2 Construction of neutron star models

If rapidly rotating neutron stars were nonaxisymmetric, they would emit gravitational waves in a very short time scale and settle down to axisymmetric configurations. Moreover, the gravity of typical neutron star is strong because

$$\frac{2 G M_{\rm ns}}{c^2 R_{\rm ns}} \sim 0.4,$$ (1)

where G is the gravitational constant, and $M_{\rm ns} \sim 1.4\ M_{\hbox{$\odot$}}$and $R_{\rm ns} \sim 10\ {\rm km}$ are the mass and the radius of a typical neutron star. Therefore, we need to solve for rotating and axisymmetric configurations in the framework of general relativity.

For the matter and the spacetime we make the following assumptions:

1.

The matter distribution and the spacetime are axisymmetric.

2.

The matter and the spacetime are in a stationary state.

3.

The matter has no meridional motions. The only motion of the matter is a circular one that is represented by the angular velocity.

4.

The angular velocity $\Omega$ is constant, as seen by a distant observer at rest.

5.

The matter can be described as a perfect fluid.

Under these assumptions, the metric can be expressed as follows (Papapetrou 1966; Carter 1969):

$$\begin{eqnarray} {\rm d}s^2 & = & - {\rm e}^{2\nu} {\rm d}t^2 + {\rm e}^{2\alpha... ...heta^2) + {\rm e}^{2\psi} ({\rm d} \varphi - \omega {\rm d}t)^2, \end{eqnarray}$$ (2) (3) (4)

  where t is the time coordinate and the polar coordinates $(r, \theta, \varphi)$ are used. The metric depends on four metric functions (or metric potentials) which are functions of r and $\theta$ only. Different authors have used the set of functions $(\nu, \omega, \alpha, \beta)$,$(\nu, \omega, \zeta, B)$ or $(\nu, \omega, \mu, \psi)$, which are related to each other through (2). In (2) and throughout the text we use gravitational units (c=G=1), unless otherwise stated.

The energy momentum tensor $T^{\rm ab}$ is expressed as

$$T^{\rm ab} = (\varepsilon + p) u^{\rm a}u^{\rm b} + p g^{\rm ab},$$ (5)

where $\varepsilon$, p and $u\rm ^a$ are the energy density, the pressure and the four-velocity, respectively. In the coordinate basis defined by (2), the components of the four-velocity are

$$u^{\rm a} = \frac{{\rm e}^{-\nu}}{\sqrt{1 - v^2}} (1, 0, 0, \Omega),$$ (6)

where the proper velocity v with respect to a local zero angular momentum observer is defined by

$$v \equiv r \sin \theta {\rm e}^{\beta- \nu} (\Omega - \omega).$$ (7)

Using the metric and the energy momentum tensor mentioned above, we can write down the Einstein equations for the metric components. Although we omit detailed expressions for the Einstein equations here, one can easily derive them by straightforward calculations or consult the papers by Butterworth & Ipser (1976), Komatsu et al. (1989a) and Bonazzola et al. (1993).

The equation of hydrostationary equilibrium can be derived from the equations of motion and takes the following form:  

$$\frac{1}{\varepsilon + p} \nabla p + \nabla \nu - \frac{1}{2} \nabla \ln(1-v^2) = 0.$$ (8)

This equation can be integrated, if we specify the equation of state which relates the energy density to the pressure. For a given EOS, a model is uniquely specified by fixing two parameters, such as the central energy density $\epsilon_c$ and the ratio of the polar to the equatorial coordinate radius, $r_{\rm p}/r_{\rm e}$. Then, four Einstein field equations and the equation of hydrostationary equilibrium must be solved with appropriate boundary conditions, to yield the four metric functions and the density distribution. The available codes for obtaining relativistic rotating neutron star models differ basically in the choice and method of integration of the four field equations and in the finite grid and finite difference scheme used for the integration.