free spacepermittivity constant ɛ 0 yieldingρ e= ∇ · E = −∇ · ∇φ − ∂∇ · A , (34)ɛ 0 ∂twhere the second term of Eq. (29) vanishes via Eq. (4).Then we set the curl of Eq. (29) equal to a suitable (magnetic)three-vector current density j m scaled by the freespace permeability constant µ 0 yielding−µ 0 j m = ∇ × E = −∇ × ∇ × K − ∂∇ × A , (35)∂twhere the first term of Eq. (29) vanishes via Eq. (3).Then we take the partial time derivative of Eq. (29) yielding∂E∂t = −∇∂φ ∂t − ∂∇ × K − ∂2 A∂t ∂t 2 . (36)In contrast to Eqs. (34) and (35) it is not evident whatsource density should be equated to the left-hand side ofEq. (36). In the present context of a time-varying electromagnetismwith both electric and magnetic charges, sucha third source relation would be inconsistent because theremaining two sources, ρ m and j e , would need to be accountedfor by the compatible relation (30). In addition,although Eq. (34) is adequate for the time-varying case,Eq. (35) is adequate only for static fields [compare withEq. (11) in Ref. 4]. If there are no magnetic sources,j m , ρ m , ξ, and K vanish and then Eq. (35) also lacks asource to uniquely specify it (or is reduced to no morethan the static case by setting j m = 0). Equation (30)suffers from similar difficulties with relations analogousto Eqs. (35) and (36). Consequently, it is problematicfor there to be a time-dependent version of theorem U1based on Eqs. (29) and (30) [or Eq. (26)] for the mostgeneral electromagnetic case.IV. TIME-DEPENDENT THREE-VECTORAND SCALAR FIELD IDENTITIESIn Sec. III it was shown that the use of the timedependentgeneralization (26) of the (static) Helmholtzidentity (13) does not appear to be adequate to obtaintime-dependent versions of uniqueness theorems U1 orH1. However, an “extended curl” 5 approach for obtaininga uniqueness theorem for a pair of coupled timedependentthree-vector fields in terms of two scalar andtwo three-vector source fields, and in terms of two auxiliaryfield definitions such as Eqs. (29) and (30), wasformulated in Ref. 22. This latter theorem was referredto there as a generalized Helmholtz theorem. 22 From ourpoint of view the first part of this two part theorem is oftype U1, and the second part is of the Helmholtz (projection)type H1, although the concepts of irrotationaland solenoidal are omitted. The proof is based on theexistence of nontrivial solutions of the associated ellipticalor hyperbolic differential equations. In the hyperboliccase the field equations can be used to treat the electromagneticcase with both electric and magnetic charges.As interesting as this theorem is, its equations are notderived from first principles, but are postulated in theirentirety. Secondly, it is a theorem for three-vectors in R 3with the time-dependence put in by hand, that is, it is notmanifestly covariant. It will be shown here that a manifestlycovariant approach 6 in R 3+1 can yield the desiredtime-varying uniqueness theorems that do not appear tofollow from first principles in R 3 .It has been previously shown (by taking a static Newtonianlimit) that the spatial components of the identityEq. (69) of theorem II of Ref. 6 [see Eq. (37)] is a Minkowskispace generalization of the (Euclidean three-space)Helmholtz identity (13), but more generally in a finitevolume of R 3 . In addition, the fourth or scalar field componentof this identity [see (Eq. 45)] was shown to yield ascalar field identity, Eq. (96) of Ref. 6, in a similar staticNewtonian limit. To obtain the full time-dependent resultswe first take the space components of this identity(that is, where the four-space index µ is replaced by thethree-space index j):[ ∫A j (x) = −∂ j ∂ νA ′ ν (x ′ )G(x, x ′ )d 4 x ′V 4′](A∮Σ ν (x ′ )n ′ ν)G(x, x ′ )dΣ ′′−[ ∫− ∂ α (∂ ′α A j (x ′ ) − ∂ ′j A α (x ′ ))G(x, x ′ )d 4 x ′V 4′]+ (A∮Σ α (x ′ )n ′j − A j (x ′ )n ′α )G(x, x ′ )dΣ ′ , (37)′where the four-volume region V 4 ′ of Minkowski space-timeis bounded by the three-surface Σ ′ , and the unprimedderivatives have been factored out of the primed coordinateintegrals. If we adopt three-vector notation andallow the four-volume region V 4 ′ to expand to includeall Minkowski space-time so that the three-surface integralscan be dropped [under the assumption that thefour-vector field A µ (x) = (φ/c, A) vanishes sufficientlyrapidly at infinity], we can simplify Eq. (37) to∫A(x) = −∇− 1 ∫∂c ∂t∫+ ∇ ×V 4′V 4′V ′4(∇ ′ · A(x ′ ) + 1 ∂φ(x ′ ))c 2 ∂t ′(− 1 ∂A(x ′ )c ∂t ′ − ∇′ φ(x ′ )cG(x, x ′ )d 4 x ′)G(x, x ′ )d 4 x ′(∇ ′ × A(x ′ ))G(x, x ′ )d 4 x ′ , (38)where the first four-volume integral of Eq. (38) followsfrom the first four-volume integral of Eq. (37), the secondfour-volume integral of Eq. (38) follows from the α = 06