The Invisibility of Diffeomorphisms

17 Apr 2018

I examine the relationship between $(d+1)$-dimensional Poincar\'e metrics and $d$-dimensional conformal manifolds, from both mathematical and physical perspectives. The results have a bearing on several conceptual issues relating to asymptotic symmetries, in general relativity and in gauge-gravity duality, as follows: (1: Ambient Construction) I draw from the remarkable work by Fefferman and Graham (1985, 2012) on conformal geometry, in order to prove two propositions and a theorem that characterise the classes of diffeomorphisms that qualify as gravity-invisible. I define natural notions of gravity-invisibility (strong, weak, and simpliciter) which apply to the diffeomorphisms of Poincar\'e metrics in any dimension. (2: Dualities) I apply the notions of invisibility to gauge-gravity dualities: which, roughly, relate Poincar\'e metrics in $d+1$ dimensions to QFTs in $d$ dimensions. I contrast QFT-visible vs. QFT-invisible diffeomorphisms: those gravity diffeomorphisms that can, respectively cannot, be seen from the QFT. The QFT-invisible diffeomorphisms are the ones which are relevant to the hole argument in Einstein spaces. The results on dualities are surprising, because the class of QFT-visible diffeomorphisms is larger than expected, and the class of QFT-invisible ones is smaller than expected, or usually believed, i.e. larger than the PBH diffeomorphisms in Imbimbo et al. (2000). I also give a general derivation of the asymptotic conformal Killing equation, which has not appeared in the literature before.