16 Gravitational Green’s functions

16.1 Equations of linearized gravity

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

gαβ = gαβ + hαβ. (16.1 )
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 hα β. To simplify the expressions we use the trace-reversed potentials γαβ defined by
1( γδ ) γαβ = hαβ − 2 g hγδ gαβ, (16.2 )
and we impose the Lorenz gauge condition,
α β γ ;β = 0. (16.3 )
In this equation, and in all others below, indices are raised and lowered with the background metric g αβ. Similarly, the connection involved in Eq. (16.3View Equation), and in all other equations below, is the one that is compatible with the background metric. If T αβ is the perturbing energy-momentum tensor, then by virtue of the linearized Einstein field equations the perturbation field obeys the wave equation
αβ α β γδ αβ ā–”γ + 2Rγ δ γ = − 16πT , (16.4 )
in which ā–” = gαβ∇ ∇ α β is the wave operator and R γαδβ the Riemann tensor. In first-order perturbation theory, the energy-momentum 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′ ′āˆ˜ ---′4 ′ γ (x ) = 4 G γ′δ′(x,x )T (x ) − gd x , (16.5 )
in terms of a Green’s function G αβ ′′(x, x′) γδ that satisfies [161]
αβ ′ α β γδ ′ (α ′ β) ′ ′ ā–”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 ), (16.6 )
where gα′(x,x′) γ is a parallel propagator and δ4(x, x′) an invariant Dirac functional. The parallel propagators are inserted on the right-hand side of Eq. (16.6View 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 Eq. (16.6View Equation), the perturbation field of Eq. (16.5View Equation) satisfies the wave equation of Eq. (16.4View Equation). Once γ αβ is known, the metric perturbation can be reconstructed from the relation h = γ − 1(gγδγ )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 −αβγ′δ′(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 background spacetime.

16.2 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±αβγ′δ′(x,x ′) = U αβγ′δ′(x,x′)δ±(σ) + V αβγ′δ′(x, x′)šœƒ± (− σ ), (16.7 )
where šœƒ±(− σ), δ±(σ) are the light-cone distributions introduced in Section 13.2, 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 šœ–. The Green’s functions are then recovered by the taking the limit of

Gšœ–± αβγ′δ′(x,x′) = U αβγ′δ′(x, x′)δ± (σ + šœ–) + V αβγ′δ′(x,x′)šœƒ±(− σ − šœ–)

as šœ– → 0+. When we substitute this into the left-hand side of Eq. (16.6View Equation) and then take the limit, we obtain

αβ α β γδ ′ αβ ′ { αβ γ γ αβ } ā–”G ± γ′δ′ + 2R γ δ G± γ′δ′ = − 4πδ4(x, x)U γ′δ′ + δ±(σ) 2U γ′δ′;γσ + (σ γ − 4)U γ′δ′ { αβ α β αβ α β γδ } + δ±(σ ) − 2V γ′δ′;γσγ + (2 − σγγ)V γ′δ′ + ā–”U γ′δ′ + 2R γ δ U γ′δ′ { } + šœƒ±(− σ) ā–”V αβγ′δ′ + 2R γα δ βV γδγ′δ′
after a routine computation similar to the one presented at the beginning of Section 14.2. Comparison with Eq. (16.6View Equation) returns: (i) the equations
[ ] [ αβ ] (α β) (α′ β′) U γ′δ′ = g γ′g δ′ = δ γ′δδ′ (16.8 )
2U αβγ′δ′;γσ γ + (σ γγ − 4)U αβγ′δ′ = 0 (16.9 )
that determine αβ ′ U γ′δ′(x, x); (ii) the equation
αβ γ 1 γ αβ 1( αβ α β γδ )|| VĖ‡ γ′δ′;γσ + 2(σ γ − 2)Ė‡V γ′δ′ = 2-ā–”U γ′δ′ + 2R γ δ U γ′δ′ |σ=0 (16.10 )
that determine αβ ′ VĖ‡ γ′δ′(x,x ), the restriction of αβ ′ V γ′δ′(x,x ) on the light cone ′ σ(x,x ) = 0; and (iii) the wave equation
αβ α β γδ ā–”V γ′δ′ + 2R γ δ V γ′δ′ = 0 (16.11 )
that determines αβ ′ V γ′δ′(x,x ) inside the light cone.

