Global structure of exact cosmological solutions in the brane world

Abstract

We find the explicit coordinate transformation which links two exact cosmological solutions of the brane world which have been recently discovered. This means that both solutions are exactly the same with each other. One of two solutions is described by the motion of a domain wall in the well-known 5-dimensional Schwarzshild-AdS spacetime. Hence, we can easily understand the region covered by the coordinate used by another solution.

Figures (5)

• Figure 1: The global structure for $\mu>0$, $K=+1$. The world volume of the brane starts at $r=0$ and ends at $r=0$. The function $r_{min}(t)$ increases from zero to $r_h$, and decreases to zero. Accordingly, the hypersurface $w=w_{min}(t)$ starts at $r=0$ with the brane, passes through the bifurcating point of the horizon, and ends at $r=0$ with the brane. Thus, the region covered by the Gaussian normal coordinate should be the shaded region. Two dashed lines in the figure are a constant-$t$ hypersurface and a constant-$w$ hypersurface.
• Figure 2: The global structure for $\mu>0$, $K=0,-1$. The world volume of the brane starts at $r=0$ and ends at $r=\infty$. The function $r_{min}(t)$ starts with the value zero and ends with the value $r_{min}(\infty)$, where $r_{min}(\infty)=r_h$ for $K=0$ and $r_{min}(\infty)=(l^2\mu)^{1/4}<r_h$ for $K=-1$. (We have assumed that $\Lambda_4=0$ and $\rho a^2\to 0$ in the limit $a\to\infty$.) Thus, the region covered by the Gaussian normal coordinate should be the shaded region. Note that hypersurfaces $w=w_{min}(t)$ and $\psi(t,w)=0$ start at the point where the brane starts, and end at the point where $r=r_{min}(\infty)$ comes in contact with the horizon. Two dashed lines in the figure are a constant-$t$ hypersurface and a constant-$w$ hypersurface.
• Figure 3: The global structure for $\mu=0$, $K=+1$. The world volume of the brane starts at $r=0$ and ends at $r=0$. The minimum of $r$ for a fixed $t$ is always zero. Thus, the region covered by the Gaussian normal coordinate should be the shaded region.
• Figure 4: The global structure for $\mu=0$, $K=0,-1$. The world volume of the brane starts at $r=0$ and ends at $r=\infty$. The minimum of $r$ for a fixed value of $t$ is always zero. Thus, the region covered by the Gaussian normal coordinate should be the shaded region.
• Figure 5: The global structure for $\mu<0$, $K-1$. The world volume of the brane starts at $r=0$ and ends at $r=\infty$. The minimum of $r$ for a fixed $t$ becomes zero in the limits $a\to 0$ and $a\to\infty$. Thus, the region covered by the Gaussian normal coordinate should be the shaded region. Note that hypersurfaces $w=w_{min}(t)$ and $\psi(t,w)=0$ start at the point where the brane starts, and end at $r=0$. Two dashed lines in the figure are a constant-$t$ hypersurface and a constant-$w$ hypersurface.