arxiv: 2604.00078 · v2 · submitted 2026-03-31 · 🌀 gr-qc

Recognition: 2 theorem links

· Lean Theorem

Regular Black Strings and BTZ Black Hole in Unimodular Gravity Supported by Maxwell Fields

G. Alencar , V. H. U. Borralho

Authors on Pith no claims yet

Pith reviewed 2026-05-13 22:50 UTC · model grok-4.3

classification 🌀 gr-qc

keywords unimodular gravityblack stringsBTZ black holeMaxwell fieldsregular black holesvariable cosmological constantvacuum energyenergy-momentum non-conservation

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The pith

In unimodular gravity the cosmological constant becomes a radially dependent vacuum energy that supports regular black strings and BTZ black holes sourced by Maxwell fields.

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

Unimodular gravity imposes a fixed spacetime volume element, which restricts diffeomorphisms and turns the cosmological constant into an integration constant rather than a fixed term in the action. Because the energy-momentum tensor is no longer conserved, this constant is allowed to vary with radius and is reinterpreted as a position-dependent vacuum contribution. The authors use this freedom to construct regular black-string and BTZ solutions whose source is Maxwell electrodynamics, then verify consistency by defining a geometric function H(r) that determines the required Λ(r). A reader would care because the construction shows how a simple volume constraint can generate variable vacuum energy without adding new fields or higher-derivative terms.

Core claim

By imposing det(g_μν) = g0, the field equations of unimodular gravity yield an integration constant that the non-conservation of the energy-momentum tensor converts into a function Λ(x). For the chosen black-string and BTZ metrics this function is evaluated as a radial vacuum contribution Λ(r) whose form is fixed once the geometric function H(r) is chosen so that the Maxwell stress-energy satisfies the modified Einstein equations.

What carries the argument

The geometric function H(r), which encodes the metric ansatz and determines both the Maxwell field strength and the supporting vacuum term Λ(r).

If this is right

Regular black-string geometries exist with Maxwell sources once the vacuum energy is permitted to vary radially.
The BTZ black-hole metric is recovered as a consistent solution supported by the same radially dependent Λ(r).
The cosmological constant is reinterpreted as an emergent, coordinate-dependent vacuum contribution rather than a fixed parameter.
Maxwell electrodynamics remains a valid matter source under the volume-preserving diffeomorphism constraint.

Where Pith is reading between the lines

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

The same mechanism could be tested in other static, spherically symmetric ansätze to see whether regularity is preserved when Maxwell fields are replaced by other matter.
Because Λ(r) is fixed by the geometry, its profile might be compared with effective dark-energy densities inferred from cosmological observations in the same theory.
The non-conservation of the stress-energy tensor implies that energy can flow between the Maxwell field and the vacuum term, an exchange that could be searched for in numerical simulations of collapse.

Load-bearing premise

The spacetime volume element stays strictly constant so that the non-conservation of the energy-momentum tensor legitimately turns the integration constant into a physically acceptable radially varying vacuum energy without extra instabilities or constraints.

What would settle it

A direct check that the Maxwell field strength derived from H(r) fails to satisfy the unimodular field equations for the chosen black-string or BTZ metric, or that the resulting Λ(r) produces negative energy density or violates the null energy condition everywhere outside the horizon.

Figures

Figures reproduced from arXiv: 2604.00078 by G. Alencar, V. H. U. Borralho.

read the original abstract

In this work, we obtain a Maxwell source for regular black string and BTZ black hole using the framework of unimodular gravity. This type of alternative to general relativity imposes an additional condition on the spacetime volume element, namely that it is constant, $\det(g_{\mu\nu}) = g_0$, and thus restricts diffeomorphism invariance to volume-preserving transformations. In this procedure, the cosmological constant does not appear directly in the action, but rather as an integration constant of the field equations. By using the non-conservation of the energy-momentum tensor, we show that the integration constant becomes a function $\Lambda(x)$, which is interpreted as a vacuum contribution depending on the radial coordinate, in our case. From the definition of the geometric function $H(r)$, we verify the validity of Maxwell electrodynamics as a source for the solutions and compute the vacuum contribution $\Lambda(r)$ that supports the 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.

Desk Editor's Note private letter to a colleague

The non-conservation step that turns the integration constant into a radial Λ(r) is inconsistent with traceless Maxwell sources, so the claimed solutions rest on shaky ground.

read the letter

The main thing to know is that the paper's central move—using non-conservation of the energy-momentum tensor to promote the cosmological constant to a function Λ(r)—does not work for Maxwell fields. In unimodular gravity the trace-free equations plus Bianchi identities give ∇^μ T_μν = (1/4) ∇_ν T. For Maxwell, T vanishes identically, so the right-hand side is zero and ordinary conservation is recovered. Nothing in the unimodular condition det(g) = constant supplies an extra degree of freedom that would let the integration constant pick up radial dependence while still satisfying the Maxwell equations. The radial Λ(r) therefore looks like it is defined by the metric ansatz rather than derived from the dynamics, which is the circularity the stress-test flags. The abstract presents this as a feature, but the logic does not close. What the paper does supply is an explicit metric ansatz for regular black strings and BTZ black holes together with a check through the geometric function H(r) that the Maxwell equations hold in the chosen coordinates. That part is straightforward and could serve as a template for other sources. The soft spot is not minor; it sits at the load-bearing step. The authors do not appear to verify the Bianchi identities or energy conditions in enough detail to rescue the claim, and the volume-preserving restriction alone does not create the advertised freedom. Readers who work on black-hole solutions in modified gravity might still find the metric forms useful as a starting point for numerical checks or different matter content. A specialist who wants to see whether the same ansatz can be made consistent with a non-traceless source would get some value. For everyone else the inconsistency makes the result unreliable. I would not send this to peer review in its present form. The conservation issue needs to be fixed or the non-conservation language dropped before the solutions can be taken seriously.

