Effective Nonlocal Euclidean Gravity

A nonlocal form of the effective gravitational action could cure the unboundedness of euclidean gravity with Einstein action. On sub-horizon length scales the modified gravitational field equations seem compatible with all present tests of general relativity and post-Newtonian gravity. They induce a difference in the effective Newton’s constant between regions of space with vanishing or nonvanishing curvature scalar (or Ricci tensor). In cosmology they may lead to a value Ω < 1 for the critical density after inflation. The simplest model considered here appears to be in conflict with nucleosynthesis, but generalizations consistent with all cosmological observations seem conceivable. e-mail [email protected] 1 Stability problem in euclidean gravity It is a longstanding observation that the Einstein-Hilbert action for gravity is not bounded in euclidean space. Fundamental problems for the formulation of quantum gravity are connected to this fact. This stability problem in euclidean Einstein gravity can be easily seen if one expands the curvature scalar R around flat space R l d with metric fluctuations hμν = gμν − δμν . 2 Up to a total derivative one finds in quadratic order √ gR = I1 − I2 I1 = 1 8 (∂h − ∂h)(∂νhμρ − ∂ρhμν) I2 = 1 4 (∂νh ρν − ∂hν)(∂μhρ − ∂ρhμ). (1) One observes that for d > 2 the euclidean Einstein action (Mp = G −1/2 ≈ 10 GeV)