Topological classes of higher-dimensional black holes in massive gravity

Abstract

Abstract In this paper, we study topological numbers for five-, six- and seven-dimensional anti-de Sitter black holes in the ghost-free massive gravity. We find that when the black holes are charged, they have the same topological number. The topological numbers for the uncharged black holes are 0 or 1, and the specific values are determined by the values of the black holes’ parameters. Since k and $$c_ 0^2c_2 m^2 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>c</mml:mi> <mml:mn>0</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> <mml:msub> <mml:mi>c</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:msup> <mml:mi>m</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> appear together in the generalized free energy in the form of $$k +c_ 0^2c_2 m^2 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:msubsup> <mml:mi>c</mml:mi> <mml:mn>0</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> <mml:msub> <mml:mi>c</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:msup> <mml:mi>m</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> , where k characterizes the horizon curvature and $$c_2 m^2 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>c</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:msup> <mml:mi>m</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> is the coefficient of the second term of massive potential associated with the graviton mass, this result is applicable to the black holes with the spherical, Ricci flat or hyperbolic horizons. This work shows that the parameters of the ghost-free massive gravity play an important role in topological classes of black holes.