Eq. (16.9View Equation) can be integrated along the unique geodesic β that links ′ x to x. The initial conditions are provided by Eq. (16.8View Equation), and if we set U αβγ′δ′(x,x ′) = g(αγ′g βδ)′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 αβ′′(x,x′) = g(α′(x,x′)gβ)′(x,x′)Δ1 āˆ•2(x,x ′), (16.12 ) γδ γ δ
which reduces to
( ) U αβ′′ = g(α′g β)′ 1 + O (λ3 ) (16.13 ) 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
( ) U αβ′ ′ = 1-g(α′gβ)′gšœ–′šœ– Rα′γ′šœ–′ι′δβ′′ + Rαδ′′šœ–′ι′δβ′′ σ ι′ + O (λ2 ), (16.14 ) γ δ;šœ– 2 α β ( δ )γ αβ 1- (α β) α′ β′ α′ β′ ι′ 2 U γ′δ′;šœ–′ = 2 g α′g β′ R γ′šœ–′ι′δ δ′ + R δ′šœ–′ι′δ γ′ σ + O (λ ), (16.15 )
and eventually,
[ αβ ā–”U γ′δ′] = 0; (16.16 )
this last result follows from the fact that [U αβ ] γ′δ′;šœ–ι is antisymmetric in the last pair of indices.

Similarly, Eq. (16.10View 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. (16.8View Equation), (16.16View 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′ . (16.17 )
With the characteristic data obtained by integrating Eq. (16.10View Equation), the wave equation of Eq. (16.11View Equation) admits a unique solution.

To summarize, the retarded and advanced gravitational Green’s functions are given by Eq. (16.7View Equation) with U αβ′′(x,x′) γ δ given by Eq. (16.12View Equation) and V αβ′′(x,x′) γδ determined by Eq. (16.11View Equation) and the characteristic data constructed with Eqs. (16.10View Equation) and (16.17View 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 ′.

16.3 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 ). (16.18 )
The derivation of this result is virtually identical to what was presented in Sections 14.3 and 15.3. A direct consequence of the reciprocity relation is the statement
V γ′δ′αβ(x ′,x) = V αβγ′δ′(x, x′). (16.19 )

The Kirchhoff representation for the trace-reversed gravitational perturbation γ αβ is formulated as follows. Suppose that γαβ(x) satisfies the homogeneous version of Eq. (16.4View 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 ∫ ( ′ ′′ ′′ ′ ) γαβ(x) = − --- G +αβγ′δ′(x,x′)∇ šœ–γγ δ(x′) − γ γδ (x ′)∇ šœ–G +αβγ′δ′(x,x′) dΣ šœ–′, (16.20 ) 4π Σ
where dΣšœ–′ = − nšœ–′dV is a surface element on Σ; nšœ–′ is the future-directed unit normal and dV is the invariant volume element on the hypersurface. The derivation of Eq. (16.20View Equation) is virtually identical to what was presented in Sections 14.4 and 15.3.

16.4 Relation with electromagnetic and scalar Green’s functions

In a spacetime that satisfies the Einstein field equations in vacuum, so that R = 0 αβ everywhere in the spacetime, the (retarded and advanced) gravitational Green’s functions satisfy the identities [144Jump To The Next Citation Point]

αβ α G ± γ′δ′;β = − G± (γ′;δ′) (16.21 )
γ′δ′ αβ αβ g G± γ′δ′ = g G ±, (16.22 )
where G α ′ ± β are the corresponding electromagnetic Green’s functions, and G ± the corresponding scalar Green’s functions.

To prove Eq. (16.21View Equation) we differentiate Eq. (16.6View Equation) covariantly with respect to β x, use Eq. (13.3View Equation) to work on the right-hand side, and invoke Ricci’s identity to permute the ordering of the covariant derivatives on the left-hand side. After simplification and involvement of the Ricci-flat condition (which, together with the Bianchi identities, implies that R ;β = 0 αγβδ), we arrive at the equation

( ) ā–” − Gαβγ′δ′;β = − 4πg α(γ′∂ δ′)δ4(x, x′). (16.23 )
Because this is also the differential equation satisfied by α G (β′;γ′), and because the solutions are chosen to satisfy the same boundary conditions, we have established the validity of Eq. (16.21View Equation).

