Optical cavity resonator in an expanding universe

Abstract

We study evolution of frequency of a standing electromagnetic (EM) wave in a resonant optical cavity placed to the expanding manifold described by the Robertson-Walker metric. One builds a local coordinate system in which spacetime is locally Minkowskian. However, due to the conformal nature of the Robertson-Walker metric the conventional transformation to the local inertial coordinates introduces ambiguity in the physical interpretation of the local time coordinate. Therefore, contrary to a common-sense expectation, a straightforward implementation of EEP alone does not allow us to decide whether atomic clocks ticks at the same rate as the clocks based on EM modes of a cavity. To resolve the ambiguity we analyzed the cavity rigidity and the oscillation of its EM modes in an expanding universe by employing the Maxwell equations. We found out that both the size of the cavity and the EM frequency experience an adiabatic drift in conformal coordinates as the universe expands. We set up the oscillation equation for the EM modes, solve it by the WKB approximation, and reduce the coordinate-dependent quantities to their counterparts measured by a local observer who counts time with atomic clock. The solution shows that there is a perfect cancellation of the adiabatic drift of cavity's frequency by the transformation to local coordinates, and the time counted by the clocks based on EM modes of cavity has the same rate as that of atomic clocks. We conclude that there should be no cosmological drift of frequency of a standing EM wave oscillating in the cavity resonator as compared to the frequency of atomic clocks. Continuous comparison of the frequency of the optical cavity resonator against that of atomic clock yields a powerful null test of the local isotropy of the Hubble expansion and the Einstein equivalence principle in cosmology.