content_image/420646.jpg content_image/420637.jpg content_image/420638.jpg content_image/420639.jpg content_image/420640.jpg content_image/420636.jpg content_image/420641.jpg content_image/420642.jpg content_image/420643.jpg content_image/420644.jpg content_image/420645.jpg
## 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. 

## A. Perturbations of the Einstein curvature 

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 

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. ∗ − 

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 

<img src='content_image/420646.jpg'>

on M$_{0}$ such that 

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 

<img src='content_image/420637.jpg'>

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]: 

¯ 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. b b 1 b 

We expand the Einstein tensor ¯ G a := ¯ R a − $_{2}$δ a ¯ R on M$_{λ}$ as 

<img src='content_image/420638.jpg'>

As shown in Appendix B, each order perturbation of the Einstein tensor is given by 

where 

<img src='content_image/420639.jpg'>

<img src='content_image/420640.jpg'>

<img src='content_image/420636.jpg'>

<img src='content_image/420641.jpg'>

<img src='content_image/420642.jpg'>

<img src='content_image/420643.jpg'>

and 

<img src='content_image/420644.jpg'>

<img src='content_image/420645.jpg'>

