Toward practical/realistic Randall-Sundrum Brane world scenario.

The Paradigm change by Randall-Sundrum brane world scenario in higher-dimensional gravitation can be summarized the following .

	$\displaystyle S=$	$\displaystyle S_{\textrm{bulk}}+S_{\textrm{brane}}+S_{\textrm{brane'}}$
	$\displaystyle(\textrm{with})$
	$\displaystyle S$	$\displaystyle=\int d^{4}x\int_{-\pi}^{+\pi}d\phi\sqrt{G}[2M^{3}R-\Lambda]$
		$\displaystyle\left<\begin{array}[]{ll}S_{\textrm{brane}}=\int d^{4}x\sqrt{G_{\textrm{brane}}}[L_{\textrm{brane}}-V_{\textrm{brane}}],\\ S_{\textrm{brane`}}=\int d^{4}x\sqrt{G_{\textrm{brane`}}}[L_{\textrm{brane`}}-V_{\textrm{brane`}}]\end{array}\right.$
	where
		$\displaystyle g_{\mu\nu}^{\textrm{brane}}(x^{\mu})=G_{\mu\nu}(x^{\mu},\phi=0)$
		$\displaystyle g_{\mu\nu}^{\textrm{brane`}}(x^{\mu})=G_{\mu\nu}(x^{\mu},\phi=\pi)$

with $G_{AB},A,B=\mu,\phi$ is the 5-dim. bulk metric
Next, the 5-dim Einstein equation that one upon extremizing this action for our system reads;

	$\displaystyle\sqrt{G}[R_{AB}-\frac{1}{2}RG_{AB}]=$
	$\displaystyle\frac{-1}{4M^{3}}[\sqrt{G}\Lambda G_{AB}$	$\displaystyle+V_{\textrm{brane}}\sqrt{g}g_{\mu\nu}^{\textrm{brane}}\delta_{A}^{\mu}\delta_{B}^{\nu}\delta(\phi)$
		$\displaystyle+V_{\textrm{brane'}}\sqrt{g}g_{\mu\nu}^{\textrm{brane'}}\delta_{A}^{\mu}\delta_{B}^{\nu}\delta(\phi-\pi)]$

when

$$\sigma(\phi)=kr_{c}|\phi|=k|y|$$

we now look for a solution corresponding to a ”non-factorizable” metric with ”warp-factor”

	$\displaystyle ds^{2}$	$\displaystyle=e^{-2\sigma(\phi)}\gamma_{\mu\nu}(x)dx^{\mu}dx^{\nu}+r_{c}^{2}d\phi^{2}$
		$\displaystyle=e^{-2\sigma(y)}\gamma_{\mu\nu}(x)dx^{\mu}dx^{\nu}+dy^{2}$

where $y\equiv r_{c}\phi$ is the 5th dimension coordinate transverse to the brane.

$$V_{\textrm{brane}}=-V_{\textrm{brane`}}=24M^{3}k,\Lambda=-24M^{3}k^{2}$$

when

$$\sigma(\phi)=kr_{c}|\phi|=k|y|$$

Finally, the bulk metric solution is given by

	$\displaystyle ds^{2}$	$\displaystyle=e^{-2\sigma(\phi)}\gamma_{\mu\nu}(x)dx^{\mu}dx^{\nu}+r_{c}^{2}d\phi^{2}$
		$\displaystyle=e^{-2\sigma(y)}\gamma_{\mu\nu}(x)dx^{\mu}dx^{\nu}+dy^{2}$

(where) $\hat{R}_{\mu\nu}(r)=0$ and we assume that $K<M$(with $M$ being the fundamental Planck scale in 5-dim.) so that the bulk curvature ($R\sim\Lambda=-24M^{3}k^{2}$) is ”small” compared to higher dimensional Planck scale($\sim M^{5}$) and we trust our solution as a classical one.

