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

volume and its associated finite volume Helmholtz theoremis satisfied. 6,9,10 Equation (13) has also been used ina time-varying context to demonstrate its compatibilitywith electromagnetism in the Coulomb gauge, and to derivethe Aharonov-Bohm transverse vector potential. 13–16The Helmholtz theorem was recently used to prove a vectoridentity for the volume integral of the square of avector field which could be time-varying, and Eq. (13)was used to derive two expressions for the energy of theelectromagnetic field. 17III. A TIME-DEPENDENT HELMHOLTZIDENTITYA logical next step for a three-vector field that is requiredto propagate via an inhomogeneous hyperbolicwave equation in terms of its three-vector (current)source is to derive an extension of the three-vector Helmholtzidentity by choosing the d’Alembertian wave equationoperator as the differential operator used to obtainan associated delta function identity. If we assume Cartesiancoordinates, it is sufficient to solve an inhomogeneousscalar d’Alembertian wave equation in terms of atwo-point scalar Green’s function G(x ν , x ′ν ) which connectsits unit delta function source located at the spacetimesource point x ′ν to a measurement at the space-timefield point x ν = (ct, x, y, z) in R 3+1 . That is,□G(x ν , x ′ν ) =(∇ 2 − 1 ∂ 2 )c 2 ∂t 2 G(x ν , x ′ν ) (17a)= −δ(x 0 − x ′0 )δ 3 (r − r ′ ) = −δ (4) (x ν − x ′ν ). (17b)An example of a Green’s function that satisfies Eq. (17b)assuming timelike causality is the familiar retardedGreen’s function as derived for example by Cushing 18or Jackson, 19 which for the metric signature (− + ++)used in this article isG ret (r, r ′ ; t, t ′ ) = 14πδ(|r − r ′ | − c(t − t ′ ))|r − r ′ , (18)|for t > t ′ . The retarded Green’s function (18) is derivedvia a spectral decomposition of the delta function,taking into account homogeneous boundary conditionson a closed spatial surface, as well as timelikecausality with the initial conditions G(r, r ′ ; t, t ′ ) = 0 and∂G(r, r ′ ; t, t ′ )/∂t = 0 for t < t ′ , and the Green’s functionsymmetry relationG(r, r ′ ; t, t ′ ) = G(r ′ , r; −t ′ , −t). (19)For a relativistically invariant version of Eq. (18), thatis, G ret (x, x ′ ), and for the advanced Green’s functionG adv (x, x ′ ), see for example Ref. 6. (In what followsthe usual shorthand notation is adopted, and the superscriptsin the functional dependencies are droppedfor brevity.) The spectral decomposition of the fourspacedelta function of Eq. (17b) also yields certain requiredcovariant and contravariant derivative propertiesof G(x, x ′ ) in Minkowski space: 6∂ µ G(x, x ′ ) = − ∂ ′ µG(x, x ′ ),∂ µ G(x, x ′ ) = − ∂ ′µ G(x, x ′ ).(20a)(20b)With ∂ µ = ((1/c)∂/∂t, ∇) and ∂ µ = (−(1/c)∂/∂t, ∇) weobtain the space and time components of Eq. (20a) as∂ i G(x, x ′ ) = − ∂ iG(x, ′ x ′ ),∂ 0 G(x, x ′ ) = − ∂ 0G(x, ′ x ′ ),(21a)(21b)with equivalent relations for the contravariant derivatives.Note that Eq. (21a) is analogous to the relation∇(1/r) = −∇ ′ (1/r) used in R 3 which can be rewrittenas ∇G(r, r ′ ) = −∇ ′ G(r, r ′ ). In R 3+1 there is the additionaltime derivative property (21b). For the presentanalysis Eqs. (21a) and (21b) can be written simply as∇G(x, x ′ ) = −∇ ′ G(x, x ′ ),(22a)∂G(x, x ′ )= − ∂G(x, x′ )∂t∂t ′ . (22b)The delta function property (17b) can be applied forany well-behaved function F(x ν ) as an integration overthe unbounded four-volume V 4 ′ of R 3+1 as follows:∫F(x) = F(x ′ )δ (4) (x − x ′ ) d 4 x ′(23a)∫=V 4′V ′4F(x ′ )□(−G(x, x ′ )) dV ′ c dt ′ .(23b)It is not necessary at this point to specify the choiceof a particular Green’s function G(x, x ′ ), other thanthat it satisfies Eq. (17b). In fact, there is a distinctadvantage to not choosing a retarded (advanced)Green’s function for doing the time integration at thispoint because the resulting retarded (advanced) threevectorfields are much more difficult with which to work.In contradistinction, the space and time derivatives ofa three-vector with unmixed arguments, for example,F(x ′ ) = F(ct ′ , x ′ , y ′ , z ′ ), needs no special treatment.Consequently, the entire (spatial) derivation of the Helmholtzidentity (13), in view of the equivalent form of relation(22a), can be used with Eq. (23b), yielding∫F(x) = −∇ ∇ ′ · F(x ′ )G(x, x ′ ) dV ′ c dt ′V 4′∫+ ∇ ×+ 1 c 2 ∂∂tV 4′∫V ′4∇ ′ × F(x ′ )G(x, x ′ ) dV ′ c dt ′∂(F(x ′ )G(x, x ′ ))∂tdV ′ c dt ′ , (24)where one of the unprimed partial time derivatives hasbeen taken outside of the integration over the primedtime derivative, and the other has been put in front ofF(x ′ ). The application of the time derivative in the integrandof the third term of Eq. (24) gives∂(F(x ′ )G(x, x ′ ))∂t= −F(x ′ ) ∂G(x, x′ )∂t ′(25a)= − ∂(F(x′ )G(x, x ′ ))∂t ′ + ∂F(x′ )∂t ′ G(x, x ′ ), (25b)4