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4.5 Gravitational Green’s functions

4.5.1 Equations of linearized gravity

We are given a background spacetime for which the metric gαβ satisfies the Einstein field equations in vacuum. We then perturb the metric from gαβ to

gαβ = gαβ + hαβ. (346 )
The metric perturbation hαβ is assumed to be small, and when working out the Einstein field equations to be satisfied by the new metric gαβ, we work consistently to first order in h αβ. To simplify the expressions we use the trace-reversed potentials γαβ defined by
1-( γδ ) γ αβ = hαβ − 2 g hγδ gα β, (347 )
and we impose the Lorenz gauge condition,
α β γ ;β = 0. (348 )
In this equation, and in all others below, indices are raised and lowered with the background metric g αβ. Similarly, the connection involved in Equation (348View Equation), and in all other equations below, is the one that is compatible with the background metric. If Tαβ is the perturbing stress-energy tensor, then by virtue of the linearized Einstein field equations the perturbation field obeys the wave equation
αβ α β γδ αβ □γ + 2Rγ δ γ = − 16πT , (349 )
in which □ = gαβ∇ ∇ α β is the wave operator and R γαδβ the Riemann tensor. In first-order perturbation theory, the stress-energy tensor must be conserved in the background spacetime: αβ T ;β = 0.

The solution to the wave equation is written as

∫ αβ αβ ′ γ′δ′ ′∘ --′- 4 ′ γ (x) = 4 G γ′δ′(x, x)T (x) − g d x , (350 )
in terms of a Green’s function G αβ ′′(x, x′) γδ that satisfies [53]
αβ ′ α β γδ ′ (α ′ β) ′ ′ □G γ′δ′(x,x ) + 2R γ δ (x)G γ′δ′(x,x ) = − 4πg γ′(x, x)g δ′(x, x)δ4(x,x ), (351 )
where gα (x,x′) γ′ is a parallel propagator and δ (x, x′) 4 an invariant Dirac functional. The parallel propagators are inserted on the right-hand side of Equation (351View Equation) to keep the index structure of the equation consistent from side to side; in particular, both sides of the equation are symmetric in α and β, and in γ′ and δ′. It is easy to check that by virtue of Equation (351View Equation), the perturbation field of Equation (350View Equation) satisfies the wave equation of Equation (349View Equation). Once γαβ is known, the metric perturbation can be reconstructed from the relation h = γ − 1(g γδγ )g αβ αβ 2 γδ αβ.

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

4.5.2 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±αβγ′δ′(x,x ′) = U αβγ′δ′(x,x′)δ±(σ) + V αβγ′δ′(x, x′)θ± (− σ ), (352 )
where θ±(− σ), δ±(σ) are the light-cone distributions introduced in Section 4.2.2, and where U αβ (x,x′) γ′δ′, Vαβ (x,x ′) γ′δ′ 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± αβγ′δ′(x,x′) = U αβγ′δ′(x, x′)δ± (σ + ε) + V αβγ′δ′(x,x′)θ±(− σ − ε)

as + ε → 0. When we substitute this into the left-hand side of Equation (351View Equation) and then take the limit, we obtain

αβ α β γδ ′ αβ ′ { αβ γ γ αβ } □G ± γ′δ′ + 2Rγ δ G ± γ′δ′ = − 4 πδ4(x,x )U γ′δ′ + δ± (σ ) 2U γ′δ′;γσ + (σ γ − 4)U γ′δ′ { αβ γ γ αβ αβ α β γδ } + δ±(σ ) − 2V γ′δ′;γσ + (2 − σ γ)V γ′δ′ + □U γ′δ′ + 2R γ δ U γ′δ′ { αβ α β γδ } + θ±(− σ) □V γ′δ′ + 2R γ δ V γ′δ′
after a routine computation similar to the one presented at the beginning of Section 4.3.2. Comparison with Equation (351View Equation) returns

