  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.
