Renormalization of the Einstein-Hilbert action

Abstract

We examine how the Einstein-Hilbert action is renormalized by adding the usual counterterms and additional corner counterterms when the boundary surface has corners. A bulk geometry asymptotic to H d+1 can have boundaries S k × H d−k and corners for 0 ≤ k < d. We show that the conformal anomaly when d is even is independent of k. When d is odd the renormalized action is a finite term that we show is independent of k when k is also odd. When k is even we were unable to extract the finite term using the counterterm method and we address this problem using instead the Kounterterm method. We also compute the mass of a two-charged black hole in AdS 7 and show that background subtraction agrees with counterterm renormalization only if we use the infinite series expansion for the counterterm.