]], a
scheme using Green’s functions by Komatsu et al. [183, 184], a minimal surface scheme due to
Neugebauer and Herold [234], and spectral-method schemes by Bonazzola et al. [47, 46] and by
Ansorg et al. [18]. Below we give a description of each method and its various implementations
(codes).
To order 
, the structure of a star changes only by quadrupole terms and the equilibrium equations
become a set of ordinary differential equations. Hartle’s [143, 148] method computes rotating stars in this
slow rotation approximation, and a review of slowly rotating models has been compiled by Datta [82].
Weber et al. [317, 319] also implement Hartle’s formalism to explore the rotational properties of four new
EOSs.
Weber and Glendenning [318] improve on Hartle’s formalism in order to obtain a more accurate
estimate of the angular velocity at the mass-shedding limit, but their models still show large discrepancies
compared to corresponding models computed without the slow rotation approximation [260]. Thus, Hartle’s
formalism is appropriate for typical pulsar (and most millisecond pulsar) rotational periods, but it is not the
method of choice for computing models of rapidly rotating relativistic stars near the mass-shedding
limit.
The BI scheme [55] solves the four field equations following a Newton–Raphson-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.
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 
. Angular derivatives are
approximated by high-accuracy 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 [55, 54].
Friedman et al. [113, 114] (FIP) extend the BI code to obtain a large number of rapidly rotating models
based on a variety of realistic EOSs. Lattimer et al. [193] used a code that was also based on the BI scheme
to construct rotating stars using “exotic” and schematic EOSs, including pion or kaon condensation and
strange quark matter.
In the KEH scheme [183, 184], the same set of field equations as in BI is used, but the three elliptic-type
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 first
order 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 [183, 184, 95] 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 that maps the semi-infinite region 
to the closed region ![[0,1]](article159x.gif){kind=small}
. 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 [69] and polytropic and realistic models are computed in [71]
and [70].
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 first order differential equation for one of the metric functions. This
derivative was introducing a numerical error of 1 – 2% 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 [294] and in [292]. It is available as a public domain code, named rns, and can be downloaded
from [291].
In the BGSM scheme [47], the field equations are derived in the 3+1 formulation. All four chosen equations
that describe the gravitational field are of elliptic type. This avoids the problem with the second order
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 [260, 261] 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 [132, 43] (see Section 2.7.7).
While the field equations used in the BI and KEH schemes assume a perfect fluid, isotropic stress-energy
tensor, the BGSM formulation makes no assumption about the isotropy of 
. Thus, the BGSM code can
compute stars with a magnetic field, a solid crust, or a solid interior, and it can also be used to construct
rotating boson stars.
Bonazzola et al. [46] have improved the BGSM spectral method by 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 divergent 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 nearly free
of Gibbs phenomena at the surface. The same method is also suitable for constructing quasi-stationary
models of binary neutron stars. The new method has been used in [133] for computing models of rapidly
rotating strange stars and it has also been used in 3D computations of the onset of the viscosity-driven
instability to bar-mode formation [129].
A new multi-domain spectral method has been introduced in [18, 19]. The method can use several domains
inside the star, one for each possible phase transition. Surface-adapted coordinates are used and
approximated by a two-dimensional Chebyshev expansion. Requiring transition conditions to be
satisfied at the boundary of each domain, the field and fluid equations are solved as a free
boundary value problem by a Newton–Raphson method, starting from an initial guess. The field
equations are simplified by using a corotating reference frame. Applying this new method to
the computation of rapidly rotating homogeneous relativistic stars, Ansorg et al. achieve near
machine accuracy, except for configurations at the mass-shedding limit (see Section 2.7.8)! The
code has been used in a systematic study of uniformly rotating homogeneous stars in general
relativity [264].
Equilibrium configurations in Newtonian gravity satisfy the well-known virial relation
This can be used to check the accuracy of computed numerical solutions. In general relativity, a different identity, valid for a stationary and axisymmetric spacetime, was found in [39]. More recently, two
relativistic virial identities, valid for general asymptotically flat spacetimes, have been derived
by Bonazzola and Gourgoulhon [132, 43]. The 3-dimensional virial identity (GRV3) [132] is
the extension of the Newtonian virial identity (28
) to general relativity. The 2-dimensional
(GRV2) [43] virial identity is the generalization of the identity found in [39] (for axisymmetric
spacetimes) to general asymptotically flat spacetimes. In [43], the Newtonian limit of GRV2, in
axisymmetry, is also derived. Previously, such a Newtonian identity had only been known for spherical
configurations [59].
