Gravity as emergent phenomena for spherically symmetric black hole accretion of multi-component flow with relativistic equation of state

Aishee Chakraborty; Souvik Ghose; Tapas K. Das; Tuhin Paul

REVIEW 2 major objections 8 minor 42 references

Gravity as emergent phenomena for spherically symmetric black hole accretion of multi-component flow with relativistic equation of state

T0 review · 2 major / 8 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Linear perturbations of spherically symmetric multi-component relativistic accretion onto a Schwarzschild black hole generate a black-hole-like acoustic spacetime whose sound horizon traps acoustic perturbations, and the associated…

desk verdict Solid incremental extension of acoustic-metric accretion to a multi-species EoS, but the surface-gravity curves rest on a formula that doesn't follow from the paper's own metric. read the letter →

arxiv 2501.15676 v3 pith:5FR3T6W5 submitted 2025-01-26 gr-qc astro-ph.HE

Tuhin Paul , Aishee Chakraborty , Souvik Ghose , Tapas K. Das This is my paper · ORCID

classification gr-qcastro-ph.HE

keywords analoguegravityacousticmetricblackholeaccretionmulti-speciesequationofstatetransonicflowsonichorizonsurfaceCarter-Penrosediagram

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish that the ordinary radial infall of a multi-component relativistic gas onto a non-rotating black hole carries a hidden curved spacetime within it. When the steady transonic accretion solution is slightly disturbed, the disturbance obeys the same wave equation as a massless scalar field moving on an effective metric, the acoustic metric, whose null surfaces are sonic horizons. Carter-Penrose diagrams built from the stationary solutions show that this sonic point is a genuine causal horizon that acoustic perturbations cannot cross. The paper then computes the acoustic surface gravity at that horizon and finds it increases with the conserved specific energy of the flow and with the proton fraction of the accreting gas. On this view, real astrophysical accretion flows become analogue-gravity systems in which horizon physics is an emergent, fluid-based phenomenon rather than a purely gravitational one.

What carries the argument

The central object is the acoustic metric, obtained by matching the linearized perturbation equation $\partial_\mu(f^{\mu\nu}\partial_\nu\tilde{\psi})=0$ to the curved-spacetime scalar wave equation $\partial_\mu(\sqrt{-g}\,g^{\mu\nu}\partial_\nu\Phi)=0$, which fixes $g_{\mu\nu}$ up to a conformal factor and yields the line element $ds^2 = -(\rho_0/c_{s0})[-c_{s0}^2 dt^2 + (dx^i-u_0^i\,dt)\delta_{ij}(dx^j-u_0^j\,dt)]$, a Painlevé-Gullstrand form. Its null structure is charted with Carter-Penrose coordinates, and the acoustic surface gravity is extracted from the Killing-vector formula $\kappa = \left|\sqrt{\psi^\mu\psi_\mu/(-g_{rr})}\,(1-c_s^2)^{-1}\,(du/dr-dc_s/dr)\right|$ at $r_c$, simplified to Eq. (83). The multi-species equation of state enters through the variable polytropic index $N$ and adiabatic index $\Gamma$, which make the sound speed, density, and critical-point data functions of temperature and composition. This machinery converts the stability question for the steady flow into a statement about the causal geometry of an emergent spacetime.

What would settle it

Launching a sound pulse just inside the sonic radius in a numerical integration of the linear perturbation equations would settle the trapping claim: if any part of the pulse crosses the sonic radius outward, the acoustic horizon is not an event horizon. Recomputing $\kappa$ at the sonic point with the exact Schwarzschild metric instead of the pseudo-Newtonian potentials would settle the surface-gravity formula: if the resulting band of $\kappa$ values differs significantly from the four-potential results, Eq. (83) is the weak point.

Watch

Extended reading notes

Core claim

The paper claims that, for spherically symmetric accretion described by any of the four post-Newtonian pseudo-Schwarzschild potentials and by a relativistic multi-species equation of state with a radially varying adiabatic index, the linear stability analysis of the stationary transonic solution yields an acoustic metric of Painlevé-Gullstrand form. The sonic point is a saddle-type critical point that becomes a null hypersurface of this acoustic spacetime, so the sound horizon behaves like an event horizon for acoustic perturbations. The acoustic surface gravity, evaluated from Eq. (83), rises monotonically with the specific energy $E_c$ at the critical point and with the proton fraction $\xi$. The paper presents this as a classical analogue-gravity model of black-hole accretion with a more realistic thermodynamics than the polytropic or isothermal equations of state used in earlier work.

