Pith. sign in

REVIEW 2 major objections 3 minor 1 cited by

Proca stars in excited states

T0 review · 2 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper claims that the first two excited families of spherically symmetric Proca stars are always unstable under small perturbations, and that their evolutions end in black-hole collapse, dissipation, or migration to a lower-mass…

desk verdict Solid spherical-symmetry result on excited Proca stars; the instability conclusion is credible, but the abstract oversells the 'even very small perturbations' and 'always' claims. read the letter →

arxiv 2411.09032 v2 pith:SDU5K23R submitted 2024-11-13 gr-qc

Carlos Joaquin , Miguel Alcubierre This is my paper · ORCID
classification gr-qc PACS 04.20.Ex04.25.Dm95.30.Sf
keywords Procastarsexcitedstatesbosonicnumericalrelativitysphericalsymmetrydynamicalstabilityblackholeformationbindingenergy
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks whether excited states of Proca stars—self-gravitating balls of massive complex vector field—can be stable. The answer, for the first two excited families in spherical symmetry, is no: no stable branch exists, and even tiny perturbations drive the star to collapse to a black hole, disperse, or settle onto the stable branch of the ground state. The importance of the claim is that excited configurations have been proposed as intermediate stages in the formation of exotic compact objects; if they are universally unstable, only their transient dynamics matter. The paper also identifies where migration is possible: a small corner of parameter space with negative binding energy and low mass, where the star ejects roughly a quarter to a third of its mass and lands on a known ground-state solution.

What carries the argument

The central object is the self-gravitating Proca field in spherical symmetry, reduced by a harmonic ansatz $\phi=\varphi(r)e^{-i\omega t}$, $a_r=i a(r)e^{-i\omega t}$, $E^r=e(r)e^{-i\omega t}$ to a first-order ODE system for the metric functions $A,\alpha$ and the field profiles $F=\alpha\varphi$ and $a$, solved as an eigenvalue problem for $\omega$ by shooting. Excited states are labelled by nodes in the vector potential $a(r)$. Stability is probed by adding a small Gaussian perturbation to the scalar potential, re-solving the Hamiltonian and Gauss constraints to get consistent initial data, and evolving with a BSSN code adapted to spherical symmetry; the end state is diagnosed through the central lapse (collapse to zero signals horizon formation), the total mass at the boundary, and a fast Fourier transform of the scalar potential at the origin whose dominant late-time frequency is matched to the stationary ground-state family.

What would settle it

Run the same perturbed first- and second-excited configurations in a full 3+1 evolution without imposing spherical symmetry: if any such configuration relaxes to a stable excited Proca star, or if a migrating star settles on a non-spherical ground state instead of the spherical one, the paper's 'always unstable, migration to the spherical ground state' claim fails.

Watch

Extended reading notes

Core claim

Within spherical symmetry, the paper constructs stationary families of Proca stars in the ground, first-excited, and second-excited states by solving the Einstein–Proca ODE system with a shooting method, then evolves perturbed configurations with a fully non-linear BSSN code. It finds that the first and second excited families have no stable branch analogous to the ground state: configurations with negative binding energy to the left of the maximum mass, which one might expect to be stable, are only metastable. Under a 5% Gaussian perturbation, the low-mass ones migrate to the stable branch of the ground state, identified by matching the final dominant frequency $\omega_f$ and final total mass $M_f$ against the ground-state family, while higher-mass ones collapse to a black hole; configurations with positive binding energy dissipate. The paper therefore concludes that excited Proca stars are always unstable against small perturbations in spherical symmetry, with a three-way final fate determined by where the initial configuration sits in the family.

Load-bearing premise

The load-bearing premise is that spherical symmetry is preserved throughout the evolution, so the three-way classification (black-hole collapse, dissipation, migration to the spherical ground state) captures the true dynamical attractors; the paper itself notes, citing reference [25], that the spherical ground state is unstable to non-spherical perturbations, so the endpoint in the full theory could differ.

