## 1. IntroductionIn many recent astrophysical applications of the theory of dense matter it is necessary to investigate the properties of rapidly rotating compact objects within general relativity theory. The reason for this development is the hope that changes in the internal structure of the dense matter, e.g. during phase transitions, could have observable consequences for the dynamics of the rotational behavior of these objects. Particular examples are the observations of glitches and postglitch relaxation in pulsars which are discussed as signals for superfluidity in nuclear matter (Pines & Ali Alpar 1992) and the suggestion that the braking index is remarkably enhanced when a quark matter core occurs in the centre of a pulsar during its spin-down evolution (Glendenning et al. 1997). Further constraints for the nuclear equation of state come from the observation of quasi-periodic brightness oscillations (QPO's) in low-mass X-ray binaries which entail mass and radius limits for rapidly rotating neutron stars, see Lamb et al. (1998) , Li et al. (1999). The problem of rotation in the general relativity theory was and remains one of the central and complicated problems (Glendenning 1997). Compared to the modern methods of numerical solutions to this problem (Friedman et al. 1986; Salgado et al. 1994) the method of perturbation theory (Hartle 1967; Hartle & Thorne 1968; Sedrakian & Chubarian 1968) is a physically transparent and systematic approach to the solution of the problem of stationary gravitational fields and their sources. This approach has been applied successfully in general relativity as well as in alternative theories of gravitation (Grigorian & Chubarian 1985). From a practical point of view in the definition of the integral characteristics of the astrophysical objects, it is important to analyze the asymptotical expansion of the metric tensor at large distances from the stars, to be able to compare the results with observational data. One can of course introduce the physical parameters of the configuration using the symmetry properties of the object and the gravitational field by expressing them in terms of conserved quantities. In this work we use the definition of the projection of the angular momentum of the nonspherical rotating star (z is the axis of the star's rotation and symmetry of the gravitational field) as a conserved integral of motion. It is a well known integral of the non diagonal element of the energy-momentum tensor in the frame of spherical coordinates. Using the method of perturbation theory, we are going to calculate the total mass, angular momentum and shape deformation from the solution of the gravitational field equations in case of hydrodynamical, thermodynamical and chemical equilibrium for a given total baryon number and angular velocity of the object. According to Sedrakian & Chubarian (1968)the expansion parameter is of the order of the ratio of the rotational and gravitational energy. It has been shown that, for compact objects in the stationary rotating regime without matter flux, the first two terms of the series expansion give a sufficiently good approximation (see also Glendenning 1997; Weber 1999). The evolution of the rotating stars could have different origins and scenarii. Our aim in this work is to discuss possible signals for a deconfinement phase transition during the evolution of a rotating compact object on the basis of solutions for the dependence of the moment of inertia. © European Southern Observatory (ESO) 2000 Online publication: June 5, 2000 |