2 Classical Dynamics

The stress-energy tensor for a distribution of matter localized on an hypersurface $\Sigma$ (the $p$-brane we mentioned above) can be written in the form $S_{\mu\nu} \delta(\eta)$, where $\delta$ is a Dirac delta, and $\eta$ can be thought as a transverse coordinate to $\Sigma$. In $N$-dimensional General Relativity it is possible to write down the equations of motion for this infinitesimally thin distribution of matter by splitting Einstein equations in the tangential and transverse part (see [4] for the $4$-dimensional case; it can be extended to higher dimensions). Israel junction conditions, then, are

$$\left[ {\cal K}_{ij} - h_{ij} {\cal K} \right] \propto S_{ij} ,$$

where ${\cal K}_{ij}$ and $\cal K$ are, respectively, the extrinsic curvature tensor and its trace and $h_{ij}$ is the induced metric on $\Sigma$. Here we introduced the standard notation $[A] = \lim_{\eta\rightarrow0^+} \{A(\eta) - A(-\eta)\}$. Israel junction conditions describe how the $(N-1)$-brane is embedded in the (in principle different) geometries of the two spacetime domains that it separates. For our purposes, we are going to write down these equations for a spherical brane with surface stress energy tensor $S _{ij} = k h _{ij}$ separating two de Sitter spacetimes. This can be done explicitly in terms of the radius $R$ of the brane⁶. Then Israel junction conditions reduce to the single differential equation

$${\cal H}(R,\dot{R})=\left[ \epsilon \sqrt{\dot{R}^2 + f(R)} \right]R^{(N-3)} - kR^{N-2} = 0 ;$$ (1)

$k$ is the constant tension of the brane, $\epsilon$ are signs to be determined by the equation itself [5], and $f(R)= 1 - \Lambda R^2$ is the metric function appearing in the static line element adapted to the spherical symmetry for the (anti-)de Sitter spacetime⁷. For suitable values of the cosmological constants equation (1) has two types of solutions: the first is $R \equiv 0$, while the second represents a brane collapsing and re-expanding from and to infinity. For our purpose it is also important to note that equation (1) can also be obtained by an effective action, which in the $N$-dimensional case can be written as

$$S_{\mathrm{eff.}} =\int \left\{ R^{N-3} \dot{R} \left[ \chi ... ...f(R)}} \right) \right] - {\cal H}(R,\dot{R}) \right \} d\tau ,$$ (2)

with the additional constraint ${\cal H} = 0$ that has to be imposed on the solutions of the corresponding Euler-Lagrange equation [5].