Curvature Effect on the Speed of Sound

Abstract

The speed of sound refers to the rate at which information travels from one point to another. It is a positive quantity and bounded by causality. It is defined as the rate of change of pressure with respect to the system's density. In this article, we derive a covariant equation for the sound wave and demonstrate how the wave equation is modified in the general relativistic formalism. One can then define an effective speed of sound by attenuating the usual definition of sound speed with the gravitational metric potential. The general relativistic curvature effect is observed to reduce the speed of sound when computed inside a neutron star. This effectively makes the star relatively softer (according to the equation of state). The change in the effective sound speed can be easily visualised if one redefines the non-radial modes in terms of it. The modes do not change, but the space-time curvature reduces the amplitude of the oscillation modes. The formalism is suited for studying astrophysical compact objects.

Figures (5)

• Figure 1: The left panel illustrates the speed of sound square ($c_s^2$)regions spanned by the monotonic (green) and non-monotonic (purple) classes of EOSs. The solid lines represent the specific EOSs, which are used to plot metric potential variation with respect to radial distance in the next figure. The maximum central densities achieved by the maximum mass stars of these sets are also marked in orange. The right panel of the plot shows the regions spanned by the monotonic (green) and non-monotonic (purple) EOS classes.
• Figure 2: Variation of the metric potentials as functions of the radial distance (r) from the centre of a neutron star, corresponding to the solid-line EoS in Fig. \ref{['EOS']}. The solid, dotted, and dashed curves represent the metric potentials indicated in the legend. The green and purple curves correspond to the monotonic and non-monotonic EoSs, respectively. All profiles are for 2 $M_\odot$ stars with central energy densities of 980 $MeV/fm^3$ (monotonic) and 820 $MeV/fm^3$ (non-monotonic). The vertical dashed lines denote the radius of the monotonic star ($R_m$) and the non-monotonic star ($R_{nm}$).
• Figure 3: Comparison of $c_s^2$ and $(c_s^2)_{eff}$ for the monotonic set of EOS. The comparison for the ensemble of 1.4 $M_{\odot}$ stars is shown in the left panel, while for the 2.0 $M_{\odot}$ stars is shown in the right panel.
• Figure 4: Same comparison of $c_s^2$ and $(c_s^2)_{eff}$ for the non-monotonic set of EOS.
• Figure 5: Left: Effect of the effective sound speed $(c_s)_{eff}$ on the pressure perturbation ($\delta p$) inside the star. The strength of the perturbation is found to be significantly greater than that obtained when considering only the standard sound speed $c_s$. Right: The next three higher modes of the perturbation are shown. The $k$ values denote the number of nodes within the star. It is also observed that higher modes produce larger perturbations at the centre of the star, indicating the greater energy associated with these modes.