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Nonlinear Scalarization of Reissner-Nordstr\"om Black Holes in Scalar-Torsion-Maxwell Teleparallel Gravity

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Joaquín Robledo, Yerko Vásquez, P. A. González

General Relativity and Quantum Cosmology · Submitted 8 Nov 2026 · 71 references

Pith's verdict: Solid numerical construction of nonlinearly scalarized RN black holes in teleparallel gravity; the main soft spot is the untested convergence of the shooting series, but the results are likely reliable.

T0 · AI review

How this review was made: An AI referee (deepseek/deepseek-v4-flash) read the paper on November 10, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

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arXiv ID2611.11686 (version 1)
Pith IDNVU2CWSC, Pith's permanent record of this paper and its review
CitationJoaqu\'in Robledo, P. A. Gonz\'{a}lez, Yerko V\'asquez. "Nonlinear Scalarization of Reissner-Nordstr\"om Black Holes in Scalar-Torsion-Maxwell Teleparallel Gravity." arXiv:2611.11686 (2026). https://pith.science/paper/NVU2CWSC
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The paper

The paper claims

Nonlinear scalar hair on charged black holes is realized in teleparallel gravity through a sixth-order coupling that saturates without linear instability.

The reading

This paper demonstrates that Reissner-Nordström black holes in scalar-torsion-Maxwell teleparallel gravity can acquire scalar hair through a purely nonlinear mechanism, without relying on linear tachyonic instabilities. The authors use polynomial coupling functions f(ϕ)=$ηϕ^{4}$−$γϕ^{6}$ that satisfy df/dϕ(0)=0 and d²f/dϕ²(0)=0, ensuring the trivial RN solution remains linearly stable. In the decoupling limit, they show that the quartic coupling alone leads to unbounded growth, but adding a negative sixth-order term creates a double-well effective potential that traps the scalar field into stationary configurations. By numerically solving the fully coupled field equations, they construct asymptotically flat, static, spherically symmetric black hole solutions with scalar hair, classify them by node number, and map how the sixth-order coupling modifies the branch structure—creating additional degenerate branches at weak coupling and suppressing higher-node branches at strong coupling.

What carries the argument

The polynomial coupling function f(ϕ)=ηϕ⁴−γϕ⁶ and the effective potential V_eff(r,ϕ)=2(q²r₊²−2r²)/r⁴·f(ϕ) derived from the Klein-Gordon equation on the fixed RN background. The double-well structure of V_eff for γ>0, with minima at ϕ=±√(2η/3γ), provides the nonlinear saturation mechanism that permits stationary scalar configurations. The numerical construction uses a shooting method with compactified radial coordinate z=1−r₊/r, 10-term near-horizon and asymptotic expansions, and matching at z=0.5 with tolerance δ=10⁻⁷.

Extended reading notes

Core claim

The authors show that asymptotically flat, static, spherically symmetric black hole solutions with scalar hair exist in scalar-torsion-Maxwell teleparallel gravity for the coupling function f₂(ϕ)=ηϕ⁴−γϕ⁶ (γ>0) that avoids linear instabilities. In the decoupling limit, while the purely quartic coupling f₁(ϕ)=ηϕ⁴ causes a nonlinear runaway for large perturbations, the sixth-order term introduces two nontrivial local minima in the effective potential at ϕ_c=±√(2η/3γ), providing a saturation mechanism. For the fully coupled system, the numerical solutions reveal multiple branches distinguished by node number n; the sixth-order coupling initially breaks degeneracy of the n=1 and n=2 branches found at γ=0, eventually suppressing higher-node branches as γ grows, until only the fundamental nodeless solution remains, and finally eliminates all scalarized solutions for sufficiently large γ.

Load-bearing premise

The numerical solutions obtained via the shooting method with 10-term near-horizon and asymptotic expansions matched at z=0.5 and integration tolerance ε=10⁻⁸ and shooting tolerance δ=10⁻⁷ accurately represent true solutions of the full field equations; no convergence test or error estimate beyond these tolerances is reported.

Editorial extensions

If this is right

  • Scalarized black hole solutions now exist in teleparallel gravity that do not rely on the standard tachyonic instability, extending the known examples beyond curvature-based theories.
  • The qualitative similarity to Einstein-scalar-Gauss-Bonnet gravity [33] suggests that the nonlinear scalarization mechanism depends more on the structure of the coupling function than on the geometric formulation (curvature vs. torsion).
  • The branching structure controlled by the sixth-order coupling parameter γ provides a testable taxonomy: weak γ enriches the solution space, strong γ reduces it, which could be linked to thermodynamic stability and phase transitions.
  • These solutions offer concrete backgrounds for studying strong-field deviations in the electromagnetic and gravitational sectors, potentially relevant for shadow and ringdown analyses.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The double-well effective potential suggests that the scalarized branches observed here may correspond to distinct vacuum states of the scalar field, and the existence of multiple degenerate branches at finite γ hints at a possible first-order phase transition between hairless and hairy configurations.
  • The suppression of higher-node branches at large γ implies that the sixth-order term acts as a regulator that limits the spatial complexity of the scalar hair; this may be generic for higher-order polynomial couplings in teleparallel theories.
  • Because the scalar field decays as ϕ∼ϕ₀r₊/r at infinity, the scalar charge ϕ₀r₊ is a new observable that could be constrained through binary pulsar timing or gravitational-wave observations of charged black hole mergers, though such tests are not performed here.
Original abstract

