Prof. José Figueroa-O'Farrill (School of Mathematics, Univ. of Edinburgh): Spacetime G-structures II
Channel: Centrum Fizyki Teoretycznej PAN
Date: 09/10/2023

Prof. José Figueroa-O'Farrill (School of Mathematics, University of Edinburgh).
Spacetime G-structures II.
Abstract: Ever since Minkowski introduced his eponymous spacetime as the arena for Einstein’s theory of special relativity, physics has modelled spacetime as a Lorentzian manifold.  However this was not always the case: Newtonian mechanics takes place in Galilei spacetime, which one can now understand as the limit of Minkowski spacetime in which the speed of light goes to infinity.  

In 1965 Lévy-Leblond took the opposite limit (speed of light tending to zero) arriving at what he termed Carroll spacetime.   These three spacetimes (Carroll, Galilei and Minkowski) are Klein models for carrollian, galilean and lorentzian geometries.  Carrollian geometry describes the geometry of null hypersurfaces in lorentzian manifolds, whereas galilean geometry describes the “null” Kaluza-Klein reductions of lorentzian manifolds.  They are today the subject of intense study due to their relation to many areas of physics: from black holes (whose horizons are carrollian) and flat-space holography (the conformal boundary of asymptotically flat spacetimes have carrollian-like structures) to fluid dynamics.

However these talks are not concerned with the physical applications, but with the underlying geometries.  In these three talks I would like to introduce the Klein models, the associated G-structures and discuss their intrinsic torsion, resulting in a coarse but useful classification.  If there is time (and interest) I will discuss generalisations of these geometries, which will reveal the Carroll and Galilei G-structures as two members of a family of G-structures associated to extended objects appearing in string theory (p-branes).

This is a bi-weekly seminar for the Grieg collaboration project SCREAM: Symmetry, Curvature Reduction, and EquivAlence Methods, funded by the Norwegian Financial Mechanism 2014-2021. The registration number of the project is 2019/34/H/ST1/00636. 

The focus of the project is on a broad class of differential geometric structures known as Cartan and parabolic geometries, which include conformal, projective, CR, and ODE geometry, (2, 3, 5)-distributions, parabolic contact structures, and many more besides. Moreover, we explore the interaction of these structures with a variety of subjects such as mechanical systems and geometric robots, the theory of integrable systems, and Penrose's Conformal Cyclic Cosmology programme. 

More information can be found at www.cft.edu.pl/grieg.