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4.1 Scalar Green’s functions in flat spacetime

4.1.1 Green’s equation for a massive scalar field

To prepare the way for our discussion of Green’s functions in curved spacetime, we consider first the slightly nontrivial case of a massive scalar field Φ (x ) in flat spacetime. This field satisfies the wave equation

2 (□ − k )Φ(x) = − 4π μ(x), (233 )
where □ = η αβ∂α∂β is the wave operator, μ (x) a prescribed source, and where the mass parameter k has a dimension of inverse length. We seek a Green’s function G (x, x′) such that a solution to Equation (233View Equation) can be expressed as
∫ Φ (x ) = G (x,x′)μ(x′)d4x ′, (234 )
where the integration is over all of Minkowski spacetime. The relevant wave equation for the Green’s function is
2 ′ ′ (□ − k )G(x,x ) = − 4πδ4(x − x ), (235 )
where δ4(x − x′) = δ(t − t′)δ(x − x′)δ(y − y′)δ(z − z′) is a four-dimensional Dirac distribution in flat spacetime. Two types of Green’s functions will be of particular interest: the retarded Green’s function, a solution to Equation (235View Equation) with the property that it vanishes if x is in the past of x′, and the advanced Green’s function, which vanishes when x is in the future of x ′.

To solve Equation (235View Equation) we use Lorentz invariance and the fact that the spacetime is homogeneous to argue that the retarded and advanced Green’s functions must be given by expressions of the form

Gret(x, x′) = θ (t − t′)g(σ ), Gadv(x, x′) = θ(t′ − t)g(σ), (236 )
where 1 ′α ′β σ = 2ηαβ (x − x ) (x − x ) is Synge’s world function in flat spacetime, and where g (σ ) is a function to be determined. For the remainder of this section we set x ′ = 0 without loss of generality.

4.1.2 Integration over the source

The Dirac functional on the right-hand side of Equation (235View Equation) is a highly singular quantity, and we can avoid dealing with it by integrating the equation over a small four-volume V that contains x ′ ≡ 0. This volume is bounded by a closed hypersurface ∂V. After using Gauss’ theorem on the first term of Equation (235View Equation), we obtain ∮ G;αdΣ α − k2∫ G dV = − 4π ∂V V, where dΣ α is a surface element on ∂V. Assuming that the integral of G over V goes to zero in the limit V → 0, we have

∮ lim G;αdΣ α = − 4π. (237 ) V→0 ∂V
It should be emphasized that the four-volume V must contain the point x′.

To examine Equation (237View Equation) we introduce coordinates (w,χ, θ,φ) defined by

t = w cosχ, x = w sinχ sin θ cos φ, y = w sin χ sinθ sinφ, z = w sinχ cosθ,

and we let ∂V be a surface of constant w. The metric of flat spacetime is given by

ds2 = − cos2χ dw2 + 2w sin 2χ dw dχ + w2 cos2 χdχ2 + w2sin2 χdΩ2

in the new coordinates, where 2 2 2 2 dΩ = dθ + sin θd φ. Notice that w is a timelike coordinate when cos 2χ > 0, and that χ is then a spacelike coordinate; the roles are reversed when cos2 χ < 0. Straightforward computations reveal that in these coordinates σ = − 12w2 cos2χ, √ --- − g = w3 sin2χ sin θ, gww = − cos2 χ, gwχ = w −1sin2 χ, gχχ = w− 2cos2 χ, and the only nonvanishing component of the surface element is 3 2 dΣw = w sin χ dχ dΩ, where dΩ = sin θdθ dφ. To calculate the gradient of the Green’s function we express it as 1 2 G = θ(±t )g(σ) = θ(±w cosχ )g (− 2w cos 2χ), with the upper (lower) sign belonging to the retarded (advanced) Green’s function. Calculation gives G;αdΣ α = θ(± cos χ)w4 sin2 χ g′(σ )dχ dΩ, with a prime indicating differentiation with respect to σ; it should be noted that derivatives of the step function do not appear in this expression.

