Emergent gravity from Michel flow with position dependent adiabatic index

This paper investigates spherically symmetric general relativistic Bondi accretion with a position-dependent adiabatic index by classifying its transonic solutions via dynamical systems theory, demonstrating their linear stability, and deriving the corresponding emergent acoustic spacetime and its causal structure to bridge astrophysical, dynamical, and analogue gravity perspectives.

The Gist

Imagine a river flowing toward a waterfall. Far upstream, the water moves slowly and calmly. As it gets closer to the edge, it speeds up, eventually rushing over the falls faster than the speed of sound (if water could make sound waves in that way). In the universe, black holes act like these waterfalls, pulling in gas and dust. This paper studies exactly that process: gas falling into a black hole, but with a special twist.

Here is a simple breakdown of what the researchers did and found, using everyday analogies.

1. The Setup: A River with Changing Rules

Usually, when scientists model gas falling into a black hole (called "Michel flow"), they assume the gas behaves like a simple, unchanging fluid. They assume its "stiffness" (how hard it is to compress) stays the same everywhere.

The Twist: The authors realized that in the real universe, gas near a black hole gets incredibly hot. Far away, it's cool and behaves one way; close to the black hole, it's scorching hot and behaves differently.

• The Analogy: Imagine driving a car where the rules of physics change depending on your location. In the suburbs, the car handles normally. But as you approach the city center, the car suddenly becomes lighter and faster to steer. The authors built a model where the "rules" of the gas change as it gets closer to the black hole, making the model more realistic.

2. The Critical Point: The "Waterfall Edge"

The gas starts far away moving slower than sound (subsonic) and ends up moving faster than sound (supersonic) right before it disappears into the black hole. Somewhere in between, it hits a "critical point" where its speed exactly matches the speed of sound.

• The Analogy: Think of a skier going down a hill. At the top, they are slow. At the bottom, they are fast. There is one specific spot where they are going exactly 20 mph. The researchers mapped out this journey. They found that for the gas to flow smoothly from slow to fast without breaking or stopping, it must pass through this specific "critical point."
• The Finding: Using math tools usually reserved for studying complex systems (like weather patterns or stock markets), they proved that this critical point acts like a "saddle." Just like a horse saddle has a high point in the middle that curves up one way and down the other, the flow is stable in some directions but unstable in others. This confirms that the flow is physically possible and behaves as expected.

3. The Big Discovery: A "Shadow" Black Hole Inside the Gas

This is the most fascinating part. The researchers didn't just study the gas; they studied what happens if you poke the gas. If you create a small ripple (a sound wave) in the falling gas, how does that ripple move?

• The Analogy: Imagine the gas is a giant, invisible trampoline. If you drop a marble (a sound wave) on it, the marble rolls. But because the gas is falling so fast toward the black hole, the trampoline itself is tilted.
• The Result: The researchers found that the ripples in the gas behave exactly like light rays moving near a real black hole.
  • The Sonic Horizon: Just as a real black hole has an "event horizon" (a point of no return for light), the falling gas has a "sonic horizon." Once a sound wave crosses this point, it is swept inward faster than it can swim outward. It is trapped.
  • The "Emergent" Gravity: The paper calls this "emergent gravity." It means that even though the gas is just normal matter, the way the sound waves move looks and acts exactly like they are moving in a curved spacetime created by gravity. The gas creates its own miniature, fake black hole for sound waves to fall into.

4. Testing the Stability: Will the Wave Break?

The researchers wanted to know: Is this "fake black hole" stable? If you shake the gas, does the sound wave explode, or does it settle down?

• The Analogy: Imagine balancing a pencil on its tip. If you nudge it, it falls. That's unstable. Now imagine a marble in a bowl. If you nudge it, it wobbles but stays in the bowl. That's stable.
• The Finding: They proved that these sound waves are like the marble in the bowl. Whether the wave is standing still (like a standing wave on a guitar string) or traveling far away, it remains stable. It doesn't blow up or disappear; it just flows along with the gas.

5. The Map of the "Shadow" Universe

To visualize this, the authors drew a "Carter-Penrose diagram."

• The Analogy: This is like a map of a city that shows you can't go back once you cross a certain bridge. They mapped out the "sonic spacetime" and showed that it has two distinct regions:
  1. The Outside: Where sound can travel in any direction.
  2. The Inside: Where sound is dragged inward so fast it can never escape.
     This map proves that the "fake black hole" inside the gas has the exact same structure as a real black hole.

Summary

In short, this paper takes the complex math of gas falling into a black hole, adds realistic details about how the gas heats up, and discovers something amazing: The falling gas creates its own miniature universe for sound waves.

Inside this gas, sound waves get trapped by a "sonic horizon" that mimics a real black hole's event horizon. The researchers proved this "fake gravity" is stable and behaves mathematically just like the real thing, offering a way to study the mysteries of black holes using the physics of flowing fluids.

Technical Summary

Technical Summary: Emergent Gravity from Michel Flow with Position-Dependent Adiabatic Index

Problem Statement
The paper addresses the dynamics of spherically symmetric, general relativistic accretion (Michel flow) onto a non-rotating Schwarzschild black hole. While previous studies of analogue gravity in accretion systems often assumed a polytropic equation of state (EoS) with a constant adiabatic index ($\Gamma$), this work recognizes that for large-scale astrophysical flows under strong gravity, $\Gamma$ is unlikely to remain invariant. Due to thermal momentum deposition by outgoing photons and the transition from non-relativistic to thermal-relativistic regimes, $\Gamma$ varies with radial distance. The authors aim to construct stationary transonic solutions for such a flow using a multi-component EoS (electrons, positrons, and protons) where $\Gamma$ is a function of radial position, and subsequently investigate the emergent acoustic geometry and stability of these solutions.

