For gravitational lensing of gravitational waves, the phase of the amplification factor at low frequency is fixed by the magnitude through a Kramers-Kronig relation, up to a Blaschke product ambiguity.
On arrival time difference between lensed gravitational waves and light
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It is known that geometrical optics no longer applies to the gravitational lensing if the wavelength of a propagating wave becomes comparable to or larger than the Schwarzshild radius of a lensing object. We investigate the propagation of gravitational waves in wave optics, particularly focusing on the difference between their arrival time and the arrival time of light. We argue that, contrary to the observation in the previous work, gravitational waves never arrive at an observer earlier than light when both gravitational waves and light are emitted from a same source simultaneously.
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On the Phase-Magnitude Relation in Gravitational Lensing: Reformulation and Applications of the Kramers-Kronig relation
For gravitational lensing of gravitational waves, the phase of the amplification factor at low frequency is fixed by the magnitude through a Kramers-Kronig relation, up to a Blaschke product ambiguity.