Recent research has considerably advanced our understanding of these objects. There now exist several independent numerical codes for obtaining accurate models of rotating neutron stars in full general relativity. Three of these codes have been shown to agree with each other to remarkable accuracy, and one code is available as public domain for use by other researchers.

The numerically constructed maximum mass models, for different proposed equations of state, differ by as much as a factor of two in mass, radius and angular velocity, a factor of five in central density and a factor of eight in the moment of inertia. These large uncertainties show that our understanding of the properties of matter at very high densities is currently rather poor.

Despite the different maximum rotation rates, corresponding to different candidates for the equation of state of neutron-star matter, one can place an absolute upper limit on the rotation of relativistic stars 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.

Although observed magnetic fields in neutron stars have a negligible effect on neutron-star structure, a sufficiently strong magnetic field acts as a centrifugal force on a relativistic star, flattening its shape and increasing the maximum mass and rotation rate for a given equation of state. The magnetic field strength of a stationary configuration has been shown to have an upper limit of G.

Rapidly rotating proto-neutron stars are shown to have an extended envelope, due to their high temperature and the presence of trapped neutrinos. If the equation of state is softened, as the neutron star cools, by a large amplitude phase transition, then the nascent neutron star may collapse to a black hole. A surprising result is that a supramassive proto-neutron star, even though it contracts during cooling, evolves to a cold neutron star of smaller angular velocity.

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) are yet to be obtained. The effect of rotation on the quasi-normal modes of oscillation is to couple polar and axial modes and to shift their frequencies and damping times, causing some modes to become unstable.

Nonaxisymmetric instabilities in rotating stars can be driven by the emission of gravitational waves (CFS-instability) or by viscosity. The onset of the CFS-instability has now been computed for fully relativistic, rapidly rotating stars. Relativity has a strong influence on the onset of the instability, allowing it to occur for less rapidly rotating stars than was suggested by Newtonian computations.

Contrary to what was previously thought, nascent neutron stars
can be subject to the
*l*
=2 bar mode CFS-instability, emitting strong gravitational waves.
The frequency of the waves sweeps downward through the optimal
LIGO sensitivity window, and first estimates show that it could
be detectable out to the distance of 140 Mpc by the advanced LIGO
detector.

The viscosity-driven instability is not favored by general relativity but, as a new relativistic computation shows, is absent in rotating neutron stars, unless the equation of state is unexpectedly stiff.

Axial fluid modes in rotating stars (*r*
-modes) received renewed attention since it was discovered that
they are generically unstable to the emission of gravitational
waves. The
*r*
-mode instability can slow down a newly-born rapidly rotating
neutron star to Crab-like rotation rates. First results show that
the gravitational waves from the spin-down (directly, or as a
stochastic background) could be detectable by the advanced LIGO
or VIRGO detectors.

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

Rotating Stars in Relativity
Nikolaos Stergioulas
http://www.livingreviews.org/lrr-1998-8
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