Towards a Consistent Calculation of the Lunar Response to Gravitational Waves

The recent increasing interest in detecting gravitational waves (GWs) by lunar seismic measurement urges us to have a clear understanding of the response of the moon to passing GWs. In this paper, we clarify the relationship between two seemly different response functions which have been derived previously using two different methods, one taking the field-theory approach and the other using the tidal force induced by GWs. We revisit their derivation and prove, by both analytical arguments and numerical calculations, that the two response functions are equivalent. Their apparent difference can be attributed to the choice of different coordinates. Using the correct response function, we calculate the sensitivities (to GWs) of several designed lunar seismometers, and find that the sensitivity curves between $10^{-3}$ and $0.1$ Hz are much flatter than the previous calculations based on normal-mode model. Our results will help clarifying the scientific objectives of lunar GW observation, as well as provide important constraints on the design of lunar GW detectors.