$$G_{\mu\nu}=e^{-2k(y)}\gamma_{\mu\nu}(x)+h_{\mu\nu}(x,y)$$

Let $h(x,y)=e^{ipx}\psi(y)$ (namely general fluctuations can be written as superpositions of modes) and use $P^{2}=P^{\alpha}P_{\alpha}=-(P^{0})^{2}+\vec{P}^{2}=-m^{2},$then

$$[-\frac{m^{2}}{2}e^{2k|y|}-2\frac{1}{2}\partial_{y}^{2}+2k^{2}-2k\delta(y)]\psi(y)=0$$

$\Rightarrow$

$$[-\frac{1}{2}\frac{d^{2}}{dz^{2}}+V(z)]\hat{\psi}(z)=\frac{m^{2}}{2}(z)$$

with

$$V(z)=\{\frac{15k^{2}}{8(k|z|+1)^{2}}-\frac{3k}{2}\delta(z)\}$$

BKL oscillatory singularity on the Ricci flat brane
Belinskii-Khalatnikov-Lifshitz(BKL) oscillatory singularity representing anisotropic cosmology approaching initial time singularity at the early universe
Initial time singularity(singularity theorem):
”-if Einstein equation holds, and
-weak/strong energy condition is satisfied, but without
-the emergence of closed timelike curves, then as we move backward in time, we approach the initial time singluarity where the proper volume element of a spacelike hyper-surface vanishes!”

Hawking and Penrose(1965 1970) $\rightarrow$ they mathematically prove the singularity theorem in terms of topology employing the Raychaudhuri’s equation.
B-K-L(1968)$\rightarrow$ they physically exhibited the singularity theorem in terms of geometry by explicitly solving the vacuum Einstein equation employing the Bianchi type-IX cosmology model
more on Belinskii-Khalatnikov-Lifshitz(BKL) oscillatory singularity
B-K-L version of initial time singularity where the proper volume element of a spacelike hypersurface, which goes like $\sim$t, vanishes!

B-K-L(1968) $\Rightarrow$ they discovered that if we model the early universe by employing the Bianchi type-IX cosmology metric, then as the initial time singularity is approached(t$\rightarrow$ 0.), one inevitably encounters successive series of Kasner regimes during each of which the distances along two of the three spatial axes oscillate, while they fall/drop off monotonically along the third and the spatial volume element drops off approximately as$\sim$t. And in going from one Kasner regime to the next, the direction along which there is a monotonic drop off of distances shifts from one spatial direction to another. And as the initial time singularity is approached, the order of these shifts takes on the character of a random process, namely,a ”chaotic map”

at early stages of the universe, we employ the Bianchi type-IX(mixmaster) cosmology model representing a homogeneous but not necessarily isotropic, spatially-closed geometry ¡homogeneous but anisotropic universe model¿

	$\displaystyle ds^{2}=$	$\displaystyle-N^{2}(t)dt^{2}+e^{2\alpha(t)}(e^{2\beta(t))}_{ij}\sigma^{i}\otimes\sigma^{j}$
	$\displaystyle=$	$\displaystyle-N^{2}dt^{2}+e^{2\alpha}[e^{2\beta_{1}}\sigma_{1}^{2}+e^{2\beta_{2}}\sigma_{2}^{2}+e^{2\beta_{3}}\sigma_{3}^{2}]$
	$\displaystyle=$	$\displaystyle-N^{2}dt^{2}+[A^{2}(t)\sigma_{1}^{2}+B^{2}(t)\sigma_{2}^{2}+C^{2}(t)\sigma_{3}^{2}]$

where $N(t)$ is the lapse function, $e^{\alpha(t)}$ is the scale factor of the spatial section and the symmetric, traceless matrix $\beta_{ij}(t)$ satisfying $\det(e^{2\beta)}=1$ (or$tr(\beta)=0$) parametrizes the anisotropy. This matrix $\beta_{ij}$ may be chosen to be and it is convenient to define them explicitly by

	$\displaystyle\beta_{11}\equiv$	$\displaystyle\beta_{1}=\beta_{+}+\sqrt{3}\beta_{-},$
	$\displaystyle\beta_{22}\equiv$	$\displaystyle\beta_{2}=\beta_{+}-\sqrt{3}\beta_{-},$
	$\displaystyle\beta_{33}\equiv$	$\displaystyle\beta_{3}=-2\beta_{+},$

