Black hole solution in third order Lovelock gravity has no Gauss-Bonnet limit

Abstract

We revisit the spherically symmetric third order Lovelock black hole solution in seven dimensions. We show that the general solution for the metric function does not admit the Gauss-Bonnet (GB) limit. This is not expected due to the linear superposition of the second (GB) and third order Lovelock Lagrangians in the general action. It is found that the two branches of the GB solutions are indeed the limit of the other two complex solutions of the field equations in the third order Lovelock gravity. These two complex solutions could not be accepted as the solutions of Einstein's field equations which are supposed to be real value functions on the entire real $r$ axis. A new solution, which is valid only if the third order Lovelock parameter is small, is introduced which can be considered as the natural extension of general relativity to the third order Lovelock modified theory of gravity. We also generalize the discussion to the higher dimensional third order Lovelock gravity coupled to matter sources with a cosmological constant.