This work belongs to the context of the standard-model extension, in which a Lorentz symmetry violation is induced by a bumblebee field as it acquires a nonzero vacuum expectation value. The mathematical formulation of a generic bumblebee model and its associated dynamical equations are presented. Then, these equations are considered for the vacuum and a substantial simplification is performed for the particular case of a gradient bumblebee field at its vacuum expectation value. After some further manipulation, a method to easily find solutions to the model is developed, in which the exact effect on the spacetime description due to the presence of this bumblebee field is explicitly provided. As some examples, the method is applied to determine the implications of the bumblebee field on the Schwarzschild spacetime and also on a rotating one. A previously published solution is recovered and some new ones are obtained. In the rotating situation, a simple solution is found which contains both the Kerr solution and the already published one as special cases. It is also shown its distinguished surfaces are still given by the same corresponding expressions for the Kerr solution. In conclusion, the mathematical improvement made is considered to be a significant contribution to the theory as a powerful tool to investigate its many aspects and consequences.
Lorentz-violating modifications to particle dynamics, thermodynamics and vacuum energy in bumblebee gravity
Universidade Federal da Paraíba（帕拉伊巴联邦大学）；Universidade Federal de Campina Grande（坎皮纳格兰德联邦大学）；Khazar University（哈扎尔大学）；Eastern Mediterranean University（东地中海大学）；University of Hradec Králové（赫拉德茨-克拉洛韦大学）；Al-Farabi Kazakh National University（阿里-法拉比哈萨克国立大学）
Universidade Federal de Mato Grosso（马托格罗索联邦大学）；Universidade Federal da Paraíba（帕拉伊巴联邦大学）；Universidade Federal de Campina Grande（坎皮纳格兰德联邦大学）；Khazar University（阿塞拜疆可汗大学）
Constrain from shadows of $M87^*$ and $Sgr A^*$ and quasiperiodic oscillations of galactic microquasars on a black hole arising from metric-affine bumblebee model
Holographic description of RN-like black hole in bumblebee gravity
Faculty of Science, Kunming University of Science and Technology（昆明理工大学理学院）；Institute of Fundamental Physics and Quantum Technology & Zhejiang Key Laboratory of Extreme Universe & School of Physical Science and Technology, Ningbo University（宁波大学物理科学与技术学院、基础物理学与量子技术研究所及浙江省极端宇宙重点实验室）；Center for Joint Quantum Studies and Department of Physics, School of Science, Tianjin University（天津大学理学院物理系及联合量子研究中心）
A no-go theorem in bumblebee vector-tensor cosmology
University of Sheffield（谢菲尔德大学）；Institute for Basic Science (IBS)（韩国基础科学研究院）；Observatoire de Paris, Université PSL, CNRS, LNE, Sorbonne Université（巴黎天文台、巴黎文理研究大学、法国国家科学研究中心、法国国家天文台、索邦大学）；Center for Quantum Spacetime, Sogang University（延世大学量子时空中心）；Institute of Science Tokyo（东京科学大学）；Yonsei University（延世大学）
Constraining Lorentz symmetry breaking in bumblebee gravity with extreme mass-ratio inspirals
Hangzhou Institute for Advanced Study, University of Chinese Academy of Sciences（中国科学院大学杭州高等研究院）；Hunan Normal University（湖南师范大学）；Beijing Normal University（北京师范大学）；Yangzhou University（扬州大学）
Gravitational Wave Standard Sirens as Probes of Lorentz Violation in Bumblebee Gravity
University of KwaZulu–Natal（夸祖鲁-纳塔尔大学）；University of Tashkent for Applied Sciences（塔什干应用科学大学）；Central Asian University（中亚大学）；Ulugh Beg Astronomical Institute（乌鲁伯格天文学研究所）；Universidad de La Frontera（拉弗龙特拉大学）；Kimyo International University in Tashkent（塔什干化学国际大学）；Alfraganus University（阿尔法拉甘努斯大学）