Integration of G;αdΣ α with respect to dΩ is immediate, and we find that Equation (237View Equation) reduces to

∫ π lim θ (± cosχ)w4 sin2χ g′(σ)dχ = − 1. (238 ) w→0 0
For the retarded Green’s function, the step function restricts the domain of integration to 0 < χ < π∕2, in which σ increases from − 1w2 2 to 1w2 2. Changing the variable of integration from χ to σ transforms Equation (238View Equation) into
∫ ε ∘ ------ ′ 1-+-ξ- liε→m0 ε −εw (σ∕ε)g (σ)dσ = − 1, w (ξ) ≡ 1 − ξ, (239 )
where ε ≡ 12w2. For the advanced Green’s function, the domain of integration is π∕2 < χ < π, in which σ decreases from 1w2 2 to − 1w2 2. Changing the variable of integration from χ to σ also produces Equation (239View Equation).

4.1.3 Singular part of g (σ)

We have seen that Equation (239View Equation) properly encodes the influence of the singular source term on both the retarded and advanced Green’s function. The function g(σ ) that enters into the expressions of Equation (236View Equation) must therefore be such that Equation (239View Equation) is satisfied. It follows immediately that g (σ ) must be a singular function, because for a smooth function the integral of Equation (239View Equation) would be of order ε, and the left-hand side of Equation (239View Equation) could never be made equal to − 1. The singularity, however, must be integrable, and this leads us to assume that g′(σ) must be made out of Dirac δ-functions and derivatives.

We make the ansatz

g(σ) = V (σ)θ(− σ) + Aδ(σ ) + B δ′(σ) + Cδ′′(σ ) + ..., (240 )
where V(σ ) is a smooth function, and A, B, C, …are constants. The first term represents a function supported within the past and future light cones of x′ ≡ 0; we exclude a term proportional to θ(σ ) for reasons of causality. The other terms are supported on the past and future light cones. It is sufficient to take the coefficients in front of the δ-functions to be constants. To see this we invoke the distributional identities
σδ(σ ) = 0 → σδ′(σ ) + δ (σ ) = 0 → σδ′′(σ) + 2δ′(σ) = 0 → ..., (241 )
from which it follows that 2 ′ 3 ′′ σ δ(σ ) = σ δ (σ) = ⋅⋅⋅ = 0. A term like f(σ)δ(σ ) is then distributionally equal to f(0)δ(σ ), while a term like ′ f(σ )δ (σ ) is distributionally equal to ′ ′ f(0) δ(σ ) − f (0)δ(σ), and a term like f(σ) δ′′(σ ) is distributionally equal to f(0)δ′′(σ) − 2f ′(0)δ′(σ ) + 2f ′′(0)δ(σ); here f(σ ) is an arbitrary test function. Summing over such terms, we recover an expression of the form of Equation (241View Equation), and there is no need to make A, B, C, …functions of σ.

Differentiation of Equation (240View Equation) and substitution into Equation (239View Equation) yields

∫ [ ∫ ] ε ′ ε ′ A B C (3) ε w (σ∕ε)g (σ) dσ = ε V (σ )w(σ∕ ε) dσ − V (0)w(0) − -εw˙(0) + ε2 ¨w(0) − ε3w (0) + ... , −ε −ε

where overdots (or a number within brackets) indicate repeated differentiation with respect to ξ ≡ σ ∕ε. The limit ε → 0 exists if and only if B = C = ⋅⋅⋅ = 0. In the limit we must then have A w˙(0) = 1, which implies A = 1. We conclude that g(σ ) must have the form of

g(σ) = δ(σ ) + V (σ)θ(− σ), (242 )
with V (σ) being a smooth function that cannot be determined from Equation (239View Equation) alone.