Load-bearing premise

The load-bearing premise is that a surface-gravity formula borrowed from earlier analogue-accretion work, simplified by identifying the pseudo-Newtonian potential with part of the acoustic metric and setting a metric component to one, remains valid for the relativistic multi-species equation of state; if that identification fails, the specific $\kappa(E_c,\xi)$ values shown in the figures are unsupported, although the existence of the acoustic horizon itself would not be at stake.

Editorial extensions

If this is right

For all four pseudo-Newtonian potentials considered, the stationary transonic solutions are stable and yield one saddle-type sonic point outside the horizon, so the acoustic-horizon result does not depend on which potential model is chosen.
The Carter-Penrose construction shows the sound horizon is a null hypersurface of the acoustic metric, so acoustic perturbations created inside the horizon cannot escape to infinity.
The acoustic surface gravity $\kappa$ increases with the conserved specific energy $E_c$ and with the proton fraction $\xi$, so hotter or more proton-rich flows possess a larger analogue Hawking temperature.
Standing-wave analysis for subsonic flows and WKB traveling-wave analysis for supersonic black-hole flows both give non-divergent perturbation amplitudes, supporting linear stability of the steady states.
In spherical symmetry only one acoustic horizon forms; obtaining an acoustic white hole would require axisymmetric flow with angular momentum, as the paper notes in its conclusion.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the surface-gravity identification survives contact with full general relativity, the analogue Hawking temperature $T_H=\kappa/2\pi$ for each pseudo-potential could be compared with the actual Schwarzschild Hawking temperature, giving a quantitative measure of how much horizon thermodynamics is emergent in accretion flows.
The monotonic rise of $\kappa$ with proton fraction $\xi$ suggests that acoustic surface gravity could in principle serve as a composition diagnostic for accreting gas, though detecting an analogue temperature directly is far beyond current instruments.
Applying the same perturbation machinery to pseudo-Kerr potentials with angular momentum should produce multiple sonic points and a shock-induced acoustic white hole; testing that prediction would extend the same method without changing its core mechanism.
Replacing the pseudo-Newtonian potentials by the exact Schwarzschild metric in Eq. (83) would provide a direct check of whether the four-potential results bracket the true relativistic surface gravity; the paper does not perform this comparison.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know before you spend time on it. The paper's core—applying the standard acoustic-metric formalism to spherically symmetric Bondi accretion with the Ryu–Chattopadhyay multi-species relativistic EoS—is a solid, incremental step and the derivation in Secs. V–VI holds up. But the surface-gravity results in Figs. 5–6, which are the paper's main quantitative claims, rest on a borrowed formula (Eq. 83) that is not connected to the acoustic metric the paper itself constructs. Those κ(E_c, ξ) curves should be treated as unsupported unless the authors re-derive them.

What's good: the EoS with variable adiabatic index is genuinely new to this analogue-gravity setting. The critical-point analysis for the four pseudo-Newtonian potentials is standard and appears consistent; the acoustic metric (Eq. 58) is the usual Unruh metric for a nonrelativistic fluid, and the Carter–Penrose diagram correctly shows the sonic horizon as a null surface. The perturbation equations in Sec. V follow the familiar route and the stability conclusions are not obviously wrong.

The soft spot is real and it's in Sec. VII. Eq. (82) is quoted from Ref. [9] without derivation, and Eq. (83) is obtained by substituting sqrt(ψψ)=sqrt(1+φ) and g_rr=-1. But the acoustic metric has g_rr = -ρ0/c_s0 and g_tt = (ρ0/c_s0)(c_s0^2-u0^2), which vanishes at the sonic point. So the Killing-norm factor in Eq. (82) should vanish there, and the sqrt(1+φ) term has no origin in the acoustic metric. This looks like an unjustified conflation of the pseudo-Newtonian background potential with the emergent spacetime. The existence of the acoustic horizon and the causal structure are unaffected, but the quantitative κ values are not.

If I were refereeing, I'd ask for a self-contained derivation of κ from Eq. (58), perhaps using the standard Killing-horizon definition adapted to the conformally related acoustic metric. The standing/traveling-wave section adds little and can be trimmed. There are also minor sign and notation slips (e.g., the metric prefactor in Eq. 57 vs. Eq. 58) that need cleanup.

Who this is for: researchers working on analogue gravity in astrophysical accretion, who will find the EoS extension useful. It's not a breakthrough, but it's a legitimate incremental contribution. I'd send it to peer review with a request for major revision on the surface-gravity part.

Recommendation: engage with it; fix the κ derivation and it's publishable.

