9 III. PERTURBATIONS OF THE FIELD EQUATIONS In terms of the gauge invariant variables defined last section, we derive the field equations, i.e., Einstein equa- tions and the equation for a matter field. To derive the perturbation of the Einstein equations and the equation for a matter field (Klein-Gordon equation), first of all, we have to derive the perturbative expressions of the Ein- stein tensor[9]. This is reviewed in Sec. III A. We also derive the first- and the second-order perturbations of the energy momentum tensor for a scalar field and the Klein- Gordon equation[12] in Sec. III B. Finally, we consider the first- and the second-order the Einstein equations in Sec. III C. where ω$_{a}$ is an arbitrary one-form on M$_{0}$ . From the property (3.1) of the covariant derivative operator ¯ ∇$_{a}$ on M$_{λ}$ , the tensor field C c ab on M$_{0}$ is given by where ¯ g ab is the inverse of ¯ g$_{ab}$ (see Appendix B). We note that the gauge dependence of the covariant derivative ¯ ∇$_{a}$ appears only through C $^{c}$$_{ab}$. The Riemann curvature ¯ R d abc on M$_{λ}$ , which is also pulled back to M$_{0}$ , is given by [24]: The relation between the curvatures associated with the metrics on the physical spacetime M$_{λ}$ and the back- ground spacetime M$_{0}$ is given by the relation between the pulled-back operator X ∗ λ ¯ ∇$_{a}$ ( X − 1 λ ) ∗ of the covariant derivative ¯ ∇$_{a}$ associated with the metric ¯ g$_{ab}$ on M$_{λ}$ and the covariant derivative ∇$_{a}$ associated with the met- ric g$_{ab}$ on M$_{0}$ . The pulled-back covariant derivative X ∗ λ ¯ ∇$_{a}$ ( X − 1 λ ) ∗ depends on the gauge choice X$_{λ}$ . The property of the derivative operator X ∗ λ ¯ ∇$_{a}$ ( X − 1 λ ) ∗ as the covariant derivative on M$_{λ}$ is given by Since the derivative operator ¯ ∇$_{a}$ (= X ∗ ¯ ∇$_{a}$ ( X − $^{1}$) $^{∗}$) may be regarded as a derivative operator on M$_{0}$ that satisfies the property (3.1), there exists a tensor field C c ab where As shown in Appendix B, each order perturbation of the Einstein tensor is given by We expand the Einstein tensor ¯ G b a := ¯ R b a − 1 $_{2}$δ b a ¯ R on M$_{λ}$ as ¯ R d abc = R d abc − 2 ∇$_{[}$$_{a}$ C d b ] c + 2 C e c [ $_{a}$C d b ] $_{e}$, (3.4) where R d abc is the Riemann curvature on M$_{0}$ . The per- turbative expression for the curvatures are obtained from the expansion of Eq. (3.4) through the expansion of C $^{c}$$_{ab}$. The first- and the-second order perturbations of the Riemann, the Ricci, the scalar, the Weyl curvatures, and the Einstein tensors on the general background space- time are summarized in Ref. [9]. We also derived the perturbative form of the divergence of an arbitrary ten- sor field of second rank to check the perturbative Bianchi identities in Ref. [9]. In this paper, we only present the perturbative expression for the Einstein tensor, and its derivations in Appendix B. and A. Perturbations of the Einstein curvature on M$_{0}$ such that ¯ ∇$_{a}$ ω$_{b}$ = ∇$_{a}$ ω$_{b}$ − C $^{c}$$_{ab}$ω$_{c}$ , (3.2) C c ab = 1 2 ¯ g $^{cd}$( ∇$_{a}$ ¯ g$_{db}$ + ∇$_{b}$ ¯ g$_{da}$ − ∇$_{d}$ ¯ g$_{ab}$ ) , (3.3) ¯ G b a = G b a + λ $^{(1)}$G b a + 1 2 λ $^{2(2)}$G b a + O ( λ $^{3}$) . (3.5) $^{(1)}$G b a = $^{(1)}$G b a [ H ] + £$_{X}$ G b a , (3.6) $^{(2)}$G b a = $^{(1)}$G b a [ L ] + $^{(2)}$G b a [ H , H ] +2 £$_{X}$ (1) ¯ G b a + { £$_{Y}$ − £ 2 X } G b a , (3.7) $^{(1)}$G b a [ A ] := $^{(1)}$Σ b a [ A ] − 1 2 δ b a $^{(1)}$Σ c c [ A ] , $^{(1)}$Σ b a [ A ] := − 2 ∇$_{[}$$_{a}$ H bd d ] [ A ] − A $^{cb}$R$_{ac}$ , (3.8) $^{(2)}$G b a [ A,B ] := $^{(2)}$Σ b a [ A,B ] − 1 2 δ b a $^{(2)}$Σ c c [ A,B ] , (3.9) $^{(2)}$Σ b a [ A,B ] := 2 R$_{ad}$B ( b c A d ) c + 2 H de [ a [ A ] H b d ] $_{e}$[ B ] + 2 H de [ a [ B ] H b d ] $_{e}$[ A ] +2 A d e ∇$_{[}$$_{a}$ H be d ] [ B ] + 2 B d e ∇$_{[}$$_{a}$ H be d ] [ A ] + 2 A b c ∇$_{[}$$_{a}$ H cd d ] [ B ] + 2 B b c ∇$_{[}$$_{a}$ H cd d ] [ A ] , (3.10) H c ab [ A ] := ∇$_{(}$$_{a}$ A c b ) − 1 2 ∇ $^{c}$A$_{ab}$ , (3.11) H$_{abc}$ [ A ] := g$_{cd}$H d ab [ A ] , H bc a [ A ] := g $^{bd}$H c ad [ A ] , H b $_{a c}$[ A ] := g$_{cd}$H bd a [ A ] . (3.12) X ∗ λ ¯ ∇$_{a}$ ( X − 1 λ ) $^{∗}$X ∗ $_{λ}$¯ g$_{ab}$ ) = 0 , (3.1) where X ∗ $_{λ}$¯ g$_{ab}$ is the pull-back of the metric on M$_{λ}$ , which is expanded as Eq. (2.22). In spite of the gauge depen- dence of the operator X ∗ λ ¯ ∇$_{a}$ ( X − 1 λ ) $^{∗}$, we simply denote this operator by ¯ ∇$_{a}$ , because our calculations are carried out only on M$_{0}$ in the same gauge choice X$_{λ}$ . Further, we denote the pulled-back metric X ∗ $_{λ}$¯ g$_{ab}$ on M$_{λ}$ by ¯ g$_{ab}$ , as mentioned above.