Emergent gravity from nonlinear perturbation of spherical accretion with variable adiabatic index

This paper demonstrates that gravity-like phenomena are not merely artifacts of linear perturbation but emerge from nonlinear higher-order perturbations in spherically accreting astrophysical systems with variable adiabatic indices, resulting in a dynamic effective acoustic spacetime where the acoustic horizon shifts in response to fluctuations in density, temperature, and mass accretion rate.

The Gist

Imagine you are standing by a river. Usually, when we study how water flows, we look at the big picture: how fast the river moves, how deep it is, and where it bends. But what if we wanted to study tiny ripples on the surface of that water?

In the world of physics, there's a fascinating idea called "Analogue Gravity." It suggests that if you look closely at how sound waves move through a flowing fluid (like that river), they behave exactly like light waves moving through the warped space around a black hole. The fluid creates a "fake" gravity, complete with an "acoustic horizon"—a point where the water flows so fast that sound waves can't swim upstream against it, just like light can't escape a black hole.

For a long time, scientists studied these ripples using linear perturbations. Think of this like studying a single, tiny, perfect ripple on a calm pond. It's a simple, straight-line approximation. It works well for small disturbances, but it assumes the water is perfectly calm and the ripple doesn't change the water's behavior.

What This Paper Does
The authors of this paper, Rohit Ghosh and his team, asked a bold question: What happens if the ripple isn't tiny? What if the water is churning, and the ripple is big enough to actually change the flow itself?

They decided to stop looking at just the simple, straight-line ripples and instead looked at non-linear perturbations. In everyday language, this means they studied "big waves" that interact with the river's current in complex ways, rather than just floating passively on top of it.

The Setup: A Cosmic Kitchen
To do this, they imagined a specific cosmic scenario: gas falling into a black hole (accretion). But they didn't use a simple model. They used a "multi-component" soup, meaning the gas is made of different particles (electrons, positrons, and protons) and is extremely hot. In this hot soup, the "stiffness" of the gas (called the adiabatic index) changes depending on the temperature. It's like cooking a sauce where the thickness changes as it heats up, making the math much harder.

The Big Discovery: The Horizon Moves
Here is the main result, explained simply:

1. The "Fake" Gravity is Alive: In the old, simple models, the "acoustic horizon" (the point where sound gets trapped) was a fixed, static line. It was like a painted line on a road. But when the authors added these complex, non-linear effects, they found that the horizon is dynamic. It's more like a living boundary that can wiggle, shift inward, or shift outward.
2. Why it Moves: The position of this horizon depends on a tug-of-war between three things:
   • How much gas is falling in (density).
   • How hot the gas is (temperature).
   • How fast the gas is being sucked in (accretion rate).
     If the temperature fluctuates or the flow rate changes, the "point of no return" for the sound waves moves. The geometry of this fake spacetime isn't static; it breathes and shifts.

The Math Behind the Magic
The team used a mathematical tool called the "acoustic metric." You can think of this as a map that tells sound waves how to travel through the fluid.

• Linear (Old Way): The map was a flat, unchanging grid.
• Non-Linear (New Way): The map itself gets distorted by the ripples. The ripples change the map, and the new map changes how the ripples travel. It's a feedback loop.

Stability Check
The authors also checked if these complex, shifting waves would cause the system to explode or collapse.

• Standing Waves: If the object is a solid star (like a neutron star), the waves bounce back and forth. They found these are stable, like a guitar string vibrating safely.
• Traveling Waves: If the object is a black hole, the waves get sucked in. They found these traveling waves are also stable, provided they are small enough. They behave like a train moving on a track that is slightly shifting but still holds the train on course.

Real-World Connection
To prove their model makes sense, they applied it to Sagittarius A*, the supermassive black hole at the center of our Milky Way galaxy.

• They calculated where the "acoustic horizon" would be for the hot gas falling into it.
• They found it sits very close to the actual event horizon (the real point of no return for light), which matches what we expect from observations.
• They also calculated the temperature of the gas at this horizon. It came out to be incredibly hot (trillions of degrees), which matches what astronomers expect to see in the ionized gas around black holes.

The Takeaway
This paper tells us that the "analogue gravity" we see in fluids isn't just a trick of simple, small ripples. Even when the fluid is churning, hot, and complex, the laws of "fake gravity" still hold up. However, the "landscape" of this gravity is not a rigid stage; it is a dynamic, shifting stage that reacts to the very waves moving across it. This gives scientists a more realistic way to study how black holes and accretion flows behave in the real, messy universe.

Technical Summary

Technical Summary: Emergent Gravity from Nonlinear Perturbation of Spherical Accretion with Variable Adiabatic Index

Problem Statement
The paper addresses a limitation in the existing framework of emergent (analogue) gravity. While it is well-established that linear perturbations of transonic fluid flows can mimic scalar fields propagating in an effective curved spacetime (acoustic geometry), most prior studies have relied on linear perturbation theory and simplified thermodynamic descriptions (e.g., polytropic or isothermal equations of state). The authors aim to demonstrate that gravity-like effects are not merely artifacts of linearization but emerge even through nonlinear higher-order perturbations. Furthermore, they seek to incorporate more realistic astrophysical conditions, specifically relativistic temperatures and multi-component fluid compositions (electrons, positrons, and protons), where the adiabatic index $\Gamma$ is a function of temperature rather than a constant.

