Lorentzian manifold

Builds on Pseudo-Riemannian manifold Signature of a quadratic form

Definition

Let $(M, g)$ be a pseudo-Riemannian manifold of dimension $n \ge 2$ . It is a Lorentzian manifold if, at each point $p \in M$ , the form $g_p$ has signature $(n - 1, 1)$ . That is, in a suitable basis of $T_pM$ the matrix of $g_p$ is $\operatorname{diag}(-1, 1, \ldots, 1)$ .

Remarks

Example: Minkowski space is $\R^n$ with coordinates $(t, x_1, \ldots, x_{n-1})$ and $g = -dt^2 + dx_1^2 + \cdots + dx_{n-1}^2$ . Its matrix is $\operatorname{diag}(-1, 1, \ldots, 1)$ at every point, with one negative and $n - 1$ positive entries.

Non-example: $\R^4$ with $g = -dt^2 - dx^2 + dy^2 + dz^2$ is pseudo-Riemannian, but its signature has two negative entries, so it is not Lorentzian.

Many physics texts use the opposite convention, signature $(1, n - 1)$ , that is $\operatorname{diag}(1, -1, \ldots, -1)$ ; the two conventions differ by replacing $g$ with $-g$ . A nonzero tangent vector $v$ is called timelike if $g(v, v) < 0$ , null if $g(v, v) = 0$ , and spacelike if $g(v, v) > 0$ (in the convention used here).

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Tag	`D1SYF8F8RQ`
URI	`m2://D1SYF8F8RQ`
Kind	definition
Writing	W0 stub
MSC	53C50, 53B30
Defines	Lorentzian manifold
Uses	Pseudo-Riemannian manifold, Signature of a quadratic form
Changed	2 November 2026

Lorentzian manifold

Definition

Remarks

Reading path

Further reading

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