Equation (354View Equation) can be integrated along the unique geodesic β that links ′ x to x. The initial conditions are provided by Equation (353View Equation), and if we set Uαβγ′δ′(x, x′) = g(αγ′gβδ)′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

αβ ′ (α ′ β) ′ 1∕2 ′ U γ′δ′(x,x ) = gγ′(x,x )g δ′(x,x )Δ (x,x ), (357 )
which reduces to
U αβ = g(αgβ) (1 + 𝒪 (λ3 )) (358 ) γ′δ′ γ′ δ′
near coincidence, with λ denoting the affine-parameter distance between x′ and x; there is no term of order λ2 because, by assumption, the background Ricci tensor vanishes at x ′ (as it does in the entire spacetime). Differentiation of this relation gives
αβ 1 (α β) ε′( α′ β′ α′ β′) ι′ 2 U γ′δ′;ε = -g α′g β′g ε R γ′ε′ι′δ δ′ + R δ′ε′ι′δ γ′ σ + 𝒪 (λ ), (359 ) 2 ( ) U αβ′ ′′ = 1g (α′gβ)′ R αγ′′ε′ι′δβ′′ + R α′δ′ε′ι′δβ′′ σ ι′ + 𝒪(λ2), (360 ) γ δ;ε 2 α β δ γ
and eventually,
[ ] □U αβ′ ′ = 0; (361 ) γ δ
this last result follows from the fact that [U αβ′ ′ ] γ δ;ει is antisymmetric in the last pair of indices.

Similarly, Equation (355View 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 (353View Equation), (361View Equation), and the additional relation [σγγ] = 4. We arrive at

[ ] ( ) αβ 1- α′ β′ β′ α′ V γ′δ′ = 2 R γ′ δ′ + R γ′ δ′ . (362 )
With the characteristic data obtained by integrating Equation (355View Equation), the wave equation of Equation (356View Equation) admits a unique solution.

To summarize, the retarded and advanced gravitational Green’s functions are given by Equation (352View Equation) with αβ ′ U γ′δ′(x,x ) given by Equation (357View Equation) and αβ ′ V γ′δ′(x,x ) determined by Equation (356View Equation), and the characteristic data constructed with Equations (355View Equation) and (362View 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 ′.

4.5.3 Reciprocity and Kirchhoff representation

The (globally defined) gravitational Green’s functions satisfy the reciprocity relation

G −γ′δ′αβ(x ′,x) = G+αβγ′δ′(x,x′). (363 )
The derivation of this result is virtually identical to what was presented in Sections 4.3.3 and 4.4.3. A direct consequence of the reciprocity relation is the statement
V γ′δ′αβ(x ′,x) = V αβγ′δ′(x, x′). (364 )

The Kirchhoff representation for the trace-reversed gravitational perturbation γ αβ is formulated as follows. Suppose that γαβ(x) satisfies the homogeneous version of Equation (349View Equation) and that initial values γ α′β′(x′), nγ′∇ γ′γα′β′(x ′) are specified on a spacelike hypersurface Σ. Then the value of the perturbation field at a point x in the future of Σ is given by

1 ∫ ( αβ ′ ′′ ′ ′ ′ αβ ) γ αβ(x) = − --- G + γ′δ′(x,x′)∇ εγγ δ(x′) − γ γδ(x′)∇ εG + γ′δ′(x,x′) dΣε′, (365 ) 4π Σ
where dΣε′ = − nε′dV is a surface element on Σ; nε′ is the future-directed unit normal and dV is the invariant volume element on the hypersurface. The derivation of Equation (365View Equation) is virtually identical to what was presented in Sections 4.3.4 and 4.4.3.

4.5.4 Singular and radiative Green’s functions

We shall now construct singular and radiative Green’s functions for the linearized gravitational field. The treatment here parallels closely what was presented in Sections 4.3.5 and 4.4.4.

