  We study the linearised perturbation of the constant mass aspect function
foliation in a Schwarzschild black hole spacetime. In particular, we
investigate the linearised perturbation of the asymptotic geometry of the
foliation at null infinity. We show that there is a 4-dimensional linear space
for the linearised perturbation of the initial leaf inside the event horizon,
corresponding to which the linearised perturbation of the asymptotic geometry
of the foliation at null infinity is preserved to be round. For such a
linearised perturbation of the initial leaf in this 4-dimensional linear space,
we calculate the corresponding linearised perturbation of the energy-momentum
vector and the Bondi mass at null infinity. We show that the linearised
perturbations of the Bondi energy and the Bondi mass both vanish, and every
possible linearised perturbation of the linear momentum can be achieved by a
linearised perturbation of the initial leaf in the 4-dimensional linear space.
