A bstract It is well-known that blackhole and cosmological horizons in equilibrium situations are well-modeled by non expanding horizons (NEHs) [1–3]. In the first part of the paper we introduce multipole moments to characterize their geometry, removing the restriction to axisymmetric situations made in the existing literature [4]. We then show that the symmetry group $$ \mathfrak{G} $$ G of NEHs is a 1-dimensional extension of the BMS group $$ \mathfrak{B} $$ B . These symmetries are used in a companion paper [5] to define charges and fluxes on NEHs, as well as perturbed NEHs. They have physically attractive properties. Finally, it is generally not appreciated that $$ \mathcal{I} $$ I ± of asymptotically flat space-times are NEHs in the conformally completed space-time . Forthcoming papers will (i) show that $$ \mathcal{I} $$ I ± have a small additional structure that reduces $$ \mathfrak{G} $$ G to the BMS group $$ \mathfrak{B} $$ B , and the BMS charges and fluxes can be recovered from the NEH framework; and, (ii) develop gravitational wave tomography for the late stage of compact binary coalescences: reading-off the dynamics of perturbed NEHs in the strong field regime (via evolution of their multipoles), from the waveform at $$ \mathcal{I} $$ I + .
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Analyzing the Weyl Construction for Dynamical Cartan Subalgebras
Abstract When the reduced twisted $C^*$-algebra $C^*_r({\mathcal{G}}, c)$ of a non-principal groupoid ${\mathcal{G}}$ admits a Cartan subalgebra, Renault’s work on Cartan subalgebras implies the existence of another groupoid description of $C^*_r({\mathcal{G}}, c)$. In an earlier paper, joint with Reznikoff and Wright, we identified situations where such a Cartan subalgebra arises from a subgroupoid ${\mathcal{S}}$ of ${\mathcal{G}}$. In this paper, we study the relationship between the original groupoids ${\mathcal{S}}, {\mathcal{G}}$ and the Weyl groupoid and twist associated to the Cartan pair. We first identify the spectrum ${\mathfrak{B}}$ of the Cartan subalgebra $C^*_r({\mathcal{S}}, c)$. We then show that the quotient groupoid ${\mathcal{G}}/{\mathcal{S}}$ acts on ${\mathfrak{B}}$, and that the corresponding action groupoid is exactly the Weyl groupoid of the Cartan pair. Lastly, we show that if the quotient map ${\mathcal{G}}\to{\mathcal{G}}/{\mathcal{S}}$ admits a continuous section, then the Weyl twist is also given by an explicit continuous $2$-cocycle on ${\mathcal{G}}/{\mathcal{S}} \ltimes{\mathfrak{B}}$.
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- Award ID(s):
- 1800749
- PAR ID:
- 10451790
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- Volume:
- 2022
- Issue:
- 20
- ISSN:
- 1073-7928
- Page Range / eLocation ID:
- 15721 to 15755
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/imrn/rnab114
