  For general relativistic, inviscid, axisymmetric flow around Kerr black hole
one may choose different flow thickness. The stationary flow equations can be
solved using methods of dynamical system to get transonic accretion flows ,
i.e, flow infalling in the blackhole that turns supersonic from subsonic with
decreasing radial distance, or vice versa. This transonic flows are obtained by
choosing the particular flow passing through critical points of phase portrait.
For certain flow thickness like the one maintaining conical shape, the sonic
point coincide with the critical point. But there are certain flows maintaining
hydrostatic equilibrium, such as the one described by Novikov-Thorne, where the
sonic point is not same as the critical point. We perturb the flow for both
kind of flow and study the behaviour of linear perturbation which behaves like
massless scalar field in some curved spacetime, known as, analogue space time.
We draw the compactified causal structure, i.e, Penrose Carter diagram for both
kind of analogue metric and prove that for both cases critical points are the
acoustic horizons, whereas in the case where sonic points do not coincide with
critical points, the sonic points are not the acoustic horizon, as one may
expect from the definition of sound speed.
