Relativistic Stars in Randall-Sundrum Gravity

TL;DR

This work demonstrates that static, relativistic stars on a single Randall-Sundrum brane can be modeled by solving the full non-linear 5D Einstein equations with elliptic relaxation, yielding regular AdS-like bulks and brane-boundary data that uniquely determine the bulk geometry. The study reveals a dual behavior: small stars (R < l) exhibit strong non-linear bulk effects and an upper mass limit, while large stars (R ≳ l) are effectively described by 4D General Relativity on the brane, with the bulk perturbation confined near the brane. The results confirm that long-wavelength brane physics remains local and 4D-like even well into the non-linear regime, suggesting Randall-Sundrum gravity reproduces relativistic astrophysical solutions such as neutron stars and black holes without conflicting with observations. The elliptic relaxation framework provides a powerful tool to explore non-linear bulk dynamics and could extend to compact extra dimensions and dynamical scenarios in future work.

Abstract

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.

Figures (26)

• Figure 1: An illustration of core pressure, $P$, against core density, $\rho$, for configurations with $\xi = 0.3$ (see equation for density profile \ref{['eq:profile']}). The proper angular radii vary monotonically from $R = 0.30 \, - \, 0.38$ from low to high density. The behavior strongly indicates a diverging core pressure for finite density, implying that for small stars an upper mass limit for a given $R$ exists. The brane does not act to stabilize the large densities. The curve appears qualitatively similar to the usual 4-dimensional incompressible fluid star behavior. (all lattices: $dr = 0.02$, $r_{\rm max} = 2$, $dz = 0.005$, $z_{\rm max} = 4$. systematic errors from comparison with linear theory in section \ref{['sec:linear_check']} are estimated at $\sim 2 \%$)
• Figure 2: An illustration of 4-d against 5-d behavior; On the left the actual core redshift of a star, calculated from the 5-dimensional geometry, is plotted as a function of the same quantity calculated from the induced 4-dimensional effective theory. On the right, the core pressure divided by density is plotted in the same way. Three values of, $\xi = 1.5, 2, 3$ are used to generate solutions for different core densities $\rho$. $\xi$ approximately corresponds to the proper radius, $R$, of the star. One clearly sees that the larger the star, the closer the solutions lie to the '4d = 5d' $45^{o}$ line. In the data presented moving vertically down towards this line the proper radius of the star increases. Furthermore, for each $\xi$, the points fall approximately on straight lines. This indicates that the goodness of approximation of 5-dimensional theory by the effective 4-dimensional one is roughly independent of the core density, and hence non-linearity, over the range tested. The degree of approximation depends only on the star size. The most non-linear $\xi=3$ stars are at $\sim 75\%$ of their upper mass limit in the 4-dimensional effective theory. The last line plotted is the 4-dimensional linear theory prediction for the $\xi=3$ stars, again against the 4-dimensional non-linear theory. We see that the linear theory deviates strongly from this, showing that the solutions probed are fully non-linear, and beyond the reach of higher order perturbation theory. These graphs are strong evidence that the effective 4-dimensional description applies far into the non-linear regime, and probably right up to the upper mass limit. (lattices: $\xi = 1.5$: $dr = 0.10$, $r_{\rm max} = 10$, $dz = 0.02$ and $0.04$, $z_{\rm max} = 21$, $\xi = 2.0$: $dr = 0.15$, $r_{\rm max} = 15$, $dz = 0.02$ and $0.04$, $z_{\rm max} = 31$, $\xi = 3.0$: $dr = 0.20$, $r_{\rm max} = 20$, $dz = 0.03$ and $0.05$, $z_{\rm max} = 46$. two lattice $dz$ resolutions are used to extrapolate to $dz = 0$. systematic errors from comparison with linear theory in section \ref{['sec:linear_check']} are estimated to be maximum for $\xi = 3.0$ at $\sim 10 \%$.)
• Figure 3: An illustration of the asymptotic and brane boundary data.
• Figure 4: An illustration of the linear boundary conditions. The matching conditions and constraints specify $a, f, g$ on the brane itself, with the asymptotic AdS condition specifying the functions asymptotically.
• Figure 5: An illustration of the deformed top-hat defined in equation \ref{['eq:profile']} for $\rho_0 = 1, \xi = 1$. A top-hat density corresponds to an incompressible fluid and in standard GR gives the largest mass possible for a given proper angular radius. We slightly deform the top-hat to avoid numerical artifacts associated with discretization.