Editorial extensions

If this is right

  • No stable branch exists for the first two excited Proca-star families; every spherically symmetric configuration tested is unstable, so these objects cannot be long-lived equilibrium states.
  • A perturbed excited Proca star has three possible fates: collapse to a black hole, dispersal of the field, or migration to a lower-mass configuration on the stable ground-state branch.
  • Migration happens only in a narrow low-mass, negative-binding-energy region; first-excited migrants lose about 25% of their mass, second-excited migrants about 35–39%, with the ejected field carrying away the excess.
  • The final migrated state is not arbitrary: its dominant frequency and total mass match a specific ground-state solution, so the endpoint is predictable from the family curves.
  • Higher excitation makes collapse more likely; in the second excited state only very low-mass configurations show migration, and the metastable window shrinks.

Reading between the lines

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

  • If this result extends beyond the first two families, excited Proca stars in general would be transient, and searches for stable bosonic dark-matter clumps should focus on ground-state configurations.
  • Because the paper restricts to spherical symmetry and the paper itself notes (via reference [25]) that the spherical ground state is unstable to non-spherical perturbations, the migration endpoint in a full 3D evolution could be a non-spherical ground state rather than the spherical one; the 'always unstable' verdict would then be strengthened but the migration destination would need revision.
  • A testable extension is to include vector self-interactions: the paper notes these can stabilize excited boson stars, but for vector fields they risk loss of hyperbolicity, so the net effect on excited Proca stars is genuinely open.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. The paper constructs stationary Proca star solutions in the first two excited states under spherical symmetry and studies their nonlinear stability by adding Gaussian perturbations to the scalar potential and evolving the system with the OllinSphere BSSN code. The main findings are that the first two excited families have no stable branch, and that perturbed configurations either migrate to a ground-state configuration, collapse to a black hole, or disperse. The abstract states that excited Proca stars are always unstable against even very small perturbations.

Significance. If the instability claim holds, the paper fills a gap in the bosonic star literature by showing that excited Proca star families lack a stable spherical branch, in contrast to the ground state. The numerical methodology is standard and carefully tested: stationary solutions are obtained by a shooting method with Hamiltonian and Gauss constraint violations around 1e-9 and fourth-order convergence, and dynamical outcomes are diagnosed by lapse collapse, apparent horizon formation, mass loss at the boundary, and FFT frequency extraction. The paper is transparent about its restriction to spherical symmetry, and it explicitly cites the result of ref. [25] that the spherical ground state is unstable against non-spherical perturbations.

major comments (2)
  1. [Sec. IV.B and Abstract] All dynamical evolutions use a Gaussian perturbation with amplitude fixed at 5% of the maximum of the scalar potential, and the statement that 'smaller perturbations give similar results' is not quantified or shown. A 5% perturbation is finite, not 'very small,' so the abstract's claim that excited Proca stars are 'always unstable against even very small perturbations' is not supported by the presented evidence. The observation that truncation error eventually triggers the instability does not provide a controlled amplitude dependence: as resolution changes, the effective perturbation size changes, so the results cannot distinguish exponential instability from a long-lived metastable process kicked by a finite perturbation. I recommend adding a systematic amplitude study (e.g., amplitudes of 0.5%, 1%, 2%, and 5% for representative models) or a linear stability analysis of the radial perturbation equations; otherwise the abstract and conclusions should be reworded to state instability under finite perturbations.
  2. [Table II and Appendix A] The identification of the migration endpoint with a particular ground-state configuration is not quantitatively consistent for several rows. In the first excited state, the row with φ0=0.009 reports (ωf, Mf) = (0.95407±0.00024, 0.839±0.005) compared with the ground-state values (ω, M) = (0.953, 0.822); the mass differs by 0.017, more than three times the stated uncertainty, and the frequency differs by about 4.5σ. Similar discrepancies appear for the row with φ0=0.017, where Mf=0.955 versus M=0.935. The appendix text says the agreement is 'remarkable,' which overstates the actual agreement. The authors should either quantify the systematic errors in the FFT frequency and the asymptotic mass, interpolate the ground-state family to the measured (ωf, Mf), or present the endpoint as consistent within estimated systematic uncertainties if that can be justified.
minor comments (3)
  1. [Secs. I and V] The paper notes (citing ref. [25]) that the spherical ground state is unstable against non-spherical perturbations; the conclusion's phrase 'stable branch of the ground state' should be qualified as stable within spherical symmetry to avoid misleading readers outside that restricted context.
  2. [Throughout] There are several typos: the Figure 6 caption duplicates 'time,' the Figure 14 caption reads 'in for the models with with,' Section III uses 'anzats' instead of 'ansatz,' and page 12 has 'the the minimum.'
  3. [Sec. IV.B] The perturbation description says 'unit width' but does not specify whether this refers to the standard deviation or the full width at half maximum of the Gaussian; please define the width explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the instability and migration conclusions are independent dynamical outcomes compared with separately computed stationary families, not fitted or definitionally forced results.

