Global Acoustic Geometry of Michel Flow in Painlevé-Gullstrand Coordinates

Abstract

The acoustic metric of spherical accretion onto a Schwarzschild black hole (the Michel flow) in standard coordinates is singular at the gravitational horizon, and inside the acoustic horizon, the coordinate time loses its evolutionary meaning, which hinders the analysis of the supersonic shell. In the present work, the acoustic geometry is reformulated in Painlevé-Gullstrand (PG) coordinates, which are regular at both horizons. The integration of null geodesics over the entire interval from infinity to the gravitational horizon reveals a qualitative difference in the opacity mechanisms: at the acoustic horizon, the outgoing characteristic freezes, whereas at the gravitational horizon, both characteristics are finite and negative, ensuring the inward tilt of the light cone without degeneracy. The generalized Binet equation remains regular inside the supersonic zone, allowing trapped rays to be traced down to the gravitational horizon. It is established that the coordinate infall time of a radial phonon is finite and, in the cold flow limit, scales as $a_{\infty}^{-3}$. The construction is verified by the exact reproduction of the invariant quantities of the exterior region and by agreement with the tunneling formalism of analog radiation. Principal results The global acoustic geometry of the Michel flow is constructed in Painlevé-Gullstrand coordinates, providing a single smooth coordinate patch regular at both the acoustic ($r_{\mathrm{a}}$) and gravitational ($r_{\mathrm{g}}$) horizons. Two qualitatively distinct opacity mechanisms are identified: the freezing of the outgoing characteristic at $r_{\mathrm{a}}$ (acoustic horizon) versus the complete inward tilt of the light cone with finite negative characteristics at $r_{\mathrm{g}}$ (gravitational horizon). A regular generalized Binet equation is derived for the entire domain $r_{\mathrm{g}} \le r < \infty$, allowing trapped non-radial phonons to be integrated across the acoustic horizon down to $r_{\mathrm{g}}$. The coordinate infall time $T_{\mathrm{fall}}$ of a radial phonon from $r_{\mathrm{a}}$ to $r_{\mathrm{g}}$ is shown to be finite, scaling as $a_{\infty}^{-3}$ in the cold flow limit (with the effective power-law index dependent on the adiabatic index $\gamma$). The construction is verified by exact reproduction of exterior invariants (surface gravity $\kappa$, acoustic photon sphere $r_{\mathrm{aph}}$, critical impact parameter $b_{\mathrm{c}}$) and is shown to be consistent with the Parikh-Wilczek tunneling formalism for analog Hawking radiation. Companion studies in this series Acoustic geometry and sound-ray dynamics in relativistic spherical accretion Perturbation spectrum of Michel flow: acoustic, vortical, and entropy modes Files The upload contains the author's preprint in English and Russian: compiled PDF versions and LaTeX sources. Keywords Michel accretion; acoustic horizon; phonon geodesics; black holes; analogue gravity; general relativity; Painlevé-Gullstrand coordinates