Applications of Tractor Calculus in General Relativity - Part I

Explore the applications of tractor calculus in general relativity in this comprehensive lecture by Dr. Jarosław Kopiński from the Center for Theoretical Physics PAS. Delve into the fundamental concepts of tractor calculus and their relationship to conformal infinities. Learn about higher fundamental forms and their use in characterizing the conformal boundary of asymptotically de Sitter spacetimes. Examine the geometric implications of spacetime with an initial isotropic singularity. Discover a general method for matching spacetimes in Penrose's Conformal Cyclic Cosmology model. Gain insights into various topics including conformal invariance and covariance, conformal geometry, the Almost Einstein equation, tractor bundles, tractor metrics and curvature, the invariant D operator, scale tractors, and the conformal sphere. This lecture is part of the SCREAM project, focusing on Cartan and parabolic geometries and their interactions with mechanical systems, integrable systems, and cosmology.

Intro
Overview
Notation and conventions
Conformal invariance and covariance
Conformal geometry
Almost Einstein equation and tractors
Tractor bundle
Tractor metric and curvature
Invariant D operator
Scale tractor
The and scalar curvature
Example - conformal sphere