The non-holonomic basis{$\sigma^{i}\}(i=1,2,3)$ from a basis Maurer-Cartan structure equation

$$d\sigma^{i}-\frac{1}{2}\epsilon^{ijk}\sigma^{i}\Lambda\sigma^{k}$$

To summarize, the 4-Einstein equations can be put to the forms;

	$\displaystyle 2\ddot{a}=(B^{2}-C^{2})^{2}-A^{4}$		(3.1)
	$\displaystyle 2\ddot{b}=(C^{2}-A^{2})^{2}-B^{4}$		(3.2)
	$\displaystyle 2\ddot{c}=(A^{2}-B^{2})^{2}-C^{4}$		(3.3)

which, upon being added up, gives

$$2(\ddot{a}+\ddot{b}+\ddot{c})=[A^{4}+B^{4}+C^{4}-2(A^{2}B^{2}+B^{2}C^{2}+C^{2}A^{2})]$$

(3.4)

(3.4)=(3.1)+(3.2)+(3.3) and

$$\dot{ab}+\dot{bc}+\dot{ba}=\frac{1}{4}[A^{4}+B^{4}+C^{4}-2(A^{2}B^{2}+B^{2}C^{2}+C^{2}A^{2})]$$

(3.5)

	$\displaystyle A(t)$	$\displaystyle=e^{a(t)}\equiv e^{(\alpha+\beta_{1})}$
	$\displaystyle B(t)$	$\displaystyle=e^{b(t)}\equiv e^{(\alpha+\beta_{2})}$
	$\displaystyle C(t)$	$\displaystyle=e^{c(t)}\equiv e^{(\alpha+\beta_{3})}.$

Therefore, we first assume that at our starting point corresponding to some early universe stage, the(r.h.s) of egs, (3.1)$\sim$(3.4) were negligibly small and hence the metric functions essentially exhibit the behaviors

$$A\simeq\tau^{P_{l}},B\simeq\tau^{P_{m}},C\simeq\tau^{P_{n}},$$

where we introduce another time parameter $d\tau=Ndt=(ABC)dt$ and ($P_{l},P_{m},P_{n}$) are numbers satisfying the relations

$$P_{l}+P_{m}+P_{n}=1=P_{l}^{2}+P_{m}^{2}+P_{n}^{2}$$

Thus our starting point in some early universe may be regarded as a Kasner-type epoch.
Point on, we shall retain the notation ($P_{1},P_{2},P_{3}$) for he triple of numbers arranged in the order $P_{1}<P_{2}<P_{3}$ and taking on values

$$-\frac{1}{3}\leq P_{1}\leq 0\leq P_{2}\leq\frac{2}{3}\leq P_{3}\leq 1$$

Note that these numbers can be written in parametric form as

$$P_{1}(u)=\frac{-u}{1+u+u^{2}},P_{2}(u)=\frac{u+1}{1+u+u^{2}},P_{3}(u)=\frac{u(u+1)}{1+u+u^{2}}$$

(3.6)

then all the different value of $P_{1},P_{2},P_{3}$ (preserving the assumed order) are obtained if the parameter ”$u$”runs in the range $u\geq 1$, The values for $u\leq 1$ are reduced to this same region as fallows

$$P_{1}(\frac{1}{u})=P_{1}(u),P_{2}(\frac{1}{u})=P_{3}(u),P_{3}(\frac{1}{u})=P_{2}(u)$$

(3.7)

[Concluding Remarks]

Our claim in the present work can be summarized as follows. The 4-dim. gravity(graviton) gets localized (trapped) on the Poincare- invariant, 4-dim. time-like Lorentzian submanifold(i.e brane world”)-(L. Randall & R. Sundrum)

Can be generalized (relaxed) to a more generic case where

The 4-dim. gravity (graviton) gets localized (trapped) on the Ricci-flat, 4- dim. time-like Lorentzian submanifold (i.e., brane world”)

A result

The original version of RS brane world is a cold, empty universe ?[unphysical]

The RS brane world could now be promoted to a much more our real expanding realistic/physical universe having structures (e.g., black holes/GW)