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Astron. Astrophys. 357, 968-976 (2000)

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2. Self-consistent set of field equations for stationary rotating stars

2.1. Einstein equations for axial symmetry

The general form of the metric for an axial symmetric space-time manifold is

[EQUATION]

written in a spherical symmetric coordinate system in order to obtain the Schwarzschild solution as a limiting case. This line element is time-translational and axial-rotational invariant; all metric functions are dependent on the coordinate distance from the coordinate center r and azimuthal angle [FORMULA] between the radius vector and the axis of symmetry.

Reversal symmetry of the time and polar angle [FORMULA] require that all metric coefficients except [FORMULA] must be even functions of the angular velocity

[EQUATION]

of the star, the gravitational field of which is described by the Eq. (1). The physical characteristics of the rotating object depend on the centrifugal forces in the local inertial frame of the observer. In general relativity, due to the Lense-Thirring law, rotational effects are described by [FORMULA] the difference of the frame dragging frequency [FORMULA] and the angular velocity [FORMULA]

[EQUATION]

The energy momentum tensor of stellar matter can be approximated by the expression of the energy momentum tensor of an ideal fluid

[EQUATION]

where [FORMULA] is the 4-velocity of matter, p the pressure and [FORMULA] the energy density.

We assume that the star due to high viscosity (ignoring the super-fluid component of the matter) rotates stationarily as a solid body with an angular velocity [FORMULA] that is independent of the spatial coordinates. The time scales for changes in the angular velocity which we will consider in our applications are well separated from the relaxation times at which hydrodynamical equilibrium is established, so that the assumption of a rigid rotator model is justified.

Therefore there are only two non-vanishing components of the velocity

[EQUATION]

The equation of state which we will use for our investigation of the deconfinement transition in rotating compact stars will be introduced in Sect. 3. Once the energy-momentum tensor (4) is fixed by the choice of the equation of state for stellar matter, the unknown metric functions [FORMULA],[FORMULA], µ, [FORMULA] can be determined by the set of Einstein field equations for which we use the following four combinations.

There are three Einstein equations for the determination of the diagonal elements of the metric tensor

[EQUATION]

and one for the determination of the non diagonal element

[EQUATION]

Here G is the gravitational constant and [FORMULA] the Einstein tensor.

We use also one equation for the hydrodynamical equilibrium (Euler equation)

[EQUATION]

where the gravitational enthalpy H thus introduced is a function of the energy and/or pressure distribution.

The parameters of the theory are the angular velocity of the rotation [FORMULA] and the central energy density [FORMULA] of the star configuration.

2.2. Perturbative approach to the solution

The problem of the rotation can be solved iteratively by using a perturbation expansion of the metric tensor and the physical quantities in a Taylor series with respect to the angular velocity. As a small parameter for this expansion we use a dimensionless quantity which is the ratio of the rotational energy to the gravitational one for a homogeneous Newtonian star [FORMULA], where [FORMULA] with the mass density [FORMULA] at the center of the star. This expansion gives sufficiently correct solutions already at [FORMULA], since the expansion parameter is limited to values [FORMULA] by the condition of mechanical stability of the rigid rotation. This can easily be seen by considering as an upper limit for attainable angular velocities the so called Kepler one [FORMULA] with M being the total mass and [FORMULA] the equatorial radius. For homogeneous Newtonian spherical stars [FORMULA] which is fulfilled in particular also for the configurations with a deconfinement transition which we are going to discuss later in this paper, see Fig. 4.

The expansion of the metric tensor is given by

[EQUATION]

According to the symmetries of the metric coefficients introduced in Eq. (1) we have even orders [FORMULA] for the diagonal elements 1

[EQUATION]

and odd orders only for the frame dragging frequency [FORMULA]

[EQUATION]

The distributions of pressure, energy density and "enthalpy" introduced in Eq. (8) are also included in the scheme of this perturbation expansion

[EQUATION]

All functions with superscript (0) denote the solution of the static configuration and therefore they are only functions of the distance from the center r, the others are the corrections corresponding to the rotation.

This series expansion allows one to transform the Einstein equations into a coupled set of equations for the coefficient functions which can be solved by recursion. At zeroth order we recover the nonlinear problem of the static spherically symmetric star configuration (Tolman-Oppenheimer-Volkoff equations), see the next Sect. 2.3. The first recursion step is to solve Eq. (7) in order to obtain the dragging frequency in [FORMULA] and to define the moment of inertia for the spherically symmetric configuration. In Sect. 2.5 we will consider the second order contribution in the [FORMULA]- expansion (10), (12) where the [FORMULA] corrections to the moment of inertia can be found. The next terms in the expansion which are of [FORMULA] correspond to corrections of the frame dragging frequency and will be neglected since they go beyond the approximation scheme adopted in the present paper.

