Carrollian and celestial spaces at infinity

TL;DR

Classify and geometrize the asymptotic infinities of Minkowski space by viewing Spi, Ti, Ni, and I as homogeneous Poincaré spaces embedded in $\mathbb{E}^{d+1,2}$. Identify invariant Carrollian, pseudo-carrollian, and novel doubly-carrollian structures, relate Ni to the bundle of scales over $\mathscr{I}$, and demonstrate holographic reconstruction of Minkowski space from these asymptotic geometries. Provide Grassmannian realizations and detailed Klein-pair analyses, showing that the symmetry algebras match expected asymptotic symmetries (e.g., BMS) and clarifying how these spaces interrelate with null infinity, the light cone, and the celestial sphere. The results offer a unified, symmetry-centered framework for flat-space holography and illuminate connections to (A)dS holography, Cartan geometries, and potential lower-dimensional dual descriptions.

Abstract

We show that the geometry of the asymptotic infinities of Minkowski spacetime (in $d+1$ dimensions) is captured by homogeneous spaces of the Poincaré group: the blow-ups of spatial (Spi) and timelike (Ti) infinities in the sense of Ashtekar--Hansen and a novel space Ni fibering over $\mathscr{I}$. We embed these spaces à la Penrose--Rindler into a pseudo-euclidean space of signature $(d+1,2)$ as orbits of the same Poincaré subgroup of O$(d+1,2)$. We describe the corresponding Klein pairs and determine their Poincaré-invariant structures: a carrollian structure on Ti, a pseudo-carrollian structure on Spi and a "doubly-carrollian" structure on Ni. We give additional geometric characterisations of these spaces as grassmannians of affine hyperplanes in Minkowski spacetime: Spi is the (double cover of the) grassmannian of affine lorentzian hyperplanes; Ti is the grassmannian of affine spacelike hyperplanes and Ni fibers over the grassmannian of affine null planes, which is $\mathscr{I}$. We exhibit Ni as the fibred product of $\mathscr{I}$ and the lightcone over the celestial sphere. We also show that Ni is the total space of the bundle of scales of the conformal carrollian structure on $\mathscr{I}$ and show that the symmetry algebra of its doubly-carrollian structure is isomorphic to the symmetry algebra of the conformal carrollian structure on $\mathscr{I}$; that is, the BMS algebra. We show how to reconstruct Minkowski spacetime from any of its asymptotic geometries, by establishing that points in Minkowski spacetime parametrise certain lightcone cuts in the asymptotic geometries. We include an appendix comparing with (A)dS and observe that the de Sitter groups have no homogeneous spaces which could play the rôle that the celestial sphere plays in flat space holography.

Figures (6)

• Figure 1: The Penrose diagram of Minkowski spacetime $\mathbb{M}$ with its hyperbolic slicing. We have also illustrated how $\mathsf{Ti}$ and $\mathsf{Spi}$ arise as the blow-ups of, respectively, timelike and spacelike infinities, while $\mathsf{Ni}$ fibers over $\mathscr{I}$ and can be understood as the bundle of scales of the conformal carrollian structure of $\mathscr{I}$.
• Figure 2: An embedding of $(d+1)$-dimensional Minkowski spacetime $\mathbb{M}_{d+1}$ as the intersection $\mathscr{Q}_0 \cap \mathscr{N}_1$ in the ambient space $\mathbb{E}^{d+1,2}$
• Figure 3: The lightcone $\mathscr{L}$ at $p \in \mathbb{M}$ as the intersection $\mathbb{L}_p \cap \mathbb{M}$
• Figure 4: $\mathsf{Ti} \cong \mathsf{AdSC}$ fibering over hyperbolic space.
• Figure 5: $\mathsf{Spi}/\mathbb{Z}_2$ fibering over elliptic de Sitter spacetime.