Gravitational dynamics in $s+1+1$ dimensions

We present the concomitant decomposition of an $(s+2)$-dimensional space-time both with respect to a timelike and a spacelike direction. The formalism we develop is suited for the study of the initial value problem and for canonical gravitational dynamics in braneworld scenarios. The bulk metric is replaced by two sets of variables. The first set consists of one tensorial (the induced metric $g_{ij}$), one vectorial ($M^i$) and one scalar ($M$) dynamical quantity, all defined on the $s$ space. Their time evolutions are related to the second fundamental form (the extrinsic curvature $K_{ij}$), the normal fundamental form (${\mathcal{K}}^i$) and normal fundamental scalar ($\mathcal{K}$), respectively. The nondynamical set of variables is given by the lapse function and the shift vector, which however has one component less. The missing component is due to the externally imposed constraint, which states that physical trajectories are confined to the $(s+1)$-dimensional brane. The pair of dynamical variables ($g_{ij}$, $K_{ij}$), well known from the Arnowitt-Deser-Misner decomposition is supplemented by the pairs ($M^i$, ${\mathcal{K}}^i$) and ($M$, $\mathcal{K}$) due to the bulk curvature. We give all projections of the junction condition across the brane and prove that for a perfect fluid brane neither of the dynamical variables has jump across the brane. Finally we complete the set of equations needed for gravitational dynamics by deriving the evolution equations of $K_{ij}$, ${\mathcal{K}}^i$ and $\mathcal{K}$ on a brane with arbitrary matter.

Figure 1

The two foliations with unit normal vector fields $n$ and $l$. The unit vector field $m$ belongs to the tangent space spanned by $n$ and $l$ and it is perpendicular to $n$.

Figure 2

Decomposition of the time-evolution vector $\partial /\partial t$, when the two foliations are not perpendicular (Fig. 3).

Figure 3

Decomposition of the off-brane evolution vector $\partial /\partial \chi$ for nonperpendicular foliations.

Figure 4

Decomposition of the temporal and off-brane evolution vectors $\partial /\partial t$ and $\partial /\partial \chi$ for perpendicular foliations.