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

**Authors:** Marcelo E. Rubio, Áron D. Kovács, M. Herrero-Valea, Miguel Bezares, Enrico Barausse  
**arXiv:** [2307.13041](https://arxiv.org/abs/2307.13041)  
**Date:** 2023-07-24  
**Summary page:** https://www.summarizepaper.com/en/paper/2307.13041v1/

## Key points

- Paper explores well-posed evolution of field theories with anisotropic scaling
- Focuses on Lifshitz scalar field in black hole space-time
- Anisotropic scaling refers to scaling between space and time that is different from usual isotropic scaling
- Considers partial differential equations with anisotropic scaling, such as those in Horava-Lifshitz gravity
- Equations have dispersive nature, containing higher-order spatial derivatives but remaining second order in time
- Convergence challenges arise when using standard explicit schemes for long-time numerical evolutions
- Authors develop stable and second-order accurate implicit evolution scheme to overcome convergence issues
- Scheme's effectiveness demonstrated through numerical evolution of Lifshitz scalar field on spherically symmetric black hole space-time
- Study investigates evolution of static pulse and approximately ingoing wave-packet for different strengths of Lorentz-breaking terms
- Takes into account effect of angular momentum eigenvalue and resulting effective centrifugal barrier
- Results show accumulation of modes between Killing and universal horizons, suggesting possible instability of universal horizon
- Provides insights into how anisotropic scaling affects field theories
- Offers novel implicit evolution scheme for equations with anisotropic scaling
- Findings contribute to understanding Horava-Lifshitz gravity and similar theories with anisotropic scaling properties.

## Summary

This paper explores the well-posed evolution of field theories with anisotropic scaling, focusing on the Lifshitz scalar field in a black hole space-time. Anisotropic scaling refers to a scaling between space and time that is different from the usual isotropic scaling. The authors specifically consider partial differential equations that exhibit this anisotropic scaling, such as those found in Horava-Lifshitz gravity. These equations have a dispersive nature, meaning they contain higher-order spatial derivatives but remain second order in time. This characteristic poses challenges for performing long-time numerical evolutions using standard explicit schemes, as convergence is not maintained unless very small time steps are used. To overcome this drawback, the authors develop an implicit evolution scheme that is stable and second-order accurate. They demonstrate the effectiveness of this scheme by studying the numerical evolution of a Lifshitz scalar field on top of a spherically symmetric black hole space-time. They investigate the evolution of both a static pulse and an approximately ingoing wave-packet for different strengths of Lorentz-breaking terms. The authors also take into account the effect of the angular momentum eigenvalue and resulting effective centrifugal barrier. Their results show that the dispersive terms in these equations lead to a cascade of modes that accumulate in the region between the Killing and universal horizons. This accumulation suggests a possible instability of the universal horizon. Overall, this study provides insights into how anisotropic scaling affects field theories and offers a novel implicit evolution scheme that overcomes convergence issues associated with these types of equations. The findings contribute to our understanding of Horava-Lifshitz gravity and have implications for other theories exhibiting similar anisotropic scaling properties.