Methodology
The authors model a spherically symmetric accretion flow onto a compact astrophysical object (modeled with a Newtonian potential $\Phi = -1/r$). The fluid is treated as a multi-species system with a relativistic equation of state (EoS) derived from Ryu et al. [10] and applied to accretion by Chattopadhyay et al. [11, 12].

1. Fluid Dynamics: The system is governed by the continuity and Euler equations. The authors define the mass accretion rate $f = \rho v r^2$ and derive a wave equation for fluctuations in $f$ and density $\rho$.
2. Acoustic Metric Construction: By extending the acoustic metric formalism beyond the linear regime, they derive an inverse acoustic metric $g^{\mu\nu}$ and a corresponding wave equation $\partial_\mu (g^{\mu\nu} \partial_\nu f) = 0$.
3. Nonlinear Perturbation Scheme: The authors perform a perturbative expansion of the mass accretion rate $f$ and density $\rho$ in powers of a dimensionless parameter $\epsilon$:
   $$f = f_0 + \sum \epsilon^i f_i, \quad \rho = \rho_0 + \sum \epsilon^i \rho_i$$
   They systematically derive the metric components $g^{\mu\nu}_{(n)}$ up to arbitrary order $n$. Crucially, they show that the $n$-th order fluctuation propagates on an effective background metric composed of the sum of metric components up to order $n-1$.
4. Horizon Analysis: The location of the acoustic horizon ($r_H$) is determined by the condition $g_{rr} = 0$ (or $g_{tt} = 0$). The authors derive an expression for the relative shift in the horizon radius ($\delta r_H / r_H$) based on fluctuations in mass accretion rate ($\delta f$), density ($\delta \rho$), and temperature ($\delta \Theta$).
5. Stability Analysis: The stability of the system is analyzed for both standing waves (relevant for objects with physical surfaces like neutron stars) and travelling waves (relevant for black holes) using the WKB approximation for high-frequency modes.

Key Contributions and Results

• Dynamical Analogue Spacetime: The primary theoretical result is that nonlinear perturbations render the analogue geometry dynamical. Unlike linear perturbations which yield a stationary acoustic metric, nonlinear corrections cause the effective metric to evolve. Consequently, the acoustic horizon is not fixed; it can shift inward or outward depending on the relative amplitudes of density, temperature, and mass accretion rate fluctuations.
• Variable Adiabatic Index: The study incorporates a temperature-dependent adiabatic index $\Gamma(\Theta)$, characteristic of relativistic multi-component flows. This introduces additional terms in the metric components (specifically $g_{rr}$) that depend on the derivative of the sound speed with respect to temperature.
• Horizon Shift Conditions: The authors derive specific inequalities (Eqs. 55 and 56) determining whether the acoustic horizon recedes or advances. The shift is driven by the interplay between the relative change in accretion rate and the combined effects of density and temperature fluctuations.
• Stability of Perturbations:
  • Standing Waves: For subsonic flows (relevant to neutron stars), the analysis shows that standing waves are stable under linear perturbations, yielding real oscillation frequencies.
  • Travelling Waves: For black hole accretion (supersonic flow), the WKB analysis demonstrates that travelling waves possess finite amplitude and are stable. The solution reveals two types of components: phase-related and amplitude-related modes.
• Astrophysical Consistency: The authors apply their model to the supermassive black hole Sagittarius A* (Sgr A*). Using representative parameters, they calculate an acoustic horizon at $r_H \approx 2.131 GM/c^2$. This is slightly larger than the Schwarzschild radius ($r_{Sch} \approx 2 GM/c^2$), consistent with the condition that the sound speed is less than the speed of light. The calculated temperature at the acoustic horizon ($\sim 9 \times 10^{11}$ K) aligns with the expected ion temperature range for radiatively inefficient accretion flows, though the authors note this represents an effective ion-dominated temperature rather than the electron temperature constrained by current observations.

Significance and Claims
The paper claims to provide a more realistic framework for investigating the dynamics of analogue spacetimes in astrophysically relevant accretion flows. By moving beyond linear perturbations and constant adiabatic indices, the work bridges the gap between simplified analogue models and the complex thermodynamics of real accretion flows near compact objects.

The authors modestly conclude that while the stability of stationary solutions for general orders of perturbation remains inconclusive due to time-dependent terms, the specific case of linear perturbations with a relativistic EoS is stable. The work establishes that the "emergent gravity" phenomenon is robust enough to persist in nonlinear regimes, where the acoustic geometry becomes dynamic and the horizon location becomes a fluctuating quantity dependent on the fluid's thermodynamic state.