Deriving an action from a metric I try to find out how in this paper
https://arxiv.org/abs/hep-ph/9905221
the authors derived an effective action from the metric.
The paper I study is related to string theory and modified gravity theories topics.
As they say in page 5

“The four-dimensional effective theory now follows by substituting Eq. (13) into the original action, Eq. (4)”.

I wonder how did they drive a 4-dimensional effective metric, equation (13)?
Then how they did substitute into the higher dimensional action to get the 4-dimensional effective one?
I have spent many times now reading lecture notes and string theory books to find out what's going on but nothing helpful till now!
So any help is appricated!
 A: The Gauss-Codazzi equations relate the intrinsic & extrinsic curvature of a submanifold $\Sigma$ to (the projection of) the curvature of the manifold in which it is embedded.  Specifically, it can be shown from the Gauss-Codazzi equations that
$$
R = \bar{R} + (\bar{K}^2 - \bar{K}_{\mu \nu} \bar{K}^{\mu \nu}) + 2 \nabla_M\left( n^N \nabla_N n^M - n^M \nabla_N n^N\right) \tag{$\star$}
$$
where $\bar{R}$ is the intrinsic Ricci scalar on $\Sigma$, $\bar{K}_{\mu \nu}$ is the extrinsic curvature of $\Sigma$, $\bar{K} \equiv \bar{K}_\mu {}^\mu$, and $n^N$ is the unit normal of $\Sigma$.  A full derivation of this result can be found in Chapter 3 of A Relativist's Toolkit by Eric Poisson (2004).
It is also possible to show from the metric (13) that
$$
\det (-G) = \det(-\bar{g}) e^{-8 k T |\phi|} T^2.
$$
(The factor of 8 comes from a factor of $e^{-2 k T |\phi|}$ for each brane dimension.)  Thus, the 5-D Einstein-Hilbert action includes the term
$$
S = \int d^4x \int d\phi \sqrt{-G} (2 M^3 R) \supset \int d^4 x \int d\phi \sqrt{-\bar{g}} \left( 2 M^3 e^{-4 k T |\phi|} |T| \bar{R} \right)
$$
and if we assume that $T$ takes on its VEV of $r_c$, we obtain the low-energy effective gravitational action in (15).*
The extrinsic curvature terms in the starred equation above would (likely) manifest as some kind of Kaluza-Klein modes;  I would expect that they are examined in detail in the subsequent paper that was "in preparation" at the time that this preprint was posted (reference [9] in the preprint.)  The last term is, of course, a total derivative and does not affect the classical equations of motion.

*Almost.  It seems to me that the 2 in the factor of $e^{-2 k r_c |\phi|}$ in (15) should be a 4 instead (or, more generally, the dimension of the submanifold.)  But it's possible that there's an explanation for it that I'm not seeing right now.  In any event, it does not make a major difference for their subsequent hierarchy arguments.
