Out of the ten components of the field equations that describe
the geometry of a rotating relativistic star, only four are
independent; one has the freedom to choose which four components
to use. After choosing four field equations, there are different
methods one can use to solve them. First models were obtained by
Wilson [19] and Bonazzola and Schneider [20]. Here we will review the following methods: Hartle's slow
rotation formalism, the NewtonRaphson linearization scheme due
to Butterworth and Ipser [21], a scheme using Green's functions by Komatsu et al. [22,
23], a minimal surface scheme due to Neugebauer and Herold [47]
[A1a], and two spectral methods by Bonazzola et al. [25,
26]. Below we give a description about each method and its various
implementations (codes).
To
the structure of a star changes only by quadrupole terms, and
the equilibrium equations become a set of ordinary differential
equations. Hartle's [27,
28] method computes rotating stars in this slowrotation
approximation; a review of slowly rotating models has been
compiled by Datta [29]. Weber et al. [30], [31] also implement Hartle's formalism to explore the rotational
properties of four new EOSs.
Weber and Glendenning [32] attempt to improve on Hartle's formalism in order to obtain a
more accurate estimate of the angular velocity at the
massshedding limit, but their models show large discrepancies
compared to corresponding models computed with fully rotating
schemes [9]. Thus, Hartle's formalism cannot be used to compute models of
rapidly rotating relativistic stars with sufficient accuracy.
The BIscheme [21] solves the four field equations following a NewtonRaphson like
linearization and iteration procedure. One starts with a
nonrotating model and increases the angular velocity in small
steps, treating a new rotating model as a linear perturbation of
the previously computed rotating model. Each linearized field
equation is discretized, and the resulting linear system is
solved. The four field equations and the hydrostationary
equilibrium equation are solved separately and iteratively until
convergence is achieved.
The space is truncated at a finite distance from the star, and
the boundary conditions there are imposed by expanding the metric
potentials in powers of 1/
r
. Angular derivatives are approximated by highaccuracy formulae,
and models with density discontinuities are treated specially at
the surface. An equilibrium model is specified by fixing its rest
mass and angular velocity.
The original BI code was used to construct uniform density
models and polytropic models [21,
33]. Friedman et al. [34,
35] extend the BI code to obtain a large number of rapidly rotating
models based on a variety of realistic EOSs. Lattimer et al. [36] used a code which was also based on the BI scheme to construct
rotating stars using recent ``exotic'' and schematic EOSs,
including pion or Kaon condensation and selfbound strange quark
matter.
In the KEH scheme [22,
23], the same set of field equations as in BI is used, but the
three elliptictype field equations are converted into integral
equations using appropriate Green's functions. The boundary
conditions at large distance from the star are thus incorporated
into the integral equations, but the region of integration is
truncated at a finite distance from the star. The fourth field
equation is an ordinary firstorder differential equation. The
field equations and the equation of hydrostationary equilibrium
are solved iteratively, fixing the maximum energy density and the
ratio of the polar radius to the equatorial radius, until
convergence is achieved. In [22,
23] and [37] the original KEH code is used to construct uniformly and
differentially rotating stars for both polytropic and realistic
EOSs.
Cook, Shapiro and Teukolsky (CST) improve on the KEH scheme by
introducing a new radial variable which maps the semiinfinite
region
to the closed region [0,1]. In this way, the region of
integration is not truncated and the model converges to a higher
accuracy. Details of the code are presented in [38] and polytropic and realistic models are computed in [39] and [40].
Stergioulas and Friedman (SF) implement their own KEH code
following the CST scheme. They improve on the accuracy of the
code by a special treatment of the second order radial derivative
that appears in the source term of the firstorder differential
equation for one of the metric functions. This derivative was
introducing a numerical error of
in the bulk properties of the most rapidly rotating stars
computed in the original implementation of the KEH scheme. The SF
code is presented in [41] and in [42]. It is available as a public domain code, named
rns, and can be downloaded from [43].
[A1b
] The scheme by Neugebauer and Herold [47] implements the minimal surface formalism for rotating
axisymmetric spacetimes [44,
45,
46], in which Einstein's field equations are equivalent to the
minimal surface equations in an abstract Riemannian potential
space with a welldefined metric, whose coordinates are the four
metric functions of the usual stationary, axisymmetric metric. A
finite element technique is used, and the system of equations is
solved by a NewtonRaphson method. Models based on realistic EOSs
are presented in [47,
24]. The NH scheme has been used to visualize rapidly rotating
stars by embedding diagrams and 4Draytracing pictures (See [48] for a review.).
