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4.4 Electromagnetic Green’s functions

4.4.1 Equations of electromagnetism

The electromagnetic field tensor F αβ = ∇ αA β − ∇ βA α is expressed in terms of a vector potential Aα. In the Lorenz gauge α ∇ αA = 0, the vector potential satisfies the wave equation

α α β α □A − R βA = − 4 πj , (314 )
where □ = gαβ∇ ∇ α β is the wave operator, R α β the Ricci tensor, and jα a prescribed current density. The wave equation enforces the condition α ∇ αj = 0, which expresses charge conservation.

The solution to the wave equation is written as

∫ ---- Aα (x ) = G α ′(x,x ′)jβ′(x′)∘ − g ′d4x ′, (315 ) β
in terms of a Green’s function α ′ G β′(x, x) that satisfies
α ′ α β ′ α ′ ′ □G β′(x,x ) − R β(x)G β′(x,x ) = − 4πg β′(x, x )δ4(x,x ), (316 )
where gα ′(x,x ′) β is a parallel propagator and δ (x, x′) 4 an invariant Dirac distribution. The parallel propagator is inserted on the right-hand side of Equation (316View Equation) to keep the index structure of the equation consistent from side to side; because α ′ ′ g β′(x, x)δ4(x,x ) is distributionally equal to [gαβ′]δ4(x,x′) = δαβ′′δ4(x,x′), it could have been replaced by either δα′β′ or δαβ. It is easy to check that by virtue of Equation (316View Equation), the vector potential of Equation (315View Equation) satisfies the wave equation of Equation (314View Equation).

We will assume that the retarded Green’s function α ′ G +β′(x, x), which is nonzero if x is in the causal future of x ′, and the advanced Green’s function G−αβ′(x,x ′), which is nonzero if x is in the causal past of x ′, exist as distributions and can be defined globally in the entire spacetime.

4.4.2 Hadamard construction of the Green’s functions

Assuming throughout this section that x is in the normal convex neighbourhood of x ′, we make the ansatz

G ±αβ′(x,x ′) = U αβ′(x,x′)δ±(σ) + V αβ′(x, x′)θ± (− σ ), (317 )
where θ±(− σ), δ±(σ) are the light-cone distributions introduced in Section 4.2.2, and where U αβ′(x,x ′), Vαβ′(x,x ′) are smooth bitensors.

To conveniently manipulate the Green’s functions we shift σ by a small positive quantity ε. The Green’s functions are then recovered by the taking the limit of

Gε± αβ′(x,x′) ≡ U αβ′(x,x′)δ± (σ + ε) + V αβ′(x,x′)θ±(− σ − ε)

as + ε → 0. When we substitute this into the left-hand side of Equation (316View Equation) and then take the limit, we obtain

□G α′ − R αG β′ = − 4πδ4(x,x ′)U α′ + δ′(σ){2U α′ σγ + (σγ − 4)U α′} ±β β ±β { β ± β ;γ γ β } + δ± (σ) − 2Vαβ′;γσγ + (2 − σγγ)V αβ′ + □U αβ′ − R αβU β′ { } β + θ (− σ ) □V α′ − R αV β′ ± β β β
after a routine computation similar to the one presented at the beginning of Section 4.3.2. Comparison with Equation (316View Equation) returns

Equation (319View Equation) can be integrated along the unique geodesic β that links ′ x to x. The initial conditions are provided by Equation (318View Equation), and if we set α ′ α ′ ′ U β′(x, x) = g β′(x,x )U(x,x ), we find that these equations reduce to Equations (272View Equation) and (271View Equation), respectively. According to Equation (273View Equation), then, we have

