We investigate mass ladder operators for the static BTZ-like black hole in Einstein-bumblebee gravity and probe the quasinormal frequencies of the mapped modes using mass ladder operators for a scalar perturbation und...We investigate mass ladder operators for the static BTZ-like black hole in Einstein-bumblebee gravity and probe the quasinormal frequencies of the mapped modes using mass ladder operators for a scalar perturbation under Dirichlet and Neumann boundary conditions.We find that the mass ladder operators depend on the Lorentz symmetry breaking parameter,and the imaginary parts of the frequencies shifted by the mass ladder operators increase with the increase in the Lorentz symmetry breaking parameter under the two boundary conditions.Note that,under the Neumann boundary condition,the mapped modes caused by the mass ladder operator D_(0,k_(+))are unstable.Moreover,the mass ladder operators do not change the Breitenlohner-Freedman bound for the scalar modes,as in the case of the usual BTZ black hole.These results could aid us in further understanding the conformal symmetry and Lorentz symmetry breaking in Einstein-bumblebee gravity.展开更多
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 Ei...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 η^(2),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+c M ,where M is a black hole mass parameter.We observe,by increasing the parameter c(η^(2)) 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 numberlwith 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.展开更多
基金Supported by the National Key Research and Development Program of China(2020YFC2201400)the National Natural Science Foundation of China(12275078,12275079,12035005)+1 种基金Hunan Province College Students Research Learning and Innovative Experiment Project(S202210542197)the Innovative Research Group of Hunan Province,China(2024JJ1006)。
文摘We investigate mass ladder operators for the static BTZ-like black hole in Einstein-bumblebee gravity and probe the quasinormal frequencies of the mapped modes using mass ladder operators for a scalar perturbation under Dirichlet and Neumann boundary conditions.We find that the mass ladder operators depend on the Lorentz symmetry breaking parameter,and the imaginary parts of the frequencies shifted by the mass ladder operators increase with the increase in the Lorentz symmetry breaking parameter under the two boundary conditions.Note that,under the Neumann boundary condition,the mapped modes caused by the mass ladder operator D_(0,k_(+))are unstable.Moreover,the mass ladder operators do not change the Breitenlohner-Freedman bound for the scalar modes,as in the case of the usual BTZ black hole.These results could aid us in further understanding the conformal symmetry and Lorentz symmetry breaking in Einstein-bumblebee gravity.
基金supported by the National Natural Science Foundation of China(Grant Nos.11705054,11881240252,12035005)the Hunan Provincial Natural Science Foundation of China(Grant No.2022JJ30367)the Scientific Research Fund of Hunan Provincial Education Department(Grant No.22A0039)。
文摘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 η^(2),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+c M ,where M is a black hole mass parameter.We observe,by increasing the parameter c(η^(2)) 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 numberlwith 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.