full rationale

The central claims rest on two independent computations. The stationary Proca-star families are generated by integrating the Einstein-Proca ODEs with the frequency omega found by shooting until the scalar potential decays at infinity; omega is an eigenvalue fixed by boundary conditions, not a parameter fitted to the dynamical evolutions. The dynamical runs start from self-consistent initial data built by adding a Gaussian perturbation to phi (or F), re-solving the Hamiltonian, polar-slicing, and Gauss constraints, and then evolving with the BSSN/OllinSphere code. The final states are read off from the lapse behavior, apparent horizons, total mass at the boundary, and FFT dominant frequencies, and only afterwards compared with the independently computed ground-state family. That comparison is a consistency check, not a fit: the ground-state M(omega) relation is not constructed from the migration data. Two limitations are explicitly flagged in the paper and weighed here, but neither is circular: (i) Sec. IV.B states all runs use a 5% Gaussian perturbation, with 'smaller perturbations give similar results' asserted but not quantified, so the abstract's 'even very small perturbations' is an extrapolation; (ii) Sec. I notes [25] showed the spherical ground state is unstable against non-spherical perturbations and the paper only considers spherical symmetry, which limits the generality of the migration endpoint but does not make the conclusion an input. The migration endpoint is identified by comparing final (omega_f, M_f) with the nearest sampled ground-state solution (Table II), and for some rows the mass match is outside the stated uncertainty; this is an identification accuracy issue, not circularity. No derivation step reduces to its own inputs, and no load-bearing self-citation or imported uniqueness theorem was found.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the Einstein-Proca model and the assumption that spherically symmetric evolutions capture the relevant dynamics; the paper itself flags the spherical-symmetry limitation. No new entities are introduced. The 5% perturbation is a numerical convention.

free parameters (2)
  • Perturbation amplitude = 5% of max |phi|
    Chosen by hand to trigger the instability in reasonable computational time; the paper states smaller perturbations give similar results but are slower, so this choice is a numerical convention rather than a load-bearing parameter.
  • Gaussian perturbation width = 1 (unit width)
    Chosen without detailed justification; used to perturb the scalar potential around the origin in all simulations.
assumptions (4)
  • domain assumption The Einstein-Proca system with minimal coupling and no self-interactions is the correct model for these compact objects.
    The action in Eq. (2) defines the theory; the paper notes self-interactions are excluded and that vector self-interactions can introduce pathologies (refs [48-53]).
  • domain assumption Spherical symmetry is preserved during evolution and the perturbed initial data are spherically symmetric.
    Section I restricts to spherical symmetry and cites ref [25] showing the spherical ground state is unstable against non-spherical perturbations; the final-state classification may not persist in 3D.
  • standard math The BSSN formulation with 1+log slicing and vanishing shift is a faithful discretization of the Einstein-Proca equations.
    Section II.B and IV describe the formalism and cite prior tests of OllinSphere; convergence is checked but the code is not shipped.
  • standard math The shooting algorithm's eigenvalue omega and the rescaling of the lapse are valid for constructing stationary solutions.
    Section III.A describes the shooting procedure and the rescaling of alpha and omega; boundary conditions are standard.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Proca stars in excited states." pith.science (2026). https://pith.science/paper/SDU5K23R

@misc{pith2026241109032,
  author       = {Pith},
  title        = {Pith review of: Proca stars in excited states},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SDU5K23R}},
  note         = {Machine review of arXiv:2411.09032}
}
read the original abstract

In this paper we consider families of solutions for excited states of Proca stars in spherical symmetry. We focus on the first two excited configurations and perform a series of fully non-linear dynamical simulations in order to study their properties and stability. Our analysis reveals that excited Proca stars are always unstable against even very small perturbations, and their dynamical evolution can lead to three different final states: collapse to a black hole, dissipation, or migration to a different configuration in the ground state. We find that migration to the ground state can only occur in a small region of the parameter space of solutions with negative binding energy.

