Kramers-Kronig relation in gravitational lensing

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.