  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.
