6 May 2019 Wed 12:15-13:45 REALF A036
8 May 2019 Fri 10:15-11:45 REALF A036

José Figueroa-O'Farrill (University of Edinburgh)

"Kinematical spacetimes: beyond Lorentzian geometry".

Over 50 years ago, Bacry and Lévy-Leblond asked which were the possible spacetime kinematics. Known examples at that time were the maximally symmetric lorentzian spacetimes (Minkowski, de Sitter and anti de Sitter) and the galilean spacetime of the Newtonian universe. They introduced the notion of a kinematical Lie algebra in four-dimensional spacetime and gave a classification subject to some “by no means compelling” assumptions, which were relaxed twenty years later by Bacry and Nuyts arriving at a classification of kinematical Lie algebras with three-dimensional space isotropy. They also observed that each such Lie algebra acts transitively on some four-dimensional homogeneous spacetime and arrived at 11 classes of kinematical spacetimes.

Recently with Stefan Prohazka we have refined and extended this pioneering work to arrive at a classification of simply-connected homogeneous kinematical spacetimes in arbitrary dimension and, in collaboration with my student Ross Grassie, we have also studied the local geometry and the Lie algebra of symmetries of these spacetimes, which unlike the isometry Lie algebras of a (pseudo)riemannian space are typically infinite-dimensional. In this lectures I would like to introduce this topic and focus on some examples.

The lectures this week will cover the following topics:

L1: Maximally symmetric spacetimes (slides).
L2: Contraction and kinematical Lie algebras (slides).

Syllabus:

• Riemannian and Lorentzian manifolds and their isometries
• Killing transport
• Maximally symmetric spaces and their isometry Lie algebras
• Geometric limits and Lie algebra contractions: non- and ultra-relativistic limits, zero- and infinite-curvature limits
• Kinematical Lie algebras