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

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

□H αβγ′δ′(x, x′) + 2R γα δ β(x )H γδγ′δ′(x, x′) = 0; (366 )

  Gr.H2: α β ′ H γ′δ′(x,x ) is symmetric in its indices and arguments,

′ ′ H γ′δ′αβ(x ,x) = H αβγ′δ′(x,x ); (367 )

  Gr.H3: H αβγ′δ′(x, x′) agrees with the retarded Green’s function if x is in the chronological future of x′,

αβ ′ αβ ′ + ′ H γ′δ′(x,x ) = G+ γ′δ′(x,x ) when x ∈ I (x ); (368 )

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

αβ ′ αβ ′ − ′ H γ′δ′(x,x ) = G − γ′δ′(x, x) when x ∈ I (x ). (369 )

It is easy to prove that Property Gr.H4 follows from Property Gr.H2, Property Gr.H3, and the reciprocity relation (363View 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 αβ H γ′δ′(x,x′) we define the singular Green’s function to be

[ ] G αβ (x,x ′) = 1- G α β (x, x′) + G αβ (x,x′) − H αβ (x, x′) . (370 ) S γ′δ′ 2 + γ′δ′ − γ′δ′ γ′δ′
This comes with the properties

  Gr.S1: αβ G S γ′δ′(x,x′) satisfies the inhomogeneous wave equation,

□G αβ′′(x,x′) + 2R α β(x)G γδ ′′(x, x′) = − 4πg (α′(x,x′)gβ)′(x,x′)δ4(x, x′); (371 ) S γ δ γ δ S γδ γ δ

  Gr.S2: αβ ′ G S γ′δ′(x,x ) is symmetric in its indices and arguments,

GS (x′,x) = GS (x,x′); (372 ) γ′δ′αβ αβγ′δ′

  Gr.S3: αβ G S γ′δ′(x,x′) vanishes if x is in the chronological past or future of x′,

G αβ′′(x,x′) = 0 when x ∈ I± (x ′). (373 ) S γδ

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

The radiative Green’s function is then defined by

G αβ′′(x,x′) = G αβ′ ′(x,x ′) − G αβ′ ′(x,x ′), (374 ) R γδ + γ δ S γ δ
and it comes with the properties

  Gr.R1: αβ ′ G R γ′δ′(x,x ) satisfies the homogeneous wave equation,

□G Rαβγ′δ′(x, x′) + 2R γα δ β(x )G Rγδγ′δ′(x,x ′) = 0; (375 )

  Gr.R2: G αβ′′(x,x ′) R γ δ agrees with the retarded Green’s function if x is in the chronological future of ′ x,

G αβ (x,x′) = G αβ (x, x′) when x ∈ I+(x′); (376 ) R γ′δ′ + γ′δ′

  Gr.R3: αβ ′ G R γ′δ′(x,x ) vanishes if x is in the chronological past of ′ x,

αβ ′ − ′ G R γ′δ′(x,x ) = 0 when x ∈ I (x). (377 )

Those follow immediately from Properties Gr.S1, Gr.S2, and Gr.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

αβ αβ H γ′δ′(x, x′) = V γ′δ′(x,x′), (378 ) 1 1 G αS βγ′δ′(x, x′) = -U αβγ′δ′(x,x′)δ(σ ) − -V αβγ′δ′(x,x′)θ (σ ), (379 ) 2 2 [ ] G α β (x, x′) = 1U αβ (x,x′)[δ (σ) − δ (σ)] + Vαβ (x,x ′) θ (− σ) + 1θ(σ ) . (380 ) R γ′δ′ 2 γ′δ′ + − γ′δ′ + 2
From these we see clearly that the singular Green’s function does not distinguish between past and future (Property Gr.S2), and that its support excludes I±(x′) (Property Gr.S3). We see also that the radiative Green’s function coincides with G+αβγ′δ′(x,x ′) in I+ (x′) (Property Gr.R2), and that its support does not include I− (x ′) (Property Gr.R3).

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