[Paper Review] Lagrangian Description of Accreting Black Hole Systems in the Context of Emergent Spacetime

Satadal Datta, Tapas K. Das|arXiv (Cornell University)|Oct 13, 2019

Astrophysical Phenomena and Observations9 references4 citations

TL;DR

This paper presents a Lagrangian framework for analyzing accreting black hole systems in emergent spacetime scenarios, using Lagrangian Perturbation Theory (LPT) to show that only barotropic flows can support locally irrotational motion necessary for emergent gravity. It demonstrates that ultra-short-wavelength eikonal waves propagate like waves in a static uniform medium from the perspective of co-moving observers, and derives the null geodesic condition for sound rays in the geometric acoustics limit, confirming the emergence of a Lorentzian acoustic metric.

ABSTRACT

We make use of the Lagrangian description of fluid motion to highlight certain features in the context of spacetime geometry as emergent phenomena in fluid systems. By using Lagrangian Perturbation Theory (LPT), we find that if the flow is not barotropic it will not be locally irrotational, and as a result, emergent gravity can not be realised for such flow. Our work gives a new perspective of examining the perturbations in a fluid from a different approach (other than Eulerian approach) which is the way of using Lagrangian Perturbation Theory. We make use of Lagrangian description of motion to examine the propagation of Eikonal wave (wave having very short wavelength) from the reference frame of the observer moving with the background flow. We find that waves of ultra short wavelength propagate similar to waves in a static uniform medium in the near vicinity of the observer. We restrict ourselves to nonrelativistic flows in astrophysical black hole accretion.

Motivation & Objective

• To examine emergent spacetime phenomena in accreting black hole systems using a Lagrangian description instead of the standard Eulerian approach.
• To investigate the conditions under which emergent gravity can arise in fluid-based analog models, particularly focusing on rotational and irrotational flow features.
• To analyze the propagation of high-frequency, short-wavelength (eikonal) waves in a moving fluid from the co-moving frame of reference.
• To derive the acoustic metric and null geodesic condition in the geometric acoustics limit using Lagrangian perturbation theory.
• To establish that only barotropic flows allow for locally irrotational motion, a necessary condition for realizing emergent gravity in physical acoustics.

Proposed method

• Employ Lagrangian Perturbation Theory (LPT) to relate Eulerian perturbations in density, pressure, and velocity to Lagrangian displacements of fluid elements.
• Define the Lagrangian displacement field δ(x,t) to express perturbations in fluid quantities as δ-dependent corrections to the background fields.
• Use the barotropic condition p′/ρ′ = cₛ₀² to relate perturbations in pressure and density, ensuring thermodynamic consistency.
• Derive the wave equation for the velocity potential in the eikonal approximation, assuming a high-frequency, short-wavelength limit.
• Apply the eikonal approximation by setting Ψ′ = a e^{iϕ}, with slowly varying amplitude a and rapidly varying phase ϕ, to obtain the ray equation and dispersion relation.
• Derive the effective acoustic metric h_{μν} from the wave equation and show that in the eikonal limit, the phase fronts obey k_μ k^μ = 0 and ∂_μ k^μ = 0, leading to null geodesics.

Experimental results

Research questions

• RQ1 Under what conditions can emergent gravity be realized in fluid-based analog models of black hole spacetime?
• RQ2 How does the Lagrangian description of fluid motion differ from the Eulerian approach in analyzing wave propagation and emergent geometry?
• RQ3 Why is the barotropic condition essential for the emergence of a locally irrotational flow and, consequently, for the realization of emergent gravity?
• RQ4 How do ultra-short-wavelength waves behave in a moving fluid from the perspective of a co-moving observer?
• RQ5 What is the form of the emergent spacetime metric in the geometric acoustics (eikonal) limit, and how is it derived from the Lagrangian framework?

Key findings

• Only barotropic flows can be locally irrotational; non-barotropic flows inherently possess vorticity, which prevents the realization of emergent gravity in the physical acoustics framework.
• In the eikonal limit, waves of ultra-short wavelength propagate as if in a static uniform medium from the viewpoint of an observer co-moving with the background flow.
• The dispersion relation for sound waves in the eikonal limit is ω = ±cₛ₀k + v₀·k, with the '+' sign corresponding to the physical wave mode, indicating Doppler-shifted phase propagation.
• The group velocity of eikonal waves is given by dxᵢ/dt = cₛ₀ ñ + v₀, showing that sound rays follow a velocity addition rule consistent with Galilean relativity.
• The effective spacetime metric emerges as ds² = -(cₛ₀² - v₀²)dt² - 2v₀·dx dt + dx², which describes a null geodesic structure independent of the conformal factor.
• The wave equation in the eikonal limit leads to k_μ k^μ = 0 and ∂_μ k^μ = 0, confirming that sound rays follow null geodesics in the emergent acoustic spacetime.