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arxiv: 2604.23592 · v1 · submitted 2026-04-26 · 🌀 gr-qc

Gravitational Collapse of an Inhomogeneous Fluid in Rastall Theory

Akbar Jahan , Naser Sadeghnezhad , Amir Hadi Ziaie This is my paper

Pith reviewed 2026-05-08 05:43 UTC · model grok-4.3

classification 🌀 gr-qc
keywords gravitational collapseRastall gravitynonsingular solutionsbounceanisotropic fluidinhomogeneous spacetimeapparent horizonweak energy condition
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The pith

Rastall gravity yields exact nonsingular solutions for inhomogeneous fluid collapse that bounce without apparent horizons.

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

The paper studies spherically symmetric collapse of an inhomogeneous fluid with anisotropic pressures in Rastall gravity, using linear equations of state for radial and tangential components. By fixing the Rastall parameter so the effective radial pressure vanishes, the authors derive exact time-dependent solutions in which collapsing shells reach a minimum radius and reverse into expansion. These solutions lack the singularities that appear in the homogeneous limit and avoid formation of trapped surfaces, so the bounce remains uncovered by any apparent horizon. The weak energy condition holds for the constructed metrics and density profiles.

Core claim

The paper finds a class of exact nonsingular solutions in which the matter shells undergo a collapse process in a contracting regime, reach a bounce point, and then enter an expanding phase. It is found that for the obtained solutions, the trapped surface formation can be avoided and consequently, the bounce event is not covered by the apparent horizon.

What carries the argument

The Rastall parameter tuned to nullify effective radial pressure, which permits closed-form metric functions describing contraction to a finite minimum radius followed by expansion.

If this is right

  • Collapsing shells reach a finite minimum radius and subsequently expand.
  • No apparent horizon forms during the evolution, leaving the bounce region untrapped.
  • The weak energy condition remains satisfied for the inhomogeneous profiles and chosen parameters.
  • The solutions are regular at the bounce and contain no spacetime singularity.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same parameter tuning could be tested in homogeneous Rastall collapse to confirm consistency with the inhomogeneous case.
  • Avoidance of trapped surfaces leaves open the possibility that the bounce dynamics could in principle produce observable exterior effects.

Load-bearing premise

The Rastall parameter must be chosen to make effective radial pressure vanish and must stay consistent with the given inhomogeneous density profiles and energy conditions for the entire evolution.

What would settle it

Explicit computation of the apparent-horizon radius as a function of time for the derived scale factors, checking whether it ever coincides with or lies inside the matter-shell radius at or before the bounce.

Figures

Figures reproduced from arXiv: 2604.23592 by Akbar Jahan, Amir Hadi Ziaie, Naser Sadeghnezhad.

Figure 1
Figure 1. Figure 1: FIG. 1: Plot of mass function for view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Evolution of the area radius for view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Evolution of collapse velocity for the same values of model parameters as of Fig.(2). view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Evolution of the difference between physical radii of the shells for the same values of model parameters as of Fig.(2). view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Behavior of the bounce and shell crossing times against radial coordinate. view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Behavior of the ratio view at source ↗
read the original abstract

We study spherically symmetric gravitational collapse of an inhomogeneous fluid with anisotropic energy momentum tensor (EMT) in Rastall gravity. Considering a linear equation of state (EoS) for the fluid profiles, i.e., $p_r=w_r\rho$ and $p_\theta=w_\theta\rho$, we try to build and investigate non-singular collapse scenarios for which, the spacetime singularity that appears in the homogeneous case~\cite{ahz2019}, is absent. We therefore set the Rastall parameter in such a way that the effective radial pressure vanishes. This helps us to obtain a class of exact nonsingular solutions in which the matter shells undergo a {collapse} process in a contracting regime, reach a bounce point, and then enter an expanding phase. We further investigate formation of trapped surfaces during the dynamical evolution of the collapsing body. {It is found} that for the obtained solutions, {the} trapped surface formation can be avoided and consequently, the bounce event is not covered by the apparent horizon. Validity of weak energy condition (WEC) is also examined for the obtained solutions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 3 minor

Summary. The manuscript studies spherically symmetric gravitational collapse of an inhomogeneous anisotropic fluid with linear equations of state p_r = w_r ρ and p_θ = w_θ ρ in Rastall gravity. By fixing the Rastall parameter to make the effective radial pressure vanish, the authors construct a class of exact nonsingular solutions in which collapsing shells reach a bounce and then expand, with no trapped surfaces or apparent horizons forming and with the weak energy condition satisfied.

Significance. If the consistency of the fixed Rastall parameter with the inhomogeneous density profiles and energy conditions can be rigorously established across the full evolution, the work would supply concrete, explicit examples of bounce dynamics in modified gravity that avoid both singularities and horizons. This is of interest for exploring how inhomogeneity and anisotropy modify collapse outcomes relative to the homogeneous case in general relativity.

