Recently, the dynamical evolution of rapidly rotating stars has become possible in numerical relativity. In the framework of the 3+1 split of the Einstein equations [283], a stationary axisymmetric star can be described by a metric of the standard form

where is the lapse function, is the shift three-vector, and is the spatial three-metric, with . The spacetime has the following properties:- The metric function in (5) describing the dragging of inertial frames by rotation is related to the shift vector through . This shift vector satisfies the minimal distortion shift condition.
- The metric satisfies the maximal slicing condition, while the lapse function is related to the metric function in (5) through .
- The quasi-isotropic coordinates are suitable for numerical evolution, while the radial-gauge coordinates [25] are not suitable for nonspherical sources (see [47] for details).
- The ZAMOs are the Eulerian observers, whose worldlines are normal to the t = const. hypersurfaces.
- Uniformly rotating stars have in the coordinate frame. This can be shown by requiring a vanishing rate of shear.
- Normal modes of pulsation are discrete in the coordinate frame and their frequencies can be obtained by Fourier transforms (with respect to coordinate time ) of evolved variables at a fixed coordinate location [106].

Crucial ingredients for the successful long-term evolutions of rotating stars in numerical relativity are the conformal ADM schemes for the spacetime evolution (see [233, 275, 28, 4]) and hydrodynamical schemes that have been shown to preserve well the sharp rotational profile at the surface of the star [106, 293, 105].

4.1 Numerical evolution of equilibrium models

4.1.1 Stable equilibrium

4.1.2 Instability to collapse

4.1.3 Dynamical bar-mode instability

4.2 Pulsations of rotating stars

4.3 Rotating core collapse

4.3.1 Collapse to a rotating black hole

4.3.2 Formation of rotating neutron stars

4.1.1 Stable equilibrium

4.1.2 Instability to collapse

4.1.3 Dynamical bar-mode instability

4.2 Pulsations of rotating stars

4.3 Rotating core collapse

4.3.1 Collapse to a rotating black hole

4.3.2 Formation of rotating neutron stars

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