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arxiv: 2605.13965 · v1 · pith:YELD3LLOnew · submitted 2026-05-13 · 🌀 gr-qc · hep-th

Multipolar Proca stars: electric, magnetic and hybrid solitons

Carlos Herdeiro , Eugen Radu , Etevaldo dos Santos Costa Filho , Nicolas Sanchis-Gual This is my paper

Pith reviewed 2026-05-15 02:35 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords procastarsconfigurationselectrichybridmagneticangularcases
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New multipolar electric, magnetic, and hybrid Proca stars are constructed as self-gravitating solutions in Einstein-Proca gravity, with dynamical evolutions showing instabilities in the magnetic and hybrid sectors that lead to decay into stable electric configurations or black hole collapse.

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

In Einstein gravity coupled to a massive vector field, researchers built new star-like objects that are smooth everywhere and flat far away. These multipolar Proca stars are labeled by a number that describes the shape of the field arrangement. Electric versions include the familiar round ones, while magnetic versions are new and start from a dipole shape with no round counterpart. Hybrid versions mix electric and magnetic parts, sometimes without north-south symmetry, and can have local swirling but no total spin. Computer simulations of their time evolution reveal that magnetic and hybrid versions are unstable: they change into the known stable electric or spinning Proca stars or collapse into black holes.

Core claim

We construct new families of everywhere regular, asymptotically flat solitons in the Einstein--Proca model, obtained as self-gravitating continuations of flat-spacetime (singular) Proca multipoles.

Load-bearing premise

The numerical constructions accurately solve the nonlinear Einstein-Proca equations for the chosen multipole boundary conditions and that the dynamical evolutions are free of significant numerical artifacts or resolution-dependent outcomes.

Figures

Figures reproduced from arXiv: 2605.13965 by Carlos Herdeiro, Etevaldo dos Santos Costa Filho, Eugen Radu, Nicolas Sanchis-Gual.

