Gravity from Dirac Eigenvalues

TL;DR

This paper explores a novel formulation of Euclidean general relativity using Dirac eigenvalues as invariant variables, analyzing their Poisson brackets, dynamics, and coupling to fermions, revealing deep connections with Einstein's equations.

Contribution

It introduces a new invariant variable framework for gravity based on Dirac eigenvalues, computes their Poisson brackets, and links their dynamics to Einstein's equations.

Findings

Poisson brackets of eigenvalues expressed via Einstein propagator

Eigenspinors' energy-momentum relates to metric change of variables

Modified spectral action yields Einstein's equations through eigenvalue scaling

Abstract

We study a formulation of euclidean general relativity in which the dynamical variables are given by a sequence of real numbers $\lambda_{n}$, representing the eigenvalues of the Dirac operator on the curved spacetime. These quantities are diffeomorphism-invariant functions of the metric and they form an infinite set of ``physical observables'' for general relativity. Recent work of Connes and Chamseddine suggests that they can be taken as natural variables for an invariant description of the dynamics of gravity. We compute the Poisson brackets of the $\lambda_{n}$'s, and find that these can be expressed in terms of the propagator of the linearized Einstein equations and the energy-momentum of the eigenspinors. We show that the eigenspinors' energy-momentum is the Jacobian matrix of the change of coordinates from the metric to the $\lambda_{n}$'s. We study a variant of the Connes-Chamseddine spectral action which eliminates a disturbing large cosmological term. We analyze the corresponding equations of motion and find that these are solved if the energy momenta of the eigenspinors scale linearly with the mass. Surprisingly, this scaling law codes Einstein's equations. Finally we study the coupling to a physical fermion field.