2.3. Zeroth order: Static spherically symmetric star models

The functions of the spherically symmetric solution in Eqs. (10) and (12) can be found from Eq. (6) and Eq. (8) in zeroth order of the [FORMULA]- expansion.

They obey the following equation (Tolman-Oppenheimer-Volkoff)

[EQUATION]

where [FORMULA] is the distribution of accumulated mass

[EQUATION]

within a sphere of radius r. For the gravitational potentials we have

[EQUATION]

[FORMULA] is the spherical radius of the star, which is defined by [FORMULA]. The set of Eq. (13) and Eq. (14) fulfils the following conditions at the center of the configuration: [FORMULA] and [FORMULA]. The central energy density [FORMULA] is the parameter of the spherical configuration. The total mass of the spherically distributed matter in the selfconsistent gravitational field is [FORMULA].

2.4. Moment of inertia

In the first order of the approximation we are solving Eq. (7), where the unknown function [FORMULA] defined by Eq. (11) is independent of the angular velocity. Using the static solutions Eqs. (13)-(15), and the representation of [FORMULA] by the series of the Legendre polynomials,

[EQUATION]

we find the equations for the coefficients [FORMULA]. It is proved that [FORMULA] is a function of the distance r only, i.e. that [FORMULA] for [FORMULA], see Hartle (1967), Hartle & Thorne (1968), Sedrakian & Chubarian (1968).

Let us write down the equations for [FORMULA], which is more suitable for the solution of the resulting equation in first order

[EQUATION]

which corresponds to Ref. Hartle (1967), where it was obtained using a different representation of the metric. Here, we use the notation [FORMULA], for which outside of configuration [FORMULA].

By definition, the angular momentum of the star in the case of stationary rotation is a conserved quantity and can be expressed in invariant form

[EQUATION]

where [FORMULA] is the invariant volume and [FORMULA]. For the case of slow rotation where the shape deformation of the rotating star can be neglected and using the definition of the moment of inertia [FORMULA] accumulated in the sphere with radius r, we obtain from Eq. (19)

[EQUATION]

Using this equation one can reduce the second order differential equation (18) to the first order one

[EQUATION]

and solve (18) as a coupled set of first order differential equations, one for the moment of inertia (20) and the other (21) for the frame dragging frequency [FORMULA].

This system of equations is valid inside and outside the matter distribution. In the center of the configuration [FORMULA] and [FORMULA]. The finite value [FORMULA] has to be defined such that the dragging frequency [FORMULA] smoothly joins the outer solution

[EQUATION]

at [FORMULA], and approaches [FORMULA] in the limit [FORMULA]. In the external solution (22) the constant [FORMULA] is the total moment of inertia of the slowly rotating star and [FORMULA] is the corresponding angular momentum. In this order of approximation, [FORMULA] is a function of the central energy density or the total baryon number only. Explicit dependences of the moment of inertia on the angular velocity occur in the second order of approximation.

2.5. Second order corrections to the moment of inertia

Due to the rotation in [FORMULA]-approximation the shape of the star is an ellipsoid, and each of the equal-pressure (isobar) surfaces in the star is an ellipsoid as well. All diagonal elements of the metric and energy-momentum tensors could be represented as a series expansion in Legendre polynomials

[EQUATION]

It has been shown that the only non vanishing solutions obeying the continuity conditions on the surface are those with [FORMULA].

The deformation of the isobaric surfaces due to the rotation can be parametrized by the shift [FORMULA] which describes the deviation from the spheric distribution as a function of the radius r in the given polar angle [FORMULA] and is completely determined by

[EQUATION]

since the expansion coefficients of the deformation [FORMULA] can be calculated from the pressure corrections

[EQUATION]

[FORMULA] is the polynomial index in the angular expansion. The function [FORMULA] is the distance of the star surface from the center of the configuration in the direction with the angle [FORMULA] to the polar axis. In particular, we can define the equatorial radius [FORMULA] and the polar radius [FORMULA] and the eccentricity [FORMULA].

The correction to the momentum of inertia [FORMULA] can be represented in the form

[EQUATION]

The first three contributions can be expressed by integrals of the angular averaged modifications of the matter distribution, the shape of the configuration and the gravitational fields, in the form

[EQUATION]

where

[EQUATION]

respectively, which have to be determined from the Eq. (6) in second order approximation. The contribution of the change of the rotational energy to the moment of inertia

[EQUATION]

includes the frame dragging contribution. In this expansion we neglect the influence of the change of the frame dragging frequency, since it corresponds to the next order of the perturbative expansion [FORMULA]. For a more detailed description of the solutions of the field equations in the [FORMULA] approximation we refer to the works of Hartle (1967), Hartle & Thorne (1968) as well as Sedrakian & Chubarian (1968).

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© European Southern Observatory (ESO) 2000

Online publication: June 5, 2000
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