On the spin dependence of the emergent gravity phenomena as observed in axially symmetric black hole accretion with spatially varying adiabatic index

This article investigates stationary, low-angular-momentum, axially symmetric accretion onto a black hole with a spatially varying adiabatic index and shows that the resulting multi-transonic flow supports stable stationary shock waves as well as an emergent acoustic geometry featuring both black-hole and white-hole horizons, whose surface gravities are determined by local variations in the speed of sound.

The Gist

Imagine a black hole not merely as a cosmic vacuum cleaner, but as a massive, rotating funnel. Around this funnel swirls a vortexing river of hot gas (composed of electrons, protons, and positrons) falling inward. This study examines precisely how this gas behaves as it is drawn in, but with some special twists that make the story far more interesting.

Here is a breakdown of their findings using simple analogies:

1. The gas is "picky" about its temperature

In many earlier studies, scientists assumed the gas behaved like a simple, uniform fluid whose "stiffness" (the adiabatic index) remained constant everywhere.

• The twist in the work: The authors realized that as the gas gets closer to the black hole, it becomes hotter and its internal chemistry changes. It is like a crowd running down a hill: at the top, they walk calmly; halfway down, they jog; at the bottom, they sprint and sweat. Their "stiffness" changes depending on location. The authors developed a model where this property changes as the gas approaches the black hole, making the simulation more realistic.

2. The "speed brake" (shocks)

Normally, gas falls in smoothly, accelerating until it breaks the "sound barrier" (becomes supersonic).

• The twist in the work: Because the black hole rotates and the gas is "picky" about its temperature, the flow does not simply accelerate smoothly. It can get stuck, hit a "speed brake," and suddenly slow down before accelerating again.
• The analogy: Imagine a car driving down a steep hill. It accelerates, hits a sudden patch of mud (the shock), brakes drastically, and then must accelerate again to finish the hill. The work maps exactly where these "mud patches" (shocks) occur and how the black hole's rotation influences them.
  • Rotation effect: The faster the black hole rotates, the farther outward the "mud patch" appears. The rotation acts like a centrifugal force, pushing the gas outward and forcing the shock to occur farther from the center.

3. The "traffic lights" (critical points)

To understand where the gas accelerates or decelerates, the authors searched for "critical points."

• The analogy: Think of these as traffic lights on the highway of space.
  • Saddle points: These are like green lights, where the flow can smoothly transition from slow (subsonic) to fast (supersonic).
  • Center points: These are like red lights or roundabouts, where the flow gets stuck in a loop and cannot pass through smoothly.
• The finding: The work shows that under the right conditions, the gas flow can encounter three of these traffic lights. It passes the outer one, gets stuck at the middle one, and then passes the inner one. This creates a complex "multi-transonic" flow where the gas accelerates, decelerates, and accelerates again.

4. The "map of sound" (emergent gravity)

This is the most mind-bending part. The authors investigated how tiny waves (sound waves) move through this swirling gas flow.

• The analogy: Imagine the gas as a river. If you throw a stone in, ripples (sound) spread through the water. If the river flows faster than the waves can swim upstream, the waves get trapped and carried downstream.
• The discovery: The authors found that the swirling gas flow creates its own "map" of space and time for these sound waves.
  • Acoustic black holes: At points where the gas flows faster than sound, sound waves cannot escape. These act exactly like the event horizon of a black hole, but for sound instead of light.
  • Acoustic white holes: At the "mud patch" (the shock), the gas suddenly slows down. This creates a barrier from which sound waves can only emerge out, but cannot enter. This is the opposite of a black hole; it is a "white hole" for sound.

5. The "shadow" of the black hole (causal structure)

Finally, the authors drew a map (a Carter-Penrose diagram) to show how these sound waves connect different parts of the universe.

• The result: They found that the flow generates a four-part structure remarkably similar to the theoretical map of a black hole, but with an additional "white hole" section in the middle.
  • Region 1: The quiet outer world.
  • Region 2: The fast-flowing zone before the shock (trapped).
  • Region 3: The compressed zone after the shock (where sound can escape).
  • Region 4: The innermost zone falling into the black hole (forever trapped).

Summary

The work claims that if you model the accretion disk of a rotating black hole with a realistic, varying gas temperature:

1. The gas flow becomes complex, with multiple accelerations and decelerations.
2. The rotation of the black hole pushes the "shock waves" farther outward.
3. These flows create a hidden, "acoustic" universe within the gas, where sound behaves exactly like light near a real black hole, complete with "black holes for sound" and "white holes for sound."

They achieved this by mathematically proving that these solutions are stable (they do not decay) and by mapping the "sound horizons" using the same tools astronomers use to map real black holes.

