Instability of non-supersymmetric smooth geometries

TL;DR

The paper tackles the stability of non-supersymmetric, horizonless JMaRT solitons by analyzing a massless scalar field in their backgrounds. Using WKB, matched asymptotic expansions, and numerical methods, it identifies growing modes associated with negative pattern speeds $\Sigma_{\psi}$, derives their spectra and growth timescales, and demonstrates stability in the supersymmetric limit $m=n+1$. The results indicate a generic ergoregion instability for non-BPS microstate geometries and argue that the instability drives evolution toward supersymmetric endpoints, with important implications for the fuzzball proposal. Overall, the work shows that smooth horizon-free non-BPS geometries with ergoregions are classically unstable, shaping how microstate geometries may describe non-extremal black holes.

Abstract

Recently certain non-supersymmetric solutions of type IIb supergravity were constructed [hep-th/0504181], which are everywhere smooth, have no horizons and are thought to describe certain non-BPS microstates of the D1-D5 system. We demonstrate that these solutions are all classically unstable. The instability is a generic feature of horizonless geometries with an ergoregion. We consider the endpoint of this instability and argue that the solutions decay to supersymmetric configurations. We also comment on the implications of the ergoregion instability for Mathur's `fuzzball' proposal.

Figures (8)

• Figure 1: Qualitative shape of the potentials $V_+$ and $V_-$ for the case in which an instability is present. An example of data that yields this kind of potentials is $(m=14\,,n=10\,,a_1=32\,,c_1=5\,,c_p=5)$. The unstable modes are those whose pattern speed $\Sigma_{\psi}$ is negative and approach the minimum of $V_+$ from above. Thus, they are nearly bound states of the potential well in $V_+$ that can however tunnel out to infinity through $V_-$. Choosing $\lambda=0$, the potentials $V_+$ and $V_-$ approach zero as $x \rightarrow \infty$, which makes a tunnelling through $V_-$ easier.
• Figure 2: An example solution showing vanishing as both $x\rightarrow 0$ and $x \rightarrow \infty$.
• Figure 3: On the left we choose the lowest harmonic and vary $l=m_{\psi}$ from 2 to 13 from upper right to lower left. The solid circles represent the numeric solutions, while the triangles are the results of the WKB analysis and the unfilled circles correspond to the matched expansion. On the right we fix $l=m_{\psi}=4$ and vary the harmonic from 0 to 4 from upper left to lower right.
• Figure 4: Damped modes are those that have positive $\Sigma_{\psi}$.
• Figure 5: Qualitative shape of the potentials $V_+$ and $V_-$ when $\omega^2-\lambda^2<0$. These are the purely bound states that are discussed in Appendix \ref{['sec:A2']}.