Methodology
The study employs a multi-faceted approach combining general relativistic hydrodynamics, dynamical systems theory, and linear perturbation analysis:

1. Mathematical Formulation: The authors utilize the CR EoS (Chattopadhyay & Ryu), a closed-form analytic approximation to the Chandrasekhar–Synge EoS, which accounts for multi-species fluids (electrons, positrons, protons) and allows $\Gamma$ to vary as a function of the dimensionless temperature parameter $\Theta$. The governing equations include the continuity equation, energy-momentum conservation, and the Euler equation in a static, spherically symmetric spacetime.
2. Stationary Solutions: The system of coupled first-order differential equations for fluid velocity ($u$) and temperature ($\Theta$) is solved numerically using the Runge-Kutta method. Boundary conditions are set at the critical point ($r_c$), where the flow velocity equals the local sound speed ($u = c_s$).
3. Dynamical Systems Analysis: The transonic flow is analyzed as a dynamical system. The nature of the critical points is classified by linearizing the equations of motion around the critical point and examining the eigenvalues of the resulting Jacobian matrix.
4. Analogue Gravity Formalism: To study emergent gravity, the mass accretion rate ($\psi$) is perturbed linearly ($\psi = \psi_o + \epsilon \psi'$). This perturbation is shown to satisfy a covariant wave equation, allowing the identification of an effective acoustic metric ($G_{\mu\nu}$) embedded within the accreting fluid.
5. Causal Structure and Stability: The causal structure of the emergent spacetime is visualized using Carter-Penrose diagrams. The acoustic surface gravity ($\kappa$) is calculated, and the Riemann curvature tensor is computed to verify the curvature of the acoustic manifold. Finally, the stability of the background flow is tested against both standing and traveling wave perturbations.

Key Contributions and Results

• Transonic Solutions with Variable $\Gamma$: The authors successfully constructed stationary integral transonic solutions for Michel flow where the adiabatic index varies radially. Phase portraits in the Mach number–radial distance plane demonstrate the transition from subsonic to supersonic flow.
• Critical Point Classification: Using dynamical systems theory, the critical points of the flow are classified as saddle points. This is confirmed by the trace of the Jacobian matrix being zero and the determinant being negative ($\Delta_c < 0$), yielding real, opposite eigenvalues. This result is consistent with the behavior of inviscid transonic flows.
• Emergent Acoustic Geometry: The linear perturbation of the mass accretion rate yields a wave equation identical to that of a massless scalar field in a curved spacetime. The resulting acoustic metric is of the Painlevé–Gullstrand type.
  • The acoustic horizon is identified precisely at the sonic surface ($r = r_c$) of the background flow.
  • The Carter-Penrose diagram reveals a two-region structure: an exterior subsonic region (Region I) and an interior supersonic region (Region II), separated by the acoustic horizon, mimicking the causal structure of a Schwarzschild black hole.
• Acoustic Surface Gravity and Curvature:
  • The acoustic surface gravity ($\kappa$) is derived as a function of the conserved specific energy ($E$) and fluid composition ($\xi$). The analysis shows that $\kappa$ increases monotonically with $E$.
  • Decomposition of $\kappa$ reveals contributions from geometric, kinematic, and thermodynamic terms, with the thermodynamic contribution (arising from the position-dependent EoS) being the most significant.
  • The calculation of the Riemann curvature tensor confirms that the emergent acoustic spacetime is a genuinely curved manifold, distinct from the background spacetime geometry.
• Stability Analysis: The stationary solutions are found to be stable under linear radial perturbations.
  • Standing waves: The analysis of the eigenvalue $\omega^2$ for standing waves in the subsonic region yields $\omega^2 > 0$, indicating stability.
  • Traveling waves: A WKB approximation for high-frequency traveling waves shows that the perturbation amplitude remains non-divergent and exhibits a monotonically shrinking behavior as the fluid approaches the origin, further confirming stability.

Significance and Claims
The paper claims to investigate relativistic black hole accretion from three distinct perspectives: astrophysical, dynamical systems, and analogue gravity. By incorporating a position-dependent adiabatic index via the CR EoS, the work provides a more realistic model for multi-component accretion than constant-$\Gamma$ models.

The primary significance lies in demonstrating that the analogue gravity phenomenon—where fluid perturbations mimic fields in curved spacetime—persists even when the equation of state is complex and radially varying. The study establishes that the acoustic horizon corresponds exactly to the physical transonic surface and that the emergent spacetime possesses non-trivial curvature and surface gravity dependent on thermodynamic properties.

The authors modestly frame their work as an extension of existing analogue gravity literature. They note that while analogue models are typically derived via linear perturbation, recent work suggests this is not merely an artifact of linearity. Consequently, this paper serves as a foundational step for their future work, which will extend this analysis to nonlinear perturbations of the Michel flow with variable $\Gamma$. The current results confirm the robustness of the stationary solutions and the validity of the emergent acoustic metric under linear perturbations within the general relativistic framework.