Referee Report

2 major / 8 minor

Summary. The paper models spherically symmetric, inviscid, irrotational accretion onto a non-rotating black hole in a pseudo-Newtonian framework, using the four standard pseudo-potentials of Paczynski-Wiita, Artemova-Bjornsson-Novikov (two versions), and Nowak-Wagoner, with the Ryu-Chattopadhyay multi-species relativistic equation of state (electron-positron-proton composition xi, radially varying adiabatic index). It constructs stationary transonic solutions, classifies the sonic (critical) points as saddle-type fixed points via a dynamical-system analysis, and derives the linearized perturbation equation for the velocity potential, which yields the standard Unruh-Visser acoustic metric. Carter-Penrose diagrams are constructed and identify the sonic point as a null acoustic horizon. The acoustic surface gravity is then evaluated with a formula quoted from earlier work (Eq. (83)) as a function of the specific energy E_c and of xi for all four potentials (Figs. 5-6). Standing-wave (globally subsonic) and WKB traveling-wave (transonic) analyses are used to argue stability of the stationary solutions.

Significance. If the quantitative surface-gravity results were fully supported, the paper's main value would be to extend analogue-gravity treatments of spherical accretion from polytropic/isothermal flows to a multi-species relativistic equation of state with variable Gamma, with a comparison of the sonic-point structure across the four pseudo-Newtonian potentials. The positive content is real: the transonic solutions and critical-point conditions for this EoS are derived consistently and in enough detail to reproduce the phase portraits; the acoustic wave equation (49) and the metric (57)-(58) follow the standard Unruh-Visser framework; and the causal-structure identification of the sonic point as an acoustic Killing horizon is independent of the disputed surface-gravity formula. The main weakness is that the headline quantitative predictions, the kappa(E_c, xi) curves, are evaluated through a quoted formula whose connection to the metric derived in Sec. 6 is not demonstrated; the stability claim additionally relies on a WKB argument that is not uniform at the sonic point.

major comments (2)

[§7, Eq. (83); Figs. 5–6] The quantitative results of the paper — the acoustic surface gravity as a function of E_c and xi shown in Figs. 5 and 6 — rest entirely on Eq. (83), quoted from Refs. [9,38] and evaluated with the substitutions sqrt(psi^mu psi_mu) = sqrt(1+phi) and g_rr = -1. These substitutions are not compatible with the acoustic metric derived in Sec. 6. For the line element (58), g_tt = (rho0/c_s0)(c_s0^2 - u0^2), g_tr = (rho0/c_s0)u0, and g_rr = -rho0/c_s0; hence sqrt(-g_rr) is a conformal factor, not 1, and the Killing-vector norm sqrt(psi^mu psi_mu) = sqrt(|g_tt|) vanishes at the sonic point u0^2 = c_s0^2, so the surface gravity must come from the standard limiting procedure for Killing horizons. The factor sqrt(1+phi) is the pseudo-Newtonian background potential from the Bernoulli integral (18)-(29); it does not appear anywhere in (58), so substituting it for the Killing-norm factor conflates the background gravitational model with the emergent acoustic spacetime. I note that the manuscript's own Carter-Penrose construction uses the quantity kappa = (u0' - c_s0')|_{rc} (Eq. (71)), which arises naturally from the leading-order expansion (65) of the metric function, whereas Eq. (83) adds the unjustified factor sqrt(1+phi)/(1-c_s^2). The authors should derive the surface gravity directly from the metric (58), recompute Figs. 5-6, and either justify or drop the extra factors; the existence of the acoustic horizon and the causal-structure results do not depend on this formula.
[§8B, Eqs. (90)–(99)] The traveling-wave stability argument is incomplete at the sonic point. The WKB phase integrals in Eq. (95), k_{-1} = i∫dr/(u0 ∓ c_s0), have a logarithmically divergent integrand at rc because one of the two denominators u0 ± c_s0 vanishes for infall (u0 = -c_s0), so the ordering check (97)-(99), performed only for large r, does not control the expansion near the sonic point; the standard connection-formula treatment of the singular point of Eq. (90) is missing. Since Sec. 8 concludes that 'in both cases ... the stability of the steady state solution is ensured', the stated stability claim is not established by the material presented, although the linearized wave equation (49) itself is not in question.

minor comments (8)