4.1.4 Smooth part of g(σ )

To determine V (σ) we must go back to the differential equation of Equation (235View Equation). Because the singular structure of the Green’s function is now under control, we can safely set x ⁄= x′ ≡ 0 in the forthcoming operations. This means that the equation to solve is in fact 2 (□ − k )g(σ ) = 0, the homogeneous version of Equation (235View Equation). We have ′ ∇ αg = g σα, ′′ ′ ∇ α∇ βg = g σα σβ + gσ αβ, and □g = 2σg′′ + 4g′, so that Green’s equation reduces to the ordinary differential equation

2σg′′ + 4g′ − k2g = 0. (243 )
If we substitute Equation (242View Equation) into this we get
− (2V + k2)δ(σ ) + (2σV ′′ + 4V ′ − k2V )θ(− σ) = 0,

where we have used the identities of Equation (241View Equation). The left-hand side will vanish as a distribution if we set

′′ ′ 2 1-2 2σV + 4V − k V = 0, V (0) = − 2k . (244 )
These equations determine V (σ) uniquely, even in the absence of a second boundary condition at σ = 0, because the differential equation is singular at σ = 0 and V is known to be smooth.

To solve Equation (244View Equation) we let V = F(z)∕z, with √ ----- z ≡ k − 2σ. This gives rise to Bessel’s equation for the new function F:

2 2 z Fzz + zFz + (z − 1)F = 0.

The solution that is well behaved near z = 0 is F = aJ1(z), where a is a constant to be determined. We have that J1(z) ∼ 12z for small values of z, and it follows that V ∼ a∕2. From Equation (244View Equation) we see that a = − k2. So we have found that the only acceptable solution to Equation (244View Equation) is

k ( √ ----) V (σ) = − √------J1 k − 2σ . (245 ) − 2σ

To summarize, the retarded and advanced solutions to Equation (235View Equation) are given by Equation (236View Equation) with g(σ ) given by Equation (242View Equation) and V (σ ) given by Equation (245View Equation).

4.1.5 Advanced distributional methods

The techniques developed previously to find Green’s functions for the scalar wave equation are limited to flat spacetime, and they would not be very useful for curved spacetimes. To pursue this generalization we must introduce more powerful distributional methods. We do so in this Section, and in the next we shall use them to recover our previous results.

Let θ+(x,Σ ) be a generalized step function, defined to be one if x is in the future of the spacelike hypersurface Σ, and defined to be zero otherwise. Similarly, define θ (x,Σ ) ≡ 1 − θ (x,Σ ) − + to be one if x is in the past of the spacelike hypersurface Σ, and zero otherwise. Then define the light-cone step functions

θ (− σ) = θ (x,Σ )θ(− σ), x ′ ∈ Σ, (246 ) ± ±
so that θ+(− σ) is one if x is an element of I+ (x ′), the chronological future of x ′, and zero otherwise, and θ− (− σ ) is one if x is an element of I− (x ′), the chronological past of x ′, and zero otherwise; the choice of hypersurface is immaterial so long as Σ is spacelike and contains the reference point ′ x. Notice that θ+ (− σ ) + θ − (− σ) = θ(− σ). Define also the light-cone Dirac functionals
δ (σ) = θ (x,Σ )δ(σ), x ′ ∈ Σ, (247 ) ± ±
so that δ+(σ), when viewed as a function of x, is supported on the future light cone of x′, while δ− (σ ) is supported on its past light cone. Notice that δ+ (σ ) + δ− (σ) = δ(σ). In Equations (246View Equation) and (247View Equation), σ is the world function for flat spacetime; it is negative if x and x′ are timelike related, and positive if they are spacelike related.

