Ergoregion instability in rotating gravastars

The ergoregion instability is known to affect very compact objects that rotate very rapidly and do not possess a horizon. We present here a detailed analysis on the relevance of the ergoregion instability for the viability of gravastars. Expanding on some recent results, we show that not all rotating gravastars are unstable. Rather, stable models can be constructed also with $J/M^2\sim 1$, where $J$ and $M$ are the angular momentum and mass of the gravastar, respectively. The genesis of gravastars is still highly speculative and fundamentally unclear if not dubious. Yet, their existence cannot be ruled out by invoking the ergoregion instability. For the same reason, not all ultracompact astrophysical objects rotating with $J/M^2\lesssim 1$ are to be considered necessarily black holes.

Figure 1

Typical example for the dragging of inertial frames $\omega (r)$, for a gravastar with $\mu =0.45$, $\delta /M=0.4$, and $J/M^2=1$ ($\Omega /{\Omega }_K=0.82$). We can see that $\omega =\mathrm{const}$ in the interior ($r<r_1$) and $\omega \to 0$ in the exterior region ($r>r_2$).

Figure 2

Typical example for the potentials $V_{\pm }$, for a gravastar with $\mu =0.45$, $\delta /M=0.4$, and $J/M^2=1$ ($\Omega /{\Omega }_K=0.82$). The first three unstable modes with negative energy “trapped” in the potential well are depicted, as well as the points $r_a$, $r_b$, and $r_c$ for the $\ell =m=1$ mode.

Figure 3

Left panel: Change in the size of the ergoregion for fixed compactness $\mu =0.45$, several different values for $\delta /M$ (from top to bottom $\delta /M=0.1$, 0.2, 0.3, and 0.4) and increasing angular velocity $\Omega$, until the Keplerian limit ${\Omega }_K$. A given value of $\Omega$ on the vertical axis determines the inner and outer radii of the ergoregion, while the vertical line shows the location of the radius of the gravastar for all the models: the ergoregion generally starts in the interior of the gravastar and goes up to a radius exterior to the radius $r_2$ of the gravastar. Right panel: Same as the left panel, but for fixed thickness of the shell $\delta /M=0.4$ and different values for $\mu$ (from top to bottom $\mu =0.42$, 0.43, 0.45, and 0.47). In this case we do not show the surface radius, since each model has a different radius $r_2$.

Figure 4

Minimum angular velocity necessary for the existence of an ergoregion, as a function of both $\mu$ and $\delta$. If an ergoregion is present, the instability will set in for a high enough value of $m$. In the gray area at the bottom, an angular velocity $\Omega$ larger than the mass-shedding limit ${\Omega }_K$ would be required for an ergoregion to develop. The gray area at the top shows a constraint on the possible (nonrotating) gravastar solutions found in Ref. 11.

Figure 5

Minimum angular momentum necessary for the existence of an ergoregion, as a function of both $\mu$ and $\delta$. Gravastars with parameters $(\mu ,\delta )$ given below a curve labeled with some value of $J_{\min }/M^2$ will be stable if rotating with angular momentum smaller or equal to $J_{\min }$. Note that the $\Omega$ is not constant along the lines and increases in the direction of the region labeled ${\Omega }_{\min }(J_{\min })>{\Omega }_K$.

Figure 6

Maximum compactness ${\mu }_{\max }$ that a stable gravastar can have in terms of the angular momentum $J/M^2$. For a given value of $J$, gravastars with $\mu >{\mu }_{\max }(J)$ will be unstable. The maximum value for the compactness is obtained for $\delta \to 0$. This figure should be compared with the corresponding Fig. 1 of Ref. 11.

Figure 7

Top row, left panel: Time scale $\tau$ of the instability (with $\ell =m=1$) as a function of the angular momentum $J$ of the gravastar, for fixed thickness of the shell $\delta /M=0.4$ and different compactnesses $\mu$ (from left to right $\mu =0.47$, 0.45, 0.44, and 0.43). The horizontal line shows a time scale of 1 s for gravastars with $M=10M_{\odot }$. Top row, right panel: Frequency $\Sigma$ of the instability as a function of $J$ for the same cases in the left panel. Bottom row: The same as in the top row, but for gravastars with fixed compactness $\mu =0.45$ and varying thickness of the shell $\delta$ (from left to right $\delta /M=0.6$, 0.5, 0.4, 0.3, and 0.2).