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Shapiro time delay is one of the fundamental tests of general relativity andpost-Newtonian theories of gravity. Consequently, its measurements can be usedto probe the parameter $\gamma$ which is related to spacetime curvatureproduced by a unit mass in the post-Newtonian formalism of gravity. To date allmeasurements of time delay have been conducted on astronomical scales. It wasasserted in 2010 that gravitational wave detectors on Earth could be used tomeasure Shapiro delay on a terrestrial scale via massive rotating systems.Building on that work, we consider how measurements of Shapiro delay can bemade using next-generation gravitational wave detectors. We perform an analysisfor measuring Shapiro delay with the next-generation gravitational wavedetectors Cosmic Explorer and Einstein Telescope to determine how precisely theeffect can be measured. Using a rotating mass unit design, we find that CosmicExplorer and Einstein Telescope can measure the Shapiro delay signal withamplitude signal to noise ratios upwards of $\sim28 $ and $\sim43$ in 1 year ofintegration time, respectively. By measuring Shapiro delay with this technique,next-generation interferometers will allow for terrestrial measurements of$\gamma$ in the paramaterized post-Newtonian formalism of gravity withsub-percent precision.

Can we use next-generation gravitational wave detectors for terrestrial precision measurements of Shapiro delay?