From the Lorentz Group to the Celestial Sphere

2015/08/04 by Blagoje Oblak, Oblak, Blagoje · 1 voice · 3 citations
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #FOS: Mathematics #FOS: Physical sciences #Group Theory (math.GR) #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Mathematics and Applications #Physics Education (physics.ed-ph) #Relativity and Gravitational Theory #hep-th #math-ph #math.GR #math.MP #physics.ed-ph

paper · pdf · doi:10.48550/arxiv.1508.00920

55 pages, 12 figures. Undergraduate-level lecture notes published in the proceedings of the seventh Brussels Summer School in Mathematics. v2: Minor cosmetic changes. v3: fixed a typo in section 4.2.2

openalex publication_date 2015/08/04 · arxiv published 2015/08/04 · arxiv created 2018/05/04 · arxiv updated 2018/05/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In these lecture notes we review the isomorphism between the (connected) Lorentz group and the set of conformal transformations of the sphere. More precisely, after establishing the main properties of the Lorentz group, we show that it is isomorphic to the group SL(2,C) of complex 2 by 2 matrices with unit determinant. We then classify conformal transformations of the sphere, define the notion of null infinity in Minkowski space-time, and show that the action of Lorentz transformations on the celestial spheres at null infinity is precisely that of conformal transformations. In particular, we discuss the optical phenomena observed by the pilots of the "Millenium Falcon" during the jump to lightspeed.

Citations

• Gravitational waves in general relativity, VII. Waves from axi-symmetric isolated system
• Infinite conformal symmetry in two-dimensional quantum field theory
• The apparent shape of a relativistically moving sphere
• The world as a hologram

Cited by

• Integrability in Asymptotic Symmetries of Spacetime: the BMS₃ scenario
• Associativity of celestial OPE, higher spins and self-duality
• Three point energy correlators in the collinear limit: symmetries, dualities and analytic results

Discussions

• C'est ce que nous explique ce papier arxiv.org/abs/1508.00920 qui présente une étude détaillée (et mathématiquement rigoureuse) du groupe de Lorentz. [bsky, 0 points, 1 comments]

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