[§5, Eqs. (52)–(54)] The statement that the perturbed mass accretion rate Sigma-tilde(r,t) obeys the wave equation (53) with the coefficient matrix (54) is asserted without derivation, and the matrix (54), proportional to (u0/Sigma0), is not the same as f^mu-nu in Eq. (50), proportional to (rho0/c_s0^2); the two differ by a non-constant factor unless additional identities are supplied. Since the acoustic metric is derived from the psi-tilde equation (49), this step should be proven, attributed, or removed.
[§5–§7, notation] The symbol psi is used for the velocity potential from Eq. (46) onward and again for the Killing vector in Eq. (81), and the potential Phi in Eq. (82) becomes phi in Eq. (83) without comment; introducing xi^mu for the Killing vector and a single symbol for the pseudo-potential would remove the ambiguity.
[§3–§4] Eqs. (29) and (34) are identical and both appear in the text; one of them should be deleted.
[§6A, Fig. 4] The caption of Fig. 4 refers to a 'green, dashed region' and a 'pink, solid region', but the figure as printed is a monochrome line drawing without such labels; the regions should be marked directly in the figure, and the flow parameters (E_c, xi, potential) used for the diagram should be stated.
[§6A, Eqs. (59)–(77)] The derivation of A_± and of the null coordinates drops the conformal factor rho0/c_s0 of the metric (58) without comment; a single sentence noting that null geodesics (and hence the causal structure) are unchanged by the conformal factor would make this step explicit.
[§2–§3, numerics] The numerical scheme behind Figs. 1-3 and 5-6 is not described: the integration domain, the outer-boundary value of Theta (equivalently, how E_c fixes the initial data), and the treatment of the critical-point boundary conditions are not stated, which hampers reproducibility.
[Eqs. (41), (B7)] The matrix entries in Eqs. (41) and (B7) contain rendering inconsistencies (Theta appears as theta in the (1,2) entries), and the final trace and determinant expressions in Eqs. (43) and (B9) should be re-checked against these matrices before publication.
[References] Refs. [8] and [32] are the same paper (T. K. Das, Class. Quantum Grav. 21, 5253 (2004)) and are cited twice under two numbers.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: acoustic-metric and horizon derivations are self-contained; the self-cited surface-gravity formula in Eq. (83) is a validity caveat, not a circular reduction.

full rationale

The derivation chain is largely self-contained: the multi-species EoS (Sec. II) and the four pseudo-potentials are stated inputs; the stationary transonic solutions and critical-point conditions are derived from conservation of energy and mass (Eqs. 18-27); the perturbation equation (48) yields the acoustic metric (56)-(58) by direct identification with the massless scalar-field equation; the Carter-Penrose causal structure uses the standard null-coordinate construction with kappa from Eq. (71); and the standing/traveling-wave stability analysis is an independent eigenvalue/WKB calculation. The sonic horizon is located at the critical point u=c_s by the standard definition, not by a hidden assumption. The only caveat is the quantitative surface-gravity result: Eq. (83) is imported from self-cited Refs. [9,38] with substitutions sqrt(psi psi)=sqrt(1+2Phi) and g_rr=-1 that do not follow from the paper's own line element (58), where g_rr=-rho0/c_s0 and psi psi=g_tt vanishes at the sonic point. This undermines the numerical kappa(E_c, xi) curves, but it is an unsupported/incorrect import rather than a circular reduction: no fitted parameter is renamed as a prediction, and the acoustic-horizon/causal-structure claim is independent of Eq. (83). Hence no significant circularity; score 2 only for the minor self-citation used for that imported formula.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claim depends on the pseudo-Newtonian modeling of gravity, the assumption that the multi-species flow remains adiabatic and barotropic, and the borrowed surface gravity formula. The energy E_c and composition xi are input parameters that are varied to produce the parameter scans; they are not fitted to external data.

free parameters (2)

xi (electron-positron-proton composition) = xi = 1 in Fig. 5; xi = 0.3 to 1.0 in Fig. 6
The ratio of proton to electron number density is a model input that controls the equation of state (Eqs. 2-4); its variation drives the surface gravity trend in Fig. 6.
E_c (specific energy at critical point) = E_c range 1.000 to 1.025; demonstrative E_c = 1.0007
The conserved specific energy selects the stationary integral solution; surface gravity is plotted against it in Fig. 5. It is an input parameter, not fitted to data.

assumptions (4)

domain assumption Pseudo-Newtonian potentials (PW, ABN1, NW, ABN2) adequately represent the Schwarzschild gravitational field for the accretion flow
The entire stationary flow and acoustic metric are computed using these potentials (Eqs. 14-17) instead of the full Schwarzschild metric; no direct comparison with the exact metric is made.
domain assumption The flow is inviscid, irrotational, and barotropic, so pressure perturbations relate to density perturbations via delta p = c_s^2 delta rho
This is needed to turn the linearized continuity and Euler equations into the acoustic wave equation (Eq. 48) and the mass accretion rate perturbation equation (Eq. 53).
domain assumption Charge neutrality with constant composition xi is maintained throughout the flow
Equation (2) enforces n_e- = n_e+ + n_p+, and xi is treated as fixed, which gives the mass density (Eq. 4) and the EoS (Eq. 5).
domain assumption The acoustic surface gravity formula of Refs. [9,38] (Eq. 82) remains valid for the multi-species relativistic EoS
The quantitative results (Figs. 5-6) depend on this formula, which the paper adopts without re-deriving for the variable-gamma EoS.

