arxiv: 2605.13920 · v1 · submitted 2026-05-13 · 🌀 gr-qc

Recognition: no theorem link

No-go theorem for spontaneous vectorization

Hsu-Wen Chiang , Sebastian Garcia-Saenz , Aofei Sang

Authors on Pith no claims yet

Pith reviewed 2026-05-15 02:58 UTC · model grok-4.3

classification 🌀 gr-qc

keywords spontaneous vectorizationno-go theoremvector-tensor theoriesblack hole hairghost instabilitiesKerr black holesgeneral relativity

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The pith

Spontaneous vectorization cannot occur on hairless black holes without ghost or gradient instabilities

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

The paper proves that in generalized vector-tensor theories a vector field cannot develop a negative effective mass squared to trigger spontaneous hair growth if the initial state is a hairless black hole. This holds for essentially all stationary and axisymmetric spacetimes of interest in general relativity. The proof shows that negative mass squared must be accompanied by ghost- or gradient-type instabilities. Demanding the absence of instabilities produces bounds on the coupling constants in terms of the black hole parameters. For Kerr black holes this implies instability above a critical spin value.

Core claim

We demonstrate that spontaneous vectorization is not possible if the initial state is a hairless black hole. The appearance of a negative effective mass squared for the vector field must necessarily be accompanied by ghost- or gradient-type instabilities. This result applies to essentially all stationary and axisymmetric solutions of interest in general relativity and translates into interesting bounds on the coupling constants as functions of the black hole parameters, including a critical spin value above which a Kerr black hole becomes unstable.

What carries the argument

The no-go theorem that negative effective mass squared for the vector field implies ghost or gradient instabilities in stationary axisymmetric spacetimes

If this is right

Bounds on coupling constants are set by the black hole parameters to avoid instabilities.
A Kerr black hole becomes unstable above a critical spin value.
Spontaneous vectorization cannot produce hairy black holes from hairless initial states without instabilities.
The result covers essentially all stationary and axisymmetric black hole solutions in general relativity.

Where Pith is reading between the lines

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

Hairy black holes in these theories, if they exist, must form from initial states that already contain vector fields rather than through spontaneous growth.
The critical spin threshold for Kerr instability offers a concrete limit that could be checked against astrophysical observations of spinning black holes.
The theorem leaves open whether spontaneous vectorization could occur in non-stationary or non-axisymmetric cases.

Load-bearing premise

The spacetime is stationary and axisymmetric with an initial hairless black hole background.

What would settle it

Discovery of a stable vector-haired black hole that grows from a hairless initial state in a vector-tensor theory with negative effective mass squared would contradict the theorem.

Figures

Figures reproduced from arXiv: 2605.13920 by Aofei Sang, Hsu-Wen Chiang, Sebastian Garcia-Saenz.

**Figure 3.** Figure 3: B. Accelerated black hole spacetime As a second example we consider the so-called Cmetric solution of GR [65]. It describes a pair of accelerating black holes connected by a line defect, representing a simple set-up to study the effects of acceleration on black hole horizons. For us, it serves as a toy model to assess how acceleration may affect black hole stability in theories with non-minimally coupled … view at source ↗

**Figure 2.** Figure 2: FIG. 2. Stability region for a Kerr-Newman black hole. Graphs show instability regions for the couplings [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Stability region for an extremal Kerr-Newman [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Stability region for an accelerated black hole of [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Stability region for a Schwarzschild-(A)dS black [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. Stability region for an extremal Kerr-(A)dS black [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7. Stability region for a Taub-NUT black hole of [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗

read the original abstract

Generalized vector-tensor theories of gravity have drawn attention for admitting hairy black hole solutions, thereby circumventing the standard no-hair theorems. It remains an open question, however, how such black holes may form starting from reasonable initial conditions. It has been suggested that vector hair may grow spontaneously as a result of the field developing a negative effective mass squared $-$ the so-called spontaneous vectorization mechanism. We demonstrate that this is not possible if the initial state is a hairless black hole, a result that applies to essentially all stationary and axisymmetric solutions of interest in general relativity. More precisely, we prove that the appearance of a negative effective mass squared for the vector field must necessarily be accompanied by ghost- or gradient-type instabilities. Demanding the absence of such instabilities translates into interesting bounds on the coupling constants of the theory as functions of the black hole parameters. In particular, we discover that a Kerr black hole may become unstable when the spin increases above a certain critical value.

