Arrival time differences between gravitational waves and electromagnetic signals due to gravitational lensing

In this study, we demonstrate that general relativity predicts arrival time differences between gravitational wave (GW) and electromagnetic (EM) signals caused by the wave effects in gravitational lensing. The GW signals can arrive $earlier$ than the EM signals in some cases if the GW/EM signals have passed through a lens, even if both signals were emitted simultaneously by a source. GW wavelengths are much larger than EM wavelengths; therefore, the propagation of the GWs does not follow the laws of geometrical optics, including the Shapiro time delay, if the lens mass is less than approximately $10^5 {\rm M}_\odot (f/{\rm Hz})^{-1}$, where $f$ is the GW frequency. The arrival time difference can reach $\sim 0.1 \, {\rm s} \, (f/{\rm Hz})^{-1}$ if the signals have passed by a lens of mass $\sim 8000{\rm M}_\odot (f/{\rm Hz})^{-1}$ with the impact parameter smaller than the Einstein radius; therefore, it is more prominent for lower GW frequencies. For example, when a distant super massive black hole binary (SMBHB) in a galactic center is lensed by an intervening galaxy, the time lag becomes of the order of $10$ days. Future pulsar timing arrays including SKA (the Square Kilometre Array) and X-ray detectors may detect several time lags by measuring the orbital phase differences between the GW/EM signals in the SMBHBs. Gravitational lensing imprints a characteristic modulation on a chirp waveform; therefore, we can deduce whether a measured arrival time lag arises from intrinsic source properties or gravitational lensing. Determination of arrival time differences would be extremely useful in multimessenger observations and tests of general relativity.