4 Rotating Stars in Numerical Relativity

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 [284], 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 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 t) 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 [234, 277, 32, 4]) and hydrodynamical schemes that have been shown to preserve well the sharp rotational profile at the surface of the star [106, 294, 105].

• 4.1 Numerical evolution of equilibrium models
• 4.2 Pulsations of rotating stars
• 4.3 Rotating core collapse

Rotating Stars in Relativity Nikolaos Stergioulas http://www.livingreviews.org/lrr-2003-3 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei.mpg.de