Superradiant instability of large radius doubly spinning black rings

TL;DR

This paper shows that 5D large-radius doubly spinning black rings are subject to a robust superradiant bound-state instability, driven by KK momentum along the ring which creates a confining potential for bound states that can be amplified in the ergoregion. By mapping the large-radius limit to a boosted Kerr black string and employing matched asymptotic expansions, the authors derive the frequency spectrum and instability timescales, and identify the endpoint as a dynamical spin-down toward reduced rotation in the S^2 direction. They argue that finite-radius rings inherit a KK-quantization-induced bound, yielding an upper bound on the S^2 rotation that can be estimated using the Hovdebo–Myers black ring model, and show that higher-dimensional rings in D>5 are stable against this mechanism due to the absence of bound states. The results imply a dynamical constraint on the rotation of doubly spinning black rings and provide insight into the stability landscape of higher-dimensional black objects.

Abstract

We point out that 5D large radius doubly spinning black rings with rotation along S^1 and S^2 are afflicted by a robust instability. It is triggered by superradiant bound state modes. The Kaluza-Klein momentum of the mode along the ring is responsible for the bound state. This kind of instability in black strings and branes was first suggested by Marolf and Palmer and studied in detail by Cardoso, Lemos and Yoshida. We find the frequency spectrum and timescale of this instability in the black ring background, and show that it is active for large radius rings with large rotation along S^2. We identify the endpoint of the instability and argue that it provides a dynamical mechanism that introduces an upper bound in the rotation of the black ring. To estimate the upper bound, we use the recent black ring model of Hovdebo and Myers, with a minor extension to accommodate an extra small angular momentum. This dynamical bound can be smaller than the Kerr-like bound imposed by regularity at the horizon. Recently, the existence of higher dimensional black rings is being conjectured. They will be stable against this mechanism.

Figures (3)

• Figure 1: In the large radius limit, the doubly spinning black ring yields a boosted Kerr black string.
• Figure 2: a)Qualitative shape of the potentials $V_+$ and $V_-$ for a boosted Kerr black string in which an instability is present. An example of data that yields this kind of potentials is $(2M=1\,,a=0.499999\,,\tanh\sigma=1/\sqrt{2}\,,\kappa=1.85\,,l=m=1)$. Potentially unstable modes are those whose frequency satisfies $a_{\rm m}<\omega< \kappa$. Thus, they are bound states of the potential well in $V_+$ that satisfy the superradiant condition $\omega< \omega_{\rm sup}$. b) Qualitative shape of the potentials $V_+$ and $V_-$ for a case in which the Kerr black string is stable. An example of data that yields this kind of potentials is $(2M=1\,,a=0.4\,,\tanh\sigma=1/\sqrt{2}\,,\kappa=1.85\,,l=m=1)$. The bound state modes with $a_{\rm m}<\omega< \kappa$ are stable because they do not satisfy the superradiant condition $\omega< \omega_{\rm sup}$, where $\omega_{\rm sup}$ is defined in (\ref{['wsup']}).
• Figure 3: a) General qualitative shape of the potentials $V_+$ and $V_-$ for a boosted Myers-Perry black string. In this case the geometry is always stable, but superradiant scattering is possible. An example of data that yields this kind of potentials is $(2M=1\,,a=0.9\,, \tanh\sigma=1/\sqrt{2}\,,\kappa=1.5\,,l=m=1,\, j=0, \, n=1)$. There are no bound state modes since $V_+$ approaches $\kappa$ at infinity from above. This absence of bound states for $n\geq 1$, as oppose to the $n=0$ case, annihilates the possibility of existing an instability for $n\geq 1$. Since $\omega< \omega_{\rm sup}$, superradiant scattering is possible. b) Qualitative shape of the potentials $V_+$ and $V_-$ for a Myers-Perry black string. In this case the geometry is stable, and moreover superradiant scattering is not possible. An example of data that yields this kind of potentials is $(2M=1\,,a=0.5\,, \tanh\sigma=1/\sqrt{2}\,,\kappa=2\,,l=m=1,\, j=0, \, n=1)$. There are no bound state modes and superradiant scattering is not possible (since $\omega> \omega_{\rm sup}$).