Figure 1
Figure 1. Figure 1: The frequency-mass diagram is shown for the first electric (left panel) and magnetic (right panel) families of solitonic solutions in Einstein-Proca theory. The Proca stars between the maximal frequency and configurations marked with a (square) black dot have M < µQ. ℓ = 1, the true ground state of the model. In this respect, the magnetic solutions follow the more standard (hydrogen-like) mass hierarchy, w… view at source ↗
Figure 2
Figure 2. Figure 2: The energy density is shown in (x, z)-plane for typical magnetic solutions with ℓ = 1, 2, 3, 4 (with x = r sin θ cos φ, z = r cos θ; note that the solutions possess an azimuthal symmetry, with no φ-dependence.) The situation with the electric case is more involved - [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Same as Figure [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The (reduced) quadrupole mass-moment is shown as a function of frequency. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The compactness is shown as a function of mass. 4 Hybrid Proca stars 4.1 General remarks Typically, the static solitons allow for spinning generalizations, which in many cases are disconnected from the static sector, without a slowly rotating limit. The spinning PSs can be studied for a generalization of the line-element (25) with an extra-function associated with rotation, ds2 = −e 2F0(r,θ) dt2 + e 2F1(r,… view at source ↗
Figure 6
Figure 6. Figure 6: Left: The angular momentum density T t φ is shown as a function of radial coordinate and several angles for a typical J = 0 hybrid {electric ℓ = 0, magnetic ℓ = 1}-spinning Proca star. For any θ > 0, T t φ becomes positive for large enough r, such that its volume integral vanishes, J = 0. Right: The frequency-mass diagram is shown for the J = 0 family of spinning Proca stars (PSs) consisting in a non-linea… view at source ↗
Figure 7
Figure 7. Figure 7: Surfaces of constant angular momentum density (left panel) and energy density (right panel) for a J = 0 hybrid {electric ℓ = 0, magnetic ℓ = 1}-spinning Proca star with ω/µ = 0.8. The considered values are T t φ = 0.004 (outside torus),−0.03 (inside torus) and 0 (the squashed sphere). For energy densities we consider are −T t t = 0.05, 0.02, 0.005 (from inside to outside). Similar solutions exist for other… view at source ↗
Figure 8
Figure 8. Figure 8: Left: The Noether charge density j t is shown as a function of angular variable θ and several values of the radial coordinate for a typical J = 0 hybrid {electric ℓ = 1, magnetic ℓ = 1}-spinning Proca star. One notices the absence of a reflection symmetry with respect to θ = π/2. Right: The frequency-mass diagram is shown for the J = 0 family of spinning Proca stars (PSs) consisting in a non-linear superpo… view at source ↗
Figure 9
Figure 9. Figure 9: Left: Snapshots of the time evolution of the Proca Komar energy density in the xy and xz planes for the magnetic Proca star with ω/µ = 0.90. Time increases from left to right and from top to bottom. Right: Same as in the left panels, but for the real part of the scalar potential Xϕ. In addition, some ejection of Proca field energy is observed during the evolution, which can produce a recoil (kick) in the s… view at source ↗
Figure 10
Figure 10. Figure 10: Left: Snapshots of the Proca Komar energy density in the xz-plane for hybrid (ℓ = 0 electric and ℓ = 1 magnetic) Proca star with ω/µ = 0.95. Middle: Same for the angular momentum density. Right: Same for the real part of the scalar potential. 5.3.2 ℓ = 1 electric + ℓ = 1 magnetic We now consider configurations constructed from the superposition of ℓ = 1 electric and ℓ = 1 magnetic Proca star solutions. Th… view at source ↗
Figure 11
Figure 11. Figure 11: Left: Snapshots of the Proca Komar energy density in the xz-plane for hybrid (ℓ = 1 electric + ℓ = 1 magnetic) Proca star with ω/µ = 0.95. Time increases from left to right and from top to bottom. Right: Same as in the left panels, but for the real part of the scalar potential Xϕ. Finally, in [PITH_FULL_IMAGE:figures/full_fig_p023_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Time evolution of the Hamiltonian constraint for all Proca star configurations considered in this section, illustrating the control of constraint violations throughout the simulations. 6 Further remarks While solitonic solutions of the Einstein–Klein–Gordon model have been investigated for more than five decades, the exploration of analogous configurations supported by a massive complex spin–1 field remai… view at source ↗
read the original abstract

We construct new families of everywhere regular, asymptotically flat solitons in the Einstein--Proca model, obtained as self-gravitating continuations of flat-spacetime (singular) Proca multipoles. First we consider static and axially symmetric solutions, organized by a multipole number $\ell$. Two distinct classes arise: electric-type configurations, which include the spherical Proca stars as the $\ell=0$ case, and magnetic-type configurations, which have no spherical counterpart and start at $\ell=1$. Then we construct hybrid solutions as nonlinear superpositions of electric and magnetic multipoles. These have non-vanishing local angular momentum density but vanishing total angular momentum, and in some cases have no north-south $\mathbb{Z}_2$-symmetry. By performing dynamical evolutions of Proca stars in the new magnetic and hybrid sectors, we show they are unstable, decaying to the (static) prolate Proca stars or the (stationary) spinning Proca stars, previously identified as dynamically robust, electric sector configurations. In some cases, they can also collapse into a black hole.

Editorial analysis

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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 / 2 minor

Summary. The paper constructs new families of everywhere-regular, asymptotically flat solitons in the Einstein-Proca model by numerically continuing flat-spacetime Proca multipoles. It identifies static axially symmetric electric-type solutions (including spherical Proca stars at ℓ=0), magnetic-type solutions (starting at ℓ=1 with no spherical limit), and hybrid electric-magnetic superpositions that carry local angular momentum density but zero total angular momentum. Dynamical evolutions demonstrate that the magnetic and hybrid families are unstable, decaying to the known stable prolate or spinning electric configurations or collapsing to black holes.

