ZONES OF DYNAMICAL INSTABILITY FOR ROTATING STRING LOOPS

The equations of first-order perturbations are derived directly in a particular gauge for a stationary rotating string ring in a flat background. The perturbations are decoupled into equatorial (i.e., in the plane of the loop) and azimuthal (i.e., perpendicular to the plane of the loop) plane waves with quantified wavelengths. A polynomial eigenvalue equation for the perturbations defining the pulsation of the plane waves is then written and, after simplification, reduces the condition of stability to the reality of the roots of a third degree polynomial with real coefficients. This condition is equivalent to the positivity of a generalized discriminant and relies only on two parameters which are the longitudinal and transverse characteristic speeds and depends on the internal structure of the string. It is found that, although the azimuthal, axisymmetric, and lowest nonaxisymmetric perturbations are stable, there exist configurations of instability in the equatorial perturbations for all the other spatial modes, especially for classical and ultrarelativistic strings. The whole range of parameters of the problem is then explored analytically and numerically, giving a complete solution to the problem. So, the stability of a particular model of strings can be checked easily. A rate of dynamical decay of the equilibrium state is also defined and calculated in some interesting cases to verify the effective cosmological instability of the loops. © 1994 The American Physical Society.