Technical Summary

Technical Summary: Spin-Dependence of Emergent Gravity in Axially Symmetric Accretion onto Rotating Black Holes with Spatially Varying Adiabatic Index

Problem Statement
This work investigates the dynamics of low-angular-momentum, axially symmetric accretion onto rotating (Kerr) black holes. While previous studies on multi-transonic, shock-bearing accretion often assumed a barotropic equation of state with a constant adiabatic index ($\Gamma$), this work addresses the more realistic scenario where $\Gamma$ varies as a function of radial distance. This variation arises from radiative processes and the presence of multi-component flows (electrons, positrons, and protons). The aim of the study is to characterize stationary transonic solutions, their stability, and the resulting phenomena of "emergent gravity"—specifically the acoustic spacetime geometry embedded within the fluid flow. The authors employ a post-Newtonian pseudo-Kerr potential (Artemova-Bjornsson-Novikov) to mimic the effects of the Kerr metric on the equatorial plane while preserving algebraic tractability.

Methodology
The authors formulate the stationary equations for mass conservation, Euler dynamics, and the first law of thermodynamics for a frictionless multi-component flow. The system is analyzed using three distinct geometric configurations for the accretion disk:

1. Vertical Equilibrium (VE): Flow in hydrostatic equilibrium in the vertical direction.
2. Conical Flow (CF): Flow with a constant opening angle.
3. Constant Height (CH): Flow with a fixed disk thickness.

The underlying equations are reduced to a system of coupled first-order nonlinear differential equations for temperature ($\Theta$) and radial velocity ($\vartheta$). The authors apply dynamical systems theory to perform an analysis of critical points and identify sonic points where the flow transitions between subsonic and supersonic regimes. They derive conditions for the existence of single or multiple critical points and classify their nature (saddle point or center) using Jacobian matrices and eigenvalue analyses.

To investigate stability and emergent gravity, the stationary solutions are subjected to linear radial perturbations. This process leads to a second-order wave equation for the perturbation variable, which is rewritten in covariant form $\partial_\mu (\Sigma^{\mu\nu} \partial_\nu f') = 0$. This enables the extraction of an effective "acoustic metric" ($g_{\mu\nu}$), treating fluid perturbations as massless scalar fields propagating on a curved acoustic spacetime. The causal structure of this emergent geometry is subsequently analyzed using Kruskal-like extensions and Carter-Penrose diagrams.

Main Contributions and Results

• Multi-Critical Parameter Space: The study confirms that for specific ranges of specific energy ($E$) and specific angular momentum ($\lambda$), the accretion flow admits up to three critical points. The parameter space is divided into regions supporting single critical points (O- and I-regions) and regions supporting three critical points (A- and W-regions). In the multi-critical regions, the inner and outer critical points are identified as saddle points (allowing transonic solutions), while the intermediate point is a center.

• Shock Formation and Spin-Dependence: Within the multi-critical range, a subset of parameters permits global accretion solutions that traverse both the outer and inner sonic points while undergoing a standing Rankine-Hugoniot shock. The work systematically analyzes the effect of the black hole's spin parameter ($a$):

  • An increase in the spin parameter shifts the shock position ($r_{sh}$) to larger radii due to enhanced frame-dragging effects (centrifugal support).
  • The shock strength (ratio of Mach numbers before and after the shock), the density compression ratio, and the temperature jump all decrease monotonically as the spin increases.
  • These trends hold for all three geometric models (VE, CF, CH), although quantitative values vary depending on the specific disk geometry.

• Linear Stability: The linear perturbation analysis shows that the stationary transonic solutions are stable. The perturbation series converges for sufficiently high frequencies, and the amplitude remains finite throughout the flow domain. Near the sonic points, the perturbations exhibit a logarithmic divergence in the Kruskal coordinate, analogous to behavior near a black hole event horizon.

• Emergent Acoustic Spacetime and Causal Structure: The linear perturbation analysis reveals that the multi-transonic, shock-bearing flow possesses an embedded acoustic metric with three distinct horizons:

  1. An acoustic black hole horizon at the outer sonic point ($x_{out}$), where outgoing acoustic rays are frozen.
  2. An acoustic white hole horizon at the shock position ($r_{sh}$), where perturbations can emerge from the compressed region behind the shock but cannot enter.
  3. A second acoustic black hole horizon at the inner sonic point ($x_{in}$), which irreversibly captures inward-propagating perturbations.

  The resulting Carter-Penrose diagram for the conical flow model reveals a causal topology with four regions. This structure is formally identical to the maximally extended Schwarzschild geometry, supplemented by a white hole sector. The shock acts as a past (white hole) horizon, while the two sonic points act as nested future event horizons.

• Surface Gravity: The authors derive a generalized expression for the acoustic surface gravity ($\kappa$) that accounts for the spatial variation of the local speed of sound, generalizing the original Unruh formula which assumed a constant speed of sound.

Significance
The work demonstrates that accretion onto rotating black holes with a spatially varying adiabatic index provides a robust physical environment for analog gravity. It shows that the spin of the central black hole modulates both the hydrodynamic structure (shock position and thermodynamic jumps) and the causal geometry of the emergent acoustic spacetime (surface gravities and Penrose diagram topology). The authors suggest that these spin-dependent acoustic features could, in principle, leave imprints in observational signatures such as quasi-periodic oscillations (QPOs) and spectral features of accreting black holes. The work provides a generalized template for investigating such flows, capable of accommodating future improvements to pseudo-Kerr potentials without requiring a fundamental restructuring of the solution scheme.