## 2. Self-consistent set of field equations for stationary rotating stars## 2.1. Einstein equations for axial symmetryThe general form of the metric for an axial symmetric space-time manifold is 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 Reversal symmetry of the time and polar angle require that all metric coefficients except must be even functions of the angular velocity 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 the difference of the frame dragging frequency and the angular velocity The energy momentum tensor of stellar matter can be approximated by the expression of the energy momentum tensor of an ideal fluid where is the 4-velocity of
matter, 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 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 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
,,
There are three Einstein equations for the determination of the diagonal elements of the metric tensor and one for the determination of the non diagonal element Here We use also one equation for the hydrodynamical equilibrium (Euler equation) where the gravitational enthalpy The parameters of the theory are the angular velocity of the rotation and the central energy density 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 ,
where with the mass density
at the center of the star. This
expansion gives sufficiently correct solutions already at
, since the expansion parameter is
limited to values 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 with The expansion of the metric tensor is given by According to the symmetries of the metric coefficients introduced
in Eq. (1) we have even orders
for the diagonal
elements and odd orders only for the frame dragging frequency The distributions of pressure, energy density and "enthalpy" introduced in Eq. (8) are also included in the scheme of this perturbation expansion All functions with superscript (0) denote the solution of the
static configuration and therefore they are only functions of the
distance from the center 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 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 - expansion (10), (12) where the corrections to the moment of inertia can be found. The next terms in the expansion which are of 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 modelsThe 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 - expansion. They obey the following equation (Tolman-Oppenheimer-Volkoff) where is the distribution of accumulated mass within a sphere of radius is the spherical radius of the star, which is defined by . The set of Eq. (13) and Eq. (14) fulfils the following conditions at the center of the configuration: and . The central energy density is the parameter of the spherical configuration. The total mass of the spherically distributed matter in the selfconsistent gravitational field is . ## 2.4. Moment of inertiaIn the first order of the approximation we are solving Eq. (7), where the unknown function defined by Eq. (11) is independent of the angular velocity. Using the static solutions Eqs. (13)-(15), and the representation of by the series of the Legendre polynomials, we find the equations for the coefficients
. It is proved that
is a function of the distance
Let us write down the equations for , which is more suitable for the solution of the resulting equation in first order which corresponds to Ref. Hartle (1967), where it was obtained using a different representation of the metric. Here, we use the notation , for which outside of configuration . 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 where is the invariant volume and
. 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
accumulated in the sphere with
radius Using this equation one can reduce the second order differential equation (18) to the first order one 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 . This system of equations is valid inside and outside the matter distribution. In the center of the configuration and . The finite value has to be defined such that the dragging frequency smoothly joins the outer solution at , and approaches in the limit . In the external solution (22) the constant is the total moment of inertia of the slowly rotating star and is the corresponding angular momentum. In this order of approximation, 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 inertiaDue to the rotation in -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 It has been shown that the only non vanishing solutions obeying the continuity conditions on the surface are those with . The deformation of the isobaric surfaces due to the rotation can be
parametrized by the shift which
describes the deviation from the spheric distribution as a function of
the radius since the expansion coefficients of the deformation can be calculated from the pressure corrections is the polynomial index in the angular expansion. The function is the distance of the star surface from the center of the configuration in the direction with the angle to the polar axis. In particular, we can define the equatorial radius and the polar radius and the eccentricity . The correction to the momentum of inertia can be represented in the form 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 where 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 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 . For a more detailed description of the solutions of the field equations in the approximation we refer to the works of Hartle (1967), Hartle & Thorne (1968) as well as Sedrakian & Chubarian (1968). © European Southern Observatory (ESO) 2000 Online publication: June 5, 2000 |