Acoustic Geometry and Sound-Ray Dynamics in Relativistic Spherical Accretion

Abstract

In analogue gravity, sound waves in a moving fluid travel along the null geodesics of an effective curved spacetime, giving a controlled setting for studying horizon physics in astrophysical accretion. The relativistic acoustic metric for spherical accretion is well known and has been used in stability and quasi-normal-mode studies, but the geometric optics of sound rays in Michel flow has received little systematic attention. The acoustic metric for a perfect fluid accreting onto a Schwarzschild black hole (Michel flow) is constructed, and high-frequency sound rays are treated as its null geodesics. From the linearized continuity–Euler–energy system, the causal surfaces of the geometry are identified-the acoustic horizon (which coincides with the sonic point), the stationary-limit surface, and the ergoregion-and a generalized Binet equation is derived that separates ray deflection into Einstein, refractive, and acoustic-wind parts. A local slowdown factor allows the comparison of acoustic and vacuum Shapiro delays, the location of the acoustic photon sphere, and the relation of the analogue Hawking scale to hydrodynamic gradients. Results for polytropic indices $\gamma=4/3$ and $5/3$ show how the thermodynamic and flow parameters of the accretion enter acoustic lensing, time delay, and the analogue Hawking temperature, providing a geometric basis for future studies of finite-frequency acoustic, vortical, and entropy modes. Principal results The acoustic event horizon is shown to coincide exactly with the sonic critical point of the Michel flow: substitution of the critical-point conditions $\upsilon_{\mathrm{c}}^{2} = 1/(4r_{\mathrm{c}})$, $a_{\mathrm{c}}^{2} = \upsilon_{\mathrm{c}}^{2}/(1-3\upsilon_{\mathrm{c}}^{2})$ into the radial core component yields $\eta^{rr}(r_{\mathrm{c}}) = 0$, while the stationary-limit surface remains distinct, bounding a genuine acoustic ergoregion. The generalized Binet equation $\mathrm{d}^{2}\varrho/\mathrm{d}\phi^{2} + \varrho = F_{\mathrm{E}} + F_{\mathrm{R}} + F_{\mathrm{W}}$ decomposes ray deflection into the Einstein (vacuum curvature), refractive (sound-speed gradients), and acoustic-wind (radial accretion) forces; numerical integration for $a_{\infty} = 0.1$ demonstrates that the wind dominates near the horizon, refraction is repulsive throughout, and the capture-barrier maximum fixes the acoustic photon sphere. The acoustic photon sphere is strongly displaced outward relative to the vacuum value $r_{\mathrm{ph}} = 3/2$: $r_{\mathrm{aph}} \approx 6.10$ for $\gamma = 5/3$ and $r_{\mathrm{aph}} \approx 20.0$ for $\gamma = 4/3$; the ratio $r_{\mathrm{aph}}/r_{\mathrm{a}} \approx 1.39$ ($\gamma = 5/3$) and $\approx 1.42$ ($\gamma = 4/3$) remains nearly constant across $a_{\infty} \in [0.05,\, 0.20]$, so the acoustic shadow tracks the horizon at a fixed multiple set mainly by the equation of state. The acoustic Shapiro delay, diagnosed by the local slowdown factor $N_{\pm} = f/|v_{\pm}|$, exhibits freeze-out of outgoing rays at the horizon ($N_{+} \sim 10^{3}$) and convergence of both branches to the geometric-optics plateau $1/a_{\infty}$ at large radii; the excess delay exceeds the vacuum counterpart by $R \sim 10^{2}$--$10^{3}$. The surface gravity at the sonic horizon is expressed through hydrodynamic gradients, $\kappa = \frac{f^{2}(r_{\mathrm{a}})}{2[f(r_{\mathrm{a}})-\beta(r_{\mathrm{a}})\upsilon^{2}(r_{\mathrm{a}})]}\,\frac{\mathrm{d}}{\mathrm{d}r}(a-\beta\vartheta\upsilon)\big|_{r=r_{\mathrm{a}}}$, so that the analogue Hawking temperature $T_{\mathrm{H}} = \kappa/2\pi$ probes the rate at which the flow crosses the sonic point; the analogue horizon is much colder than vacuum, $2\kappa = T^{\mathrm{ac}}_{\mathrm{H}}/T^{\mathrm{vac}}_{\mathrm{H}} \approx 2.76\cdot 10^{-2}$ for $\gamma = 5/3$ and $\approx 7.02\cdot 10^{-3}$ for $\gamma = 4/3$. The tabulated surface gravity reproduces the normalization of the quasi-normal spectrum of Chaverra et al. with accuracy better than $0.5\%$, yielding the convention-independent relations $\mathrm{Re}\,s \approx -0.77\,\kappa$ and $\mathrm{Im}\,s \approx (0.42 + 1.21\,\ell)\,\kappa$ for $\ell \geq 1$. Files The upload contains the author's preprint in English and Russian: compiled PDF versions and LaTeX sources. Keywords acoustic metric; analogue gravity; Michel accretion; Shapiro delay; sound rays; acoustic horizon; photon sphere; Hawking temperature