Two instabilities of Schwarzschild-AdS black holes in Einstein–Weyl-scalar theory

Abstract

Stability of Schwarzschild-AdS (SAdS) black hole is investigated in Einstein–Weyl-scalar (EWS) theory with a negative cosmological constant. Here, we introduce a quadratic scalar coupling to the Weyl term, instead of the Gauss–Bonnet term. The linearized EWS theory admits the Lichnerowicz equation for Einstein tensor as well as scalar equation. The linearized Einstein-tensor carries with a regular mass term (M2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {M}}}^2$$\\end{document}), whereas the linearized scalar has a tachyonic mass term (-3r02/m2r6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-3r_0^2/m^2r^6$$\\end{document}). Two instabilities of SAdS black hole in EWS theory are found as Gregory–Laflamme and tachyonic instabilities. It shows that the correlated stability conjecture holds for small SAdS black holes obtained from EWS theory by establishing a close relation between Gregory–Laflamme and thermodynamic instabilities. On the other hand, tachyonic instability of SAdS black hole can be used for making five branches of scalarized black holes when considering proper thermodynamic quantities of EWS theory (M2>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {M}}}^2>0$$\\end{document}).