U α ′(x,x ′) = gα ′(x, x′)Δ1 ∕2(x,x′), (322 ) β β
which reduces to
( ) U α′ = gα ′ 1 + 1-R ′′σγ′σ δ′ + 𝒪 (λ3) (323 ) β β 12 γδ
near coincidence, with λ denoting the affine-parameter distance between ′ x and x. Differentiation of this relation gives
( ) α 1 γ′ α α′ 1 α δ′ 2 U β′;γ = 2g γ g α′R β′γ′δ′ − 3g β′R γ′δ′ σ + 𝒪 (λ ), (324 ) ( ) α 1- α α′ 1- α δ′ 2 U β′;γ′ = 2 g α′R β′γ′δ′ + 3 gβ′R γ′δ′ σ + 𝒪 (λ ), (325 )
and eventually,
[ ] 1 □U αβ′ = -δ αβ′′R (x ′). (326 ) 6

Similarly, Equation (320View Equation) can be integrated along each null geodesic that generates the null cone ′ σ(x,x ) = 0. The initial values are obtained by taking the coincidence limit of this equation, using Equations (318View Equation), (326View Equation), and the additional relation γ [σ γ] = 4. We arrive at

( ) [Vα ] = − 1- Rα′ − 1δα′R ′ . (327 ) β′ 2 β′ 6 β′
With the characteristic data obtained by integrating Equation (320View Equation), the wave equation of Equation (321View Equation) admits a unique solution.

To summarize, the retarded and advanced electromagnetic Green’s functions are given by Equation (317View Equation) with U α′(x,x′) β given by Equation (322View Equation) and V α′(x, x′) β determined by Equation (321View Equation) and the characteristic data constructed with Equations (320View Equation) and (327View Equation). It should be emphasized that the construction provided in this section is restricted to 𝒩 (x′), the normal convex neighbourhood of the reference point x ′.

4.4.3 Reciprocity and Kirchhoff representation

Like their scalar counterparts, the (globally defined) electromagnetic Green’s functions satisfy a reciprocity relation, the statement of which is

G −β′α(x ′,x) = G+αβ′(x, x′). (328 )
The derivation of Equation (328View Equation) is virtually identical to what was presented in Section 4.3.3, and we shall not present the details. It suffices to mention that it is based on the identities
( ) G+αβ′(x, x′) □G −αγ′′(x, x′′) − RαγG −γγ′′(x,x ′′) = − 4πG+αβ′(x,x′)gαγ′′(x,x′′)δ4(x,x′&#x203


( γ ) G−αγ′′(x,x ′′) □G α+β′(x, x′) − R αγG +β′(x,x′) = − 4πG −αγ′′(x,x′′)gαβ′(x, x′)δ4(x,x ′).

A direct consequence of the reciprocity relation is

′ ′ V β′α(x ,x) = Vαβ′(x,x ), (329 )
the statement that the bitensor V ′(x,x′) αβ is symmetric in its indices and arguments.

The Kirchhoff representation for the electromagnetic vector potential is formulated as follows. Suppose that A α(x) satisfies the homogeneous version of Equation (314View Equation) and that initial values ′ A α(x′), n β′∇ β′Aα′(x′) are specified on a spacelike hypersurface Σ. Then the value of the potential at a point x in the future of Σ is given by

1 ∫ ( ) A α(x) = − --- G +αβ′(x,x′)∇ γ′A β′(x ′) − Aβ′(x′)∇ γ′G +αβ′(x,x′) dΣγ′, (330 ) 4π Σ
where dΣ ′ = − n ′dV γ γ is a surface element on Σ; n ′ γ is the future-directed unit normal and dV is the invariant volume element on the hypersurface. The derivation of Equation (330View Equation) is virtually identical to what was presented in Section 4.3.4.

4.4.4 Singular and radiative Green’s functions

We shall now construct singular and radiative Green’s functions for the electromagnetic field. The treatment here parallels closely what was presented in Section 4.3.5, and the reader is referred to that section for a more complete discussion.

