15 Electromagnetic Green’s functions

15.1 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 ∇ A α = 0 α, the vector potential satisfies the wave equation

α α β α ā–”A − R βA = − 4 πj , (15.1 )
where ā–” = 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

∫ α α ′ β′ ′ āˆ˜ ---′4 ′ A (x) = G β′(x, x)j (x ) − gd x , (15.2 )
in terms of a Green’s function G α ′(x, x′) β that satisfies
α ′ α β ′ α ′ ′ ā–”G β′(x,x ) − R β(x)G β′(x,x ) = − 4πg β′(x, x )δ4(x,x ), (15.3 )
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 Eq. (15.3View Equation) to keep the index structure of the equation consistent from side to side; because gαβ′(x, x′)δ4(x,x ′) is distributionally equal to [gαβ′]δ4(x,x′) = δαβ′′δ4(x,x′), it could have been replaced by either δα′β′ or δαβ.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 Eq. (15.3View Equation), the vector potential of Eq. (15.2View Equation) satisfies the wave equation of Eq. (15.1View Equation).

We will assume that the retarded Green’s function G α+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.

15.2 Hadamard construction of the Green’s functions

Assuming throughout this subsection 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′)šœƒ± (− σ ), (15.4 )
where šœƒ±(− σ), δ±(σ) are the light-cone distributions introduced in Section 13.2, and where U α (x,x ′) β′, Vα (x,x ′) β′ are smooth bitensors.

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

G šœ–± αβ′(x, x′) := U αβ′(x,x′)δ±(σ + šœ–) + Vαβ′(x,x ′)šœƒ±(− σ − šœ–)

as šœ– → 0+. When we substitute this into the left-hand side of Eq. (15.3View Equation) and then take the limit, we obtain

β { } ā–”G ±αβ′ − R αβG± β′ = − 4πδ4(x, x′)U αβ′ + δ′±(σ) 2U αβ′;γσγ + (σγγ − 4)U αβ′ { } + δ±(σ ) − 2V αβ′;γσγ + (2 − σγγ)Vαβ′ + ā–”U αβ′ − RαβU ββ′ { } + šœƒ±(− σ) ā–”V αβ′ − R αβV ββ′
after a routine computation similar to the one presented at the beginning of Section 14.2. Comparison with Eq. (15.3View Equation) returns: (i) the equations
[ α ] [ α ] α′ U β′ = gβ′ = δ β′ (15.5 )
and
2U αβ′;γσ γ + (σ γγ − 4)U αβ′ = 0 (15.6 )
that determine α ′ U β′(x, x); (ii) the equation
α γ 1 γ α 1( α α β )|| Ė‡V β′;γσ + --(σ γ − 2)VĖ‡β′ = --ā–”U β′ − R βU β′| (15.7 ) 2 2 σ=0
that determines α ′ VĖ‡β′(x,x ), the restriction of α ′ V β′(x,x ) on the light cone ′ σ(x, x) = 0; and (iii) the wave equation
β ā–”V αβ′ − R αβV β′ = 0 (15.8 )
that determines α ′ V β′(x,x ) inside the light cone.

Eq. (15.6View Equation) can be integrated along the unique geodesic β that links x ′ to x. The initial conditions are provided by Eq. (15.5View Equation), and if we set U αβ′(x,x ′) = gαβ′(x, x′)U (x,x ′), we find that these equations reduce to Eqs. (14.7View Equation) and (14.6View Equation), respectively. According to Eq. (14.8View Equation), then, we have

U α (x,x ′) = gα (x, x′)Δ1 āˆ•2(x,x′), (15.9 ) β′ β′
which reduces to
( 1 ′ ′ ) U αβ′ = gαβ′ 1 + --R γ′δ′σγ σδ + O (λ3) (15.10 ) 12
near coincidence, with λ denoting the affine-parameter distance between x′ and x. Differentiation of this relation gives
( ) U αβ′;γ = 1-gγγ′ gαα′R α′β′γ′δ′ − 1g αβ′R γ′δ′ σδ′ + O (λ2), (15.11 ) 2 ( 3 ) 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 ), (15.12 )
and eventually,
[ α ] 1- α′ ′ ā–”U β′ = 6δ β′R (x ). (15.13 )

Similarly, Eq. (15.7View 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 Eqs. (15.5View Equation), (15.13View Equation), and the additional relation [σγγ] = 4. We arrive at

[ α ] 1( α′ 1-α′ ′) V β′ = − 2 R β′ − 6δ β′R . (15.14 )
With the characteristic data obtained by integrating Eq. (15.7View Equation), the wave equation of Eq. (15.8View Equation) admits a unique solution.

To summarize, the retarded and advanced electromagnetic Green’s functions are given by Eq. (15.4View Equation) with U α ′(x,x ′) β given by Eq. (15.9View Equation) and V α′(x,x′) β determined by Eq. (15.8View Equation) and the characteristic data constructed with Eqs. (15.7View Equation) and (15.14View Equation). It should be emphasized that the construction provided in this subsection is restricted to ′ š’© (x), the normal convex neighbourhood of the reference point x ′.

15.3 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′). (15.15 )
The derivation of Eq. (15.15View Equation) is virtually identical to what was presented in Section 14.3, and we shall not present the details. It suffices to mention that it is based on the identities
( γ ) G+αβ′(x,x′) ā–”G α−γ′′(x,x′′) − R αγG −γ′′(x,x ′′) = − 4πG+αβ′(x,x′)gαγ′′(x, x′′)δ4(x,x′′&#x0029

and

( ) − ′′ α ′ α γ ′ − ′′ α ′ ′ Gα γ′′(x, x ) ā–”G +β′(x, x) − R γG+ β′(x,x ) = − 4πG αγ′′(x,x )g β′(x, x)δ4(x,x ).

