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We investigate properties of $r$-mode instability in slowly rotatingrelativistic polytropes. Inside the star slow rotation and low frequencyformalism that was mainly developed by Kojima is employed to study axialoscillations restored by Coriolis force. At the stellar surface, in order totake account of gravitational radiation reaction effect, we use a near-zoneboundary condition instead of the usually imposed boundary condition forasymptotically flat spacetime. Due to the boundary condition, complexfrequencies whose imaginary part represents secular instability are obtainedfor discrete $r$-mode oscillations in some polytropic models. It is found thatsuch discrete $r$-mode solutions can be obtained only for some restrictedpolytropic models. Basic properties of the solutions are similar to thoseobtained by imposing the boundary condition for asymptotically flat spacetime.Our results suggest that existence of a continuous part of spectrum cannot beavoided even when its frequency becomes complex due to the emission ofgravitational radiation.Comment: 10 pages, 4 figures, accepted for publlication in PR

R-mode instability of slowly rotating nonisentropic relativistic stars