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Cite this review

Pith. "Pith review of Gravity as emergent phenomena for spherically symmetric black hole accretion of multi-component flow with relativistic equation of state." pith.science (2026). https://pith.science/paper/5FR3T6W5

@misc{pith2026250115676,
  author       = {Pith},
  title        = {Pith review of: Gravity as emergent phenomena for spherically symmetric black hole accretion of multi-component flow with relativistic equation of state},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5FR3T6W5}},
  note         = {Machine review of arXiv:2501.15676}
}

read the original abstract

We investigate analogue gravity phenomena arising as a result of the linear perturbation of the spherically symmetric accretion flows onto non rotating black holes, where the gravitational field is determined by a set of post Newtonian pseudo Schwarzschild black hole potentials and the infaling matter is described by a relativistic multi-species equation of state. The stationary transonic integral accretion solutions corresponding to the steady state of aforementioned type of accreting systems are constructed and the stability analysis of such solutions are performed through the time dependent linear perturbation of the accretion flow. Such linear stability analysis leads to the formation of a black hole like sonic metric embedded within the infalling matter. The acoustic horizons are then identified by constructing the causal structure, i.e., the Carter-Penrose diagrams. The variation of the analogue surface gravity corresponding to the aforementioned sonic metric has been studied as a function of various parameters governing the accretion flow.

Figures

Figures reproduced from arXiv: 2501.15676 by the authors.

**Figure 2.** FIG. 2: Phase profile for different pseudo-potentials. There is a slight variation of the sonic points (for a [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3: Variation of ∆ with [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: FIG. 4: Carter-Penrose Diagram for analogue spacetime obtained from the stationary accretion solution with PW [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: FIG. 5: Variation of surface gravity (defined through eq. (83)) with energy [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: FIG. 6: Variation of surface gravity ( [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [9]

Barcel´ o, S

C. Barcel´ o, S. Liberati, and M. Visser, Living Reviews in Relativity 14, 3 (2011)

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[2] [1]

For the case of spherically symmetric Bondi flow, the only available velocity component is the radial velocity com- ponent which we denote simply with u

are the components of the 3-velocity u0 (u0 = −∇ψ0). For the case of spherically symmetric Bondi flow, the only available velocity component is the radial velocity com- ponent which we denote simply with u. The continuity equation for this spherically symmetric flow is given by ∂ρ ∂t + 1 r2 ∂ ∂r ρur2 = 0 (51) For steady state we get, ρur2 = constant, whic...

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[3] [2]

acoustic metric

−u1 0 −u2 0 −u3 0 −u1 0 1 0 0 −u2 0 0 1 0 −u3 0 0 0 1   (57) The corresponding line element is given by: ds2 = gµνdxµdxν = − ρ0 cs0 − cs0 2dt2 + (dxi − ui 0dt)δij(dxj − uj 0dt) (58) This line element is similar to the Schwarzschild line element in Painlev´ e-Gullstrand coordinates and has a nonzero curvature. However, since eq. (49) holds for the acous...

work page 1981

[4] [3]

i0, the Spatial Infinity , is the endpoint of the ψ image of all space-like curves in ( M , g)

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[5] [4]

J +, formally known as the Future Causal Infinity is the endpoint of the ψ image of all future directed causal curves in ( M , g)

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[6] [5]

Surface gravity

J −, Past Causal Infinity , is the endpoint of the ψ image of all past directed causal curves in ( M , g). The sound horizon can be readily identified as a null hypersurface from the fig. (4). Evidently, no acoustic perturbation, created within the sound horizon (green, dashed region in the fig. (4) can cross the sound horizon and escape to a large distan...

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[7] [6]

Moncrief, Astrophysical Journal, Part 1, vol

V. Moncrief, Astrophysical Journal, Part 1, vol. 235, Feb. 1, 1980, p. 1038-1046. 235, 1038 (1980)

work page 1980

[8] [7]

W. G. Unruh, Physical Review Letters 46, 1351 (1981)

work page 1981