Topology of Hořava–Lifshitz black holes in different ensembles

Abstract

In this paper, we study topological numbers for uncharged and charged static black holes obtained in z=3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$z=3$$\\end{document} Hořava–Lifshitz gravity theory in different ensembles, where z measures the degree of anisotropy between space and time. We first calculate the topological numbers for the uncharged black holes by changing the value of the dynamic coupling constant, and find that the black holes with spherical and flat horizons have the same topological number. When the black hole’s horizon is hyperbolic, different values of the coupling constant generate different topological numbers, which can be 1, 0 or -1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-1$$\\end{document}. This shows that the coupling constant plays an important role in the topological classification. Then we study the topological numbers for the charged black holes in different ensembles. The black hole with a spherical horizon has the same topological number in canonical and grand canonical ensembles. When the horizons are flat or hyperbolic, they have different topological numbers in canonical and grand canonical ensembles. Therefore, the topological numbers for the uncharged black holes are parameter dependent, and those for the charged black holes are ensemble dependent.