Figures

Figures reproduced from arXiv: 2411.09032 by the authors.

Figure 1
Figure 1. Left panel: Total integrated mass M (in units of M2 P lanck/m) as a function of frequency ω. We observe that as the excitation level of the star increases the total mass M also increases. Right panel: Binding energy U as a function of φ0. For each of the families U has a global minimum, which corresponds to the solutions with maximum mass of Table I. reaches a minimum, which in fact corresponds to the solutions with… view at source ↗
Figure 2
Figure 2. Left panel.: The effective compactness defined as C99 = M/R99 versus φ0. Right panel: Total mass M as a function of the effective radius R99. We observe that excited Proca stars increase in both mass and radius. In [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. We show the solutions for the ground state and the first two excited states for the same central value of the scalar [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (14 more)
Figure 4
Figure 4. Figure 4: shows a plot of the binding energy U for Proca stars in the first excited state as a function of the central value of the scalar potential φ0. In the figure we also indicate 7 different models that we considered for our dynamical evolutions of perturbed initial data. N…
Figure 5
Figure 5. Figure 5: Evolution of the central value of the lapse function [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: Apparent horizon mass as a function of time time for the three models in the first excited state that exhibit collapse to [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]
Figure 7
Figure 7. Figure 7: Evolution of the real part of the scalar potential [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: Evolution of the total mass M at the outer boundary for models in the first excited state that exhibit migration [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 9
Figure 9. Figure 9: Fourier transform of the time evolution of the real part of [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]
Figure 10
Figure 10. Figure 10: Using the data for the final dominant frequency [PITH_FULL_IMAGE:figures/full_fig_p016_10.png]
Figure 11
Figure 11. Figure 11: Binding energy U as a function of φ0 for the second excited state. The circles show the 6 models that we considered for our perturbed dynamical evolutions. As before, the color bar and the size of the circles indicate the initial total mass [PITH_FULL_IMAGE:figures/f…
Figure 12
Figure 12. Figure 12: Evolution of the central value of the lapse function [PITH_FULL_IMAGE:figures/full_fig_p017_12.png]
Figure 13
Figure 13. Figure 13: Apparent horizon mass as a function of time for the 2 collapsing models in the second excited state. The dashed [PITH_FULL_IMAGE:figures/full_fig_p017_13.png]
Figure 14
Figure 14. Figure 14: Evolution of the real part of the scalar potential [PITH_FULL_IMAGE:figures/full_fig_p018_14.png]
Figure 15
Figure 15. Figure 15: Evolution of the total mass M at the boundary for models in the second excited state that exhibit migration. We also obtain the FFT of the real part of ϕ evaluated at r = 0 for the initial and final stages of the evolution, see [PITH_FULL_IMAGE:figures/full_fig_p018_…
Figure 16
Figure 16. Figure 16: Fourier transform of the time evolution of the real part of [PITH_FULL_IMAGE:figures/full_fig_p019_16.png]
Figure 17
Figure 17. Figure 17: Using the data for the final dominant frequency [PITH_FULL_IMAGE:figures/full_fig_p019_17.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

  1. Nonrelativistic Proca stars: Spherical stationary and multi-frequency states

    gr-qc 2024-12 conditional novelty 6.0 of 10

    Nonrelativistic Proca stars have a ground state with constant polarization (linear or circular depending on the sign of the spin-spin coupling), and a symmetry-enhanced sector with λs=0 contains a continuum of multi-f...

