Speed of Sound in a High-Pressure General-Relativistic Solid

Abstract

The characteristic equation for sound wave fronts in an elastic solid is derived in terms of the formalism recently developed by Carter and Quintana for the description of a perfectly elastic solid at high pressure in general relativity. A modified elasticity tensor (analogous to that used by Hadamard for the similar purposes in classical theory) is introduced in order to express the characteristic equation in a simple form. The equation is solved explicitly for the special case of a perfect (i.e., intrinsically isotropic) solid in (or sufficiently close to) its unsheared state. It is found that the squared sound speed is given by $\frac{d\overline{p}}{d\overline{\ensuremath{\rho}}}+\frac{4\ensuremath{\mu}}{3(\overline{\ensuremath{\rho}}+\overline{p})}$ for longitudinal waves and by $\frac{\ensuremath{\mu}}{(\overline{\ensuremath{\rho}}+\overline{p})}$ for transverse waves, where $\overline{\ensuremath{\rho}}$, $\overline{p}$, and $\ensuremath{\mu}$ are the density, pressure, and modulus of rigidity, respectively.