A direct consequence of the reciprocity relation is

V β′α(x′,x) = Vαβ′(x,x′), (15.16 )
the statement that the bitensor ′ Vαβ′(x,x ) 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 Eq. (15.1View 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

∫ ( ) A α(x) = − -1- G α ′(x,x ′)∇ γ′A β′(x′) − A β′(x′)∇ γ′G α′(x,x′) d Σγ′, (15.17 ) 4π Σ + β +β
where dΣγ′ = − nγ′dV is a surface element on Σ; nγ′ is the future-directed unit normal and dV is the invariant volume element on the hypersurface. The derivation of Eq. (15.17View Equation) is virtually identical to what was presented in Section 14.4.

15.4 Relation with scalar Green’s functions

In a spacetime that satisfies the Einstein field equations in vacuum, so that Rαβ = 0 everywhere in the spacetime, the (retarded and advanced) electromagnetic Green’s functions satisfy the identities [54Jump To The Next Citation Point]

G±αβ′;α = − G ±;β′, (15.18 )
where G± are the corresponding scalar Green’s functions.

To prove this we differentiate Eq. (15.3View Equation) covariantly with respect to xα and use Eq. (13.3View Equation) to express the right-hand side as +4 π∂ β′δ4(x,x ′). After repeated use of Ricci’s identity to permute the ordering of the covariant derivatives on the left-hand side, we arrive at the equation

( ) ā–” − Gαβ′;α = − 4π ∂β′δ4(x,x ′); (15.19 )
all terms involving the Riemann tensor disappear by virtue of the fact that the spacetime is Ricci-flat. Because Eq. (15.19View Equation) is also the differential equation satisfied by G;β′, and because the solutions are chosen to satisfy the same boundary conditions, we have established the validity of Eq. (15.18View Equation).

15.5 Singular and regular Green’s functions

We shall now construct singular and regular Green’s functions for the electromagnetic field. The treatment here parallels closely what was presented in Section 14.5, 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

H1:
H α ′(x,x ′) β satisfies the homogeneous wave equation,
ā–”H α ′(x, x′) − R α (x )H β (x,x′) = 0; (15.20) β β β′
H2:
α ′ H β′(x,x ) is symmetric in its indices and arguments,
H β′α(x′,x ) = Hα β′(x,x ′); (15.21)
H3:
α ′ H β′(x,x ) agrees with the retarded Green’s function if x is in the chronological future of ′ x,
α ′ α ′ + ′ H β′(x,x ) = G +β′(x,x ) when x ∈ I (x ); (15.22)
H4:
H α ′(x,x ′) β agrees with the advanced Green’s function if x is in the chronological past of x′,
H αβ′(x, x′) = G −αβ′(x,x′) when x ∈ I− (x′). (15.23)

It is easy to prove that property H4 follows from H2, H3, and the reciprocity relation (15.15View 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 14.5.

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

α ′ 1-[ α ′ α ′ α ′] G Sβ′(x,x ) = 2 G +β′(x, x) + G −β′(x,x ) − H β′(x, x) . (15.24 )
This comes with the properties
S1:
α ′ GSβ′(x,x ) satisfies the inhomogeneous wave equation,
β ā–”G Sαβ′(x, x′) − R αβ (x )GSβ′(x,x′) = − 4 πgαβ′(x,x ′)δ4(x,x′); (15.25)
S2:
α ′ GSβ′(x,x ) is symmetric in its indices and arguments,
S ′ S ′ G β′α(x,x ) = Gαβ′(x,x ); (15.26)
S3:
α ′ GSβ′(x,x ) vanishes if x is in the chronological past or future of ′ x,
G αSβ′(x,x ′) = 0 when x ∈ I±(x′). (15.27)

These can be established as consequences of H1 H4 and the properties of the retarded and advanced Green’s functions.

The regular two-point function is then defined by

G α′(x,x′) = G α ′(x,x ′) − G α′(x,x′), (15.28 ) Rβ + β Sβ
and it comes with the properties
R1:
α ′ G Rβ′(x,x ) satisfies the homogeneous wave equation,
ā–”G αRβ′(x, x′) − R αβ(x )GRββ′(x,x ′) = 0; (15.29)
R2:
G αRβ′(x,x ′) agrees with the retarded Green’s function if x is in the chronological future of x′,
α ′ α ′ + ′ G Rβ′(x, x) = G +β′(x,x ) when x ∈ I (x); (15.30)
R3:
G α (x,x ′) Rβ′ vanishes if x is in the chronological past of x′,
α ′ − ′ G Rβ′(x, x) = 0 when x ∈ I (x ). (15.31)

Those follow immediately from S1 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

α ′ α ′ H β′(x, x ) = V β′(x,x ), (15.32 ) α ′ 1- α ′ 1- α ′ G Sβ′(x, x ) = 2U β′(x,x )δ(σ) − 2V β′(x,x )šœƒ(σ), (15.33 ) 1 [ ] [ 1 ] G Rαβ′(x, x′) = -U αβ′(x,x′) δ+(σ) − δ− (σ ) + V αβ′(x,x′) šœƒ+(− σ) +-šœƒ(σ ) . (15.34 ) 2 2
From these we see clearly that the singular Green’s function does not distinguish between past and future (property S2), and that its support excludes I± (x ′) (property S3). We see also that the regular two-point function coincides with α ′ G+ β′(x,x ) in + ′ I (x ) (property R2), and that its support does not include − ′ I (x) (property R3).
  Go to previous page Go up Go to next page