Vacuum solutions of the gravitational field equations in the brane world model

We consider some classes of solutions of the static, spherically symmetric gravitational field equations in the vacuum in the brane world scenario, in which our Universe is a three-brane embedded in a higher dimensional space-time. The vacuum field equations on the brane are reduced to a system of two ordinary differential equations, which describe all the geometric properties of the vacuum as functions of the dark pressure and dark radiation terms (the projections of the Weyl curvature of the bulk, generating nonlocal brane stresses). Several classes of exact solutions of the vacuum gravitational field equations on the brane are derived. In the particular case of a vanishing dark pressure, the integration of the field equations can be reduced to the integration of an Abel type equation. A perturbative procedure, based on the iterative solution of an integral equation, is also developed for this case. Brane vacuums with particular symmetries are investigated by using Lie group techniques. In the case of a static vacuum brane admitting a one-parameter group of conformal motions, the exact solution of the field equations can be found, with the functional form of the dark radiation and pressure terms uniquely fixed by the symmetry. The requirement of the invariance of the field equations with respect to the quasihomologous group of transformations also imposes a unique, linear proportionality relation between the dark energy and dark pressure. A homology theorem for the static, spherically symmetric gravitational field equations in the vacuum on the brane is also proven.