major comments (3)
  1. The choice of the Rastall parameter to enforce vanishing effective radial pressure is the load-bearing step for obtaining the exact nonsingular bounce solutions (as stated in the abstract and the paragraph introducing the solutions). It is necessary to demonstrate explicitly that this single fixed value satisfies the Rastall field equations identically for arbitrary inhomogeneous ρ(r,t) at every stage of the evolution (pre-bounce contraction, at the bounce, and post-bounce expansion) without forcing additional constraints on the metric functions or the anisotropy parameters w_r and w_θ.
  2. The avoidance of trapped surfaces and apparent horizons is shown only for the solutions constructed with the tuned parameter. A clearer statement is required on whether this avoidance is a generic feature of inhomogeneous fluids in Rastall theory or whether it is an artifact of setting the effective radial pressure to zero; the current presentation leaves open the possibility that the result is parameter-specific rather than a robust prediction.
  3. The weak energy condition is examined for the obtained solutions, but the check must be extended to confirm that no violations arise at the bounce point itself, where the density reaches its maximum and the anisotropy (controlled by w_r and w_θ) could potentially produce transient violations during the transition from contraction to expansion.
minor comments (3)
  1. The abstract contains a minor grammatical issue: 'the bounce event is not covered by the apparent horizon' would read more clearly as 'the bounce is not covered by an apparent horizon.'
  2. The metric ansatz and coordinate conventions should be restated explicitly at the start of the section presenting the field equations to aid readability for readers unfamiliar with the homogeneous reference case.
  3. A brief quantitative comparison (e.g., via a table or plot) between the inhomogeneous bounce solutions and the singular homogeneous solutions of the cited reference would strengthen the discussion of the role of inhomogeneity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments on our manuscript. We address each major comment below and will incorporate revisions to strengthen the presentation.

read point-by-point responses

  1. Referee: The choice of the Rastall parameter to enforce vanishing effective radial pressure is the load-bearing step for obtaining the exact nonsingular bounce solutions (as stated in the abstract and the paragraph introducing the solutions). It is necessary to demonstrate explicitly that this single fixed value satisfies the Rastall field equations identically for arbitrary inhomogeneous ρ(r,t) at every stage of the evolution (pre-bounce contraction, at the bounce, and post-bounce expansion) without forcing additional constraints on the metric functions or the anisotropy parameters w_r and w_θ.

    Authors: We agree this verification is essential. The Rastall parameter is fixed to a constant value determined only by w_r and w_θ such that the effective radial pressure vanishes identically. Substituting this choice into the Rastall field equations yields a reduced system whose solutions for the metric functions admit a family of inhomogeneous ρ(r,t) profiles. These profiles are not completely arbitrary but are constrained by the reduced equations; once chosen consistently with the metric, the original field equations are satisfied by construction at all times. We will add an explicit back-substitution of the fixed parameter and the resulting ρ(r,t) into the full set of Rastall equations, evaluated at representative times before, at, and after the bounce, to confirm no further constraints arise. revision: yes

  2. Referee: The avoidance of trapped surfaces and apparent horizons is shown only for the solutions constructed with the tuned parameter. A clearer statement is required on whether this avoidance is a generic feature of inhomogeneous fluids in Rastall theory or whether it is an artifact of setting the effective radial pressure to zero; the current presentation leaves open the possibility that the result is parameter-specific rather than a robust prediction.

    Authors: We do not claim the result is generic. Horizon avoidance in our solutions follows directly from the vanishing effective radial pressure, which keeps the expansion of outgoing null geodesics positive throughout the evolution. We will revise the relevant sections to state explicitly that this feature is tied to the specific choice of the Rastall parameter that enforces p_r^eff = 0 and is therefore not asserted to hold for arbitrary values of the parameter or for other inhomogeneous fluids in Rastall gravity. A brief remark will be added noting that other parameter choices would generally require numerical integration and lie outside the scope of the present analytic study. revision: yes

  3. Referee: The weak energy condition is examined for the obtained solutions, but the check must be extended to confirm that no violations arise at the bounce point itself, where the density reaches its maximum and the anisotropy (controlled by w_r and w_θ) could potentially produce transient violations during the transition from contraction to expansion.

    Authors: We will extend the WEC analysis as requested. At the bounce, where the scale factor reaches its minimum and density is maximum, the conditions reduce to ρ ≥ 0 (satisfied by construction), ρ + p_r ≥ 0, and ρ + p_θ ≥ 0. Because p_r^eff = 0, the first combination is simply ρ > 0. For the tangential direction the inequality holds for the range of w_θ we consider, with no sign change during the contraction-to-expansion transition. We will add an explicit evaluation of all three WEC inequalities at the bounce time t_b for representative values of w_r and w_θ, confirming that no transient violations occur. revision: yes

Circularity Check

0 steps flagged

No significant circularity; parameter choice is explicit model assumption for constructing solutions.

full rationale

The paper explicitly sets the Rastall parameter to nullify effective radial pressure as a deliberate step to enable exact nonsingular solutions with bounce behavior under the linear anisotropic EoS. This is presented transparently as a means to investigate specific scenarios rather than a hidden reduction or prediction derived from general equations. Solutions are then obtained by solving the field equations with this choice and the inhomogeneous density profiles; properties such as WEC validity and absence of trapped surfaces are verified on the resulting metrics. The self-citation to the homogeneous case is only contextual background and not load-bearing. No step equates a claimed result to an input by construction, and the derivation remains self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The claim rests on a tuned Rastall parameter and assumed linear equations of state plus spherical symmetry; these are introduced to enable the bounce without independent derivation from first principles.

free parameters (2)
  • Rastall parameter
    Explicitly set to make effective radial pressure vanish, enabling nonsingular solutions.
  • w_r and w_theta
    Constants in the linear equation of state for radial and tangential pressures.
axioms (2)
  • domain assumption Spherically symmetric spacetime metric for the collapsing body
    Standard assumption for modeling gravitational collapse in spherical symmetry.
  • domain assumption Linear equation of state p_r = w_r rho and p_theta = w_theta rho
    Chosen for the fluid profiles to allow exact solutions.

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