Almost none of the r-modes ordinarily found in rotating stars exist, if the star and its perturbations obey the same one-parameter equation of state; and rotating relativistic stars with one-parameter equations of state have no pure r-modes at all, no modes whose limit, for a star with zero angular velocity, is a perturbation with axial parity. Similarly (as we show here) rotating stars of this kind have no pure g-modes, no modes whose spherical limit is a perturbation with polar parity and vanishing perturbed pressure and density. Where have these modes gone?In spherical stars of this kind, r-modes and g-modes form a degenerate zerofrequency subspace. We find that rotation splits the degeneracy to zeroth order in the star's angular velocity Ω, and the resulting modes are generically hybrids, whose limit as Ω → 0 is a stationary current with axial and polar parts. Lindblom and Ipser have recently found these hybrid modes in an analytic study of the Maclaurin spheroids. Since the hybrid modes have a rotational restoring force, they call them "rotation modes" or "generalized r-modes".Because each mode has definite parity, its axial and polar parts have alternating values of l. We show that each mode belongs to one of two classes, axial-led or polar-led, depending on whether the spherical harmonic with lowest value of l that contributes to its velocity field is axial or polar. We numerically compute these modes for slowly rotating polytropes and for Maclaurin spheroids, using a straightforward method that appears to be novel and robust. Timescales for the gravitational-wave driven instability and for viscous damping are computed using assumptions appropriate to neutron stars. The instability to nonaxisymmetric modes is, as expected, dominated by the l = m r-modes with simplest radial dependence, the only modes which retain their axial character in isentropic models; for relativistic isentropic stars, these l = m modes must also be replaced by hybrids of the kind considered here.1 An appendix in this paper incorrectly claims that no l = m r-modes exist, based on an incorrect assumption about their radial behavior 2 Based on this equation, Kojima has argued that the spectrum is continuous, and his argument has been made precise in a recent paper of Beyer and Kokkotas (1999) (See also Kojima and Hosonuma 1999). Beyer and Kokkotas, however, also point out that the continuous spectrum they find may be an artifact of the fact that the imaginary part of the frequency vanishes in the slow-rotation limit.

We study the r-modes and rotational "hybrid" modes of relativistic stars. As in Newtonian gravity, the spectrum of low-frequency rotational modes is highly sensitive to the stellar equation of state. If the star and its perturbations obey the same one-parameter equation of state (as with isentropic stars), there exist no pure r-modes at all -no modes whose limit, for a star with zero angular velocity, is an axial-parity perturbation. Rotating stars of this kind similarly have no pure g-modes, no modes whose spherical limit is a perturbation with polar parity and vanishing perturbed pressure and density.In spherical stars of this kind, the r-modes and g-modes form a degenerate zero-frequency subspace. We find that rotation splits the degeneracy to zeroth order in the star's angular velocity Ω, and the resulting modes are generically hybrids, whose limit as Ω → 0 is a stationary current with both axial and polar parts. Because each mode has definite parity, its axial and polar parts have alternating values of l. We show that each mode belongs to one of two classes, axial-led or polar-led, depending on whether the spherical harmonic with the lowest value of l that contributes to its velocity field is axial or polar. Newtonian isentropic stars retain a vestigial set of purely axial modes (those with l = m); however, for relativistic isentropic stars we show that these modes must also be replaced by axial-led hybrids. We compute the post-Newtonian corrections to the l = m modes for uniform density stars.On the other hand, if the star is non-isentropic (or, more broadly, if the perturbed star obeys an equation of state that differs from that of the unperturbed star) the r-modes alone span the degenerate zero-frequency subspace of the spherical star. In Newtonian stars, this degeneracy is split only by the order Ω 2 rotational corrections. However, when relativistic effects are included the degeneracy is again broken at zeroth order. We compute the r-modes of a non-isentropic, uniform density model to first post-Newtonian order. *

We study the inertial modes of slowly rotating, fully relativistic compact stars. The equations that govern perturbations of both barotropic and nonbarotropic models are discussed, but we present numerical results only for the barotropic case. For barotropic stars all inertial modes are a hybrid of axial and polar perturbations. We use a spectral method to solve for such modes of various polytropic models. Our main attention is on modes that can be driven unstable by the emission of gravitational waves. Hence, we calculate the gravitational-wave growth time scale for these unstable modes and compare the results to previous estimates obtained in Newtonian gravity ͑i.e. using post-Newtonian radiation formulas͒. We find that the inertial modes are slightly stabilized by relativistic effects, but that previous conclusions concerning, e.g., the unstable r modes remain essentially unaltered when the problem is studied in full general relativity.

Numerical codes based on a direct implementation of the standard ADM formulation of Einstein's equations have generally failed to provide long-term stable and convergent evolutions of black hole spacetimes when excision is used to remove the singularities. We show that, for the case of a single black hole in spherical symmetry, it is possible to circumvent these problems by adding to the evolution equations terms involving the constraints, thus adjusting the standard ADM system. We investigate the effect that the choice of the lapse and shift has on the stability properties of numerical simulations and thus on the form of the added constraint term. To facilitate this task, we introduce the concept of quasi well-posedness, a version of well-posedness suitable for ADM-like systems involving second-order spatial derivatives.

A brief review of the stability of rotating relativistic stars is followed by a more detailed discussion of recent work on an instability of r-modes, modes of rotating stars that have axial parity in the slow-rotation limit. These modes may dominate the spin-down of neutron stars that are rapidly rotating at birth, and the gravitational waves they emit may be detectable.