Quantum Limit on Stability of Clocks in a Gravitational Field

Good clocks are of importance both to fundamental physics and for applications in astronomy, metrology and global positioning systems. In a recent technological breakthrough, researchers at NIST have been able to achieve a stability of 1 part in $10^{18}$ using an Ytterbium clock. This naturally raises the question of whether there are fundamental limits to the stability of clocks. In this paper we point out that gravity and quantum mechanics set a fundamental limit on the stability of clocks. This limit comes from a combination of the uncertainty relation, the gravitational redshift and the relativistic time dilation effect. For example, a single ion hydrogen maser clock in a terrestrial gravitational field cannot achieve a stability better than one part in $10^{22}$. This observation has implications for laboratory experiments involving both gravity and quantum theory.