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Gregory-Laflamme instability of black hole in Einstein-scalar-Gauss-Bonnet theories

Yun Soo Myung and De-Cheng Zou
Phys. Rev. D 98, 024030 – Published 16 July 2018
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  • INTRODUCTION
  • ESGB THEORY
  • INSTABILITY OF BLACK HOLE WITHOUT SCALAR…
  • STATIC SCALAR PERTURBATION
  • DISCUSSIONS
  • ACKNOWLEDGMENTS
    • References

    Abstract

    We investigate the stability analysis of a Schwarzschild black hole in Einstein-scalar-Gauss-Bonnet theory because the instability of the Schwarzschild black hole without scalar hair implies the Gauss-Bonnet black hole with scalar hair. The linearized scalar equation is compared to the Lichnerowicz-Ricci tensor equation in the Einstein-Weyl gravity. It turns out that the instability of the Schwarzschild black hole in Einstein-scalar-Gauss-Bonnet theory is interpreted as not the tachyonic instability but the Gregory-Laflamme instability of a black string. In the small mass regime of 1/λ<1.174/r+, the Schwarzschild solution becomes unstable, and a new branch of the solution with scalar hair bifurcates from the Schwarzschild one. This is very similar to finding a newly non-Schwarzschild black hole in Einstein-Weyl gravity.

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    • Received 15 May 2018

    DOI:https://doi.org/10.1103/PhysRevD.98.024030

    © 2018 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Alternative gravity theories
    Gravitation, Cosmology & Astrophysics

    Authors & Affiliations

    Yun Soo Myung1,* and De-Cheng Zou1,2,†

    • 1Institute of Basic Sciences and Department of Computer Simulation, Inje University Gimhae 50834, Korea
    • 2Center for Gravitation and Cosmology and College of Physical Science and Technology, Yangzhou University, Yangzhou 225009, China

    • *ysmyung@inje.ac.kr
    • †dczou@yzu.edu.cn

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    Issue

    Vol. 98, Iss. 2 — 15 July 2018

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    Images

    • Figure 1
      Figure 1

      The potentials as a function of r∈[1,∞) for horizon radius r+=1 and l=0. The blue (bottom), red (middle), and green (top) curves represent the potentials V(r) of the scalar for mass parameter 1/λ=1.095 (sufficient condition for instability), 1.174 (threshold of instability), and 1.2 (stable case), respectively. These all have negative regions near the horizon. However, the tachyonic potential (cyan curve) Vt(r) develops a positive region near horizon while it approaches −0.04 as r→∞ for mT=0.2.

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    • Figure 2
      Figure 2

      Ω graphs as a function of mass parameter 1/λ for small black holes of r+=1, 2, 3. Here, one reads off the thresholds of instability (1/λ)th from the points at which curves of Ω intersect the positive 1λ axis: (1/λ)th≈1.174, 0.587, 0.294. The instability range decreases as the horizon radius increases.

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    • Figure 3
      Figure 3

      Plots of unstable modes filled circle on three curves with the horizon radii r+=1, 2, 4. The y(x) axis denotes Ω in eΩt (mass m of the massive spin-2 mode). The smallest curve represents r+=4, the medium denotes r+=2, and the largest one shows r+=1. Here, the thresholds of instability are located at mth≈0.876, 0438, 0.219, which means that the instability region is smaller and smaller as the horizon radius is larger and larger.

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    • Figure 4
      Figure 4

      Graphs for scalar perturbation δϕ as a function of r/r+ for r+/λ≈1.174, 0.453, 0.280, which corresponds to the number of nodes n=0, 1, 2 in scalar profiles.

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