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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