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 , 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
- 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  are not suitable for nonspherical sources (see  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 .
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].