Metric and Coordinates

The neutron star models which we compute are assumed to be stationary, axisymmetric, uniformly rotating perfect fluid solutions of the Einstein field equations. The assumptions of stationarity and axisymmetry allow the introduction of two coordinates $\phi$ and t on which the space-time metric doesn't depend. The metric, $g_{\alpha \beta}$ can be written as

$$ds^2 = - e^{\gamma + \rho} dt^2 + e^{\gamma - \rho} \bar{r}... ...+ e^{2 \alpha} \left( d\bar{r}^2 + \bar{r}^2 d\theta^2\right),$$ (1)

where the metric potentials $\rho, \gamma, \alpha$ and $\omega$ depend only on the coordinates $\bar{r}$ and $\theta$. The function $\frac{1}{2}(\gamma + \rho)$ is the relativistic generalization of the Newtonian gravitational potential; the time dilation factor between an observer moving with angular velocity $\omega$ and an observer at infinity is $e^{\frac{1}{2}(\gamma + \rho)}$. The coordinate $\bar{r}$ is not the same as the Schwarzschild coordinate r. In the limit of spherical symmetry, $\bar{r}$ corresponds to the isotropic Schwarzschild coordinate. Circles centred about the axis of symmetry have circumference $2 \pi r$ where r is related to our coordinates $\bar{r},\theta$ by

$$r = e^{\frac{1}{2}(\gamma - \rho)} \bar{r} \sin \theta .$$ (2)

The metric potential $\omega$ is the angular velocity about the symmetry axis of zero angular momentum observers (ZAMOs) and is responsible for the Lense-Thirring effect. The fourth metric potential, $\alpha$ specifies the geometry of the two-surfaces of constant t and $\phi$. When the star is non-rotating, the exterior geometry is that of the isotropic Schwarzschild metric, with

$$e^{\frac{1}{2}(\gamma + \rho)} = \frac{1-M/2\bar{r}}{1+M/2\ba... ...)} = e^\alpha = \left( 1 + M/2\bar{r} \right)^2, \; \omega=0.$$ (3)

The program uses a compactified coordinate s which is related to $\bar{r}$ by

$$\bar{r} = \bar{r}_e \left( \frac{s}{1-s} \right),$$ (4)

where $\bar{r}_e$ is the value of $\bar{r}$ at the star's equator. This definition of s gives

$$\Leftrightarrow \bar{r} = \bar{r}_e$$ s = 0.5 (5)

$$\Leftrightarrow \bar{r} \rightarrow \infty$$ s = 1.0 (6)

The angular variable $\mu$, defined by

$$\mu = \cos \theta$$

is used by the program.