We begin by introducing the bitensor H αβ′(x,x ′) with properties

  Em.H1: α ′ H β′(x, x) satisfies the homogeneous wave equation,

α ′ α β ′ □H β′(x,x ) − R β(x)H β′(x,x ) = 0; (331 )

  Em.H2: H αβ′(x, x′) is symmetric in its indices and arguments,

′ ′ H β′α(x ,x) = H αβ′(x, x); (332 )

  Em.H3: α ′ H β′(x, x) agrees with the retarded Green’s function if x is in the chronological future of x′,

H α′(x,x′) = G α ′(x,x ′) when x ∈ I+ (x′); (333 ) β + β

  Em.H4: H α ′(x, x′) β agrees with the advanced Green’s function if x is in the chronological past of x′,

α ′ α ′ − ′ H β′(x,x ) = G −β′(x, x) when x ∈ I (x ). (334 )

It is easy to prove that Property Em.H4 follows from Property Em.H2, Property Em.H3, and the reciprocity relation (328View Equation) satisfied by the retarded and advanced Green’s functions. That such a bitensor exists can be argued along the same lines as those presented in Section 4.3.5.

Equipped with the bitensor α ′ H β′(x, x) we define the singular Green’s function to be

[ ] GSαβ′(x,x ′) = 1- G α+β′(x, x′) + G −αβ′(x,x′) − H αβ′(x, x′) . (335 ) 2
This comes with the properties

  Em.S1: α ′ G Sβ′(x, x) satisfies the inhomogeneous wave equation,

□G α (x,x′) − R α (x)G β (x, x′) = − 4πg α (x,x′)δ (x,x′); (336 ) Sβ′ β Sβ′ β′ 4

  Em.S2: G Sαβ′(x, x′) is symmetric in its indices and arguments,

S ′ S ′ G β′α(x ,x) = G αβ′(x, x); (337 )

  Em.S3: G Sαβ′(x, x′) vanishes if x is in the chronological past or future of x ′,

α ′ ± ′ G Sβ′(x,x ) = 0 when x ∈ I (x). (338 )

These can be established as consequences of Properties Em.H1, Em.H2, Em.H3, and Em.H4, and the properties of the retarded and advanced Green’s functions.

The radiative Green’s function is then defined by

α ′ α ′ α ′ G Rβ′(x,x ) = G+ β′(x,x ) − G Sβ′(x,x ), (339 )
and it comes with the properties

  Em.R1: α ′ G Rβ′(x,x ) satisfies the homogeneous wave equation,

α ′ α β ′ □G Rβ′(x,x ) − R β(x)G Rβ′(x,x ) = 0; (340 )

  Em.R2: G Rαβ′(x,x′) agrees with the retarded Green’s function if x is in the chronological future of x ′

G α′(x,x′) = G α ′(x, x′) when x ∈ I+(x′); (341 ) Rβ +β

  Em.R3: G α′(x,x′) Rβ vanishes if x is in the chronological past of x′,

G α′(x,x′) = 0 when x ∈ I− (x′). (342 ) Rβ

Those follow immediately from Properties Em.S1, Em.S2, and Em.S3 and the properties of the retarded Green’s function.

When x is restricted to the normal convex neighbourhood of ′ x, we have the explicit relations

H αβ′(x, x′) = V αβ′(x,x′), (343 ) 1 1 G αSβ′(x, x′) = -U αβ′(x,x′)δ(σ ) − -V αβ′(x,x′)θ(σ), (344 ) 2 2 [ ] α ′ 1- α ′ α ′ 1- GR β′(x, x) = 2U β′(x,x )[δ+(σ) − δ− (σ)] + V β′(x,x ) θ+(− σ) + 2θ(σ ) . (345 )
From these we see clearly that the singular Green’s function does not distinguish between past and future (Property Em.S2), and that its support excludes I±(x′) (Property Em.S3). We see also that the radiative Green’s function coincides with G α ′(x,x ′) + β in I+(x′) (Property Em.R2), and that its support does not include I− (x ′) (Property Em.R3).
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