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arxiv: 2605.17615 · v1 · pith:W4L4D56Mnew · submitted 2026-05-17 · 🌀 gr-qc

Junction Conditions and Gravitational Collapse in Scalar-Tensor-Vector Gravity

Debanjan Debnath , Anant Badal , Kaushik Bhattacharya This is my paper

Pith reviewed 2026-05-19 22:53 UTC · model grok-4.3

classification 🌀 gr-qc
keywords gravitational collapsejunction conditionsscalar-tensor-vector gravitythin shellReissner-Nordström solutionblack hole formationFLRW interior
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The pith

In Scalar-Tensor-Vector Gravity, a collapsing FLRW interior matched through an STVG-charged shell to a static RN-like exterior forms horizons in finite proper time.

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

The paper derives junction conditions for Scalar-Tensor-Vector Gravity that relate the metric and field derivatives across a thin shell. These conditions are then used to match an interior Friedmann-Lemaître-Robertson-Walker spacetime filled with baryonic matter and dark energy to an exterior static spherically symmetric Reissner-Nordström-like solution. The matching shows that the shell's STVG charge permits the collapse to continue until an RN-like horizon forms, and this occurs after only finite proper time for observers on the shell. Two explicit models are constructed, one ending at an extremal configuration and the other approaching a sub-extremal black hole as seen from infinity.

Core claim

Starting from the STVG action, the junction conditions are obtained by requiring continuity of the metric and appropriate jumps in the derivatives of the scalar, vector, and gravitational fields across the shell. When these conditions are imposed on a collapsing thin shell carrying STVG charge, the interior FLRW geometry can be joined smoothly to the exterior RN-like geometry, and the resulting dynamics allow the shell to reach the horizon radius in finite proper time.

What carries the argument

The thin-shell junction conditions that enforce continuity of the metric while fixing the jumps in the derivatives of the scalar and vector fields, with the shell's STVG charge providing the necessary surface stress-energy to match the interior FLRW to the exterior RN-like solution.

If this is right

  • Horizon formation occurs after finite proper time for the collapsing matter.
  • Both extremal and sub-extremal RN-like end states are reachable depending on the initial charge and mass parameters.
  • The presence of dark energy in the interior does not prevent the shell from crossing the would-be horizon.
  • Asymptotic observers at infinity record the formation of an event horizon after finite coordinate time in the chosen models.

Where Pith is reading between the lines

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

  • The same junction framework could be applied to other modified-gravity theories that admit static charged solutions.
  • Numerical evolution of the shell equation without the static-exterior assumption would test whether horizon formation remains timely.
  • Observational signatures of the final charge-to-mass ratio might distinguish STVG collapse from standard general-relativistic collapse.

Load-bearing premise

The exterior region is assumed to remain a static, spherically symmetric RN-like solution of the STVG equations throughout the collapse.

What would settle it

An explicit integration of the shell's proper-time equation of motion that yields infinite proper time to the horizon radius when the exterior is taken to be the derived STVG solution rather than an assumed RN-like form.

Figures

Figures reproduced from arXiv: 2605.17615 by Anant Badal, Debanjan Debnath, Kaushik Bhattacharya.

Figure 1
Figure 1. Figure 1: The above panel shows the variation of R(τ ) (Fig. [1a]), matter-energy density ρM (Fig. [1b]), energy density ρDE (Fig. [1c]) and the energy density of the shell σΣ with proper time (Fig. [1d]) in the interior spacetime region. Here we have taken the values of various constants and parameters as follows: sin χ0 = 1/ √ 2, G0 = 1, M = 1, α = 0.6 and Q2 = αG0M2 and G = G0(1 + α). The initial conditions are c… view at source ↗
Figure 2
Figure 2. Figure 2: The above figure shows the variation of the horizon function, [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The above panel shows the variation of R(τ ) (Fig. [3a]), matter-energy density ρM (Fig. [3b]), energy density ρDE (Fig. [3c]) and the energy density of the shell σΣ with proper time (Fig. [3d]) in the interior spacetime region. Here we have taken the values of various constants and parameters as follows: sin χ0 = 1/ √ 2, G0 = 1, M = 1.2, G = 1.6 and Q2 = GM2 . This corresponds to an extremal RN-like solut… view at source ↗
Figure 4
Figure 4. Figure 4: The above figures show the horizon function. For the parameters we have chosen, the horizon function [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
read the original abstract

We formulate the junction conditions for Scalar-Tensor-Vector Gravity (STVG/MOG), proposed by J.~W.~Moffat. Using these conditions, the theory of gravitational collapse is constructed. In the collapsing process, an interior Friedmann-Lema\^itre-Robertson-Walker (FLRW) spacetime with baryonic matter and dark energy is matched with an exterior static, spherically symmetric Reissner--Nordstr\"{o}m (RN)-like spacetime through a shell that carries STVG-charge. Starting from the standard STVG action, we derive the junction conditions across a boundary that relate the values of the various field quantities and their derivatives across the matching surface. Using the matching conditions and the nature of the collapsing shell, it is shown that a gravitational collapse can proceed in the present situation, and one can have RN-like horizon formation in finite proper time. We present two simplified models of gravitational collapse in this article: one ends up as an extremal RN-like black hole, and the other tends to collapse towards a sub-extremal RN-like black hole, as observed by an asymptotic observer at an infinite distance away from the collapsing system.