Significance. If the numerical constructions are accurate, the work meaningfully enlarges the known solution space of Proca stars by adding magnetic and hybrid sectors and supplies concrete stability information. The hybrid configurations with vanishing total angular momentum yet nonzero local density are a notable addition. The manuscript supplies reproducible numerical evidence for the instability results and the decay channels, which strengthens its contribution to the study of vector-field solitons.

major comments (2)
  1. [Numerical Methods] Numerical Methods section: the manuscript provides no explicit residual norms, grid-convergence tables, or comparison of the nonlinear solutions against the linearized flat-space multipoles. Without these, it is impossible to confirm that the reported regularity at the origin and the multipole asymptotics at infinity are achieved to controllable truncation error, which directly underpins the central claim that the configurations solve the full nonlinear Einstein-Proca system.
  2. [Dynamical Evolutions] Dynamical Evolutions section (around the stability results): the time-evolution runs for the magnetic and hybrid families lack reported resolution studies or constraint-violation diagnostics. This leaves open whether the observed decay to prolate/spinning stars or black-hole formation is robust or influenced by numerical artifacts, which is load-bearing for the instability conclusions.
minor comments (2)
  1. [Hybrid Solutions] The definition of the hybrid boundary conditions at infinity should be stated more explicitly, including how the electric and magnetic multipole moments are independently prescribed.
  2. [Figures] Figure captions for the energy-density plots should include the specific values of the Proca mass parameter and the multipole order ℓ used in each panel.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will incorporate the requested numerical diagnostics into the revised version.

read point-by-point responses

  1. Referee: [Numerical Methods] Numerical Methods section: the manuscript provides no explicit residual norms, grid-convergence tables, or comparison of the nonlinear solutions against the linearized flat-space multipoles. Without these, it is impossible to confirm that the reported regularity at the origin and the multipole asymptotics at infinity are achieved to controllable truncation error, which directly underpins the central claim that the configurations solve the full nonlinear Einstein-Proca system.

    Authors: We agree that explicit residual norms, grid-convergence tables, and comparisons to the linearized flat-space multipoles would strengthen the numerical validation. In the revised manuscript we will add a new subsection to the Numerical Methods section containing: (i) L2 residual norms of the Einstein-Proca equations evaluated on representative solutions at three successively refined grid resolutions, (ii) a table reporting the observed convergence order, and (iii) direct pointwise comparisons of the nonlinear solutions near the origin and in the asymptotic region against the corresponding linearized multipole expansions. These additions will demonstrate that the reported regularity and multipole decay are achieved to controllable truncation error. revision: yes

  2. Referee: [Dynamical Evolutions] Dynamical Evolutions section (around the stability results): the time-evolution runs for the magnetic and hybrid families lack reported resolution studies or constraint-violation diagnostics. This leaves open whether the observed decay to prolate/spinning stars or black-hole formation is robust or influenced by numerical artifacts, which is load-bearing for the instability conclusions.

    Authors: We acknowledge that resolution studies and constraint-violation diagnostics are necessary to substantiate the dynamical instability results. In the revised manuscript we will expand the Dynamical Evolutions section to include, for representative magnetic and hybrid initial data: (i) time evolutions performed at three different spatial resolutions, showing convergence of the maximum energy density and total angular momentum, and (ii) plots of the L2 norms of the Hamiltonian and momentum constraint violations as functions of time, confirming that violations remain at truncation-error level and decrease with resolution. These diagnostics will establish that the observed decay channels are not numerical artifacts. revision: yes

Circularity Check

0 steps flagged

Numerical constructions of multipolar Proca stars exhibit no circularity

full rationale

The paper obtains its central results by direct numerical integration of the Einstein-Proca equations subject to multipole boundary conditions taken from the known flat-space Proca multipoles. No step reduces a claimed prediction or uniqueness result to a fitted parameter or to a prior self-citation by construction. The dynamical evolutions are presented as independent stability tests rather than as outputs forced by the static solutions. Self-citations to earlier Proca-star papers exist but are not load-bearing for the new multipolar families, which rest on the field equations themselves.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard Einstein-Proca action and the existence of numerical solutions with specified multipole boundary conditions at infinity; no new free parameters or invented entities are introduced beyond the model.

axioms (1)
  • domain assumption The Einstein-Proca field equations govern the system
    The model is the standard Einstein-Proca theory with a massive vector field.

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Reference graph

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