Fixing the dynamical evolution of self-interacting vector fields

TL;DR

This paper identifies the causes of numerical instabilities in simulating self-interacting massive vector fields, attributes them to well-posedness issues, and proposes fixes for stable evolution, including conditions avoiding breakdowns and implications for black hole formation.

Contribution

It characterizes well-posedness breakdowns in vector field simulations, distinguishes between different types, and introduces methods to fix equations for stable numerical evolution.

Findings

Breakdowns are due to well-posedness issues similar to scalar-tensor theories.

Fixing the equations enables stable spherical symmetry simulations.

Certain vector self-interactions avoid Tricomi-type breakdowns.

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.