On spherically symmetric minimally coupled brane worlds
For a static, spherically symmetric brane in the framework of the RS2 concept, we study the conditions under which the 4D tensor Eμν, arising from the 5D Weyl tensor, vanishes on the brane. Gravity on the brane is then decoupled from the bulk geometry, it is the so-called minimally coupled brane world (MCBW). Assuming Eμν = 0 in the whole bulk, we try to solve the 5D Einstein equations GAB + Λ5gAB = 0 and obtain an overdetermined set of equations for functions of the radial coordinate. Some special solutions are found, among which are the well-known "black string" solution with the Schwarzschild metric on the brane and its generalizations with Schwarzschild-(A)dS on-brane metrics. It is concluded that a MCBW can be embedded, in general, in a bulk where Eμ νis not identically zero but only vanishes on the brane. We also present some previous results on the general properties of scalar fields on the brane and give an example of a wormhole supported by a scalar field in a MCBW.