  In this work, we perform linear perturbation on general relativistic
isothermal accretion onto a non-rotating astrophysical black hole to study the
salient features of the corresponding emergent acoustic metric. For spherically
symmetric accretion as well as for the axially symmetric matter flow for three
different geometric configuration of matter, we perturb the velocity potential,
the mass accretion rate, and the integral solution of the time independent part
of the general relativistic Euler equation to obtain such acoustic geometry. We
provide the procedure to locate the acoustic horizon and identify such horizon
with the transonic surfaces of the accreting matter through the construction of
the corresponding causal structures. We then discuss how one can compute the
value of the acoustic surface gravity in terms of the accretion variable
corresponding to the background flow solutions - i.e., stationary integral
transonic accretion solutions for different matter geometries. We show that the
salient features of the acoustic geometry is independent of the physical
variable we perturb, but sensitively depends on the geometric configuration of
the black hole accretion disc.
