The Kramers-Kronig relation is a well-known relation, especially in the field
of optics. The key to this relation is the causality that output comes only
after input. We first show that gravitational lensing obeys the causality in
the sense that (electromagnetic/gravitational) waves emitted from the source
arrive at an observer only after the arrival of the signal in geometrical
optics. This is done by extending the previous work which is based on the thin
lens approximation. We then derive the Kramers-Kronig relation in gravitational
lensing, as the relation between real and imaginary parts of the amplification
factor, which is the amplitude ratio of the lensed wave to the unlensed wave.
As a byproduct, we find a new relation that equates integration of the square
of the real part of the amplification factor over frequency to that for the
imaginary part of the amplification factor. We also obtain a sum rule which
relates the integral of the imaginary part of the amplification factor with the
magnification of the first arrival image in geometrical optics. Finally, we
argue that an incorrect separation of the observed gravitational waveform into
the amplification factor and the unlensed waveform generically leads to the
violation of the Kramers-Kronig relation. Our work suggests that examining the
violation of the Kramers-Kronig relation may be used for correctly extracting
the lensing signal in the gravitational wave observations.Comment: 17 pages, 5 figure