Linear perturbations of low angular momentum accretion flow in the Kerr metric and the corresponding emergent gravity phenomena

For certain geometric configurations of matter falling onto a rotating black hole, we develop a novel linear perturbation analysis scheme to perform the stability analysis of stationary integral accretion solutions corresponding to the steady state low angular momentum, inviscid, barotropic, irrotational, general relativistic accretion of hydrodynamic fluid. We demonstrate that such steady states remain stable under linear perturbation, and hence, the stationary solutions are reliable to probe the black hole spacetime using the accretion phenomena. We report that a relativistic acoustic geometry emerges out as a consequence of such a stability analysis procedure. We study various properties of that sonic geometry in detail. We construct the causal structures to establish the one to one correspondences of the sonic points with the acoustic black hole horizons and the shock location with an acoustic white hole horizon. The influence of the spin of the rotating black holes on the emergence of such acoustic spacetime has been discussed.

Figure 1

The critical points $r_{\mathrm{crit}}$ (which are transonic points of the monotransonic accretion flow) are plotted as a function of the black hole spin $a$ for the set of values $[{\xi }_0,\gamma ,{\lambda }_0]=[1.1,1.4,2.1]$.

Figure 2

Causal structure of the acoustic spacetime for monotransonic accretion. $t_{+}(r)$ vs $r$, i.e., $z=\text{constant}$ lines are represented by the solid lines, and $t_{-}(r)$ vs $r$, i.e., $w=\text{constant}$ lines are represented by the dashed lines. $t_{\pm }(r)$ are given by Eq. (56). The causal structures are plotted with $[{\xi }_0,\gamma ,{\lambda }_0]=[1.1,1.4,2.1]$ for $a=-0.9,-0.5$, 0, 0.5, 0.9, by rows, from top to bottom. It could be noticed that the acoustic horizon where the $t_{+}(r)$ lines diverges, it coincides with the critical point $r_{\mathrm{crit}}$. This further illustrates that the transonic surface of the accretion flow is indeed the acoustic horizon of the embedded acoustic spacetime.

Figure 3

Mach number $\mathcal{M}$ vs $r$ plot (on the left) and the corresponding causal structure (on the right). The parameters [${\xi }_0=1.002$, $\gamma =1.35$, ${\lambda }_0=3.05$] are the same as where the black hole spin is $a=0.5$ (top panel), $a=0.55$ (middle panel), and $a=0.6$ (bottom panel). The solid lines represents $t_{+}(r)$ vs $r$ lines, and the dashed lines represents the $t_{-}(r)$ vs $r$ lines.