2014/06/26 by Tuyen Trung Truong, Truong, Tuyen Trung
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Cosmology and Gravitation Theories #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Mathematical Physics (math-ph) #math-ph #math.MP
paper · pdf · doi:10.48550/arxiv.1406.6917
4 pages. Comments are welcome!
arxiv created 2014/06/26 · openalex publication_date 2014/06/26 · arxiv updated 2014/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let (M,g) be a spacetime. That is, M is a real manifold of dimension 4 equipped with a Lorentzian metric g. We show that any separation of time and space in M is equivalent to introducing a (non-smooth) Riemann metric h. If h is smooth, it induces a smooth line bundle Tp→ M, whose any fiber is generated by a time-like vector, called the time bundle. Whether (M,g,h) is time orientable or not corresponds to whether this line bundle is trivial or not. As well-known, the last condition is characterized by the first Stiefel-Whitney class w1(Tp)∈ H1(M,ℤ/2). We then define a partial time orientation of M as a section of the line bundle T→ M. As applications, we discuss time and space differentiations on M.