Surface terms as counterterms in the AdS-CFT correspondence

Abstract

We examine the recently proposed technique of adding boundary counterterms to the gravitational action for spacetimes which are locally asymptotic to anti-de Sitter spacetimes. In particular, we explicitly identify higher order counterterms, which allow us to consider spacetimes of dimensions dр7. As the counterterms eliminate the need of ''background subtraction'' in calculating the action, we apply this technique to study examples where the appropriate background was ambiguous or unknown: topological black holes, Taub-NUTAdS and Taub-Bolt-AdS. We also identify certain cases where the covariant counterterms fail to render the action finite, and we comment on the dual field theory interpretation of this result. In some examples, the case of a vanishing cosmological constant may be recovered in a limit, which allows us to check results and resolve ambiguities in certain asymptotically flat spacetime computations in the literature. ͓S0556-2821͑99͒07318-X͔

Black hole mass and Hamilton-Jacobi counterterms

“…This can be compared with similar expressions for pure gravitational backgrounds, as found in [6,7]. Note that g is also the inverse of the AdS length scale, −1 , which is given in terms of the constant term V 0 in the scalar potential V (φ) by:…”

“…These problems can be avoided by using the boundary counterterm approach for removing divergences from the action [6,7]. There are two common prescriptions for calculating the boundary counterterms.…”

Energy and the AdS/CFT correspondence

“…Note that, in general, the Noether current will be different from the usual choice of Eq. (25). It will give rise to a 'conserved' charge which is the canonical energy that we have discussed before.…”

“…(4) and the metrical energy which is constructed out of the conserved current Eq. (25), for the canonical energies the regular modes propagate for any real ν ≥ 0, whereas the irregular ones propagate when the constraint Eq. (20) In this case we find the constraint…”

Comments on Jacobson’s “entanglement equilibrium and the Einstein equation”

“…First, we must evaluate the on-shell bulk action but to regulate the result, we introduce a cut-off surface at z = z ǫ near the boundary. Then, we add a counterterm action that cancels the divergences [30]. Hence the total gravitational action becomes 12) where I bulk is just the action in eq.…”