2.4 Numerical Schemes

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 Newton-Raphson 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).

2.4.1 Hartle

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 slow-rotation 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 mass-shedding 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.

2.4.2 Butterworth and Ipser (BI)

The BI-scheme [21] 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.

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 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 [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 self-bound strange quark matter.

2.4.3 Komatsu, Eriguchi and Hachisu (KEH)

In the KEH scheme [22, 23], 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 [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 semi-infinite 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 first-order 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].

2.4.4 Neugebauer and Herold (NH)

[A1b ] The scheme by Neugebauer and Herold [47] implements the minimal surface formalism for rotating axisymmetric space-times [44, 45, 46], in which Einstein's field equations are equivalent to the minimal surface equations in an abstract Riemannian potential space with a well-defined 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 Newton-Raphson 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 4D-ray-tracing pictures (See [48] for a review.).

2.4.5 Bonazzola et al. (BGSM)

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 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 [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 space-times, 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 stress-energy 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 high-accuracy, unstable equilibrium models as initial data for an axisymmetric collapse computation.

2.4.6 Bonazzola et al. (BGM-98)

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 quasi-stationary models of binary neutron stars.

2.4.7 Direct Comparison of Numerical Codes

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 grid-points 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 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 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 post-Newtonian approximation is presented in [53].

Rotating Stars in Relativity Nikolaos Stergioulas http://www.livingreviews.org/lrr-1998-8 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de