The distributions θ± (− σ ) and δ±(σ) are not defined at ′ x = x and they cannot be differentiated there. This pathology can be avoided if we shift σ by a small positive quantity ε. We can therefore use the distributions θ± (− σ − ε) and δ± (σ + ε) in some sensitive computations, and then take the limit ε → 0+. Notice that the equation σ + ε = 0 describes a two-branch hyperboloid that is located just within the light cone of the reference point ′ x. The hyperboloid does not include ′ x, and θ+ (x,Σ) is one everywhere on its future branch, while θ− (x,Σ ) is one everywhere on its past branch. These factors, therefore, become invisible to differential operators. For example, θ′+ (− σ − ε) = θ+ (x, Σ)θ′(− σ − ε) = − θ+ (x,Σ)δ(σ + ε) = − δ+(σ + ε). This manipulation shows that after the shift from σ to σ + ε, the distributions of Equations (246View Equation) and (247View Equation) can be straightforwardly differentiated with respect to σ.

In the next paragraphs we shall establish the distributional identities

lim+εδ± (σ + ε) = 0, (248 ) ε→0 ′ lε→i0m+ εδ± (σ + ε) = 0, (249 ) ′′ ′ lε→i0m+ εδ± (σ + ε) = 2 πδ4(x − x ) (250 )
in four-dimensional flat spacetime. These will be used in the next Section 4.1.6 to recover the Green’s functions for the scalar wave equation, and they will be generalized to curved spacetime in Section 4.2.

The derivation of Equations (248View Equation, 249View Equation, 250View Equation) relies on a “master” distributional identity, formulated in three-dimensional flat space:

ε-- 2π- √ -2----- lε→im0+ R5 = 3 δ3(x ), R ≡ r + 2ε, (251 )
with ∘ -2----2---2- r ≡ |x | ≡ x + y + z. This follows from yet another identity, 2 −1 ∇ r = − 4πδ3(x ), in which we write the left-hand side as lim ε=0+ ∇2R −1; since R −1 is nonsingular at x = 0, it can be straightforwardly differentiated, and the result is ∇2R − 1 = − 6ε∕R5, from which Equation (251View Equation) follows.

To prove Equation (248View Equation) we must show that εδ±(σ + ε) vanishes as a distribution in the limit + ε → 0. For this we must prove that a functional of the form

∫ A ±[f] = lim εδ± (σ + ε)f (x )d4x, ε→0+

where f(x) = f(t,x ) is a smooth test function, vanishes for all such functions f. Our first task will be to find a more convenient expression for δ±(σ + ε). Once more we set x′ = 0 (without loss of generality) and we note that 2(σ + ε) = − t2 + r2 + 2 ε = − (t − R )(t + R ), where we have used Equation (251View Equation). It follows that

δ(t ∓ R ) δ± (σ + ε) = ---R----, (252 )
and from this we find
∫ f(±R, x) ∫ ε 2π ∫ A± [f ] = lim+ ε---------d3x = lim+ --5R4f (±R, x)d3x = --- δ3(x )r4f(±r, x)d3x = 0, ε→0 R ε→0 R 3

which establishes Equation (248View Equation).

The validity of Equation (249View Equation) is established by a similar computation. Here we must show that a functional of the form

∫ B± [f ] = lim εδ±′(σ + ε)f(x)d4x ε→0+

vanishes for all test functions f. We have

∫ ∫ ∫ ( ˙ ) B± [f ] = lim ε d- δ±(σ + ε)f(x)d4x = lim ε d-- f-(±R,-x-)d3x = lim ε ± -f-− -f- d3x ε→0+ dε ε→0+ dε R ε→0+ R2 R3 ∫ ( ) ∫ ( ) = lim -ε- ±R3 f˙− R2f d3x = 2π- δ3(x) ±r3f˙− r2f d3x ε→0+ R5 3 = 0,
and the identity of Equation (249View Equation) is proved. In these manipulations we have let an overdot indicate partial differentiation with respect to t, and we have used ∂R ∕∂ε = 1∕R.

