Measurement of Quantum Fluctuations in Geometry

Abstract

A particular form for the quantum indeterminacy of relative spacetime position of events is derived from the limits of measurement possible with Planck wavelength radiation. The indeterminacy predicts fluctuations from a classically defined geometry in the form of ``holographic noise'' whose spatial character, absolute normalization, and spectrum are predicted with no parameters. The noise has a distinctive transverse spatial shear signature, and a flat power spectral density given by the Planck time. An interferometer signal displays noise due to the uncertainty of relative positions of reflection events. The noise corresponds to an accumulation of phase offset with time that mimics a random walk of those optical elements that change the orientation of a wavefront. It only appears in measurements that compare transverse positions, and does not appear at all in purely radial position measurements. A lower bound on holographic noise follows from a covariant upper bound on gravitational entropy. The predicted holographic noise spectrum is estimated to be comparable to measured noise in the currently operating interferometer GEO600. Because of its transverse character, holographic noise is reduced relative to gravitational wave effects in other interferometer designs, such as LIGO, where beam power is much less in the beamsplitter than in the arms.

7 pages, 2 figures, LaTeX. Extensive rewrite of original version, including more detailed analysis. Main result is the same but the estimate of noise in strain units for GEO600, showing 1/f behavior at low f and flat at high f, is improved. To appear in Phys. Rev. D

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