---
tag: BY4QYYS788
kind: concept
name: Lorentzian manifold
msc: [53C50, 53B30]
specializes: [GH6CXWFNDJ]
sources:
  - url: https://en.wikipedia.org/wiki/Pseudo-Riemannian_manifold
    locator: Lorentzian manifold
---

## Summary

A Lorentzian manifold is a [[pseudo-Riemannian manifold|GH6CXWFNDJ]] whose metric has exactly one negative direction at each point.
It is the mathematical model of spacetime in general relativity.
Its tangent vectors split into timelike, null and spacelike vectors, which gives each tangent space a light cone.

## Explanations

### L1 human

In a Lorentzian manifold one direction plays the role of time and the others play the role of space.
The metric measures time with the opposite sign from space.
At each point the vectors of squared length zero form a double cone, the light cone, along which light travels.
Minkowski space $\R^4$ with metric $-dt^2 + dx^2 + dy^2 + dz^2$ is the flat example.
The vector $(1, 0, 0, 0)$ has squared length $-1$ and points in a time direction, while $(1, 1, 0, 0)$ lies on the light cone.