  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.
