arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 2112.03319hep-thgr-qcmath.DG
收藏

无穷远处的Carrollian与celestial空间

Carrollian and celestial spaces at infinity

José Figueroa-O'Farrill, Emil Have, Stefan Prohazka, Jakob Salzer

备注：
34+6 pages, 6 figures, v2: Appendix E added, v3: Various minor improvements, matches published version
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

AI总结：

本文利用Poincaré群的齐性空间刻画了Minkowski时空渐近无穷远（Spi、Ti及新空间Ni）的几何结构，揭示了其Carrollian不变性、Grassmannian几何特征，并证明了Ni的对称代数同构于BMS代数，且可由渐近几何重构时空。

摘要：

我们证明了Minkowski时空（在 $d+1$ 维中）的渐近无穷远的几何由Poincaré群的齐性空间所刻画：Ashtekar--Hansen意义下的空间无穷远（Spi）和类时无穷远（Ti）的blow-up，以及一个在 $\mathscr{I}$ 上纤维化的新空间Ni。我们将这些空间按照Penrose--Rindler的方式嵌入到符号差为 $(d+1,2)$ 的伪欧几里得空间中，作为O$(d+1,2)$ 的同一Poincaré子群的轨道。我们描述了相应的Klein对，并确定了它们的Poincaré不变结构：Ti上的carrollian结构，Spi上的伪carrollian结构，以及Ni上的“双重carrollian”结构。我们给出了这些空间作为Minkowski时空中的仿射超平面的grassmannian的额外几何刻画：Spi是仿射lorentzian超平面的grassmannian（的双覆盖）；Ti是仿射类空超平面的grassmannian；而Ni在仿射零平面的grassmannian（即 $\mathscr{I}$）上纤维化。我们将Ni展示为 $\mathscr{I}$ 与celestial球上的光锥的纤维积。我们还证明了Ni是 $\mathscr{I}$ 上的共形carrollian结构的标度丛的全空间，并证明其双重carrollian结构的对称代数同构于 $\mathscr{I}$ 上的共形carrollian结构的对称代数，即BMS代数。我们展示了如何从其任意渐近几何重构Minkowski时空，通过确立Minkowski时空中的点参数化了渐近几何中的某些光锥割线。我们包含了一个与(A)dS比较的附录，并观察到de Sitter群没有能扮演celestial球在平坦空间全息中角色的齐性空间。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