We investigate the nonlinear scalarization of Reissner-Nordstr\"om black holes in scalar-torsion-Maxwell teleparallel gravity, in which the scalar field is nonminimally coupled to the Weitzenb\"ock invariant. We consider polynomial coupling functions that avoid tachyonic instabilities at the linear level, thereby exploring a scalarization mechanism driven by nonlinear effects. In the decoupling limit, we study the nonlinear evolution of finite-amplitude scalar perturbations and show that the inclusion of a sixth-order coupling term allows the scalar field to dynamically evolve toward nontrivial stationary configurations. This behavior can be understood in terms of the effective potential, which develops nontrivial local minima that provide a mechanism for nonlinear saturation. By numerically solving the fully coupled field equations, we construct asymptotically flat charged black hole solutions endowed with scalar hair. We classify these solutions according to the number of nodes in the scalar field profile, identifying multiple branches whose structure and multiplicity depend on the coupling parameters. We further show that the higher-order coupling progressively suppresses higher-node branches and characterize the scalar backreaction through deviations of the metric functions and the torsion scalar from the Reissner-Nordstr\"om background.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-11-10: 1 request, 5,755 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.21 · human participants 0.01 · controlled comparison 0.18 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.26 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We investigate the nonlinear scalarization of Reissner-Nordstr\"om black holes in scalar-torsion-Maxwell teleparallel gravity, in which the scalar field is nonminimally coupled to the Weitzenb\"ock invariant. We consider polynomial coupling functions that avoid tachyonic instabilities at the linear level, thereby exploring a scalarization mechanism driven by nonlinear effects. In the decoupling limit, we study the nonlinear evolution of finite-amplitude scalar perturbations and show that the inclusion of a sixth-order coupling term allows the scalar field to dynamically evolve toward nontrivial stationary configurations. This behavior can be understood in terms of the effective potential, which develops nontrivial local minima that provide a mechanism for nonlinear saturation. By numerically solving the fully coupled field equations, we construct asymptotically flat charged black hole solutions endowed with scalar hair. We classify these solutions according to the number of nodes in the scalar field profile, identifying multiple branches whose structure and multiplicity depend on the coupling parameters. We further show that the higher-order coupling progressively suppresses higher-node branches and characterize the scalar backreaction through deviations of the metric functions and the torsion scalar from the Reissner-Nordstr\"om background.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We investigate the nonlinear scalarization of Reissner-Nordstr\"om black holes in scalar-torsion-Maxwell teleparallel gravity, in which the scalar field is nonminimally coupled to the Weitzenb\"ock invariant. contribution 0.96 0.85 0.07 asserted 0.86
s1 We consider polynomial coupling functions that avoid tachyonic instabilities at the linear level, thereby exploring a scalarization mechanism driven by nonlinear effects. method 0.75 0.90 0.07 asserted 0.93
s2 In the decoupling limit, we study the nonlinear evolution of finite-amplitude scalar perturbations and show that the inclusion of a sixth-order coupling term allows the scalar field to dynamically evolve toward nontrivial stationary configurations. result 0.62 0.96 0.26 asserted 1.00
s3 This behavior can be understood in terms of the effective potential, which develops nontrivial local minima that provide a mechanism for nonlinear saturation. interpretation 1.00 0.67 0.11 asserted 0.81
s4 By numerically solving the fully coupled field equations, we construct asymptotically flat charged black hole solutions endowed with scalar hair. result 0.52 0.97 0.34 asserted 1.00
s5 We classify these solutions according to the number of nodes in the scalar field profile, identifying multiple branches whose structure and multiplicity depend on the coupling parameters. result 0.63 0.96 0.13 asserted 1.00
s6 We further show that the higher-order coupling progressively suppresses higher-node branches and characterize the scalar backreaction through deviations of the metric functions and the torsion scalar from the Reissner-Nordstr\"om background. result 0.94 0.95 0.22 asserted 1.00

What kind of paper the classifier reports this is: simulation study 0.60 · evidence from simulation or computation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. FIG. 1: Time evolution of the scalar field on the RNBH back
Figure 2. FIG. 2: Effective potential for a scalar field on the RN black
Figure 3. FIG. 3: Bifurcation diagrams for scalarized black hole solu
Figure 4. FIG. 4: Radial profiles of the scalar field
Figure 5. FIG. 5: Radial profiles of the scalar field
Figure 7. FIG. 7: The behavior of the torsion scalar deviation

Showing 6 of the paper's 7 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:2611.11686 by the authors.

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Discussion

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