a Lorentzian manifold
/ lor-ENT-see-an /
A Riemannian metric measures lengths and angles with everything positive — every direction costs positive distance. A Lorentzian metric breaks that symmetry by making exactly one direction count with the opposite sign. That single sign flip is the mathematical fingerprint of time: it splits directions at each point into timelike (the sign-flipped 'time' directions), spacelike (ordinary 'space' directions), and the lightlike directions on the boundary between them, which travel at the speed of light. A Lorentzian manifold is the geometry of spacetime, the arena of Einstein's general relativity.
Precisely, a Lorentzian manifold is a smooth manifold M with a nondegenerate symmetric metric tensor g of signature (-, +, +, ..., +) — one minus and the rest plus (some texts use the opposite, + - - -; the convention must be stated). At each point the tangent space is a copy of Minkowski space, and the metric sorts tangent vectors v by the sign of g(v, v): timelike if negative, null/lightlike if zero, spacelike if positive. The null vectors form a double cone, the light cone, at every point, and the entire causal structure — which events can influence which — is encoded in these cones. As in Riemannian geometry there is a unique Levi-Civita connection (metric-compatible and torsion-free), a curvature tensor, and geodesics; but now geodesics come in three causal types, timelike geodesics are the worldlines of free-falling particles, null geodesics are light rays, and 'length' along a timelike curve is proper time, the time a clock actually measures. A time orientation (a consistent global choice of which half of each light cone is 'future') is extra data that need not exist.
Why it matters: Lorentzian geometry is general relativity. The Einstein field equations Ric - (1/2) S g + Lambda g = 8 pi T relate the curvature of the Lorentzian metric to the matter content T, the singularity theorems prove that under physically reasonable conditions geodesics must end (black holes, the Big Bang), and causality is a geometric notion. The honest cautions are sharp. First, the central analytic difference from Riemannian geometry: the geodesic distance is NOT a metric in the topological sense — timelike curves can have arbitrarily small or even zero proper time, there is no Hopf-Rinow theorem, completeness behaves entirely differently, and 'shortest' is replaced by causal 'longest' (timelike geodesics locally MAXIMIZE proper time). Second, a Lorentzian metric does not exist on every manifold — its existence requires a nonvanishing line field, equivalent to either noncompactness or zero Euler characteristic — so unlike Riemannian metrics (which every manifold admits), being Lorentzian is a topological restriction. Do not import Riemannian intuition wholesale; the sign change is not cosmetic.
Minkowski spacetime is R^4 with ds^2 = -dt^2 + dx^2 + dy^2 + dz^2. A vector v = (v_t, v_x, v_y, v_z) is timelike when -v_t^2 + v_x^2 + v_y^2 + v_z^2 < 0, i.e. it points more in the t-direction than in space; the curve t -> (t, 0, 0, 0) is a timelike geodesic of proper time equal to elapsed t, and it LOCALLY MAXIMIZES proper time among nearby curves between its endpoints — the twin paradox in geometric form.
Minkowski space: timelike geodesics maximize proper time, the opposite of Riemannian length-minimizing geodesics.
The Lorentzian 'distance' is not a metric and Hopf-Rinow fails: timelike geodesics locally maximize (not minimize) proper time, and geodesic completeness, causality, and the existence of a global time function are all genuinely different from the Riemannian story — importing Riemannian intuition uncritically is the classic error.