# 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!

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).*
*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.
• Hello, thanks for your answer, may I have one more question, I think now if I have a single brane in a model instead of two branes as in RS model, I should substitute by the whole: $\det (-G) = \det(-\bar{g}) e^{-8 k T |\phi|} T^2$, and the 5-D action will be $S \supset \int d^4 x \int d\phi \sqrt{-\bar{g}} \left( 2 M^3 e^{-8 k T |\phi|} T^2 \bar{R} \right)$. Is that right, or we take the squre root in all cases ? Commented Oct 10, 2021 at 6:56
• And I think in this case, I mean in case of a single brane, it won't be $\subset$, the action simply equals $S= \int d^4 x \int d\phi \sqrt{-\bar{g}} \left( 2 M^3 e^{-8 k T |\phi|} T^2 \bar{R} \right)$ ? Commented Oct 10, 2021 at 9:18