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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.