The various Green's functions are now defined,
Gretaa, (x, z)
Gadvaa'(x, z)
Gaa' (x, z)
20 [e(x), z] Guaa(X, z),
20 [z, E(x)] Gaa,(x, z),
Gadvaa'(X, z) Gretaa,(X, Z),
where E(x) is an arbitrary spacelike hypersurface containing x, and 0 [E(x), z]
1 0 [z, E(x)] is equal to 1 when z lies to the past of E(x) and vanishes when z lies
to the future. These Green's functions satisfy the equations
a (Gret + Gadvaa)
2aa -' F (Lr
(2-87)
V2 Gaa RobGaao
V2 Gret aa
V2Gadvaa,
Rab Gretaa,
SabGadvaa,
g- 1/2 ga (4),
V2 Gaa RabGaa, = 0.
Also, they have the symmetry properties
Gaa (x, z)
G 'et,(x, z)
Gaai(x, z)
Vaa,(X, z)
-Gaia(Z, x),
Gadva'a(z, X),
Gala(Z, x),
Va'a(Z, x).-
Finally, one can note that the substitution of Eq. (2-69) into Eq. (2-76) via
Eq. (2-72) leaves wo0 arbitrary in the solution for Waa,, which corresponds to
adding to G(1) any singularity-free solution of the wave equation. However, this
arbitrariness disappears in the solution for the symmetric Green's functions as it is
evident from Eq. (2-83).
(2-84)
(2-85)
(2-86)
(2-88)
(2-89)
(2-90)
(2-91)
(2-92)
(2-93)