Temperature of McVittie Black Holes in Cavities

Abstract

In this paper, we obtain the temperature of the McVittie black hole and the charged McVittie black hole in a cavity with radius $r_c$. The cavity temperature is derived from the black hole Hawking temperature and Kodama vector by using the Tolman relation. We also obtain the cavity temperature of the de Sitter black hole and the charged de Sitter black hole respectively by setting $H^2=Λ/3$ from the above results, which are the same as those from the action method. By the way, we also find that the Kodama vector of the McVittie and charged McVittie black hole is just $\partial/\partial t$, which is very remarkable and interesting. It shows that this method to obtain the cavity temperature is consistent and applicable. It is obvious that this method is much easier than the action method and can be used to similar investigations. It also shows that Kodama vector is very important and useful for dynamical spacetimes.