  Realistic black holes are usually dynamical, noticeable or sluggish. The
Vaidya metric is a significant and tractable model for simulating a dynamical
black hole. In this study, we consider scalar perturbations in a dynamical
Vaidya black hole, and explore the quasinormal modes by employing the matrix
method. We find the proper boundary conditions of the quasinormal modes from
physical analysis in the background of a dynamical black hole for the first
time. The results show that the eigenfrequencies become different at the
apparent horizon and null infinity, because the physical interactions propagate
with finite velocity in nature. Any variation of the hole does not affect the
boundary condition at null infinity in a finite time. The quasinormal modes
originated around the horizon would not immediately come down to, but slowly
goes to the final state following the mass accretion process of the hole. The
precision of the matrix method is quite compelling, which reveals more details
of the eigenfrequencies of the quasinormal mode of perturbations in the Vaidya
spacetime.

General Relativity and Quantum Cosmology