Quasinormal modes and late time tails of perturbation fields on a Schwarzschild-like black hole with a global monopole in the Einstein-bumblebee theory

In this paper we complete a systematic study on quasinormal modes (QNMs) and late time tails for scalar, Dirac and Maxwell fields on a spherically symmetric Schwarzschild-like black hole with a global monopole in the Einstein-bumblebee theory. To look for QNMs, we solve the equations of motion for all perturbation fields considered herein numerically, by employing both the matrix and the WKB methods, and find good agreements for numeric data obtained by these two techniques in the regime when both are valid. The impact of the bumblebee parameter c, the monopole parameter η2, and the multipole number ℓ on the fundamental quasinormal frequency are analyzed in detail. Our results are shown in terms of the quasinormal frequency measured by 1+cM\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt {1 + c} \,M$$\end{document}, where M is a black hole mass parameter. We observe, by increasing the parameter c (η2) with fixed first few ℓ, that the real part of QNMs increases for all spin fields; while the magnitude of the imaginary part decreases for scalar and Dirac fields but increases for Maxwell fields. By increasing the multipole number ℓ with fixed other parameters, we disclose that the real part of QNMs for all spin fields increases while the magnitude of the imaginary part decreases for scalar and Dirac fields but increases for Maxwell fields. In the eikonal limit (ℓ ≫ n), QNMs for all spin fields coincide with each other and the real part scale linearly with ℓ. In particular, the asymptotic QNMs approach the corresponding results given by the first order WKB formula, and only the real part of QNMs is dependent on the bumblebee and monopole parameters. In addition, it is shown that the late time behavior is determined not only by the multipole number but also by the bumblebee and monopole parameters, and is distinct for bosonic and fermonic fields. Moreover, the presence of the bumblebee (monopole) field makes the spin fields decay faster. Our results indicate, both in the context of QNMs and late time tails, that the bumblebee field and the monopole field play the same role in determining the dynamic evolution of perturbation fields.