2016/02/29 by Eric Woolgar · 1 citation
Mathematics · Physics and Astronomy · #math.DG #gr-qc #hep-th
paper · pdf · doi:10.1007/jhep03(2017)104
published as JHEP 1703 (2017) 104 · Statement and proof of Lemma 3.1 corrected, other minor changes
arxiv created 2017/02/27 · arxiv updated 2018/04/02
The "new positive energy conjecture" Horowitz and Myers (1999) probes a possible nonsupersymmetric AdS/CFT correspondence. We consider a version formulated for complete, asymptotically Poincaré-Einstein Riemannian metrics (M,g) with bounded scalar curvature R≥ -n(n-1). This version then asserts that any such (M,g) must have mass not less than the mass m0 of a metric g0 induced on a time-symmetric slice of a certain AdS soliton spacetime. The conjecture remains unproved, having so far resisted standard techniques. Little is known other than that the conjecture is true for metrics which are sufficiently small perturbations of g0. We pose another test for the conjecture. We assume its validity and attempt to prove as a corollary the corresponding scalar curvature rigidity statement, that g0 is the unique asymptotically Poincaré-Einstein metric with mass m=m0 obeying R≥ -n(n-1). Were a second such metric g1 not isometric to g0 to exist, it then may well admit perturbations of lower mass, contradicting the assumed validity of the conjecture. We find that the minimum mass metric must be static Einstein, so the problem is reduced to that of static uniqueness. When n=3 the manifold is isometric to a time-symmetric slice of an AdS soliton spacetime, unless it has a non-compact horizon. En route we study the mass aspect, obtaining and generalizing known results. The mass aspect is (i) related to the holographic energy density, (ii) a weighted invariant under boundary conformal transformations when the bulk dimension is odd, and (iii) zero for negative Einstein manifolds with Einstein conformal boundary.