Astron. Astrophys. 331, 280-290 (1998)

2. Theoretical model

2.1. Basic assumptions

The general space-time line element in terms of the quantities of the (3+1)-formalism of general relativity (see e.g. York 1979for an introduction) is given by

with the lapse function N, the shift vector , and , the metric tensor induced in the spatial hypersurfaces . Before the symmetry breaking, the space-time associated with the rotating star is stationary and axisymmetric. We briefly recall the main conclusions for the case where the star matter is assumed to be constituted by a perfect fluid, the stress energy tensor having the form

The involved quantities are the fluid proper energy density e, the fluid pressure p, the fluid four-velocity , and the space-time metric tensor . Two Killing vector fields and are linked to the space-time symmetries where is time-like at least far from the star and space-like, its orbits being closed curves. In the case of rigid rotation, the space-time is circular and the two-surfaces orthogonal to both and are globally integrable (Carter 1973). The coordinates t and are associated with the both Killing vector fields and whereas the remaining coordinates r and can be chosen arbitrarily. The standard coordinates for stationary axisymmetric systems are quasi-isotropic coordinates where a conformally flat metric in the -coordinate planes is adopted. The general line element (1) specified to these coordinates reads

The shift vector has only one non-vanishing component which represents the dragging of inertial frames by the rotating star. For this coordinate choice, Bonazzola et al. (1993) have exhibited a compact set of elliptic equations for the metric potentials N, and , , the latter being defined by , . Let us also remind that quasi-isotropic coordinates satisfy both the minimal distortion coordinate condition, introduced by Smarr & York (1978), and the maximal slicing condition where K is the scalar of extrinsic curvature. When the star deviates from axisymmetry, it is no more stationary either, as gravitational radiation carries away energy and angular momentum. However, at the very beginning of the symmetry breaking, this deviation is very small, and, consequently, the losses due to gravitational radiation are negligible. Under this assumption and for rigid rotation, there exists a Killing vector field which is proportional to the fluid velocity (Carter 1979),

where is a strictly positive scalar function. In the stationary and axisymmetric case, the Killing vector is given by

The constant is the angular velocity defined as . In the non-axisymmetric case, we assume (1) the existence of a vector field which is time-like at least far from the star, (2) a vector field which is space-like everywhere, (3) a constant , such that defined by (5) is a Killing vector field and (4) the fluid velocity is proportional to . The Killing vector field is associated with the persisting helical symmetry of the space-time generated by the non-axisymmetric body which appears still static in the corotating frame.

2.2. Matter equations

Based on the assumptions made in the previous section, namely that (1) the star matter is composed of a single constituent perfect fluid, and (2) the star rotates rigidly, it is possible to derive a simple first integral of motion following the procedure outlined in Bonazzola et al. (1996). We first introduce the family of Eulerian observers whose four-velocity coincides with the future directed unit vector field orthogonal to the space-like hypersurfaces . The Lorentz factor between these local rest observers and the fluid comoving observers is given by

With the baryon chemical potential µ and the mean baryon mass , the log-enthalpy H is defined as

which is the relativistic generalization of the Newtonian specific enthalpy h. Introducing , we recover the first integral of motion

already familiar from the axisymmetric and stationary case. Note, however, that in the present case all quantities are functions of where is the azimuthal angular variable in the corotating frame . At the Newtonian limit, we have , , , and (8) approaches the classical first integral of motion where U is the Newtonian potential and the distance from the rotation axis.

2.3. Field equations

As announced in Sect.  1, we apply an approximate set of field equations derived under the assumptions that (1) the helical symmetry of space-time is conserved after deviation from the axisymmetric and stationary configuration, and (2) gravitational radiation is negligible.

