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

Carter-Penrose diagrams for emergent spacetime in axisymmetrically accreting black hole systems