Spinors, Jets, and the Einstein Equations

TL;DR

This paper uses advanced geometric and spinor techniques on jet spaces to analyze Einstein's equations, revealing fundamental properties like the non-existence of local observables and uniqueness of the gravitational symplectic structure.

Contribution

It introduces a novel approach combining jet bundle calculus with spinor parametrization to study Einstein equations, leading to new insights into symmetries and conserved quantities.

Findings

Proves non-existence of local observables for vacuum spacetimes.

Establishes a uniqueness theorem for the gravitational symplectic structure.

Analyzes generalized symmetries and characteristic cohomology of Einstein equations.

Abstract

Many important features of a field theory, {\it e.g.}, conserved currents, symplectic structures, energy-momentum tensors, {\it etc.}, arise as tensors locally constructed from the fields and their derivatives. Such tensors are naturally defined as geometric objects on the jet space of solutions to the field equations. Modern results from the calculus on jet bundles can be combined with a powerful spinor parametrization of the jet space of Einstein metrics to unravel basic features of the Einstein equations. These techniques have been applied to computation of generalized symmetries and ``characteristic cohomology'' of the Einstein equations, and lead to results such as a proof of non-existence of ``local observables'' for vacuum spacetimes and a uniqueness theorem for the gravitational symplectic structure.