Astroseismology of neutron stars from gravitational waves in the limit of perfect measurement

The oscillation spectrum of a perturbed neutron star is intimately related to the physical properties of the star, such as the equation of state. Observing pulsating neutron stars therefore allows one to place constraints on these physical properties. However, it is not obvious exactly how much can be learnt from such measurements. If we observe for long enough, and precisely enough, is it possible to learn everything about the star? A classical result in the theory of spectral geometry states that one cannot uniquely hear the shape of a drum'. More formally, it is known that an eigenfrequency spectrum may not uniquely correspond to a particular geometry; some drums’ may be indistinguishable from a normal-mode perspective. In contrast, we show that the drum result does not extend to perturbations of simple neutron stars within general relativity – in the case of axial (toroidal) perturbations of static, perfect fluid stars, a quasi-normal mode spectrum uniquely corresponds to a stellar profile. We show in this paper that it is not possible for two neutron stars, with distinct fluid profiles, to oscillate in an identical manner. This result has the information-theoretic consequence that gravitational waves completely encode the properties of any given oscillating star: unique identifications are possible in the limit of perfect measurement.