Superradiant instabilities of rotating black branes and strings

TL;DR

The work identifies a universal, rotation-triggered instability for rotating black branes caused by superradiant amplification of massless-field perturbations that are trapped by a potential well. By mapping the problem to a massive-field equation in Kerr spacetime with an effective mass mu from extra dimensions, the authors analyze both four-dimensional and higher-dimensional cases using Leaver's method to compute quasinormal modes. They find instability for Kerr_4 x R^p branes (d=4) with calculable growth rates, while Kerr_d x R^p branes with d>4 are stable due to the absence of bound states, a result tied to the lack of stable circular orbits in higher dimensions. They also demonstrate an acoustic analogue and discuss the endpoint of the instability, which presumably spins the brane down and radiates away the excess angular momentum. These findings highlight a new mechanism affecting extended rotating objects and have potential implications for higher-dimensional gravity models and laboratory analogues.

Abstract

Black branes and strings are generally unstable against a certain sector of gravitational perturbations. This is known as the Gregory-Laflamme instability. It has been recently argued that there exists another general instability affecting many rotating extended black objects. This instability is in a sense universal, in that it is triggered by any massless field, and not just gravitational perturbations. Here we investigate this novel mechanism in detail. For this instability to work, two ingredients are necessary: (i) an ergo-region, which gives rise to superradiant amplification of waves, and (ii) ``bound'' states in the effective potential governing the evolution of the particular mode under study. We show that the black brane Kerr_4 x R^p is unstable against this mechanism, and we present numerical results for instability timescales for this case. On the other hand, and quite surprisingly, black branes of the form Kerr_d x R^p are all stable against this mechanism for d>4. This is quite an unexpected result, and it stems from the fact that there are no stable circular orbits in higher dimensional black hole spacetimes, or in a wave picture, that there are no bound states in the effective potential. We also show that it is quite easy to simulate this instability in the laboratory with acoustic black branes.

Figures (7)

• Figure 1: A typical form for the effective potential, here shown for $l=m=1$ modes. We have set $M=1$, so the rotation parameter $a$ varies between $0$ (Schwarzschild limit) and $1/2$ (extremal limit). Here we plot the effective potential for the near extreme situation, $a \sim 0.5$ and for $\mu=0.7$ and $\omega=0.6878$.
• Figure 2: Details for the instability in the four dimensional case. We have set $M=1$, so the rotation parameter $a$ varies between $0$ (Schwarzschild limit) and $1/2$ (extremal limit). In the left panel we plot the imaginary part of the unstable modes as a function of $a$ for nine values of $\mu$, and for $l=m=1$. The same quantities are plotted in the right panel, but now we focus on the maximum instability region, for values of $\mu$ near unity. The instability is stronger for larger values of $a$ and for $\mu \sim 1$. In fact, our results suggest that the imaginary part of the unstable modes attains its maximum value, $10^{-6}\times{\rm Re}[\omega]$ for $\mu=0.9$ and $a=0.497$. Notice finally that as one decreases $a$ (going right on the $x$ axis in the left panel), there is a critical $a$ below which there is no instability. Although there are an infinity of unstable modes, we only show here the most unstable one.
• Figure 3: More details of the instability in the four dimensional case. Here we plot the superradiant factor ${\rm Re}[\omega]-\frac{ma}{a^2+r_+^2}$ as a function of $a$ for $l=m=1$ and for several values of $\mu$. Again, we have set $M=1$ and so the superradiant factor is, in four dimensions, ${\rm Re}[\omega]-\frac{ma}{r_+}$. the instability disappears for $a$ below a certain critical point, which is given by setting the superradiant factor to zero. On the right panel we make the same plot but on a logarithmic scale, for better visualization. Notice that the superradiant factor is negative, as it should for the instability to be triggered.
• Figure 4: The factorized potentials $V_\pm$ as a function of $r^{-1}$. The potentials are characterized by $n=0$, $l=m=1$, $j=0$, $\mu=0.7$, and $a=0.4999$.
• Figure 5: The curve of $dV_+/dr(r^{-1},\mu)=0$ for the parameter $n=0$, $l=m=1$, $j=0$, and $a=0.4999$.