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arxiv: 2312.01406 · v2 · pith:33ATRXGKnew · submitted 2023-12-03 · 🌀 gr-qc

Neutron stars more compact than black holes as a probe of strong-field gravity

Shoulong Li , H. L\"u , Yong Gao , Rui Xu , Lijing Shao , Hongwei Yu This is my paper

Pith reviewed 2026-05-24 05:20 UTC · model grok-4.3

classification 🌀 gr-qc
keywords neutron starsblack holesquasi-topological gravitystrong-field gravitygravitational wave echoescompactnessradial stability
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0 comments X

The pith

Stable stellar configurations more compact than black holes arise in quasi-topological gravity when neutron-star equations of state are embedded.

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

The paper demonstrates that embedding conventional neutron-star equations of state into quasi-topological gravity produces stable stellar objects whose compactness exceeds the black-hole limit allowed in general relativity. A sympathetic reader would care because this supplies a concrete mechanism for testing gravity modifications in the strongest-field regime, where black holes normally mark the upper bound on compactness. The authors construct explicit solutions, verify radial stability, and identify gravitational-wave echoes as a potential distinguishing signature from black holes.

Core claim

In quasi-topological gravity, neutron-star equations of state can be embedded to produce stable ultra-compact stars that exceed the black-hole compactness limit; these configurations remain stable against radial perturbations and exhibit macroscopic properties and gravitational-wave echoes that could observationally distinguish them from black holes.

What carries the argument

Embedding of neutron-star equations of state within quasi-topological gravity, which permits stable configurations beyond the general-relativity compactness bound.

If this is right

  • Ultra-compact stars exceeding black-hole compactness can be constructed using standard neutron-star equations of state.
  • These stars remain stable against radial perturbations.
  • They possess distinct macroscopic properties compared with black holes.
  • Gravitational-wave echoes provide a potential observational channel to detect such objects and thereby probe physics beyond general relativity.

Where Pith is reading between the lines

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

  • If such stars exist, strong-field tests of gravity could reveal higher-curvature corrections without requiring exotic matter.
  • Similar embeddings might be possible in other higher-curvature extensions of general relativity.
  • Targeted searches for echo signals in compact-object merger data could place bounds on the quasi-topological coupling parameters.

Load-bearing premise

Quasi-topological gravity allows neutron-star equations of state to be embedded while preserving radial stability for objects exceeding black-hole compactness.

What would settle it

A calculation or numerical simulation showing that no radially stable solutions exist above the black-hole compactness limit for any neutron-star equation of state in quasi-topological gravity.

Figures

Figures reproduced from arXiv: 2312.01406 by H. L\"u, Hongwei Yu, Lijing Shao, Rui Xu, Shoulong Li, Yong Gao.

Figure 1
Figure 1. Figure 1: FIG. 1. The upper and lower plots illustrate the numerical solutions of the functions ( [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The upper plot shows the relationship between the compactness [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The left panel shows the relation between compactness [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
read the original abstract

Probing gravity in its strongest regime is a central goal of modern physics, as the nature of the most compact objects reflects fundamental aspects of Einstein's theory of general relativity (GR). In GR, black holes are regarded as the most compact objects in the Universe. Here, for the first time, we demonstrate that stable stellar configurations more compact than black holes can arise when neutron-star equations of state are embedded in quasi-topological gravity, a class of higher-curvature extensions of GR. We construct such ultra-compact stars, analyze their macroscopic properties, and establish their stability against radial perturbations, confirming their physical plausibility. We further identify potential observational signatures to distinguish these stars from black holes, most notably gravitational-wave echoes whose detectability could provide direct evidence of physics beyond Einstein's GR in the strong-field regime.

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 / 1 minor

Summary. The paper claims that embedding standard neutron-star equations of state into quasi-topological gravity yields stable, static stellar configurations with compactness C = M/R > 1/2 (exceeding the black-hole limit), analyzes their macroscopic properties, demonstrates radial stability, and identifies gravitational-wave echoes as potential observational discriminants from black holes.

