2010/01/31 by Metin Gürses, Metin Gurses
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Cosmology and Gravitation Theories #Covariant derivative #Covariant transformation #Field (mathematics) #Geometric Analysis and Curvature Flows #Gravitation #Gravitational field #Killing vector field #Spacetime #Tensor (intrinsic definition) #Vector field #Vector potential #gr-qc #hep-th #math-ph #math.MP
paper · pdf · doi:10.1088/0264-9381/27/20/205018
published as Class.Quant.Grav.27:205018,2010 · 25 pages, some changes made and some references added, to be published in Classical and Quantum Gravity
arxiv created 2010/07/29 · openalex publication_date 2010/09/17 · arxiv updated 2014/11/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Killing vector fields in three dimensions play an important role in the construction of the related spacetime geometry. In this work we show that when a three-dimensional geometry admits a Killing vector field then the Ricci tensor of the geometry is determined in terms of the Killing vector field and its scalars. In this way we can generate all products and covariant derivatives at any order of the Ricci tensor. Using this property we give ways to solve the field equations of topologically massive gravity (TMG) and new massive gravity (NMG) introduced recently. In particular when the scalars of the Killing vector field (timelike, spacelike and null cases) are constants then all three-dimensional symmetric tensors of the geometry, the Ricci and Einstein tensors, their covariant derivatives at all orders, and their products of all orders are completely determined by the Killing vector field and the metric. Hence, the corresponding three-dimensional metrics are strong candidates for solving all higher derivative gravitational field equations in three dimensions.