Reference graph

Works this paper leans on

57 extracted references · 34 canonical work pages · cited by 1 Pith paper

  1. [25]

    Lazarte and M

    C. Lazarte and M. Alcubierre, Class. Quant. Grav. 41, 135003 (2024)

    work page 2024

  2. [1]

    The perturbation in φ (or F ) must be even with respect to reflections on the origin to maintain regularity there

    We add a small perturbation to the scalar potential φ(r) (in practice, we add it to F (r)), typically a Gaussian, leaving the vector potential a(r) unchanged. The perturbation in φ (or F ) must be even with respect to reflections on the origin to maintain regularity there. For example, if we add a Gaussian centered on a point r = r0 away from the origin, ...

    work page

  3. [2]

    We solve again the Hamiltonian constraint (38), the polar slicing condition (39), and the Gauss constraint (37) for the functions ( A, α, e). Notice that we now cannot use the explicit solution for the electric field e(r) given by (34) since it is no longer valid for a perturbed star, so we must solve for e(r) using equation (37). The procedure just descr...

    work page

  4. [3]

    In the figure we also indicate 7 different models that we considered for our dynamical evolutions of perturbed initial data

    First excited state Figure 4 shows a plot of the binding energy U for Proca stars in the first excited state as a function of the central value of the scalar potential φ0. In the figure we also indicate 7 different models that we considered for our dynamical evolutions of perturbed initial data. Notice that all these models have negative binding energy, a...

    work page

  5. [4]

    Second excited state As shown in Figure 11, for the case of the second excited states we have also selected models with negative binding energy to the left of the minimum, but in this case even closer to the origin. The reason for this is the general trend of instability for the Proca stars: for higher excited states collapse to a black hole becomes more ...

    work page

  6. [5]

    D. J. Kaup, Phys. Rev. 172, 1331 (1968)

    1968

  7. [6]

    Ruffini and S

    R. Ruffini and S. Bonazzola, Phys. Rev. 187, 1767 (1969). 21 Models ωi Mi ωf ± δωf Mf ± 0.005 ω M 1° excited Migration Initial data: gnd 0.006 0.980 1.042 0.96287 ± 0.00031 0.769 0.960 0.772 0.009 0.977 1.137 0.95407 ± 0.00024 0.839 0.953 0.822 0.013 0.973 1.232 0.94499 ± 0.00026 0.897 0.941 0.898 0.017 0.967 1.322 0.93170 ± 0.00035 0.955 0.933 0.935 2° e...

    work page 1969

  8. [7]

    Brito, V

    R. Brito, V. Cardoso, C. A. Herdeiro, and E. Radu, Physics Letters B 752, 291 (2016)

    work page 2016

Show all 57 references
  1. [8]

    S. L. Liebling and C. Palenzuela, Living Reviews in Relativity 26 (2023)

    2023

  2. [9]

    F. E. Schunck and E. W. Mielke, Classical and Quantum Gravity 20 (2003)

    2003

  3. [10]

    Seidel and W.-M

    E. Seidel and W.-M. Suen, Phys. Rev. Lett. 66, 1659 (1991)

    1991

  4. [11]

    Finster, J

    F. Finster, J. Smoller, and S.-T. Yau, Phys. Rev. D 59, 104020 (1999), arXiv:gr-qc/9801079

    1999 arXiv

  5. [12]

    S. L. Liebling and C. Palenzuela, Living Rev.Rel. 15, 6 (2012), arXiv:1202.5809 [gr-qc]

    2012 arXiv

  6. [13]

    C. A. R. Herdeiro and E. Radu, Symmetry 12, 2032 (2020), arXiv:2012.03595 [gr-qc]

    2020 arXiv

  7. [14]

    Galactic halos and rotating bosonic dark matter,

    J. C. Mourelle and C. Adam, “Galactic halos and rotating bosonic dark matter,” (2024), 2407.07839 [astro-ph.GA]

    2024 arXiv

  8. [15]

    The imitation game reloaded: effective shadows of dynamically robust spinning proca stars,

    I. Sengo, P. V. P. Cunha, C. A. R. Herdeiro, and E. Radu, “The imitation game reloaded: effective shadows of dynamically robust spinning proca stars,” (2024), 2402.14919 [gr-qc]

    2024 arXiv

  9. [16]

    J. L. Rosa and D. Rubiera-Garcia, Physical Review D 106 (2022)

    2022

  10. [17]

    Sanchis-Gual, C

    N. Sanchis-Gual, C. Herdeiro, J. A. Font, E. Radu, and F. Di Giovanni, Physical Review D 99 (2019)

    2019

  11. [18]

    Searching for vector boson-star mergers within ligo-virgo intermediate-mass black-hole merger candidates,

