content_image/338996.jpg content_image/338997.jpg content_image/338998.jpg content_image/338999.jpg content_image/339000.jpg content_image/339001.jpg content_image/339002.jpg
<img src='content_image/338996.jpg'>

<img src='content_image/338997.jpg'>

<img src='content_image/338998.jpg'>

We obtain a 5D wave-like equation for Ψ by subtracting (A14) from (A15) to give 

<img src='content_image/338999.jpg'>

Equations (A17) and (A16) comprise N +1 dynamical equations for the N +1 perturbation variables Ψ and ξ$_{a}$ . However, since these variables are connected through the constraint (A13), only N of them are independent. Therefore we may use (A13) to eliminate one of the variables in terms of the others. Choosing to eliminate the N th scalar field, ξ$_{N}$ , in terms of ξ$_{a<}$$_{N}$ and Ψ, equation (A13) gives 

<img src='content_image/339000.jpg'>

(Note that we cannot eliminate Ψ in terms of the scalar fields ξ$_{a}$ , since this requires an integration over unknown functions. This is understandable since doing so would amount to reducing the problem to one in flat spacetime, which ought to be impossible.) Substituting 

this into (A17) and (A16) and rearranging gives 

<img src='content_image/339001.jpg'>

<img src='content_image/339002.jpg'>

