Using the formalism of general relativity for axially symmetric gravitation
al fields and their sources - rotating compact stars - a perturbation theor
y with respect to angular velocity is developed and physical quantities suc
h as mass, shape, momentum of inertia and total energy of the star are defi
ned. The change of the internal structure of the star due to rotation has b
een investigated and the different contributions to the moment of inertia h
ave been evaluated separately. Numerical solutions have been performed usin
g a two-flavor model equation of state describing the deconfinement phase t
ransition as constrained by the conservation of total baryon number and ele
ctric charge. During the spin down evolution of the rotating neutron star,
below critical values of angular velocity a quark matter core can appear wh
ich might be detected as a characteristic signal in the pulsar timing. With
in the spin-down scenario due to magnetic dipole radiation it is shown that
the deviation of the braking index from n = 3 could signal not only the oc
currence but also the size of a quark core in the pulsar. A new scenario is
proposed where, due to mass accretion onto the rapidly rotating compact st
ar, a spin-down to spin-up transition might signal a deconfinement transiti
on in its interior.