2.2.2 Green's Functions in Curved Spacetime

 Looking for the solutions of the covariant vector wave equation


 V2A, RabAb = 0, (2-68)


one follows Hadamard [14], according to which an elementary solution can be

written in the form


 G),,a = (2w U + vaa, n 1 + aa), (2-69)
 (2))2 I

where the functions Uaa', Vaa', Waa' are bi-vectors. If Eq. (2-69) is substituted into

Eq. (2-68), the first function is uniquely determined, using the boundary condition

at x z,

 Uaa = A1/2 aa' (2-70)

while the other two are most easily obtained by expanding the functions in a power

series
 OO
 Vaa' = Y Vnaa'g, (2-71)
 n=0
 oo
 Waal' = 'oaO'Cn, (2-72)
 n=0

and obtaining the recurrence formulae for the coefficients. Using Eq. (2-67) for

Eq. (2-70), one obtains


 Uaa' 1 Rb'bc;b2';c + O(S3) gaa. (2-73)


By repeatedly differentiating gaa', however, one finds

 1 dl
 gaa';bc = gda'Rbca + 0(s). (2-74)


Then, differentiating Eq. (2-73) repeatedly and using Eq. (2-74), one also finds

 1
 V2U aa' gaa'R + O(s). (2-75)
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