Significance. If the central construction were viable, the result would constitute a significant extension of strong-field gravity phenomenology, offering a concrete mechanism for ultra-compact objects beyond the GR black-hole compactness bound together with falsifiable GW signatures. The explicit construction and stability analysis would strengthen the case for using such objects to test higher-curvature corrections.

major comments (2)
  1. [Abstract; construction of stellar solutions (likely §3–4)] The vacuum exterior solution in quasi-topological gravity is the Schwarzschild metric (unique asymptotically flat spherically symmetric vacuum solution). Any static stellar surface with R < 2M therefore lies inside the event horizon, where the timelike Killing vector is spacelike and no time-independent stellar configuration can exist. This geometric obstruction is independent of the interior equation of state and of the higher-curvature terms that leave the exterior unchanged; it directly undermines the central claim of stable C > 1/2 configurations.
  2. [Stability analysis (likely §5)] The radial-stability analysis and EOS embedding cannot circumvent the exterior matching problem. Even if interior solutions are found and perturbations are shown to be stable, the global spacetime cannot be static once the surface is placed inside the would-be horizon.
minor comments (1)
  1. Clarify the precise definition of compactness C used throughout and confirm that the exterior metric is indeed Schwarzschild (including any statement on the junction conditions).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for raising these important geometric issues. We address each major comment below and indicate the revisions that will be made to the next version of the paper.

read point-by-point responses

  1. Referee: [Abstract; construction of stellar solutions (likely §3–4)] The vacuum exterior solution in quasi-topological gravity is the Schwarzschild metric (unique asymptotically flat spherically symmetric vacuum solution). Any static stellar surface with R < 2M therefore lies inside the event horizon, where the timelike Killing vector is spacelike and no time-independent stellar configuration can exist. This geometric obstruction is independent of the interior equation of state and of the higher-curvature terms that leave the exterior unchanged; it directly undermines the central claim of stable C > 1/2 configurations.

    Authors: We agree with the referee that the vacuum exterior remains the Schwarzschild metric, as quasi-topological gravity does not alter the unique asymptotically flat spherically symmetric vacuum solution. This raises a substantive question about whether a static stellar surface can be placed at R < 2M while preserving a time-independent configuration. We will revise the manuscript (primarily in §§3–4) to include an explicit analysis of the Israel junction conditions at the stellar surface in quasi-topological gravity and to discuss whether the higher-curvature contributions to the interior can be consistently matched without violating the static exterior geometry. If the matching cannot be performed, we will qualify or restrict the compactness claims accordingly. revision: yes

  2. Referee: [Stability analysis (likely §5)] The radial-stability analysis and EOS embedding cannot circumvent the exterior matching problem. Even if interior solutions are found and perturbations are shown to be stable, the global spacetime cannot be static once the surface is placed inside the would-be horizon.

    Authors: We concur that radial stability of the interior alone does not resolve the global spacetime issue. The referee correctly notes that a static configuration requires consistent matching across the entire spacetime. In the revised manuscript we will add a dedicated subsection on global spacetime structure and exterior-interior matching, explicitly checking whether the proposed ultra-compact solutions remain static when the exterior is fixed to Schwarzschild. This will either strengthen the construction or lead to a more limited statement of the results. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper constructs ultra-compact stellar solutions by embedding standard neutron-star equations of state into the field equations of quasi-topological gravity, then solves for interior metrics, matches to the Schwarzschild exterior, and performs radial stability analysis. No quoted step reduces the central claim (existence of stable C > 1/2 configurations) to a fitted parameter, self-definition, or load-bearing self-citation. The derivation chain relies on independent numerical integration of the modified Einstein equations with given EOS; the target result is not presupposed by construction. This matches the reader's assessment of score 1.0 and contains none of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the claim rests on the domain assumption that quasi-topological gravity is a consistent higher-curvature extension that can host standard neutron-star equations of state in stable ultra-compact configurations. No free parameters or invented entities are identifiable from the abstract.

axioms (1)
  • domain assumption Quasi-topological gravity is a valid higher-curvature extension of GR that permits embedding of neutron-star equations of state
    Invoked to construct the ultra-compact stars and establish their stability.

pith-pipeline@v0.9.0 · 5679 in / 1228 out tokens · 36819 ms · 2026-05-24T05:20:18.202338+00:00 · methodology

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Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

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Reference graph

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