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

2 major / 2 minor

Summary. The paper derives junction conditions for Scalar-Tensor-Vector Gravity (STVG) from the standard action and uses them to model gravitational collapse. An interior FLRW spacetime containing baryonic matter and dark energy is matched across a thin shell carrying STVG charge to an exterior static, spherically symmetric RN-like solution. The authors conclude that collapse proceeds to RN-like horizon formation in finite proper time and present two simplified models, one ending as an extremal RN-like black hole and the other as a sub-extremal RN-like black hole as seen by an asymptotic observer.

Significance. If the junction conditions are correctly obtained and the static exterior remains valid, the work supplies a concrete matching framework for collapse in STVG, extending standard thin-shell techniques to a theory with an additional vector field. The two explicit models give falsifiable predictions for horizon formation timescales that could be compared with numerical simulations in the GR limit.

major comments (2)
  1. [Matching to exterior spacetime and collapse models] The central construction assumes the exterior remains exactly the static RN-like vacuum solution of the STVG equations while the charged shell collapses. Because the vector field ϕ^μ is dynamical and sourced by the matter on the shell, the moving boundary can induce time-dependent components that back-react on the metric. No explicit junction conditions for ϕ^μ or demonstration that a time-independent solution is preserved throughout the collapse are provided; this assumption is load-bearing for the claim of RN-like horizon formation in finite proper time.
  2. [Junction conditions derivation] The abstract states that the junction conditions are derived from the standard STVG action, yet the provided text gives no explicit expressions for the Israel-type conditions on the metric, scalar, or vector field (or their derivatives) across the shell. Without these equations it is impossible to verify that the matching is consistent with the modified field equations or that the interior FLRW can be joined smoothly to the assumed exterior.
minor comments (2)
  1. [Abstract] The abstract refers to an 'RN-like' exterior without quoting the explicit metric form or the value of the STVG charge parameter; adding the line element and the relation between the shell charge and the exterior parameters would improve clarity.
  2. [Simplified models] The two simplified models are described only qualitatively. Including the explicit time evolution of the shell radius or the proper-time coordinate at which the horizon is reached would make the finite-time claim easier to assess.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major points below and will revise the text to improve clarity and completeness.

read point-by-point responses

  1. Referee: [Matching to exterior spacetime and collapse models] The central construction assumes the exterior remains exactly the static RN-like vacuum solution of the STVG equations while the charged shell collapses. Because the vector field ϕ^μ is dynamical and sourced by the matter on the shell, the moving boundary can induce time-dependent components that back-react on the metric. No explicit junction conditions for ϕ^μ or demonstration that a time-independent solution is preserved throughout the collapse are provided; this assumption is load-bearing for the claim of RN-like horizon formation in finite proper time.

    Authors: We agree that explicit verification of the static exterior is important. In the thin-shell limit the exterior is the vacuum STVG solution by construction, with the conserved STVG charge on the shell and spherical symmetry ensuring no time-dependent back-reaction is induced. We will add the explicit junction condition for ϕ^μ together with a short paragraph demonstrating preservation of the static exterior throughout the collapse. revision: yes

  2. Referee: [Junction conditions derivation] The abstract states that the junction conditions are derived from the standard STVG action, yet the provided text gives no explicit expressions for the Israel-type conditions on the metric, scalar, or vector field (or their derivatives) across the shell. Without these equations it is impossible to verify that the matching is consistent with the modified field equations or that the interior FLRW can be joined smoothly to the assumed exterior.

    Authors: The junction conditions are obtained in the main text by integrating the STVG field equations across the shell. To make verification straightforward we will insert a dedicated subsection that lists the full set of explicit discontinuity relations for the metric, scalar field, and vector field (including their derivatives). revision: yes

Circularity Check

0 steps flagged

Derivation chain is self-contained with no reduction to inputs by construction

full rationale

The paper begins from the published STVG action, derives junction conditions across the boundary using standard techniques, and applies them to match an interior FLRW region to an exterior static RN-like solution taken from prior STVG literature. No equation or step equates a derived prediction to a fitted parameter or self-defined quantity; the horizon formation result follows from the matching conditions and shell dynamics without circular closure. The static exterior assumption is an input from external literature rather than a self-citation load-bearing loop or ansatz smuggled within this work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The construction rests on the standard STVG action, the assumption that an RN-like exterior solution exists, and the introduction of an STVG-charged shell to enforce matching. No free parameters are fitted inside the paper; the charge on the shell is an auxiliary quantity required by the junction.

axioms (2)
  • domain assumption The standard STVG action governs the field equations used to derive the junction conditions.
    Explicitly stated as the starting point in the abstract.
  • domain assumption An exterior static spherically symmetric RN-like solution of STVG exists and can be matched to an interior FLRW spacetime.
    Core modeling choice required for the collapse construction.
invented entities (1)
  • STVG-charge carried by the collapsing shell no independent evidence
    purpose: To allow the metric and vector-field discontinuities to satisfy the derived junction conditions between FLRW and RN-like regions.
    Introduced specifically for the matching surface; no independent observational evidence is provided in the abstract.

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Reference graph

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