A theoretical preprint reports that the point where an accreting fluid crosses from subsonic to supersonic flow can sit in very different places depending on the fluid used around a modeled deformed black hole. The authors say deformation of the Konoplya-Zhidenko (KZ) spacetime changes the radial location and structure of this critical region, with the differences depending on the fluid's equation of state, the rule linking pressure to energy density. The paper examines steady, spherical accretion onto a static KZ black hole, so its findings are mathematical model results, not observations of matter around a real black hole.
How the model defines a critical flow
In the model, a sonic point is defined where inward radial velocity equals local sound speed. It marks the transonic transition, the point at which the flow changes from subsonic to supersonic. To obtain the flow equations, the authors apply conservation laws for particle number and energy-momentum, then express fluid motion through a Hamiltonian formulation, a mathematical framework for tracking the dynamics.
The test matter is treated as a perfect fluid. Its pressure is set proportional to energy density through a linear equation of state, with a dimensionless parameter greater than zero and no greater than one. The calculations examine four named model cases: ultra-stiff, ultra-relativistic, radiation and sub-relativistic fluids. None is an empirical sample; the configurations are deterministic theoretical choices.
Different fluids, different reported points
The clearest contrast appears in the ultra-stiff case. The reported sonic point coincides with the event horizon. Its sonic radius is 2.11208, its listed velocity is 1, and its Hamiltonian value is 0.0502521. In this calculation, the transition to transonic flow therefore occurs at the modeled horizon itself.
For the ultra-relativistic fluid, the reported critical point is at a radius of 0.95469, with a velocity of -1.49358 and a Hamiltonian value of 0.0502521. The radiation-fluid calculation places its sonic point at 3.12777, with a radial velocity of 0.0765349, a three-velocity of 0.0167319 and a Hamiltonian value of 1.64172. These figures are outputs of the respective model solutions.
For the sub-relativistic case, the prose reports a sonic radius of 3.62375, a radial velocity of 2.23029, a three-velocity of 0.919143 and a Hamiltonian value of 0.62636. Taken together, the reported cases place the critical flow at different radii and with different velocity values, depending on the fluid choice.
There is an internal numerical warning in the paper. Its tabulated results repeat the ultra-stiff figures, but give the ultra-relativistic Hamiltonian as 0.505231 rather than 0.0502521. For radiation, they list a three-velocity of 0.167319 rather than 0.0167319. For the sub-relativistic case, they give a radius of 3.63275 and a three-velocity of 0.914943 rather than the values in the prose. The paper therefore contains different reported values for some of its KZ cases.
What remains a model result
The authors also compare the KZ solution with other modeled black-hole geometries. Their qualitative conclusion is that the deformation changes the radial location and structure of the critical region, but that the differences between geometries depend on the fluid equation of state. The calculations therefore do not point to one geometry-independent accretion pattern within the cases examined.
A separate calculation says the derived accretion rate becomes singular when its denominator vanishes. In the supplied evidence, this is a mathematical feature of the derivation, not an observational result. More broadly, the work remains a theoretical calculation of steady spherical flow using a perfect test fluid, and the authors state that no data were generated or analyzed.
The document is an arXiv preprint, version 1, dated 28 Aug 2026. The authors state that the study had no external funding and no conflicts of interest bearing on the results or conclusions.
Paper data and sources
Original title: Spherical Accretion onto Konoplya Zhidenko Black Holes: Sonic Point Analysis and Fluid Dynamics
Authors: M H Waheed, M Z A Moughal
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
Original paper · Full text