content_image/379815.jpg content_image/379825.jpg content_image/379822.jpg content_image/379823.jpg content_image/379821.jpg content_image/379824.jpg
<img src='content_image/379815.jpg'>

where Γ$_{0}$, Γ$_{i}$ Γ$_{ij}$ are the collection of the quadratic term of the first-order perturbations as follows: 

<img src='content_image/379825.jpg'>

<img src='content_image/379822.jpg'>

<img src='content_image/379823.jpg'>

Here, we used Eqs. (4.8), (5.12), (5.14), (5.16) and (5.18). 

The tensor part of Eq. (6.22) is given by 

<img src='content_image/379821.jpg'>

This tensor mode is also called the second-order gravitational waves. 

Further, the vector part of Eqs. (6.21) and (6.22) yields the initial value constraint and the evolution equation of the vector mode (2) ν$_{j}$ : 

<img src='content_image/379824.jpg'>