Referee Report

1 major / 1 minor

Summary. The paper claims to derive regular black string and BTZ black hole solutions in unimodular gravity sourced by Maxwell fields. By imposing a constant spacetime volume element det(g_μν)=g0, the framework restricts diffeomorphisms and yields field equations in which the cosmological constant appears as an integration constant; the authors invoke non-conservation of the energy-momentum tensor to promote this constant to a radially dependent function Λ(r), interpreted as a vacuum contribution, and verify the solutions via a geometric function H(r).

Significance. If the central mechanism were consistent, the work would supply explicit regular black-hole geometries in a volume-preserving gravity theory with electromagnetic sources, illustrating how the unimodular constraint can generate position-dependent vacuum energy without additional fields. Such constructions are of interest for singularity resolution and for exploring the restricted diffeomorphism invariance of unimodular gravity.

major comments (1)

[Abstract and field-equation derivation] Abstract and the derivation of the field equations: the assertion that non-conservation of the EMT converts the integration constant into a radially dependent Λ(r) is internally inconsistent for a traceless Maxwell source. Unimodular gravity produces the trace-free equations R_μν − (1/4)R g_μν = 8π(T_μν − (1/4)T g_μν); Bianchi identities then enforce ∇^μ T_μν = (1/4)∇_ν T. With T=0 identically for Maxwell fields, the right-hand side vanishes and standard conservation ∇^μ T_μν=0 is recovered, forcing the integration constant to remain constant rather than acquiring radial dependence while satisfying both the Maxwell equations and det(g)=g0.

minor comments (1)

[Solutions section] The explicit definition and differential properties of the geometric function H(r) should be stated at the outset of the solutions section so that the verification step can be followed without ambiguity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for identifying an important consistency issue in the derivation of the field equations. We address the major comment below and commit to revisions that strengthen the presentation.

read point-by-point responses

Referee: [Abstract and field-equation derivation] Abstract and the derivation of the field equations: the assertion that non-conservation of the EMT converts the integration constant into a radially dependent Λ(r) is internally inconsistent for a traceless Maxwell source. Unimodular gravity produces the trace-free equations R_μν − (1/4)R g_μν = 8π(T_μν − (1/4)T g_μν); Bianchi identities then enforce ∇^μ T_μν = (1/4)∇_ν T. With T=0 identically for Maxwell fields, the right-hand side vanishes and standard conservation ∇^μ T_μν=0 is recovered, forcing the integration constant to remain constant rather than acquiring radial dependence while satisfying both the Maxwell equations and det(g)=g0.

Authors: We agree with the referee that the current wording in the abstract is imprecise and potentially misleading. For a traceless Maxwell source the trace-free equations together with the Bianchi identities do enforce standard conservation ∇^μ T_μν = 0, so the integration constant cannot be promoted to a function of r by invoking non-conservation. We will revise the abstract and the section on the derivation of the field equations to remove any reference to non-conservation of the EMT. Instead, we will present an explicit integration of the trace-free equations under the unimodular constraint det(g_μν)=g0 for the chosen static, cylindrically symmetric metric ansatz, showing how the effective vacuum term Λ(r) is fixed by the requirement that the metric components satisfy both the field equations and the volume constraint. The geometric function H(r) will be used to verify that the resulting solutions remain consistent with the Maxwell equations and the trace-free Einstein equations. These changes will be made without altering the reported solutions or their regularity properties. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper derives explicit solutions for regular black strings and BTZ black holes in unimodular gravity by imposing the constant volume condition det(g_μν)=g0, which makes the cosmological constant appear as an integration constant in the trace-free field equations. It then invokes the framework's modified conservation law to allow this constant to become a radially dependent function Λ(r), computes the explicit form from the chosen metric ansatz and the geometric function H(r), and verifies that the source reduces to Maxwell electrodynamics plus this vacuum term. No step reduces a claimed prediction or result to an input by construction: the radial dependence is obtained by direct integration of the differential equations under the unimodular constraint rather than by fitting or re-labeling a presupposed quantity. No self-citation is load-bearing for the central claim, no uniqueness theorem is imported from prior work by the same authors, and the ansatz is the standard one for black-string geometries. The derivation remains self-contained against the field equations and the unimodular condition.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the unimodular volume constraint and the interpretation of the integration constant as a physical vacuum function; no new particles or forces are introduced, but the radial dependence of Λ(r) is effectively postulated to match the solutions.

free parameters (1)

Λ(r)
The radially dependent vacuum contribution is computed from the field equations for the chosen ansatz rather than predicted independently.

axioms (2)

domain assumption det(g_μν) = g0 (constant)
Invoked at the start of the unimodular gravity framework to restrict diffeomorphisms.
ad hoc to paper Non-conservation of the energy-momentum tensor is physically acceptable and converts the integration constant into Λ(x)
Used to obtain the radial dependence without further justification in the abstract.

pith-pipeline@v0.9.0 · 5463 in / 1567 out tokens · 29225 ms · 2026-05-13T22:50:06.176923+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

By using the non-conservation of the energy-momentum tensor, we show that the integration constant becomes a function Λ(x)... H(r)≡4m'(r)ℓ/r²−2m''(r)ℓ/r =−2E²(r)LF
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

R_μν − R/n g_μν = T_μν − (1/n) T g_μν ... G_μν = T_μν + Λ(x) g_μν

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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