In addition, we only retain the dominant non-axisymmetric contributions in the field equations up to order 1/2-PN, their leading relativistic order being less or equal than where

is the post-Newtonian expansion parameter. At this level of approximation, the lapse function and the shift vector have to be considered as three-dimensional quantities. The components and are genuinely three-dimensional contributions and are absent at the previous approximation level 0-PN. Corrections of higher relativistic order to the metric tensor are included for the diagonal components via the axisymmetric potentials and , their sources being essentially dominated by the fluid pressure whereas the extra-diagonal terms are again generically non-axisymmetric quantities and hence are neglected. Accordingly, the spatial metric tensor reads

The choice of as conformal factor of the (nearly flat) conformal three-metric has proven to be particularly advantageous, as has been exposed by Bonazzola et al. (1993). It isolates the lapse function as the predominant part of the gravitational fields, which is underlined by considering the weak field limit of the corresponding space-time line element, given by

which conducts to the Newtonian equation for the gravitational potential .

Having specified the space-time line element for the perturbed neutron star models, we can proceed to derive the governing field equations. The temporal evolution of is determined by

with the tensor of extrinsic curvature . For , the determinant of the spatial metric tensor normalized by that of flat space spherical coordinates , it follows immediately the relation

where denotes the trace of . Eqs. (12) and (13) enable us to derive the evolution equation of the conformal metric tensor

The lapse function N, the only three dimensional quantity involved in the product , cancels out in this term. Therefore, the temporal derivative of vanishes identically. If we further impose the maximal slicing condition , we can determine from the metric potentials according to

Furthermore, the maximal slicing condition yields an elliptic equation for the lapse function N

Here, E stands for the total energy density and S for the trace of the stress tensor , all of them measured by the Eulerian observer .

Inserting (15) into the momentum constraint equation

leads immediately to the maximal slicing-minimal distortion shift vector equation

introduced by Smarr & York (1978) where denotes the momentum density vector. Indeed, the York minimal distortion gauge condition is trivially fulfilled, since already the interior of the square brackets equals 0. Note that any coordinate system whose conformal metric tensor is time-independent automatically satisfies the minimal distortion gauge condition. This is notably the case for isotropic coordinates, but also for our choice of quasi-isotropic coordinates where the conformal metric is not that of flat space like in the Wilson scheme, but time-independent as well. As a consequence, our approximation of keeping the original form of the spatial metric except for the lapse function N, being treated as a three-dimensional quantity now, ensures the coordinates to remain maximal slicing-minimal distortion coordinates in the three dimensional case after deviation from the initial stationary and axisymmetric configuration.

The explicit field equations for our particular choice (10) of the spatial metric tensor are then derived after introduction of the auxiliary variables

following the procedure outlined in Bonazzola et al. (1993). We obtain the following elliptic equation for the logarithm of the lapse function N

where denotes the three-dimensional flat space scalar Laplacian with respect to the coordinates of the corotating frame

and where the following abridged notation

is used. We define the pseudo-physical components of the shift vector via the following relations

The shift vector equation (18) associated with this particular frame yields

with specified by (30) below. We further introduce the explicit expressions of the involved derivative operators, associated with the standard orthonormal frame of flat space spherical coordinates. The pseudo-physical components of the three-dimensional flat space vector Laplacian are specified as

We further need to compute the covariant divergence which reads

Finally, the gradient of a scalar potential U is computed according to

The pseudo-physical components of the actual source, computed by means of (18), read

Eqs. (20) and (24) together with (30) constitute the 3D-part of our field equations. The remaining gravitational potentials and are computed by means of the dynamical Einstein and the Hamiltonian constraint equations after integration over , which conducts to the original equations derived in Bonazzola et al. (1993). They are genuine 2D-equations, intimately related to the axisymmetry and stationarity of the initial configuration (see Gourgoulhon & Bonazzola (1993) for a geometrically motivated derivation of the (3+1)-equations for this case). For the potentials and , we then have

where stands for the two-dimensional flat space scalar Laplacian

and where denotes the average of A with respect to the angular variable . This ensures the consistency of the actual sources of (31) and (32) with the axisymmetry of and .

