Nonlinear r-modes in Rapidly Rotating Relativistic Stars

We present the first study of nonlinear r-modes in isentropic,rapidly rotating relativistic stars, via 3-D general-relativistic hydrodynamical evolutions (in the Cowling approximation). Our study aims at addressing several questions regarding the properties of r-modes, which are significant for the prospect of detecting gravitational waves from the r-mode instability. We find that (1) on dynamical timescales, there is no strong 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 $\alpha \sim 1$. An absolute upper limit on the amplitude (relevant for the most rapidly rotating stars) is set by causality, $\alpha_{causal} \sim 3c/\Omega R$. (2) The spectrum of r-modes and inertial modes in isentropic stars is discrete. (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.