In the BGSM scheme [25], the field equations are derived in the 3+1 formulation. All
four equations describing the gravitational field are of elliptic
type. This avoids the problem with the secondorder radial
derivative in the source term of the ODE used in BI and KEH. The
equations are solved using a spectral method, i.e. all functions
are expanded in terms of trigonometric functions in both the
angular and radial directions, and a Fast Fourier Transform (FFT)
is used to obtain coefficients. Outside the star, a redefined
radial variable is used, which maps infinity to a finite
distance.
In [49] the code is used to construct a large number of models based on
recent EOSs. The accuracy of the computed models is estimated
using two general relativistic Virial identities, valid for
general asymptotically flat spacetimes, that were discovered by
Gourgoulhon and Bonazzola [50,
51].
While the field equations used in the BI and KEH schemes
assume a perfect fluid, isotropic stressenergy tensor, the BGSM
formulation makes no assumption about the isotropy of
. Thus, the BGSM code can compute stars with magnetic field,
solid crust or solid interior, and it can also be used to
construct rotating boson stars.
Since it is based on the 3+1 formalism, the BGSM code is also
suitable for providing highaccuracy, unstable equilibrium models
as initial data for an axisymmetric collapse computation.
The BGSM spectral method has been improved by Bonazzola et al. [26] allowing for several domains of integration. One of the domain
boundaries is chosen to coincide with the surface of the star,
and a regularization procedure is introduced for the infinite
derivatives at the surface (that appear in the density field when
stiff equations of state are used). This allows models to be
computed that are free of Gibbs phenomena at the surface. The
method is also suitable for constructing quasistationary models
of binary neutron stars.
The accuracy of the above numerical codes can be estimated, if
one constructs exactly the same models with different codes and
compares them directly. The first such comparison of rapidly
rotating models constructed with the FIP and SF codes is
presented by Stergioulas and Friedman in [41]. Rapidly rotating models constructed with several EOS's agree
to
in the masses and radii and to better than
in any other quantity that was compared (angular velocity and
momentum, central values of metric functions etc.). This is a
very satisfactory agreement, considering that the BI code was
using relatively few grid points, due to limitations of computing
power at the time of its implementation.
In [41], it is also shown that a large discrepancy between certain
rapidly rotating models, constructed with the FIP and KEH codes,
that was reported by Eriguchi et al. [37], was only due to the fact that a different version of a
tabulated EOS was used in [37] than by FIP.
Recently, Nozawa et al. [16] have completed an extensive direct comparison of the BGSM, SF
and the original KEH codes, using a large number of models and
equations of state. More than twenty different quantities for
each model are compared, and the relative differences range from
to
or better, for smooth equations of state. The agreement is
excellent for soft polytropes, which shows that all three codes
are correct and compute the desired models to an accuracy that
depends on the number of gridpoints used to represent the
spacetime.
If one makes the extreme assumption of uniform density, the
agreement is at the level of
. In the BGSM code this is due to the fact that the spectral
expansion in terms of trigonometric functions cannot accurately
represent functions with discontinuous firstorder derivatives at
the surface of the star. In the KEH and SF codes, the threepoint
finitedifference formulae cannot accurately represent
derivatives across the discontinuous surface of the star.
The accuracy of the three codes is also estimated by the use
of the two Virial identities due to Gourgoulhon and Bonazzola [50,
51]. Overall, the BGSM and SF codes show a better and more
consistent agreement than the KEH code with BGSM or SF. This is
largely due to the fact that the KEH code does not integrate over
the whole spacetime but within a finite region around the star,
which introduces some error in the computed models.

Going further.
A review of spectral methods in general relativity can be
found in [52]. A formulation for nonaxisymmetric, uniformly rotating
equilibrium configurations in the second postNewtonian
approximation is presented in [53].

Rotating Stars in Relativity
Nikolaos Stergioulas
http://www.livingreviews.org/lrr19988
© MaxPlanckGesellschaft. ISSN 14338351
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