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1.3 Green’s functions in flat spacetime

To see how Equation (5View Equation) can eventually be generalized to curved spacetimes, I introduce a new layer of mathematical formalism and show that the decomposition of the retarded potential into symmetric-singular and regular-radiative pieces can be performed at the level of the Green’s functions associated with Equation (1View Equation). The retarded solution to the wave equation can be expressed as
∫ α α ′ β′ ′ ′ Aret(x) = G +β′(x, x )j (x )dV (6 )
in terms of the retarded Green’s function α ′ α ′ ′ ′ G +β′(x,x ) = δβ′δ(t − t − |x − x |)∕ |x − x |. Here x = (t,x ) is an arbitrary field point, x ′ = (t′,x ′) is a source point, and dV ′ ≡ d4x′; tensors at x are identified with unprimed indices, while primed indices refer to tensors at x′. Similarly, the advanced solution can be expressed as
∫ ′ A αadv(x ) = G −αβ′(x,x′)jβ(x ′)dV ′ (7 )
in terms of the advanced Green’s function G −αβ′(x,x′) = δαβ′δ (t − t′ + |x − x ′|)∕|x − x′|. The retarded Green’s function is zero whenever x lies outside of the future light cone of x′, and α ′ G +β′(x,x ) is infinite at these points. On the other hand, the advanced Green’s function is zero whenever x lies outside of the past light cone of x′, and G α ′(x,x ′) − β is infinite at these points. The retarded and advanced Green’s functions satisfy the reciprocity relation
G −′(x′,x) = G+ ′(x, x′); (8 ) βα αβ
this states that the retarded Green’s function becomes the advanced Green’s function (and vice versa) when x and x′ are interchanged.

From the retarded and advanced Green’s functions we can define a singular Green’s function by

G α′(x,x′) = 1-[G α ′(x,x ′) + G α ′(x, x′)] (9 ) Sβ 2 + β −β
and a radiative Green’s function by
α ′ α ′ α ′ 1-[ α ′ α ′] GR β′(x, x) = G +β′(x,x ) − GSβ′(x,x ) = 2 G +β′(x, x) − G −β′(x,x ) . (10 )
By virtue of Equation (8View Equation) the singular Green’s function is symmetric in its indices and arguments: GSβ′α(x′,x ) = GSαβ′(x,x′). The radiative Green’s function, on the other hand, is antisymmetric. The potential
∫ α α ′ β′ ′ ′ A S(x) = GSβ′(x,x )j (x )dV (11 )
satisfies the wave equation of Equation (1View Equation) and is singular on the world line, while
∫ α α ′ β′ ′ ′ A R(x) = G R β′(x,x )j (x) dV (12 )
satisfies the homogeneous equation α □A = 0 and is well behaved on the world line.

Equation (6View Equation) implies that the retarded potential at x is generated by a single event in spacetime: the intersection of the world line and the past light cone of x’ (see Figure 1View Image). I shall call this the retarded point associated with x and denote it z(u); u is the retarded time, the value of the proper-time parameter at the retarded point. Similarly we find that the advanced potential of Equation (7View Equation) is generated by the intersection of the world line and the future light cone of the field point x. I shall call this the advanced point associated with x and denote it z(v); v is the advanced time, the value of the proper-time parameter at the advanced point.

View Image

Figure 1: In flat spacetime, the retarded potential at x depends on the particle’s state of motion at the retarded point z(u) on the world line; the advanced potential depends on the state of motion at the advanced point z (v ).

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