The identity of Eq. (16.22View Equation) follows simply from the fact that gγ′δ′Gαβ γ′δ′ and gαβG satisfy the same tensorial wave equation in a Ricci-flat spacetime.

16.5 Singular and regular Green’s functions

We shall now construct singular and regular Green’s functions for the linearized gravitational field. The treatment here parallels closely what was presented in Sections 14.5 and 15.5.

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

αβ ′ H γ′δ′(x,x ) satisfies the homogeneous wave equation,
ā–”H αβ′′(x,x′) + 2R α β(x)H γδ′ ′(x,x ′) = 0; (16.24) γ δ γ δ γ δ
αβ ′ H γ′δ′(x,x ) is symmetric in its indices and arguments,
H γ′δ′αβ(x′,x ) = Hα βγ′δ′(x,x ′); (16.25)
αβ ′ H γ′δ′(x,x ) agrees with the retarded Green’s function if x is in the chronological future of x′,
H αβ (x,x ′) = G αβ (x,x′) when x ∈ I+(x ′); (16.26) γ′δ′ + γ′δ′
H αβ′′(x,x′) γ δ agrees with the advanced Green’s function if x is in the chronological past of ′ x,
H αβγ′δ′(x, x′) = G −αβγ′δ′(x,x′) when x ∈ I− (x′). (16.27)

It is easy to prove that property H4 follows from H2, H3, and the reciprocity relation (16.18View 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 αβ ′ H γ′δ′(x,x ) we define the singular Green’s function to be

αβ ′ 1-[ αβ ′ αβ ′ αβ ′] G S γ′δ′(x, x) = 2 G + γ′δ′(x, x) + G − γ′δ′(x, x ) − H γ′δ′(x, x) . (16.28 )
This comes with the properties
αβ ′ GS γ′δ′(x,x ) satisfies the inhomogeneous wave equation,
αβ α β γδ (α β) ā–”G S γ′δ′(x, x′) + 2R γ δ (x )GS γ′δ′(x,x′) = − 4πg γ′(x,x ′)g δ′(x,x ′)δ4(x,x′); (16.29)
αβ ′ GS γ′δ′(x,x ) is symmetric in its indices and arguments,
S ′ S ′ G γ′δ′αβ(x,x ) = Gαβγ′δ′(x,x ); (16.30)
αβ ′ GS γ′δ′(x,x ) vanishes if x is in the chronological past or future of ′ x,
G αS γβ′δ′(x,x ′) = 0 when x ∈ I±(x′). (16.31)

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 αβ′′(x,x′) = G αβ′ ′(x,x ′) − G αβ′′(x,x′), (16.32 ) R γδ + γ δ S γ δ
and it comes with the properties
αβ ′ G R γ′δ′(x,x ) satisfies the homogeneous wave equation,
ā–”G Rαβγ′δ′(x,x′) + 2Rγαδβ(x)G Rγδγ′δ′(x, x′) = 0; (16.33)
αβ G R γ′δ′(x,x ′) agrees with the retarded Green’s function if x is in the chronological future of x′,
αβ ′ αβ ′ + ′ G R γ′δ′(x, x) = G + γ′δ′(x,x ) when x ∈ I (x); (16.34)
G αβ (x,x ′) R γ′δ′ vanishes if x is in the chronological past of x′,
αβ ′ − ′ G R γ′δ′(x, x) = 0 when x ∈ I (x ). (16.35)

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

αβ ′ αβ ′ H γ′δ′(x, x ) = V γ′δ′(x,x ), (16.36 ) αβ ′ 1 αβ ′ 1 αβ ′ G S γ′δ′(x, x ) = 2U γ′δ′(x,x )δ(σ) − 2V γ′δ′(x,x )šœƒ(σ), (16.37 ) 1 [ ] [ 1 ] G Rαβγ′δ′(x, x′) = -U αβγ′δ′(x,x′) δ+(σ) − δ− (σ ) + V αβγ′δ′(x,x′) šœƒ+(− σ) +-šœƒ(σ ) . (16.38 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 αβ ′ G+ γ′δ′(x,x ) in + ′ I (x ) (property R2), and that its support does not include − ′ I (x) (property R3).

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