The two virial identities are an important tool for checking the accuracy of numerical models and have
been repeatedly used by several authors [47, 260, 261, 235, 18].
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 [294]. Rapidly rotating
models constructed with several EOSs agree to 0.1 – 1.2% in the masses and radii and to better than 2%
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 [294], 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. [95], resulted from the fact that Eriguchi
et al. and FIP used different versions of a tabulated EOS.
Nozawa et al. [235] 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
10–3 to 10–4 or better, for smooth equations of state. The agreement is also excellent for soft
polytropes. These checks show that all three codes are correct and successfully compute the
desired models to an accuracy that depends on the number of grid points used to represent the
spacetime.
If one makes the extreme assumption of uniform density, the agreement is at the level of 10–2. 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 first order derivatives at the surface of the star. In the KEH and SF codes, the three-point finite-difference 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. 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.
A new direct comparison of different codes is presented by Ansorg et al. [18]. Their multi-domain
spectral code is compared to the BGSM, KEH, and SF codes for a particular uniform density model of a
rapidly rotating relativistic star. An extension of the detailed comparison in [18], which includes
results obtained by the Lorene/rotstar code in [129] and by the SF code with higher resolution
than the resolution used in [235], is shown in Table 2. The comparison confirms that the virial
identity GRV3 is a good indicator for the accuracy of each code. For the particular model in
Table 2, the AKM code achieves nearly double-precision accuracy, while the Lorene/rotstar
code has a typical relative accuracy of 2 × 10–4 to 7 × 10–6 in various quantities. The SF
code at high resolution comes close to the accuracy of the Lorene/rotstar code for this model.
Lower accuracies are obtained with the SF, BGSM, and KEH codes at the resolutions used
in [235].
The AKM code converges to machine accuracy when a large number of about 24 expansion coefficients are used at a high computational cost. With significantly fewer expansion coefficients (and comparable computational cost to the SF code at high resolution) the achieved accuracy is comparable to the accuracy of the Lorene/rotstar and SF codes. Moreover, at the mass-shedding limit, the accuracy of the AKM code reduces to about 5 digits (which is still highly accurate, of course), even with 24 expansion coefficients, due to the nonanalytic behaviour of the solution at the surface. Nevertheless, the AKM method represents a great achievement, as it is the first method to converge to machine accuracy when computing rapidly rotating stars in general relativity.
Going further A review of spectral methods in general relativity can be found in [42]. A formulation for nonaxisymmetric, uniformly rotating equilibrium configurations in the second post-Newtonian approximation is presented in [22].
| AKM | Lorene/ | SF | SF | BGSM | KEH | |
| rotstar | (260 × 400) | (70 × 200) | ||||
 |
1.0 | |||||
 |
0.7 | 1 × 10–3 | ||||
 |
1.41170848318 | 9 × 10–6 | 3 × 10–4 | 3 × 10–3 | 1 × 10–2 | 1 × 10–2 |
 |
0.135798178809 | 2 × 10–4 | 2 × 10–5 | 2 × 10–3 | 9 × 10–3 | 2 × 10–2 |
 |
0.186338658186 | 2 × 10–4 | 2 × 10–4 | 3 × 10–3 | 1 × 10–2 | 2 × 10–3 |
 |
0.345476187602 | 5 × 10–5 | 3 × 10–5 | 5 × 10–4 | 3 × 10–3 | 1 × 10–3 |
 |
0.0140585992949 | 2 × 10–5 | 4 × 10–4 | 5 × 10–4 | 2 × 10–2 | 2 × 10–2 |
 |
1.70735395213 | 1 × 10–5 | 4 × 10–5 | 1 × 10–4 | 2 × 10–2 | 6 × 10–2 |
 |
–0.162534082217 | 2 × 10–4 | 2 × 10–3 | 2 × 10–2 | 4 × 10–2 | 2 × 10–2 |
 |
11.3539142587 | 7 × 10–6 | 7 × 10–5 | 1 × 10–3 | 8 × 10–2 | 2 × 10–1 |
 |
4 × 10–13 | 3 × 10–6 | 3 × 10–5 | 1 × 10–3 | 4 × 10–3 | 1 × 10–1 |
 |
 |
 |
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