Gravitational instabilities of superspinars

Superspinars are ultracompact objects whose mass $M$ and angular momentum $J$ violate the Kerr bound ($cJ/G{M}^2>1$). Recent studies analyzed the observable consequences of gravitational lensing and accretion around superspinars in astrophysical scenarios. In this paper we investigate the dynamical stability of superspinars to gravitational perturbations, considering either purely reflecting or perfectly absorbing boundary conditions at the “surface” of the superspinar. We find that these objects are unstable independently of the boundary conditions, and that the instability is strongest for relatively small values of the spin. Also, we give a physical interpretation of the various instabilities that we find. Our results (together with the well-known fact that accretion tends to spin superspinars down) imply that superspinars are very unlikely astrophysical alternatives to black holes.

Figure 1

Top: Real (left) and imaginary part (right) of unstable gravitational modes of a superspinar as a function of the spin parameter, $a/M$, for $l=m=2$ and several fixed values of $r_0$. Bottom: Real (left) and imaginary part (right) of unstable gravitational modes of a superspinar as a function of the mirror location, $r_0/M$, for $l=m=2$ and different fixed values of the spin parameter. Large dots indicate purely imaginary modes.

Figure 2

Proper volume of the ergoregion as a function of the spin $a/M$. The volume increases monotonically when $a<M$, is infinite at $a=M$ and decreases monotonically when $a>M$. The proper volumes for $a\sim 2M$ and $a\sim 0.3M$ are roughly the same. In the inset we show the azimuthal section of the ergoregion for selected values of the spin. These spins are marked by filled circles and capital Latin letters in the main plot; their numerical value is indicated in parentheses in the figure. In the limit $a/M\to \infty$ the ergoregion becomes so oblate that its proper volume shrinks to zero.

Figure 3

Left: Imaginary part of unstable gravitational modes of a superspinar as a function of the mirror location, $r_0/M$, for $a=1.1M$, $l=2$ and $m=0$, 1, 2. Right: Imaginary part of unstable gravitational modes of a superspinar as a function of the mirror location, $r_0/M$, for $l=2$, $m=0$ and several values of the spin parameter, $a$.

Figure 4

Left: Purely imaginary unstable mode as a function of the mirror location, $r_0/M<0$, for $a=1.4M$, $s=l=2$ and $m=0$. In the limit $r_0\to -\infty$, $M{\omega }_I\sim 7.07$, which perfectly agrees with results in Ref. 11. Right: Purely imaginary unstable mode as a function of the spin, $a/M$, for $r_0=-3M$, $s=l=2$, $m=0$. Numerical results (black straight line) are consistent with the quadratic fit of Eq. (13) (red dashed line). The dot marks the case considered in the left panel.

Figure 5

Top: Real (left) and imaginary part (right) of unstable gravitational modes of a superspinar as a function of the spin parameter, $a/M$, for $l=m=2$ and several fixed values of the horizon location $r_0/M$. Bottom: Real (left) and imaginary part (right) of unstable gravitational modes of a superspinar as a function of the horizon location, for $l=m=2$ and fixed values of the spin parameter. Large dots indicate purely imaginary modes.

Figure 6

Imaginary part of unstable gravitational modes of a superspinar as the spin parameter, $a$, for $l=2$, $m=0$ and several values of the horizon location, $r_0$.

Figure 7

Critical radius $r_{\mathrm{crit}}/M$ as a function of $a/M$ for $s=l=2$ and $m=0$, 2. We impose both $\mathcal{R}=0$ and $\mathcal{R}=1$ at $r=r_0$. For $m=2$ an instability occurs when $r_0<r_{\mathrm{crit}}$. For $m=0$ an instability occurs in the region delimited by the curves on the left. When $l=m\gg 1$ an instability is expected to occur also when $a\gtrsim 2.2M$ in the $\mathcal{R}=1$ case, and when $a\gtrsim 1.75M$ in the $\mathcal{R}=0$ case (cf. Figure 8). Although not shown, the region where $r_0/M<0$ is expected to be unstable for any value of $a>M$.

Figure 8

Imaginary part of the fundamental unstable mode as a function of the spin $a/M$ for a superspinar with $r_0=0$. Upper curves refer to $\mathcal{R}=1$ with $l=m=2$, 3, 4, 5. Lower curves refer to $\mathcal{R}=0$ with $l=m=2$, 3, 4, 5. Dots indicate purely imaginary modes.

Figure 9

The sign of $\xi =L_z/E$ as a function of $a/M$ and the radius $r_c/M$ of null circular orbits. When $a>M$, $\xi \to \infty$ at $r_c=M$. Stable circular null orbits exist when $a>M$ for $r_c<r_S$, and have negative energy when $r_c<M$. Regions marked with NCOs are those where no circular orbits exist. Orbits of constant radius $r_1$ are stable polar null circular orbits.