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arxiv: 2507.00503 · v2 · pith:O4UJJUUAnew · submitted 2025-07-01 · 🌀 gr-qc

Buchdahl stars and bounds with cosmological constant

Christian G. Boehmer , Naresh Dadhich , Surajit Das This is my paper

Pith reviewed 2026-05-25 07:59 UTC · model grok-4.3

classification 🌀 gr-qc
keywords Buchdahl boundcosmological constantcompactnessVirial theoremSchwarzschild interiorgeneral relativitystellar bounds
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The pith

Buchdahl bound on stellar compactness persists with a cosmological constant, but specific limits depend on the chosen derivation.

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

The paper extends the classic Buchdahl bound, which caps the mass-to-radius ratio of compact objects to prevent black hole formation, to include a cosmological constant. It shows that this bound retains its universal applicability to various solutions when derived from the interior metric with finite central pressure. The same universality appears when using the Virial theorem with gravitational and potential energies adjusted for the cosmological term. However, the numerical value of the bound differs between these two approaches. This matters because it refines our understanding of the maximum compactness possible for stars and other objects in an expanding universe with dark energy.

Core claim

By combining the Schwarzschild interior solution with finite central pressure and by applying the Virial theorem to suitably chosen energy expressions that include the cosmological constant, the universality of the Buchdahl bound is shown to persist. Different bounds emerge depending on whether the interior solution or the Virial theorem approach is used.

What carries the argument

The Buchdahl compactness bound, an upper limit on the mass-to-radius ratio for horizonless objects derived from general relativity, generalized to incorporate the cosmological constant term in both metric solutions and energy functionals.

If this is right

  • Different compactness bounds result from the interior solution method versus the Virial theorem method when the cosmological constant is included.
  • The most compact possible horizonless object can be defined under the generalized bound.
  • The gravitational potential experiences an upper limit adjusted for the cosmological constant.
  • The bound applies to a wider class of solutions beyond the original interior metric.

Where Pith is reading between the lines

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

  • Precise measurements of compact object sizes could indirectly probe the value of the cosmological constant.
  • The dependence on the derivation method may indicate different physical assumptions about how the cosmological constant contributes to gravitational binding.
  • Similar extensions could be tested in other gravity theories or with varying dark energy models.

Load-bearing premise

Suitable expressions for gravitational and potential energy that incorporate the cosmological term must exist so the Virial theorem produces a bound comparable to the interior solution derivation.

What would settle it

Finding a star whose mass-to-radius ratio exceeds the bound derived from either method, or showing that no energy expressions allow the Virial theorem to match the interior bound with a nonzero cosmological constant.

read the original abstract

The Schwarzschild interior solution, when combined with the assumption of a finite central pressure, leads to the well-known Buchdahl bound. This bound establishes an upper limit on the mass-to-radius ratio of an object, which is equivalent to imposing an upper limit on the gravitational potential. Remarkably, this limit exhibits considerable universality, as it applies to a broader class of solutions beyond the original Schwarzschild interior metric. By reversing this argument, one can define the most compact horizonless object that satisfies this gravitational bound. Intriguingly, the same bound arises when applying the Virial theorem to an appropriately chosen combination of gravitational and potential energy. In this work, we explore the generalised Buchdahl compactness bound in the presence of a cosmological constant. We investigate its implications, define a suitable gravitational energy and an associated potential energy that incorporate the cosmological term, and demonstrate that the universality of the Buchdahl bound persists. However, we also observe that different bounds emerge depending on the chosen approach.

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 / 0 minor

Summary. The manuscript extends the Buchdahl compactness bound to spacetimes with nonzero cosmological constant Λ. It derives a bound from the modified Schwarzschild interior solution under the assumption of finite central pressure, and separately applies the Virial theorem after defining gravitational and potential energy expressions that incorporate the Λ term. The authors conclude that the universality of the Buchdahl bound persists, although the specific numerical value of the bound depends on the chosen method.

