Rotating relativistic stars are of fundamental interest in physics. Their bulk properties constrain the proposed equations of state for densities greater than nuclear density. Accreted matter in their gravitational fields undergoes high-frequency oscillations that could become a sensitive probe for general relativistic effects. Temporal changes in the rotational period of millisecond pulsars can also reveal a wealth of information about important physical processes inside the stars or of cosmological relevance. In addition, rotational instabilities can produce gravitational waves, the detection of which would initiate a new field of observational asteroseismology of relativistic stars.

There exist several independent numerical codes for obtaining accurate models of rotating neutron stars in full general relativity, including one that is freely available. One recent code achieves near machine accuracy even for uniform density models near the mass-shedding limit. The uncertainty in the high-density equation of state still allows numerically constructed maximum mass models to differ by as much as a factor of two in mass, radius and angular velocity, and a factor of eight in the moment of inertia. Given these uncertainties, an absolute upper limit on the rotation of relativistic stars can be obtained by imposing causality as the only requirement on the equation of state. It then follows that gravitationally bound stars cannot rotate faster than 0.28 ms.

In rotating stars, nonaxisymmetric perturbations have been studied in the Newtonian and post-Newtonian approximations, in the slow rotation limit and in the Cowling approximation, but fully relativistic quasi-normal modes (except for neutral modes) have yet to be obtained. A new method for obtaining such frequencies is the time evolution of the full set of nonlinear equations. Frequencies of quasi-radial modes have already been obtained this way. Time evolutions of the linearized equations have also improved our understanding of the spectrum of axial and hybrid modes in relativistic stars.

Nonaxisymmetric instabilities in rotating stars can be driven by the emission of gravitational waves (CFS instability) or by viscosity. Relativity strengthens the former, but weakens the latter. Nascent neutron stars can be subject to the l = 2 bar mode CFS instability, which would turn them into a strong gravitational wave source.

Axial fluid modes in rotating stars (r-modes) have received considerable attention since it was discovered that they are generically unstable to the emission of gravitational waves. The r-mode instability could slow down newly-born relativistic stars and limit their spin during accretion-induced spin-up, which would explain the absence of millisecond pulsars with rotational periods less than 1.5 ms. Gravitational waves from the r-mode instability could become detectable if the amplitude of r-modes is of order unity. Recent 3D simulations show that this is possible on dynamical timescales, but nonlinear effects seem to set a much smaller saturation amplitude on longer timescales. Still, if the signal persists for a long time (as has been found to be the case for strange stars) even a small amplitude could become detectable.

Recent advances in numerical relativity have enabled the long-term dynamical evolution of rotating stars. Several interesting phenomena, such as dynamical instabilities, pulsation modes, and neutron star and black hole formation in rotating collapse have now been studied in full general relativity. The current studies are limited to relativistic polytropes, but new 3D simulations with realistic equations of state should be expected in the near future.

The goal of this article is to present a summary of theoretical and numerical methods that are used to describe the equilibrium properties of rotating relativistic stars, their oscillations and their dynamical evolution. It focuses on the most recently available preprints, in order to rapidly communicate new methods and results. At the end of some sections, the reader is directed to papers that could not be presented in detail here, or to other review articles. As new developments in the field occur, updated versions of this article will appear.

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