nLab pseudo-Riemannian metric

Idea

The notion of pseudo-Riemannian metric is a slight variant of that of Riemannian metric.

Where a Riemannian metric is governed by a positive-definite bilinear form, a Pseudo-Riemannian metric is governed by an indefinite bilinear form.

This is equivalently the Cartan geometry modeled on the inclusion of a Lorentz group into a Poincaré group.

Definition

A pseudo-Riemannian “metric” is a nondegenerate quadratic form on a real vector space $\mathbb{R}^n$. A Riemannian metric is a positive-definite quadratic form on a real vector space. The data of such a quadratic form may be equivalently given by a nondegenerate symmetric bilinear pairing $\langle\, , \, \rangle$ on $\mathbb{R}^n$.

A pseudo-Riemannian metric

$$Q: \mathbb{R}^n \to \mathbb{R}$$

can always be diagonalized: there exists a basis $e_1, \ldots, e_n$ such that

$$Q(\sum_{1 \leq i \leq n} x_i e_i) = x_1^2 + \ldots + x_p^2 - x_{p+1}^2 - \ldots - x_n^2$$

where the pair $(p, n-p)$ is called the signature of the form $Q$. Pseudo-Riemannian metrics on $\mathbb{R}^n$ are classified by their signatures; thus we have a standard metric of signature $(p, n-p)$ where $\{e_1, \ldots, e_n\}$ is the standard basis of $\mathbb{R}^n$.

More generally, there is a notion of pseudo-Riemannian manifold (of type $(p, n-p)$, which is an $n$-dimensional manifold $M$ equipped with a global section

$$\sigma: M \to S^2(T^* M)$$

of the bundle of symmetric bilinear forms over $M$, such that each $\sigma(x)$ is a nondegenerate form on the tangent space $T_x(M)$.

Certain theorems of Riemannian geometry carry over to the more general pseudo-Riemannian setting; for example, pseudo-Riemannian manifolds admit Levi-Civita connections, or in other words a unique notion of covariant differentiation of vector fields

$$\nabla: (X, Y) \mapsto \nabla_X(Y)$$

such that

$$X \cdot \langle Y, Z\rangle = \langle \nabla_X Y, Z\rangle + \langle Y, \nabla_X Z\rangle$$

$$[X, Y] = \nabla_X Y - \nabla_Y X$$

In that case, one may define a notion of geodesic in pseudo-Riemannian manifolds $M$, and we have a notion of “distance squared” between the endpoints along any geodesic path $\alpha: [0, 1] \to M$ (which might be a negative number of course). The term “pseudo-Riemannian metric” may refer to such distances in general pseudo-Riemannian manifolds. (I guess.)

A typical example of pseudo-Riemannian manifold is a Lorentzian manifold, where the metric is of type $(1, n-1)$. This is particularly so in the case $n = 4$, where such manifolds are the mathematical backdrop for studying general relativity and cosmological models.

• The terminology “metric” is not optimal of course: the values of the quadratic form would need to be nonnegative to avoid terminological conflict with metric as it is more commonly understood (and even in that case, the values of “metric” refer to the square of the metric rather than the metric itself). Caveat lector.

Related concepts

geometric context	gauge group	stabilizer subgroup	local model space	local geometry	global geometry	differential cohomology	first order formulation of gravity
differential geometry	Lie group/algebraic group $G$	subgroup (monomorphism) $H \hookrightarrow G$	quotient (“coset space”) $G/H$	Klein geometry	Cartan geometry	Cartan connection
examples	Euclidean group $Iso(d)$	rotation group $O(d)$	Cartesian space $\mathbb{R}^d$	Euclidean geometry	Riemannian geometry	affine connection	Euclidean gravity
	Poincaré group $Iso(d-1,1)$	Lorentz group $O(d-1,1)$	Minkowski spacetime $\mathbb{R}^{d-1,1}$	Lorentzian geometry	pseudo-Riemannian geometry	spin connection	Einstein gravity
	anti de Sitter group $O(d-1,2)$	$O(d-1,1)$	anti de Sitter spacetime $AdS^d$				AdS gravity
	de Sitter group $O(d,1)$	$O(d-1,1)$	de Sitter spacetime $dS^d$				deSitter gravity
	linear algebraic group	parabolic subgroup/Borel subgroup	flag variety	parabolic geometry
	conformal group $O(d,t+1)$	conformal parabolic subgroup	Möbius space $S^{d,t}$		conformal geometry	conformal connection	conformal gravity
supergeometry	super Lie group $G$	subgroup (monomorphism) $H \hookrightarrow G$	quotient (“coset space”) $G/H$	super Klein geometry	super Cartan geometry	Cartan superconnection
examples	super Poincaré group	spin group	super Minkowski spacetime $\mathbb{R}^{d-1,1\vert N}$	Lorentzian supergeometry	supergeometry	superconnection	supergravity
	super anti de Sitter group		super anti de Sitter spacetime
higher differential geometry	smooth 2-group $G$	2-monomorphism $H \to G$	homotopy quotient $G//H$	Klein 2-geometry	Cartan 2-geometry
	cohesive ∞-group	∞-monomorphism (i.e. any homomorphism) $H \to G$	homotopy quotient $G//H$ of ∞-action	higher Klein geometry	higher Cartan geometry	higher Cartan connection
examples			extended super Minkowski spacetime		extended supergeometry		higher supergravity: type II, heterotic, 11d