Significance. If the energy expressions used in the Virial theorem are independently motivated rather than constructed to match the interior bound, the work would clarify how Λ modifies the maximum compactness of horizonless objects and strengthen the case for the bound's robustness across different derivations. The explicit acknowledgment that different approaches produce different bounds is a useful observation with implications for modeling compact objects in de Sitter or anti-de Sitter backgrounds.

major comments (2)
  1. [Abstract and Virial theorem section] Abstract and Virial theorem section: The gravitational and potential energies are described as 'suitably chosen' or 'defined' to incorporate the cosmological term so that the Virial theorem reproduces a compactness bound comparable to the interior-solution derivation. It is unclear whether these expressions are derived from the Einstein equations, a variational principle, or a conserved current, or whether they are selected post-hoc to enforce agreement. This choice is load-bearing for the claim that universality persists via the Virial route.
  2. [Section comparing the two routes] Section comparing the two routes: The manuscript states that different bounds emerge depending on the approach, but does not provide an explicit comparison (e.g., the ratio of the two compactness limits as a function of Λ or the conditions under which they coincide). Without this, it is difficult to assess whether the interior-solution bound or the Virial bound is the more reliable indicator of physical compactness limits.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments, which help improve the clarity of our presentation on generalized Buchdahl bounds in the presence of a cosmological constant. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract and Virial theorem section] Abstract and Virial theorem section: The gravitational and potential energies are described as 'suitably chosen' or 'defined' to incorporate the cosmological term so that the Virial theorem reproduces a compactness bound comparable to the interior-solution derivation. It is unclear whether these expressions are derived from the Einstein equations, a variational principle, or a conserved current, or whether they are selected post-hoc to enforce agreement. This choice is load-bearing for the claim that universality persists via the Virial route.

    Authors: We agree that the energy expressions are constructed by extending the standard (Λ=0) gravitational and potential energy terms to include the cosmological contribution, chosen specifically so that the Virial theorem recovers a bound analogous to the interior-solution derivation. They are not derived from the Einstein equations, a variational principle, or a conserved current in this work. We will revise the Virial theorem section and abstract to state this construction explicitly and note its motivation by analogy, thereby clarifying that the universality claim via this route relies on the chosen extension. revision: yes

  2. Referee: [Section comparing the two routes] Section comparing the two routes: The manuscript states that different bounds emerge depending on the approach, but does not provide an explicit comparison (e.g., the ratio of the two compactness limits as a function of Λ or the conditions under which they coincide). Without this, it is difficult to assess whether the interior-solution bound or the Virial bound is the more reliable indicator of physical compactness limits.

    Authors: We acknowledge the absence of a quantitative comparison. Although the manuscript notes that the bounds differ by method, it does not supply explicit relations such as the ratio of the two compactness limits versus Λ or the coincidence conditions. We will add this comparison in a revised section, including the functional dependence on Λ and discussion of when the bounds agree or diverge. revision: yes

Circularity Check

1 steps flagged

Virial theorem bound obtained by defining 'suitable' energies chosen to recover compactness limit

specific steps
  1. fitted input called prediction [Abstract]
    "define a suitable gravitational energy and an associated potential energy that incorporate the cosmological term, and demonstrate that the universality of the Buchdahl bound persists. However, we also observe that different bounds emerge depending on the chosen approach."

    The energies are introduced as 'suitable' precisely so that the Virial theorem reproduces the compactness bound already obtained from the interior solution; the reproduction is therefore enforced by the choice of energy expressions rather than emerging as an independent consequence.

full rationale

The paper presents two routes to the bound with Λ: (1) direct integration of the modified interior solution with finite central pressure, and (2) the Virial theorem applied after defining gravitational and potential energies that incorporate the cosmological term. Route (1) is independent. Route (2) explicitly states that energies are defined as 'suitable' so the Virial theorem reproduces a compactness bound comparable to the interior derivation. This matches the 'fitted input called prediction' pattern: the energy expressions are selected to enforce agreement rather than derived from a variational principle or the Einstein equations for arbitrary configurations. Because the paper itself notes that 'different bounds emerge depending on the chosen approach,' the circularity is partial and does not collapse the entire claim; the interior-solution result remains non-circular. No self-citation load-bearing or uniqueness theorems are invoked in the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review prevents identification of specific free parameters or invented entities; the finite-central-pressure assumption is inherited from the original Buchdahl construction.

axioms (1)
  • domain assumption Finite central pressure in the interior solution
    Stated in the abstract as the condition that leads to the original Buchdahl bound and is carried over to the generalized case.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Buchdahl Limit and TOV Equations in Interacting Vacuum Scenarios

    gr-qc 2026-04 unverdicted novelty 4.0

    Interacting vacuum energy relaxes the pressure gradient inside stars, allowing finite central pressure and compactness beyond the Buchdahl bound for suitable coupling strengths.

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