Constant Density Stars Research Articles

Einstein constraints: Uniqueness and nonuniqueness in the conformal thin sandwich approach

We study the appearance of multiple solutions to certain decompositions of Einstein's constraint equations. Pfeiffer and York recently reported the existence of two branches of solutions for identical background data in the extended conformal thin-sandwich decomposition. We show that the Hamiltonian constraint alone, when expressed in a certain way, admits two branches of solutions with properties very similar to those found by Pfeiffer and York. We construct these two branches analytically for a constant-density star in spherical symmetry, but argue that this behavior is more general. In the case of the Hamiltonian constraint this non-uniqueness is well known to be related to the sign of one particular term, and we argue that the extended conformal thin-sandwich equations contain a similar term that causes the breakdown of uniqueness.

Spherical solutions in Einstein-aether theory: static aether and stars

The time-independent spherically symmetric solutions of general relativity coupled to a dynamical unit timelike vector are studied. We find that there is a three-parameter family of solutions with this symmetry. Imposing asymptotic flatness restricts to two parameters and requiring that the aether be aligned with the timelike Killing field further restricts to one parameter, the total mass. These ‘static aether’ solutions are given analytically up to solution of a transcendental equation. The positive mass solutions have spatial geometry with a minimal area 2-sphere, inside which the area diverges at a curvature singularity occurring at an extremal Killing horizon that lies at a finite affine parameter along a radial null geodesic. Regular perfect fluid star solutions are shown to exist with static aether exteriors, and the range of stability for constant density stars is identified.

Relativistic hydrodynamics on spacelike and null surfaces: Formalism and computations of spherically symmetric spacetimes

We introduce a formulation of Eulerian general relativistic hydrodynamics which is applicable for (perfect) fluid data prescribed on either spacelike or null hypersurfaces. Simple explicit expressions for the characteristic speeds and fields are derived in the general case. A complete implementation of the formalism is developed in the case of spherical symmetry. The algorithm is tested in a number of different situations, predisposing for a range of possible applications. We consider the Riemann problem for a polytropic gas, with initial data given on a retarded/advanced time slice of Minkowski spacetime. We compute perfect fluid accretion onto a Schwarzschild black hole spacetime using ingoing null Eddington-Finkelstein coordinates. Tests of fluid evolution on dynamic background include constant density and TOV stars sliced along the radial null cones. Finally, we consider the accretion of self-gravitating matter onto a central black hole and the ensuing increase in the mass of the black hole horizon.

Constraints in spherically symmetric classical general relativity. II. Identifying the configuration space: A moment of time symmetry.

We continue our investigation of the configuration space of general relativity begun in I (gr-qc/9411009). Here we examine the Hamiltonian constraint when the spatial geometry is momentarily static (MS). We show that MS configurations satisfy both the positive quasi-local mass (QLM) theorem and its converse. We derive an analytical expression for the spatial metric in the neighborhood of a generic singularity. The corresponding curvature singularity shows up in the traceless component of the Ricci tensor. We show that if the energy density of matter is monotonically decreasing, the geometry cannot be singular. A supermetric on the configuration space which distinguishes between singular geometries and non-singular ones is constructed explicitly. Global necessary and sufficient criteria for the formation of trapped surfaces and singularities are framed in terms of inequalities which relate appropriate measures of the material energy content on a given support to a measure of its volume. The strength of these inequalities is gauged by exploiting the exactly solvable piece-wise constant density star as a template.

Goldstone bosons to gravitons: A new gravity-wave production mechanism.

We propose a new mechanism for gravity-wave production: the conversion of Goldstone bosons to gravitons in the presence of an external gravitational field. We compute the gravity-wave luminosity for the cases in which the external fields are due to various types of black holes: Schwarzschild, Reissner-Nordstroem, and Kerr. We also consider the case of the Schwarzschild constant-density star. The Goldstone-boson-induced gravity-wave luminosities are largely independent of the mass of the black hole. For the Goldstone bosons we have considered (the omion, recently proposed by Sikivie) these luminosities are small. However this is not due to effects inherent in the conversion process, rather this smallness comes about due to the incoming Goldstone-boson spectrum.

Radiation from collapsing relativistic stars. III - Second order perturbations of collapse with rotation

We study the gravitational waves generated during the collapse of a slowly rotating, pressureless, constant density star. To first order in the rotation rate the perturbation is an odd-parity dipole and generates no gravitational waves. The first order perturbation, however, drives a quadrupole deformation which is second order in the rotation rate and which produces the radiation. Using a gauge invariant description, we derive the equations governing this second order perturbation. These equations are then applied to the case of the collapse of a slowly and uniformly rotating constant-density star following the sudden turnoff of pressure. Numerical results are given for the gravitational waveforms and energies generated.