To complete the analytic description, we add the matter related quantities for a perfect fluid whose stress-energy tensor has been defined by (2), expressed in terms of variables of the (3+1)-formalism. With the Lorentz factor , defined by (6), the "physical" fluid velocity with respect to the Eulerian observer along the a -th coordinate line is given by

where is the corresponding spatial unit vector. For our coordinates, the components then read

and the normalization condition on the fluid four-velocity yields

The matter related variables , E, and , specified to our coordinate system, take the approximate form

whereas the remaining components equal 0. By combining (30) and (39), one obtains the final form of ,

where the latter is exactly the same expression as the one presented in Bonazzola et al. (1993) except that the lapse function N, the shift vector component and the matter term are allowed to depend on now.

Let us finally mention that our analytic scheme yields the exact solution to the general field and matter equations for two limiting cases: (1) at the Newtonian limit for an arbitrary deviation from axisymmetry, and (2) in the axisymmetric case up to arbitrary relativistic order.

2.4. Stability of an axisymmetric configuration

At this point, we can summarize our analytical approach. The elliptic field equations (20), (24), (31), and (32), completed by the first integral equation (8), fully determine an axisymmetric and stationary equilibrium model, having specified e.g. the central value of the log-enthalpy and the angular velocity for some particular equation of state. The above problem can be formulated as a fixed point problem in some appropriate functional Banach space. Under reasonable physical assumptions, the induced mapping is contractive, thus a unique solution exists and the deviation of the sequence members from the fixed point is bounded by some decaying exponential function. We refer to Schaudt & Pfister (1996) for a recent proof of this statement, though, at present, restricted to weakly relativistic configurations such as white dwarfs. The solution scheme consists in solving the three-dimensional field and matter equations iteratively where as initial guess a spherical, static matter distribution with a parabolic density profile is assumed. The gravitational potentials are initially set to their flat space values. After a few iterations, rotation is switched on, and the solution converges to the stationary and axisymmetric configuration fixed by the model parameters and . A particular approximate solution, obtained from a previous axisymmetric one, remains axisymmetric. The sequence of equilibrium models is therefore restricted to the subspace of axisymmetric and stationary ones which is part of the full configuration space of three-dimensional quasi-equilibrium configurations.

At a certain iteration step , after convergence is considered to be sufficient, a small perturbation

is added to , which excites the , mode. Here denotes the central value of the log-enthalpy and is a small constant of order . The three-dimensional gravitational potentials N and respond to this perturbation via the field equations and the matter distribution via the first integral of motion. The non-axisymmetry of a particular configuration is conveniently measured by a parameter , introduced by means of the Fourier expansion of in ,

where denotes the mean stellar radius in the equatorial plane. As mentioned above, applied to axisymmetric configurations is contractive in some neighbourhood of the previously constructed axisymmetric solution. Axisymmetric perturbations will thus decay exponentially. This may be different for the non-axisymmetric perturbation (42) depending on the influence of the three-dimensional terms in the field and matter equations. The stability of the axisymmetric model is decided by inspection of the behaviour of the non-axisymmetry parameter during the continued iteration. Having in mind that we operate in the linear regime, we may introduce , the amplification factor of the non-axisymmetry parameter between two successive iterations, and so the following three cases can be distinguished:

1. : decreases exponentially and the perturbed configuration converges to the non-perturbed axisymmetric configuration - the configuration is secularly stable,
2. : does not change during the subsequent iterations - the configuration is secularly meta-stable,
3. : grows exponentially and the perturbed configuration evolves subsequently away from the unstable axisymmetric configuration towards a new stable triaxial quasi-equilibrium configuration - the configuration is secularly unstable.

To infer the actual stability of a certain configuration, one has of course to keep in mind the approximate character of our perturbed equations. In particular, for fully relativistic configurations, relativistic terms beyond the current approximation level of order 1/2-PN which are not included in the present scheme will possibly alter the stability against the triaxial secular instability in some a priori unknown sense.

© European Southern Observatory (ESO) 1998

Online publication: February 4, 1998
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