Membrane Paradigm Approach to the Johannsen-Psaltis Black Hole

Abstract: The Johannsen-Psaltis (JP) black hole is a phenomenologically viable metric obtained by judiciously deforming the Kerr black hole such that the metric is asymptotically flat and causal and is consistent with the weak field, post-Newtonian tests of gravity; however, it has additional hairs besides its mass and angular momentum. As it deviates from the Kerr black hole in the strong coupling regime, it is a useful metric to test the Kerr hypothesis that states that all astrophysical black holes are of the Kerr type. Here we give a membrane description of this black hole that effectively amounts to replacing the observable part of the black hole with a fluid with thermal properties. A timelike membrane, a stretched horizon local in time is assumed to exist. This membrane is expected to approximate the null event horizon that is highly non-local in time. We derive the energy-momentum tensor of the fluid and all the transport coefficients using the action formulation to the membrane as advocated by Parikh and Wilczek. In the fluid description, one observes that the finiteness of the transport coefficients constrains the additional hairs of the Johannsen-Psaltis black hole. Analytically continuing the pressure of the fluid to all values of the radial coordinate $r$, one obtains interesting Van der Waals-type behavior of the pressure of the fluid which diverges at the radius of the outer ergosphere, lending support, from the membrane paradigm's perspective to the claim that relativistic astrophysical jets are produced by the ergoregion of the black hole.