  The non-linear behaviour of Randall-Sundrum gravity with one brane is
examined. Due to the non-compact extra dimension, the perturbation spectrum has
no mass gap, and the long wavelength effective theory is only understood
perturbatively. The full 5-dimensional Einstein equations are solved
numerically for static, spherically symmetric matter localized on the brane,
yielding regular geometries in the bulk with axial symmetry. An elliptic
relaxation method is used, allowing both the brane and asymptotic radiation
boundary conditions to be simultaneously imposed. The same data that specifies
stars in 4-dimensional gravity, uniquely constructs a 5-dimensional solution.
The algorithm performs best for small stars (radius less than the AdS length)
yielding highly non-linear solutions. An upper mass limit is observed for these
small stars, and the geometry shows no global pathologies. The geometric
perturbation is shown to remain localized near the brane at high densities, the
confinement interestingly increasing for both small and large stars as the
upper mass limit is approached. Furthermore, the static spatial sections are
found to be approximately conformal to those of AdS. We show that the intrinsic
geometry of large stars, with radius several times the AdS length, is described
by 4-dimensional General Relativity far past the perturbative regime. This
indicates that the non-linear long wavelength effective action remains local,
even though the perturbation spectrum has no mass gap. The implication is that
Randall-Sundrum gravity, with localized brane matter, reproduces relativistic
astrophysical solutions, such as neutron stars and massive black holes,
consistent with observation.
