1. Preliminaries
Roger Penrose · Society for Industrial and Applied Mathematics eBooks · 1972
1.1. Definition. A space-time M is to be a real, four-dimensional connected C∞ Hausdorff manifold with a globally defined C∞ (or C2 would do) tensor field g of type (0, 2), which is nondegenerate and Lorentzian. By Lorentzian (or hyperbolic normal) is meant that for any x ∈ M there is a basis in Tx=Tx(M) (the tangent space to M at x) relative to which gx has the matrix diag (1, − 1, − 1, − 1).