Gravitational wave signatures of the absence of an event horizon: Nonradial oscillations of a thin-shell gravastar

Gravitational waves from compact objects provide information about their structure, probing deep into strong-gravity regions. Here we illustrate how the presence or absence of an event horizon can produce qualitative differences in the gravitational waves emitted by ultracompact objects. In order to set up a straw-man ultracompact object with no event horizon, but which is otherwise almost identical to a black hole, we consider a nonrotating thin-shell model inspired by Mazur and Mottola’s gravastar, which has a Schwarzschild exterior, a de Sitter interior and an infinitely thin shell with finite tension separating the two regions. As viewed from the external space-time, the shell can be located arbitrarily close to the Schwarzschild radius, so a gravastar might seem indistinguishable from a black hole when tests are only performed on its external metric. We study the linearized dynamics of the system, and, in particular, the junction conditions connecting internal and external gravitational perturbations. As a first application of the formalism we compute polar and axial oscillation modes of a thin-shell gravastar. We show that the quasinormal mode spectrum is completely different from that of a black hole, even in the limit when the surface redshift becomes infinite. Polar quasinormal modes depend on the equation of state of matter on the shell and can be used to distinguish between different gravastar models. Our calculations suggest that low-compactness gravastars could be unstable when the sound speed on the shell $v_s/c\gtrsim 0.92$.

Figure 1

First few axial (continuous lines) and polar (dashed lines) QNMs of a thin-shell gravastar with $v_s^2=0.1$. In the left panel we follow modes with $l=2$ as the compactness $\mu$ varies. In the right panel we do the same for modes with $l=3$. Along each track we mark by different symbols (as indicated in the legend) the points where $\mu =0.1$, 0.2, 0.3, 0.4, and 0.49. Our numerical method becomes less reliable when $2M{\omega }_I$ is large and when the modes approach the pure-imaginary axis. Numbers next to the polar and axial modes refer to the overtone index $N$ ($N=1$ being the fundamental mode).

Figure 2

Real (left) and imaginary (right) part of polar and axial QNMs with $l=2$ as functions of $\mu$ for $v_s^2=0.1$. Line styles are the same as in Fig. 1. Numbers refer to the overtone index.

Figure 3

Top row: real and imaginary part of the wave function in the interior for the first four $w$-modes. Bottom row: real and imaginary part of the wave function in the interior for the first four $w_{\mathrm{II}}$-modes. In both cases we consider polar QNMs with $l=2$ and $v_s^2=0.1$.

Figure 4

Tracks of the fundamental polar and axial $w$-modes for different values of the “sound speed” parameter $v_s$ when $v_s^2<0$ (left) and when $v_s^2>0$ (right). Different line styles correspond to different values of $v_s^2$, as indicated in the legend.

Figure 5

Left: spectrum of the weakly damped family of QNMs. The vertical line corresponds to twice the orbital frequency of a particle in circular orbit at the innermost stable circular orbit (ISCO), as we will discuss in a follow-up paper, only QNMs to the left of the line can be excited by a compact object inspiraling into the gravastar along quasicircular orbits. In the case $v_s=0.8$ the mode “turns around” describing a loop in the complex plane. For $v_s\lesssim 0.8$ the modes move clockwise in the complex plane as $\mu$ increases. For $v_s\gtrsim 0.8$ they move counterclockwise and they cross the real axis at finite compactness. To facilitate comparison, in the right panel we show again the right panel of Fig. 4 using a logarithmic scale for the imaginary part.

Figure 6

Real (left) and imaginary parts (right) of the fundamental polar $w$-mode for different values of the equation of state parameter $v_s$. Different line styles correspond to different values of $v_s^2$, as indicated in the legend. The horizontal line in the left panel corresponds to twice the orbital frequency of a particle in circular orbit at the ISCO: only modes below the line can be excited during a quasicircular inspiral.

Figure 7

Significant thresholds in the $(\mu ,v_s^2)$ plane. Thin-shell gravastars should be unstable with respect to nonradial perturbations below the solid line, corresponding to ordinary QNM frequencies whose imaginary part crosses the zero axis while their real part stays finite. The dashed line corresponds to the “vanishing point” of weakly damped QNM frequencies, i.e. to the point where both their real and imaginary parts have a zero crossing. We could not find weakly damped modes in the region above the dashed line.

Figure 8

The potential governing axial and polar perturbations for different values of the gravastar compactness $\mu \equiv M/a$, where $a$ is the location of the shell (see 57, showing a similar plot for axial perturbations of constant-density stars). The potential develops a minimum when $\mu \gtrsim 0.43$. Note that the polar and axial perturbations in the interior are both governed by the same potential, given in Eq. (3.5) below.