A new class of unstable modes of rotating relativistic stars

Abstract

The first numerical study of axial (toroidal) pulsation modes of a slowly rotating relativistic star is presented. The calculation includes terms of first order in $ε\equiv Ω\sqrt{R^3/M}<<1$ ($R$ is the radius, $M$ is the mass and $Ω$ is the rotation frequency of the star), and accounts for effects due to the coriolis force. Effects due to the centrifugal flattening of the star enter at order $ε^2$ and are not included in the analysis. It is shown that increased rotation tends to decrease the damping times for prograde modes, while retrograde become longer lived. Specifically, we show that rotation affects the axial gravitational-wave $w$-modes in this way. We also present the first relativistic calculation of the so-called $r$-modes (analogous to Rossby waves in the Earth's oceans). These have frequencies of the same order of magnitude as the rotation frequency of the star. The presented results indicate that the $r$-modes are unstable due to the emission of gravitational radiation for \underline{all} rotating perfect fluid stars. This is interesting since the previously considered gravitational-wave instability associated with (for example) the $f$-mode of the star sets in at a critical rotation rate. Because they are unstable also for the slowest rotating stars the $r$-modes may well be of considerable astrophysical importance.