W-modes: a new family of normal modes of pulsating relativistic stars

We demonstrate explicitly the existence of a new family of outgoing-wave normal modes of pulsating relativistic stars, the first such family known that has no analogue in Newtonian stars. These modes were discovered earlier by the authors in a toy model, where they were called strongly damped normal modes. Kojima then found the first examples of these modes in realistic spherical polytropic stellar models. Here we give a number of arguments that demonstrate the existence of this family unequivocally, and we calculate a large number of eigenfrequencies. Physically, the modes arise directly from the coupling of the fluid oscillations of the star to the gravitational-wave oscillations of the space-time metric. Previously studied modes of relativistic stars have been close analogues of modes of Newtonian stars, where the coupling to gravitational waves mainly generates a small imaginary part to the frequency. Such modes can be classified using the Newtonian classes: f-, p-, g- and r- modes. The present modes, by contrast, have no Newtonian analogue. They are primarily oscillations of the space-time metric in which the fluid hardly vibrates at all. We christen them w-modes (gravitational-wave modes). These modes are strongly damped, being characterized by complex frequencies with unusually large imaginary parts, comparable to their real parts. We calculate a sequence of l = 2 modes for a number of spherical polytropic stellar models. An interesting feature of w-modes is that the lowest order mode of each sequence has a frequency similar to that of the lowest order mode of a spherical black hole. For higher modes, the spectrum diverges from the black-hole spectrum, but shows remarkable similarity to that of the strongly damped modes of the toy problem. As carriers of gravitational-wave information, w-modes may be important and observable in the burst of gravitational radiation that follows the formation of a neutron star. They should also be essential in solving the problem of the completeness of the outgoing-wave normal modes of radiating systems.