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Nonlinear r-modes in Rapidly Rotating Relativistic Stars

The r-mode instability in rotating relativistic stars has been shown recently to have important astrophysical implications (including the emission of detectable gravitational radiation, the explanation of the initial spins of young neutron stars and the spin-distribution of millisecond pulsars and the explanation of one type of gamma-ray bursts), provided that r-modes are not saturated at low amplitudes by nonlinear effects or by dissipative mechanisms. Here, we present the first study of nonlinear r-modes in isentropic, rapidly rotating relativistic stars, via 3-D general-relativistic hydrodynamical evolutions. Our numerical simulations show that (1) on dynamical timescales, there is no strong nonlinear coupling of r-modes to other modes at amplitudes of order one -- unless nonlinear saturation occurs on longer timescales, the maximum r-mode amplitude is of order unity (i.e., the velocity perturbation is of the same order as the rotational velocity at the equator). An absolute upper limit on the amplitude (relevant, perhaps, for the most rapidly rotating stars) is set by causality. (2) r-modes and inertial modes in isentropic stars are predominantly discrete modes and possible associated continuous parts were not identified in our simulations. (3) In addition, the kinematical drift associated with r-modes, recently found by Rezzolla, Lamb and Shapiro (2000), appears to be present in our simulations, but an unambiguous confirmation requires more precise initial data. We discuss the implications of our findings for the detectability of gravitational waves from the r-mode instability.