General Relativity in terms of Dirac Eigenvalues

TL;DR

This paper explores the use of Dirac operator eigenvalues as invariant variables in general relativity, analyzing their Poisson brackets, relation to energy-momentum, and implications for Einstein's equations.

Contribution

It introduces a framework where Dirac eigenvalues serve as fundamental variables for gravitational dynamics, connecting spectral data with Einstein's equations.

Findings

Poisson brackets of eigenvalues expressed via eigenspinor energy-momentum and Einstein propagator

Eigenspinor energy-momentum forms the Jacobian for metric-to-eigenvalue transformation

Modified spectral action yields equations of motion consistent with Einstein's equations under specific conditions

Abstract

The eigenvalues of the Dirac operator on a curved spacetime are diffeomorphism-invariant functions of the geometry. They form an infinite set of ``observables'' for general relativity. Recent work of Chamseddine and Connes suggests that they can be taken as variables for an invariant description of the gravitational field's dynamics. We compute the Poisson brackets of these eigenvalues and find them in terms of the energy-momentum of the eigenspinors and the propagator of the linearized Einstein equations. We show that the eigenspinors' energy-momentum is the Jacobian matrix of the change of coordinates from metric to eigenvalues. We also consider a minor modification of the spectral action, which eliminates the disturbing huge cosmological term and derive its equations of motion. These are satisfied if the energy momentum of the trans Planckian eigenspinors scale linearly with the eigenvalue; we argue that this requirement approximates the Einstein equations.