Planck—Wheeler Quantum Foam as White Noise: Metric Diffusion and Congruence Focussing for Fluctuating Spacetime Geometry

Abstract

It is expected that quantum effects endow spacetime with stochastic properties near the Planck scale as exemplified by random fluctuations of the metric, usually referred to as spacetime foam or geometrodynamics. In this paper, a methodology is presented for incorporating Planck scale stochastic effects and corrections into general relativity within the ADM formalism, by coupling the Riemann 3-metric to white noise. The ADM—Cauchy evolution of a Riemann 3-metric h ij (t) induced on spacelike hypersurface C(t) can be interpreted within pure general relativity as a smooth geodesic flow in superspace, whose points consist of equivalence classes of 3-metrics. Coupling white noise to h ij gives Langevin stochastic differential equations for the Cauchy evolution of h ij, which is now a Brownian motion or diffusion in superspace. A fluctuation h′ij away from h ij is considered to be related to h ij by elements of the diffeomorphism group diff(C). Hydrodynamical Fokker—Planck continuity equations are formulated describing the stochastic Cauchy evolution of h ij as a probability flow. The Cauchy invariant or equilibrium solution gives a stationary probability distribution of fluctuations peaked around the deterministic metric. By selecting a physically viable ansatz for the scale dependent diffusion coefficient, one reproduces the Wheeler uncertainty relation for the metric fluctuations of quantum geometrodynamics. Treating h ij as a random variable, a non-linear Raychaudhuri—Langevin equation is derived describing the “geometro-hydrodynamics” of a congruence of fluid or dust matter propagating on the stochastic spacetime. For an initially converging congruence θ′>0 at s′ the singularity θ=−∞ at future proper time s=3/|θ′|$, which is expected in general relativity, is now smeared out near the Planck scale. Proper time s can be extended indefinitely (s→∞) so that intrinsic metric fluctuations can restore geodesic completeness although the geodesics remain trapped for all time: although a singularity can be removed the collapsing matter still creates a black hole. A Fokker—Planck formulation also gives zero probability that θ→−∞ for s→∞. Essentially, the short distance stochastic corrections to the deterministic equations of general relativity can remove pathologies such as singularities, conjugate points and geodesic incompleteness.