    J. C. Bustillo, N. Sanchis-Gual, S. H. W. Leong, K. Chandra, A. Torres-Forne, J. A. Font, C. Herdeiro, E. Radu, I. C. F. Wong, and T. G. F. Li, “Searching for vector boson-star mergers within ligo-virgo intermediate-mass black-hole merger candidates,” (2023), 2206.02551 [gr-qc]

    2023 arXiv

  12. [19]

    Brito, V

    R. Brito, V. Cardoso, C. F. B. Macedo, H. Okawa, and C. Palenzuela, Phys. Rev. D 93, 044045 (2016), 1512.00466

    2016 arXiv

  13. [20]

    Visinelli, Int

    L. Visinelli, Int. J. Mod. Phys. D 30, 2130006 (2021), arXiv:2109.05481 [gr-qc]

    2021 arXiv

  14. [21]

    Di Giovanni, N

    F. Di Giovanni, N. Sanchis-Gual, C. A. R. Herdeiro, and J. A. Font, Phys. Rev. D 98, 064044 (2018), arXiv:1803.04802 [gr-qc]

    2018 arXiv

  15. [22]

    Sanchis-Gual, C

    N. Sanchis-Gual, C. Herdeiro, E. Radu, J. C. Degollado, and J. A. Font, Phys. Rev. D 95, 104028 (2017), arXiv:1702.04532 [gr-qc]

    2017 arXiv

  16. [23]

    Zhang, L.-X

    R. Zhang, L.-X. Huang, and Y.-Q. Wang, (2023), arXiv:2312.15755 [gr-qc]

    2023 arXiv

  17. [24]

    Salazar Landea and F

    I. Salazar Landea and F. Garc ´ ıa, Phys. Rev. D94, 104006 (2016), arXiv:1608.00011 [hep-th]

    2016 arXiv

  18. [26]

    Balakrishna, E

    J. Balakrishna, E. Seidel, and W.-M. Suen, Physical Review D 58 (1998)

    1998

  19. [27]

    L. G. Collodel, B. Kleihaus, and J. Kunz, Physical Review D 96 (2017)

    2017

  20. [28]

    Brito, C

    M. Brito, C. Herdeiro, E. Radu, N. Sanchis-Gual, and M. Zilh˜ ao, Phys. Rev. D 107, 084022 (2023), arXiv:2302.08900 [gr-qc]

    2023 arXiv

  21. [29]

    Herdeiro, E

    C. Herdeiro, E. Radu, N. Sanchis-Gual, N. Santos, and E. dos Santos Costa Filho, Physics Letters B 852, 138595 (2024)

    2024

  22. [30]

    Alcubierre, Introduction to 3 + 1 Numerical Relativity (Oxford Univ

    M. Alcubierre, Introduction to 3 + 1 Numerical Relativity (Oxford Univ. Press, New York, 2008)

    2008

  23. [31]

    Shibata and T

    M. Shibata and T. Nakamura, Phys. Rev. D52, 5428 (1995)

    1995

  24. [32]

    T. W. Baumgarte and S. L. Shapiro, Phys. Rev. D59, 024007 (1998), gr-qc/9810065

    1998 arXiv

  25. [33]

    Alcubierre and M

    M. Alcubierre and M. D. Mendez, Gen.Rel.Grav. 43, 2769 (2011), arXiv:1010.4013 [gr-qc]

    2011 arXiv

  26. [34]

    Balakrishna, E

    J. Balakrishna, E. Seidel, and W.-M. Suen, Phys. Rev. D58, 104004 (1998), arXiv:gr-qc/9712064 [gr-qc]

    1998 arXiv

  27. [35]

    Guzman, Phys.Rev

    F. Guzman, Phys.Rev. D70, 044033 (2004), arXiv:0407054 [gr-qc]

    2004

  28. [36]

    Guzman, Revista Mexicana de Fisica 55, 321 (2009)

    F. Guzman, Revista Mexicana de Fisica 55, 321 (2009)

    2009

  29. [37]

    Seidel and W

    E. Seidel and W. Suen, Phys. Rev. D42, 384 (1990)

    1990

  30. [38]

    A. Y. Loginov, Phys. Rev. D 91, 105028 (2015)

    2015

  31. [39]

