On the Geometric Properties of AdS Instantons.pdf

a r X i v : h e p - t h / 9 9 0 5 2 3 1 v 2 1 5 J u l 1 9 9 9 hep-th 9905231 On the Geometric Properties of AdS Instantons Ali Kaya1 Center for Theoretical Physics, Texas A M University, College Station, Texas 77843, USA. Abstract According to the positive energy conjecture of Horowitz and My- ers, there is a specific supergravity solution, AdS soliton, which has minimum energy among all asymptotically locally AdS solutions with the same boundary conditions. Related to the issue of semiclassical stability of AdS soliton in the context of pure gravity with a negative cosmological constant, physical boundary conditions are determined for an instanton solution which would be responsible for vacuum de- cay by barrier penetration. Certain geometric properties of instantons are studied, using Hermitian differential operators. On a d-dimensional instanton, it is shown that there are d? 2 harmonic functions. A class of instanton solutions, obeying more restrictive boundary conditions, is proved to have d ? 1 Killing vectors which also commute. All but one of the Killing vectors are duals of harmonic one-forms, which are gradients of harmonic functions, and do not have any fixed points. 1e-mail: ali@rainbow.tamu.edu 1 Introduction Like for any other field theory, positivity of the energy is a necessary condition for ensuring the stability of general relativity. In the presence of gravity, although the notion of local energy density is not well defined, one can still talk about the total energy of a gravitating system defined in terms of the asymptotic behavior of the metric with respect to a background geometry [1],[2]. For asymptotically flat spaces, under general assumptions, the complete proof of positive energy theorem was first given in [3]. Later, a simple and elegant proof was presented in [4], using spinors. On the other hand, it is known that there are non-trivial zero [5] and negative energy [6] asymptotically flat spaces on which these proofs do not apply. Ge