I =.The action $I$ is invariant under the gauge transformations ł[gt]{} \_[kl]{} =h\_[kl]{} -(\_k \_l +\_l \_k ), where $\na$ is defined with respect to the metric $g_{ij}$. With the background metric (\[m\]), $ds^2= dy^2 +g_{\m\n}dx^\m
dx^\n$, the gauge transformations (\[gt\]) take the form ł[gt4]{} &&\_[44]{} =h\_[44]{}-2\_4[\_4]{}\
ł[gtm]{} &&\_[4]{} =h\_[4]{}-(\_\_4 +\_4 \_-2\^\_[4]{}\_)\
ł[gtmn]{} &&\_ =h\_-(D\_\_+D\_\_-2\^4\_\_4 ) ,where $D_\m$ is defined with respect to the metric $g_{\m\n}$.

To solve the equations following from the action (\[2f\]) in the region $0<y<\b{y}$, instead of the exact metric, we use the approximate metric (\[24\]). In the region $0<y<\b{y}$ we rewrite it as ł[ap]{} ds\^2 = dy\^2 +b(y)\_(x)dx\^dx\^,where $b(y)=e^{-2\m |y|}$. We impose the gauge conditions $h_{4i}=0$. The gauge condition $\t{h}_{44}=0$ can be realized by performing transformation with ł[gc1]{} \_4 (y, x\^) =\^y\_[-y]{} dy’ h\_[44]{} (y’ , x\^) Condition $\t{h}_{4\m} =0$ is imposed by taking ł[gc2]{} \_(y, x\^) = b(y)(C\_(x\^) +\^y\_[-y]{} dy’ b\^[-1]{} (y’) (h\_[4]{}-\_\_4 )(y’ ,x\^) ) . $C_\m (x)$ are arbitrary functions.

From the Einstein equations for the background metric (\[1a\]) follow the relations ł[3f]{} && R= -\^2 (y)(\_\^-4) 20\^2 +16 (y)\
&& R\_\^= \^\_(R-2Ł)+\^2 (y)(\_\^-\_\^) \^\_(-4\^2 +2(y) ) +\^2(y) \_\^, where in the radiation-dominated period we have set $\tau^\m_\m =0$. The Ricci tensor 