... body

The basic principles of the AKM-method rely on a rapidly converging Chebyshev representation of all physical and geometrical quantities within appropriate coordinates. The method is therefore applicable to arbitrary differentially rotating configurations (with some analytical rotation law). In this article we restrict ourselves to uniform rotation. In this case we may use the simpler field equations valid within the comoving frame of reference.

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... follows

Once $\nu,\omega$ and W are known, the potential $\alpha$ can be determined by a line integral (with $\alpha\to 0$ as $\rho^2+\zeta^2\to\infty$).

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... form

See, for example, Kramer et al. (1980).

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...

It is a consequence of the field equations that k' is then also differentiable.

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...=0

Note that the interior coordinate transformation introduced in Sect. 4.2.3 is not invertible at the equatorial rim of the star, but it is so at s=0.

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... e.g. 

The Chebyshev-polynomials are defined by $T_j(x)=\cos[j\arccos(x)], x\in [-1,1]$.

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... follows

In order to get m unknowns representing the surface of the star in the $m{\rm th}$-order approximation discussed in Sect. 3.3, we take the radii  $r_{\rm e},r_{\rm p}$ and (m-2) Chebyshev-coefficients for the function g.

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... surface

The continuity conditions of the fields' derivatives will be part of the set of algebraic equations in question.

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... functions

  Note that it is straightforward to calculate (i) the Chebyshev coefficients of a function from its values at the gridpoints (s_j,t_k) (or t_k in the one-dimensional case), see Eq. (42), (ii) the value of a function at an arbitrary point inside or at the boundary of I² (or I) from its Chebyshev coefficients, and (iii) the Chebyshev coefficients of the derivative and the integral of a function from its Chebyshev coefficients, see Press et al. (1992).

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... reads

  Note that the function H_U'^(m)(s=0,t) tends to a constant in the limit $m\to\infty$. Similar properties hold for all functions listed in (40), see also Sect. 3.6.

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... respectively

For a description of the equations of state corresponding to polytropic and strange star matter, see Tooper (1965) and e.g. Gourgoulhon et al. (1999) respectively.

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... body

The quantities M and J can also be taken from the exterior fields $\nu$ and $\omega$, see Eqs. (19).

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...

Note that the example listed in Table 5 has previously been calculated, see Nozawa et al. (1998).

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... stationarity

A locally stationary, axisymmetric spacetime requires the existence of a timelike linear combination of the two Killing vectors corresponding to stationarity and axisymmetry. The latter one vanishes on the symmetry axis.

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