Fixing the dynamical evolution of self-interacting vector fields

Creators: Rubio, Marcelo¹; Lara, Guillermo²; Bezares, Miguel³; Crisostomi, Marco⁴; Barausse, Enrico¹

1. International School for Advanced Studies
2. Max Planck Institute for Gravitational Physics
3. University of Nottingham
4. California Institute of Technology

Abstract

Numerical simulations of the Cauchy problem for self-interacting massive vector fields often face instabilities and apparent pathologies. We explicitly demonstrate that these issues, previously reported in the literature, are actually due to the breakdown of the well posedness of the initial-value problem. This is akin to shortcomings observed in scalar-tensor theories when derivative self-interactions are included. Building on previous work done for 𝑘-essence, we characterize the well-posedness breakdowns, differentiating between Tricomi- and Keldysh-like behaviors. We show that these issues can be avoided by “fixing the equations,” enabling stable numerical evolutions in spherical symmetry. Additionally, we show that, for a class of vector self-interactions, no Tricomi-type breakdown takes place. Finally, we investigate initial configurations for the massive vector field which lead to gravitational collapse and the formation of black holes.

Copyright and License (English)

Acknowledgement (English)

We thank Ramiro Cayuso, Luis Lehner, and Marc Schneider for discussions throughout this work. M. R. acknowledges hospitality from the Perimeter Institute for Theoretical Physics, where part of this work was carried out. M. R. and E. B. acknowledge support from the European Union’s H2020 ERC Consolidator Grant “Gravity from Astrophysical to Microscopic Scales” (Grant No. GRAMS-815673), the PRIN 2022 Grant “GUVIRP—Gravity Tests in the Ultraviolet and Infrared with Pulsar Timing,” and the EU Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie Grant Agreement No. 101007855. M. C. is funded by the European Union under the Horizon Europe’s Marie Sklodowska-Curie Project No. 101065440.

Funding (English)

European Union’s H2020 ERC Consolidator Grant “Gravity from Astrophysical to Microscopic Scales” (Grant No. GRAMS-815673), the PRIN 2022 Grant “GUVIRP—Gravity Tests in the Ultraviolet and Infrared with Pulsar Timing,” and the EU Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie Grant Agreement No. 101007855. M. C. is funded by the European Union under the Horizon Europe’s Marie Sklodowska-Curie Project No. 101065440.

Data Availability (English)

We stress that the Z3 formulation considers an additional evolution field, 𝑍𝑖, which accounts for the momentum constraint, and thus it should remain close to zero throughout the whole numerical evolution. In spherical symmetry, only the radial component 𝑍𝑟 is dynamical. We refer the reader to Refs. [81,82], where the explicit set of field equations in this formulation is displayed.

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PhysRevD.110.063015.pdf

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