    Arbona, C

    A. Arbona, C. Bona, J. Mass´ o, and J. Stela, Phys. Rev. D60, 104014 (1999), gr-qc/9902053

    1999 arXiv

  32. [40]

    Alcubierre, B

    M. Alcubierre, B. Br¨ ugmann, P. Diener, M. Koppitz, D. Pollney, E. Seidel, and R. Takahashi, Phys. Rev. D67, 084023 (2003), gr-qc/0206072

    2003 arXiv

  33. [41]

    Alcubierre, B

    M. Alcubierre, B. Br¨ ugmann, D. Pollney, E. Seidel, and R. Takahashi, Phys. Rev. D64, R61501 (2001), gr-qc/0104020

    2001 arXiv

  34. [42]

    Alcubierre and J

    M. Alcubierre and J. M. Torres, Class. Quant. Grav. 32, 035006 (2015), arXiv:1407.8529 [gr-qc]

    2015 arXiv

  35. [43]

    Alcubierre, J

    M. Alcubierre, J. Barranco, A. Bernal, J. C. Degollado, A. Diez-Tejedor, M. Megevand, D. N´ u˜ nez, and O. Sarbach, Class. 22 Quant. Grav. 36, 215013 (2019), arXiv:1906.08959 [gr-qc]

    2019 arXiv

  36. [44]

    J. C. Degollado, M. Salgado, and M. Alcubierre, Phys. Lett. B 808, 135666 (2020), arXiv:2008.10683 [gr-qc]

    2020 arXiv

  37. [45]

    Jim´ enez-V´ azquez and M

    E. Jim´ enez-V´ azquez and M. Alcubierre, Physical Review D105 (2022), 10.1103/physrevd.105.064071

    2022 doi

  38. [46]

    Jim´ enez-V´ azquez and M

    E. Jim´ enez-V´ azquez and M. Alcubierre, Physical Review D106 (2022), 10.1103/physrevd.106.044071

    2022 doi

  39. [47]

    Brito, V

    R. Brito, V. Cardoso, C. A. R. Herdeiro, and E. Radu, Phys. Lett. B 752, 291 (2016), arXiv:1508.05395 [gr-qc]

    2016 arXiv

  40. [48]

    C. A. R. Herdeiro, A. M. Pombo, and E. Radu, Phys. Lett. B 773, 654 (2017), arXiv:1708.05674 [gr-qc]

    2017 arXiv

  41. [49]

    Cardoso and P

    V. Cardoso and P. Pani, Living Rev. Rel. 22, 4 (2019), arXiv:1904.05363 [gr-qc]

    2019 arXiv

  42. [50]

    Sanchis-Gual, C

    N. Sanchis-Gual, C. Herdeiro, and E. Radu, Classical and Quantum Gravity 39, 064001 (2022)

    2022

  43. [51]

    Di Giovanni, S

    F. Di Giovanni, S. Fakhry, N. Sanchis-Gual, J. C. Degollado, and J. A. Font, Class. Quant. Grav. 38, 194001 (2021), arXiv:2105.00530 [gr-qc]

    2021 arXiv

  44. [52]

    Clough, T

    K. Clough, T. Helfer, H. Witek, and E. Berti, Physical Review Letters 129 (2022)

    2022

  45. [53]

    Coates and F

    A. Coates and F. M. Ramazano˘ glu, Physical Review Letters 129 (2022)

    2022

  46. [54]

    Aoki and M

    K. Aoki and M. Minamitsuji, Phys. Rev. D 106, 084022 (2022), arXiv:2206.14320 [gr-qc]

    2022 arXiv

  47. [55]

    Herdeiro, E

    C. Herdeiro, E. Radu, and E. dos Santos Costa Filho, Journal of Cosmology and Astroparticle Physics 2023, 022 (2023)

    2023

  48. [56]

    Brito, C

    M. Brito, C. Herdeiro, N. Sanchis-Gual, E. dos Santos Costa Filho, and M. Zilhao, Classical and Quantum Gravity 41, 195005 (2024)

    2024

  49. [57]

    M. E. Rubio, G. Lara, M. Bezares, M. Crisostomi, and E. Barausse, Phys. Rev. D 110, 063015 (2024)

    2024

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.