Quantum astrometric observables II: time delay in linearized quantum gravity

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 $\mu$, 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 $(s \ell_p/\mu)^2$, where $\ell_p$ 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.