The nonlinear behavior of Randall-Sundrum gravity with one brane is examined. Due to the noncompact extra dimension, the perturbation spectrum has no mass gap, and the long wavelength effective theory is only understood perturbatively. The full five-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 four-dimensional gravity, uniquely constructs a five-dimensional solution. The algorithm performs best for small stars (radius less than the AdS length) yielding highly nonlinear solutions, core photons being redshifted by up to $\mathcal{Z}\simeq 12.$ 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 a radius several times the AdS length, is described by four-dimensional general relativity far past the perturbative regime, the largest stars being tested up to a core redshift of $\mathcal{Z}\simeq \mathrm{2.1.}$ This indicates that the nonlinear 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.

• Received 23 November 2001

DOI:https://doi.org/10.1103/PhysRevD.65.124007