A clock synchronization thought experiment is modeled by a diffeomorphism
invariant "time delay" observable. In a sense, this observable probes the
causal structure of the ambient Lorentzian spacetime. Thus, upon quantization,
it is sensitive to the long expected smearing of the light cone by vacuum
fluctuations in quantum gravity. After perturbative linearization, its mean and
variance are computed in the Minkowski Fock vacuum of linearized gravity. The
na\"ive divergence of the variance is meaningfully regularized by a length
scale μ, the physical detector resolution. This is the first time vacuum
fluctuations have been fully taken into account in a similar calculation.
Despite some drawbacks this calculation provides a useful template for the
study of a large class of similar observables in quantum gravity. Due to their
large volume, intermediate calculations were performed using computer algebra
software. The resulting variance scales like (sℓp/μ)2, where ℓp
is the Planck length and s is the distance scale separating the ("lab" and
"probe") clocks. Additionally, the variance depends on the relative velocity of
the lab and the probe, diverging for low velocities. This puzzling behavior may
be due to an oversimplified detector resolution model or a neglected second
order term in the time delay.Comment: 30 pages, 8 figures, revtex4-1; v3: minor updates and corrections,
close to published versio