To establish Equation (250View Equation) we consider the functional

∫ ′′ 4 C ±[f] = lεi→m0+ εδ±(σ + ε)f (x )d x,

and show that it evaluates to 2πf(0,0 ). We have

2 ∫ 2 ∫ C± [f ] = lim ε-d- δ± (σ + ε)f (x)d4x = lim ε-d- f(±R,--x)-d3x ε→0+ dε2( ) ε→0+ d ε2 R ∫ ¨f ˙f f ∫ ( 1 ) = lim ε ---∓ 3---+ 3--- d3x = 2π δ3(x ) -r2 ¨f ± rf˙+ f d3x ε→0+ R3 R4 R5 3 = 2πf (0,0),
as required. This proves that Equation (250View Equation) holds as a distributional identity in four-dimensional flat spacetime.

4.1.6 Alternative computation of the Green’s functions

The retarded and advanced Green’s functions for the scalar wave equation are now defined as the limit of the functions G ε±(x,x′) as ε → 0+. For these we make the ansatz

ε ′ G± (x,x ) = δ±(σ + ε) + V (σ )θ± (− σ − ε), (253 )
and we shall prove that G ε(x,x′) ± satisfies Equation (235View Equation) in the limit. We recall that the distributions θ± and δ± were defined in the preceding Section 4.1.5, and we assume that V (σ) is a smooth function of ′ 1 ′ α ′ β σ(x,x ) = 2ηαβ(x − x ) (x − x ); because this function is smooth, it is not necessary to evaluate V at σ + ε in Equation (253View Equation). We recall also that θ+ and δ+ are nonzero when x is in the future of x′, while θ− and δ− are nonzero when x is in the past of x ′. We will therefore prove that the retarded and advanced Green’s functions are of the form
Gret(x,x ′) = lim G ε+(x,x′) = θ+(x,Σ )[δ(σ) + V(σ )θ(− σ )] (254 ) ε→0+
′ ε ′ Gadv(x,x ) = εli→m0+ G− (x,x ) = θ− (x,Σ )[δ(σ) + V (σ )θ(− σ)], (255 )
where Σ is a spacelike hypersurface that contains ′ x. We will also determine the form of the function V (σ).

The functions that appear in Equation (253View Equation) can be straightforwardly differentiated. The manipulations are similar to what was done in Section 4.1.4, and dropping all labels, we obtain 2 ′′ ′ 2 (□ − k )G = 2 σG + 4G − k G, with a prime indicating differentiation with respect to σ. From Equation (253View Equation) we obtain G ′ = δ′ − V δ + V ′θ and G′′ = δ′′ − V δ′ − 2V ′δ + V′′θ. The identities of Equation (241View Equation) can be expressed as (σ + ε)δ′(σ + ε) = − δ(σ + ε) and (σ + ε)δ′′(σ + ε) = − 2δ ′(σ + ε), and combining this with our previous results gives

(□ − k2)G ε±(x,x′) = (− 2V − k2)δ±(σ + ε) + (2σV ′′ + 4V ′ − k2V )θ±(− σ − ε) − 2εδ′′(σ + ε) + 2V εδ′(σ + ε) + 4V ′εδ (σ + ε). ± ± ±
According to Equation (248View Equation, 249View Equation, 250View Equation), the last two terms on the right-hand side disappear in the limit ε → 0+, and the third term becomes − 4πδ4(x − x′). Provided that the first two terms vanish also, we recover 2 ′ ′ (□ − k )G (x, x) = − 4π δ4(x − x ) in the limit, as required. Thus, the limit of ε ′ G ±(x, x) as + ε → 0 will indeed satisfy Green’s equation provided that V(σ ) is a solution to
2σV ′′ + 4V ′ − k2V = 0, V (0) = − 1k2; (256 ) 2
these are the same statements as in Equation (244View Equation). The solution to these equations was produced in Equation (245View Equation),
--k---- ( √ -----) V (σ) = − √ −-2σ-J1 k − 2σ , (257 )
and this completely determines the Green’s functions of Equations (254View Equation) and (255View Equation).
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