Time Delay Interferometry (TDI) is often utilized in the data pre-processing
of space-based gravitational wave detectors, primarily for suppressing laser
frequency noise. About twenty years ago, assuming armlengths remain constant
over time, researchers presented comprehensive mathematical descriptions for
the first-generation and modified first-generation TDI. However, maintaining a
steady distance between satellites is pragmatically challenging. Hence, the
operator equation that neutralizes laser frequency noise, though provided, was
deemed difficult to resolve. In this paper, we solve this equation in the
context of a non-static scenario where distances between spacecrafts vary over
time. Surprisingly, contrary to what previous researchers thought, the study
reveals that the equation has only the zero solution, which suggests that no
nonzero TDI combination can entirely suppress laser frequency noise under
time-varying armlengths. This necessitates the persistent search for
second-generation TDI combinations through alternative methods besides directly
solving the operator equation. We establish the connections between TDI
combinations of different generations and propose a search strategy for finding
higher-generation TDI combinations by using generators of lower-generation TDI.
The findings contribute to the ongoing discussion on gravitational waves and
provide a novel insight into the hurdles faced in space-based gravitational
wave detection.Comment: accepted by Physical Review