  Axion as a coherently oscillating massive scalar field is known to behave as
a zero-pressure irrotational fluid with characteristic quantum stress on a
small scale. In relativistic perturbation theory, the case was proved in the
axion-comoving gauge up to fully nonlinear and exact order. Our basic
assumption is that the field is oscillating with Compton frequency and the
Compton wavelength is smaller than the horizon scale. Here, we revisit the
relativistic proof to the linear order in the other gauge conditions. We show
that the same equation for density perturbation known in the non-relativistic
treatment can be derived in two additional gauge conditions: the zero-shear
gauge and the uniform-curvature gauge. The uniform-expansion gauge fails to get
the aimed equation, and the quantum stress term is missing in the synchronous
gauge. For comparison, we present the relativistic density perturbation
equations in the zero-pressure fluid in these gauge conditions. Except for the
comoving and the synchronous gauge, the equations strikingly differ from the
axion case. We clarify that the relativistic analysis based on time averaging
is valid for scales larger than the Compton wavelength. Below the Compton
wavelength, the field is not oscillating, and our oscillatory ansatz does not
apply. We suggest an equation valid in all scales in the comoving gauge. For
comparison, we review the non-relativistic quantum hydrodynamics and present
the Schr\"odinger equation to first-order post-Newtonian expansion in the
cosmological context.
