1996/12/13 by Giovanni Landi, Carlo Rovelli · 95 citations
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Classical mechanics #Cosmology and Gravitation Theories #Diffeomorphism #Eigenvalues and eigenvectors #Equations of motion #General covariance #General relativity #Invariant (physics) #Jacobian matrix and determinant #Mathematical analysis #Mathematical physics #Mathematics #Observable #Physics #Quantum mechanics #Relativity and Gravitational Theory #Theoretical physics #gr-qc #hep-th
paper · pdf · doi:10.1103/physrevlett.78.3051
published in Physical Review Letters 78(16), 3051-3054 (American Physical Society) · 6 pages, RevTex
arxiv created 1996/12/13 · openalex publication_date 1997/04/21 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The eigenvalues of the Dirac operator are diffeomorphism-invariant functions of the geometry, namely, ``observables'' for general relativity. Recent work by Chamseddine and Connes suggests taking them as gravity's dynamical variables. We compute their Poisson brackets, find that these can be expressed in terms of energy momenta T of the eigenspinors, and show that T is the Jacobian matrix of the transformation from metric to eigenvalues. We consider a small modification of the spectral action that gets rid of the cosmological term, and derive its equations of motion. These are solved if T scales linearly. We show that such a scaling law yields Einstein equations.