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Stable Black Strings in Anti-de Sitter Space

Takayuki Hirayama, Gungwon Kang

TL;DR

This work analyzes the classical stability of five-dimensional $AdS$/$dS$ black strings formed by foliating $AdS_4$/$dS_4$-Schwarzschild black holes. Using naive entropy arguments and a full linearized perturbation analysis in a warped Randall-Sundrum setting, it finds that $AdS$ black strings are generically unstable but become stable when the four-dimensional horizon radius satisfies $r_+ > l_4$ (specifically around $r_+ \simeq 0.20\,l_4$), due to an effectively confining KK spectrum with $m_{ ext{min}}=4/l_4$. The $dS$ and flat brane cases remain unstable, with the results broadly consistent with the Gubser-Mitra conjecture despite deviations arising from broken translational symmetry. The paper also discusses end-states and fragmentation patterns, noting a distinctive “box-like” KK potential in the AdS case that concentrates perturbations toward the conformal boundary, potentially yielding multi-black-hole configurations along the string.

Abstract

In the five-dimensional Einstein gravity with negative cosmological constant in the presence/absence of a {\it non-fine-tuned} 3-brane, we have investigated the classical stability of black string solutions which are foliations of four-dimensional $AdS/dS$-Schwarzschild black holes. Such black strings are generically unstable as in the well-known Gregory-Laflamme instability. For $AdS$ black strings, however, it turns out that they become stable if the longitudinal size of horizon is larger than the order of the $AdS_4$ radius. Even in the case of unstable black strings, the $AdS$ black strings have a very different feature of string fragmentations from that in the flat brane world. Some implications of our results on the Gubser-Mitra conjecture are also discussed.

Stable Black Strings in Anti-de Sitter Space

TL;DR

This work analyzes the classical stability of five-dimensional / black strings formed by foliating /-Schwarzschild black holes. Using naive entropy arguments and a full linearized perturbation analysis in a warped Randall-Sundrum setting, it finds that black strings are generically unstable but become stable when the four-dimensional horizon radius satisfies (specifically around ), due to an effectively confining KK spectrum with . The and flat brane cases remain unstable, with the results broadly consistent with the Gubser-Mitra conjecture despite deviations arising from broken translational symmetry. The paper also discusses end-states and fragmentation patterns, noting a distinctive “box-like” KK potential in the AdS case that concentrates perturbations toward the conformal boundary, potentially yielding multi-black-hole configurations along the string.

Abstract

In the five-dimensional Einstein gravity with negative cosmological constant in the presence/absence of a {\it non-fine-tuned} 3-brane, we have investigated the classical stability of black string solutions which are foliations of four-dimensional -Schwarzschild black holes. Such black strings are generically unstable as in the well-known Gregory-Laflamme instability. For black strings, however, it turns out that they become stable if the longitudinal size of horizon is larger than the order of the radius. Even in the case of unstable black strings, the black strings have a very different feature of string fragmentations from that in the flat brane world. Some implications of our results on the Gubser-Mitra conjecture are also discussed.

Paper Structure

This paper contains 4 sections, 20 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Black string solutions and naive stability arguments
  3. Linearized perturbation analysis
  4. Conclusion

Figures (3)

  • Figure 1: The left figure is for the $AdS$ case with $r_0=1, 2$ and $4$. The right figure is for the $dS$ case with $r_0=1, 2, 3.5$, and $3.8$. The Nariai solution corresponds to $r_0 \simeq 3.85$. The fixed $AdS$ and $dS$ radius is $l_4=10$. The straight vertical lines denote the lowest KK masses, $0.4$ for $AdS$ and $0.15$ for $dS$.
  • Figure 2: The threshold masses $m_*$ for varying $\Lambda_4$ with given $r_0=2$. The left vertical dotted line denotes the Nariai limit and the right one the critical $\Lambda_4$ predicted by the GM conjecture.
  • Figure 3: The left figure: threshold masses for varying $r_0$ with given $l_4=10$ in the $AdS$ case. The vertical dotted line denotes the critical $r_0 \simeq 7.7$ predicted from the Gubser-Mitra conjecture. The numerical data stops at $r_0 \simeq 7.0$. The right figure: same diagram for the $dS$ case. The vertical dotted line denotes the Nariai limit, $r_0 \simeq 3.85$. Note that $m_*(r_0\simeq 3.85)=0.29 > m_{\rm min.} \simeq 0.15$.