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We investigate the problem of metric fluctuations in the presence of thevacuum fluctuations of matter fields and critically assess the usual assertionthat vacuum energy implies a Planckian cosmological constant. A new stochasticclassical approach to the quantum fluctuations of spacetime is developed. Thework extends conceptually Boyer's random electrodynamics to a theory of randomgravity but has a considerably richer structure for inheriting nonlinearityfrom general relativity. Attention is drawn to subtleties in choosing boundaryconditions for metric fluctuations in relation to their dynamical consequences.Those compatible with the observed Lorentz invariance must allow forspontaneous conformal fluctuations, in addition to stochastic gravitationalwaves due to zero point gravitons. This is implemented through an effectivemetric defined in terms of the random spacetime metric modulo a fluctuatingconformal factor. It satisfies an effective Einstein equation coupled to aneffective stress-energy tensor incorporating gravitational self energy ofmetric fluctuations as well as matter fields. The effective Einstein equationis expanded up to second order nonlinearity. A UV-cutoff is introduced whosespecific value, however, does not enter into the resulting description ofrandom gravity. The averaged effective metric satisfies the empty spaceEinstein equation with an effective cosmological constant. This vanishes whenonly the massless matter fields are included. More generally, a finiteeffective cosmological constant compatible with the observational constraintscan be obtained as long as the bare masses of the massive matter fields arenearly zero, or the conformal invariance of matter is restored at some highenergy scale.

Nonlinear random gravity. I. Stochastic gravitational waves and spontaneous conformal fluctuations due to the quantum vacuum