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 paper proves that spontaneous vectorization from a hairless black hole background forces ghost or gradient instabilities in generalized vector-tensor theories.

read the letter

The main point is straightforward: spontaneous vectorization via a negative effective mass squared cannot occur on a stationary axisymmetric hairless black hole without introducing instabilities in the quadratic action. The authors show that the sign flip in the mass term necessarily makes at least one kinetic or gradient coefficient change sign, ruling out stable hair growth from those initial states. This covers Kerr and similar GR solutions of interest. They also extract explicit bounds on the coupling functions in terms of black hole parameters, including a critical spin value above which Kerr develops an instability. The argument comes from expanding the action to second order around the vacuum GR background and tracking how the vector couplings enter the perturbation equations. This is new in the vector-tensor context and directly constrains model building for hairy black holes. The assumptions are stated clearly and used without circularity or extra parameters. The logic holds together on the terms given. The main limitation is the restriction to stationary axisymmetric backgrounds with standard vector kinetics; it leaves open whether vector hair could form in fully dynamical or non-axisymmetric cases. No other gaps stand out. This is useful for anyone working on vector-tensor gravity or black hole stability in modified theories. The result is concrete enough to be checked against numerics and worth a referee's time.

Referee Report

0 major / 0 minor

Summary. The manuscript claims to prove a no-go theorem for spontaneous vectorization in generalized vector-tensor theories of gravity. It demonstrates that, on any stationary and axisymmetric hairless black-hole background satisfying the vacuum GR equations, a negative effective mass squared for the vector field necessarily induces ghost- or gradient-type instabilities in the quadratic action for linear perturbations. This precludes spontaneous growth of vector hair from such initial states and yields bounds on the coupling functions, including a critical spin value above which a Kerr black hole becomes unstable.

Significance. If the central result holds, it is significant for modified gravity and black-hole physics. The theorem rules out spontaneous vectorization as a formation channel for hairy solutions on standard GR backgrounds, supplying parameter-free constraints on vector-tensor models that aim to evade no-hair theorems. The proof rests on standard linear stability analysis around vacuum solutions and applies broadly to stationary axisymmetric spacetimes of interest; the spin-dependent instability for Kerr provides a concrete, testable prediction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The provided summary correctly captures the central no-go theorem: on stationary axisymmetric vacuum GR backgrounds, a negative effective mass squared for the vector field in generalized vector-tensor theories necessarily induces ghost or gradient instabilities, thereby ruling out spontaneous vectorization as a formation mechanism and yielding spin-dependent bounds on the couplings.

Circularity Check

0 steps flagged

No significant circularity detected in the no-go theorem

full rationale

The paper derives a no-go result by expanding the generalized vector-tensor action to quadratic order around a stationary axisymmetric vacuum GR black-hole background, extracting the effective mass-squared term for the vector perturbation from the coupling functions, and demonstrating that its negativity forces at least one kinetic or gradient coefficient in the quadratic action to flip sign. All steps follow directly from the explicit form of the action, the background equations, and the stated assumptions (stationary axisymmetric metric, standard vector kinetic structure); no parameter is fitted and then relabeled as a prediction, no self-citation supplies a load-bearing uniqueness theorem, and no ansatz is smuggled in. The theorem is therefore self-contained under its stated premises and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The theorem rests on standard assumptions of general relativity without introducing new free parameters or invented entities.

axioms (2)

domain assumption Spacetime is stationary and axisymmetric
Invoked to restrict to solutions of interest in general relativity.
domain assumption Initial state is a hairless black hole
Central premise for the no-go result on spontaneous vectorization.

pith-pipeline@v0.9.0 · 5465 in / 1190 out tokens · 36475 ms · 2026-05-15T02:58:24.812454+00:00 · methodology

discussion (0)

Reference graph

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