A Spectral Criterion for Emergent Gravity

This paper formulates a spectral criterion for emergent gravity from the energy-momentum tensor commutator. The Weinberg-Witten theorem excludes an elementary massless spin-2 particle in a Poincaré-invariant quantum field theory with a covariant conserved stress tensor. A massless spin-2 excitation, if it appears in the stress-tensor spectral function, must therefore be interpreted as a collective mode rather than as a fundamental field. Since the spectral function is defined by a commutator, additive static vacuum-energy terms do not contribute to the dynamical spin-2 spectral density. Coarse-graining in momentum space constructs, at each scale, a macroscopic rank-2 tensor field. The irreversibility of this operation promotes the scale itself to a dynamical order parameter. Its Langevin dynamics is governed by two renormalization-group fixed points, an ultraviolet repellor at the Planck scale and an infrared attractor at the Hubble scale. Entropy production and energy transfer define an effective temperature, and the fluctuation-dissipation theorem relates the corresponding noise and dissipation. The criterion advanced in this paper is whether the spin-2 spectral density can develop an isolated zero-momentum pole. Such a pole is absent in perturbation theory and represents a nonperturbative reorganization of spectral weight. If the pole is present, Weinberg's low-energy theorem fixes the infrared effective action to Einstein-Hilbert form. Newton's constant is then read from the pole residue, while the infrared fixed point supplies the natural curvature scale. The criterion is in principle testable. Lattice gauge theory