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)