Well-posed evolution of field theories with anisotropic scaling: the Lifshitz scalar field in a black hole space-time

TL;DR

This paper introduces a stable, second-order implicit numerical scheme for evolving anisotropic scaling field equations, demonstrated through Lifshitz scalar fields around black holes, revealing potential horizon instabilities.

Contribution

It develops an implicit evolution method for anisotropic PDEs with higher-order spatial derivatives, enabling stable long-term simulations.

Findings

Dispersive terms cause mode cascades near horizons.

Potential instability of the universal horizon.

Effective centrifugal barrier influences wave evolution.

Abstract

Partial differential equations exhibiting an anisotropic scaling between space and time -- such as those of Horava-Lifshitz gravity -- have a dispersive nature. They contain higher-order spatial derivatives, but remain second order in time. This is inconvenient for performing long-time numerical evolutions, as standard explicit schemes fail to maintain convergence unless the time step is chosen to be very small. In this work, we develop an implicit evolution scheme that does not suffer from this drawback, and which is stable and second-order accurate. As a proof of concept, we study the numerical evolution of a Lifshitz scalar field on top of a spherically symmetric black hole space-time. We explore the evolution of a static pulse and an (approximately) ingoing wave-packet for different strengths of the Lorentz-breaking terms, accounting also for the effect of the angular momentum eigenvalue and the resulting effective centrifugal barrier. Our results indicate that the dispersive terms produce a cascade of modes that accumulate in the region in between the Killing and universal horizons, indicating a possible instability of the latter.