background geometry gab. The perturbation begins with a background metric gab

which is a vacuum solution of the Einstein equations Gab(g) = 0. The particle then

disturbs the geometry by an amount hab(g) = O(p) which is determined by the

perturbed Einstein equations with the stress-energy tensor Tab = O(P) of the object

being the source,

 Eab(h) --8Tab + 0(2). (5-6)

Here, Eab(h) is the linear differential operator defined by

 JGab
 Eab(h)- h d, (5-7)

and Gab is the Einstein tensor of gab, so that


 2Eab(h) V2hab + VaVbh 2V(Oachb)c

 +2RaCbdhcd + gab (c dhd 2h) (5-8)

with h habgb. If hab is a solution of Eq. (5-6), then it follows from Eq. (5-7) that

gab + hab is an approximate solution of the Einstein equations with source Tab,


 Gab(g + h) 8Tab + O(2). (5-9)

 From Eq. (5-8), using the Bianchi identity, we have


 VaEab(h) 0 (5-10)

for any symmetric tensor hab. Thus, an integrability condition for Eq. (5-6) is that

the stress-energy tensor Tab be conserved in the background geometry [10]


 VaTab 0(P2). (5-11)

 The second order perturbation analysis is no more difficult than the first order.

The main difference is the integrability condition: for the second order equations

Tab must be conserved not in the background, but in the first order perturbed