Abstract
I examine the relationship between (d+1)-dimensional Poincaré 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 (Elie Cartan et les Mathématiques d’aujourd’hui, Astérisque, 1985; The Ambient Metric. Annals of Mathematics Studies, Princeton University Press, Princeton, 2012) on conformal geometry, in order to prove two propositions and a theorem that characterise which classes of diffeomorphisms qualify as gravity-invisible. I define natural notions of gravity-invisibility (strong, weak, and simpliciter) that apply to the diffeomorphisms of Poincaré metrics in any dimension. (2: Dualities) I apply the notions of invisibility, developed in (1), to gauge–gravity dualities: which, roughly, relate Poincaré metrics in d+1 dimensions to QFTs in d dimensions. I contrast QFT-visible versus 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. (Class Quantum Gravity 17(5):1129, 2000, Eq. 2.6). I also give a general derivation of the asymptotic conformal Killing equation, which has not appeared in the literature before.
Highlights
The asymptotic symmetries of gravity have been a central foundational topic in general relativity since at least the work Arnowitt et al [3,4], Sachs [34,35], Bondi et al [7], Penrose [32,33], Newman et al [31], Geroch [20], Ashtekar et al [5], and others
I analyse the case of a negative cosmological constant. (For a discussion of the other cases: see the physical motivation, below.) I will use gauge–gravity duality to argue that there is a significant, non-empty, class of diffeomorphisms— which I will, broadly speaking, call ‘visible’, in a sense that I will make precise—which act on the dual gauge theory, and which act on the physical degrees of freedom of the gravity theory
In this paper I have presented a number of results which: (i) make rigorous a number of physical intuitions about asymptotic symmetries in general relativity with a negative cosmological constant, and in gauge–gravity dualities; (ii) provide the mathematical and physical basis for the philosophical comparison of duality and gauge symmetry presented in De Haro et al [14, Sects. 5, 6]); (iii) underpin the discussion of background-independence for gauge–gravity dualities in De Haro [11, Sects. 2.3.2–2.3.4]
Summary
The asymptotic symmetries of gravity have been a central foundational topic in general relativity since at least the work Arnowitt et al [3,4], Sachs [34,35], Bondi et al [7], Penrose [32,33], Newman et al [31], Geroch [20], Ashtekar et al [5], and others. The difference between the two kinds of diffeomorphisms—those that are visible versus those that are invisible through the duality—is a crucial property of the duality map, and determines what is ‘physical’, on both sides of the duality The diffeomorphisms differ both in their physical properties and in the ways in which they can be regarded to be novel properties of the gravity theory. As stressed in De Haro [12]: the content that is invariant across the duality (the ‘common core’) is what should be regarded as physically significant for this particular theory of quantum gravity This gives us an additional argument to the effect that the diffeomorphisms which are visible to the QFT act on general relativity’s asymptotic degrees of freedom. AdS/CFT is a good case study which has already provided insights into possibilities for defining a gauge–gravity duality for spaces with a positive cosmological constant (see e.g. Maldacena [29], Strominger [38], De Haro et al [15, Sect. 8])
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